Subgroups — #28377071

Order 13952.

1124 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295748	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295736	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295731	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295709	C₂	1	2	yes	maximal, complemented
5	#20295705	C₂	1	2	yes	maximal, complemented
6	#20295700	C₂	1	2	yes	maximal, complemented
7	D₄×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295685	C₂	1	2	yes	maximal, complemented
9	#20295677	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295675	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295671	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295664	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295648	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295647	C₂	1	1	yes	maximal, characteristic, complemented
15	#20295623	C₂	1	2	yes	maximal, complemented
16	#20295618	C₂	1	2	yes	maximal, complemented
17	#20295580	C₂	1	1	yes	maximal, characteristic, complemented
18	#20295554	C₂	1	1	yes	maximal, characteristic, complemented
19	#20295535	C₂	1	2	yes	maximal, complemented
20	C₂×#16812677	C₂	1	2	yes	maximal, complemented
21	#20295505	C₂	1	1	yes	maximal, characteristic, complemented
22	C₄×D₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
23	#520×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#13182029	C₂²	1	2	yes	complemented
25	C₂×#13182028	C₂²	1	1	yes	characteristic
26	C₂²×Dic₂₁₈	C₂²	1	2	yes	complemented
27	C₂×D₄×D₁₀₉	C₂²	1	1	yes	characteristic
28	C₂²×C₄×D₁₀₉	C₂²	1	2	yes	—
29	#16812699	C₂²	1	1	yes	characteristic, complemented
30	#16812698	C₂²	1	1	yes	characteristic, complemented
31	#16812698	C₂²	1	1	yes	characteristic, complemented
32	#16812697	C₂²	1	1	yes	characteristic, complemented
33	#16812694	C₂²	1	2	yes	complemented
34	#16812694	C₂²	1	2	yes	complemented
35	#16812694	C₂²	1	1	yes	characteristic, complemented
36	#16812693	C₂²	1	2	yes	complemented
37	#16812693	C₂²	1	2	yes	complemented
38	#16812692	C₂²	1	1	yes	characteristic, complemented
39	#16812691	C₂²	1	2	yes	complemented
40	#16812689	C₂²	1	1	yes	characteristic, complemented
41	#16812689	C₂²	1	2	yes	complemented
42	#16812689	C₂²	1	2	yes	complemented
43	#16812688	C₂²	1	1	yes	characteristic
44	#16812687	C₂²	1	1	yes	characteristic
45	#16812687	C₂²	1	2	yes	complemented
46	#16812685	C₂²	1	1	yes	characteristic, complemented
47	#16812685	C₂²	1	2	yes	complemented
48	#16812684	C₂²	1	1	yes	characteristic
49	#16812684	C₂²	1	2	yes	complemented
50	#16812684	C₂²	1	2	yes	complemented
51	#16812683	C₂²	1	2	yes	—
52	#16812682	C₂²	1	2	yes	—
53	#16812682	C₂²	1	2	yes	complemented
54	#16812682	C₂²	1	2	yes	complemented
55	#16812681	C₂²	1	1	yes	characteristic
56	#16812681	C₂²	1	2	yes	complemented
57	#16812680	C₂²	1	2	yes	complemented
58	#16812679	C₂²	1	1	yes	characteristic
59	#16812678	C₂²	1	2	yes	complemented
60	#16812678	C₂²	1	2	yes	complemented
61	#16812678	C₂²	1	1	yes	characteristic, complemented
62	#16812677	C₂²	1	2	yes	—
63	#16812677	C₂²	1	2	yes	—
64	#16812677	C₂²	1	1	yes	characteristic
65	#16812677	C₂²	1	2	yes	—
66	#16812677	C₂²	1	2	yes	complemented
67	#16812677	C₂²	1	2	yes	complemented
68	#16812677	C₂²	1	2	yes	complemented
69	#16812677	C₂²	1	2	yes	complemented
70	#16812677	C₂²	1	2	yes	complemented
71	#16812676	C₂²	1	2	yes	complemented
72	#16812676	C₂²	1	2	yes	complemented
73	#16812675	C₂²	1	1	yes	characteristic
74	#16812675	C₂²	1	1	yes	characteristic, complemented
75	#16812674	C₂²	1	1	yes	characteristic
76	#16812673	C₂²	1	2	yes	—
77	#16812673	C₂²	1	1	yes	characteristic
78	#16812672	C₂²	1	1	yes	characteristic
79	#16812671	C₂²	1	2	yes	—
80	#16812671	C₂²	1	2	yes	complemented
81	#16812669	C₂²	1	2	yes	—
82	#16812667	C₂²	1	2	yes	—
83	C₄×Dic₂₁₈	C₂²	1	2	yes	complemented
84	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic, complemented
85	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic, complemented
86	#16812664	C₂²	1	1	yes	characteristic, complemented
87	C₄×D₄₃₆	C₂²	1	2	yes	—
88	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
89	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
90	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
91	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
92	#16812661	C₂²	1	1	yes	characteristic
93	C₄×C₂²⋊D₁₀₉	C₂²	1	2	yes	—
94	C₄×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic
95	C₄×C₂²⋊D₁₀₉	C₂²	1	2	yes	—
96	C₄×C₂²⋊D₁₀₉	C₂²	1	2	yes	complemented
97	C₄×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic, complemented
98	#16812659	C₂²	1	2	yes	—
99	#16812659	C₂²	1	2	yes	complemented
100	#16812656	C₂²	1	2	yes	—
101	#16812656	C₂²	1	2	yes	complemented
102	#16812656	C₂²	1	1	yes	characteristic, complemented
103	#16812655	C₂²	1	2	yes	complemented
104	C₂×#13182008	C₂²	1	2	yes	complemented
105	C₂×#13182008	C₂²	1	2	yes	complemented
106	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
107	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic, complemented
108	C₂×#13182006	C₂²	1	2	yes	—
109	C₂×#13182006	C₂²	1	2	yes	—
110	C₂²⋊C₄×D₁₀₉	C₂²	1	2	yes	—
111	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
112	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
113	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
114	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
115	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
116	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
117	C₄×D₄×C₁₀₉	C₂²	1	2	yes	complemented
118	C₄×D₄×C₁₀₉	C₂²	1	2	yes	complemented
119	#117×C₁₀₉	C₂²	1	2	yes	complemented
120	#117×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
121	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
122	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	complemented
123	C₂³×D₁₀₉	C₂³	1	2	yes	—
124	#13182029	—	4	4	no	—
125	#13182028	—	2	4	no	—
126	C₂×Dic₂₁₈	—	4	4	no	—
127	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
128	C₂×Dic₂₁₈	C₂³	1	2	yes	—
129	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
130	C₂×Dic₂₁₈	C₂³	1	2	yes	—
131	C₂×Dic₂₁₈	C₂³	1	2	yes	—
132	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
133	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
134	C₂×D₄₃₆	C₂³	1	2	yes	—
135	C₂×C₂²⋊D₁₀₉	C₂³	1	2	yes	—
136	C₂×C₂²⋊D₁₀₉	C₂³	1	2	yes	—
137	C₂×C₂²⋊D₁₀₉	C₂³	1	1	yes	characteristic
138	C₂×C₂²⋊D₁₀₉	C₂³	1	1	yes	characteristic
139	C₂×C₂²⋊D₁₀₉	C₂³	1	2	yes	—
140	D₄×D₁₀₉	—	2	4	no	—
141	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
142	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
143	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
144	C₂×C₄×D₁₀₉	—	4	4	no	—
145	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
146	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
147	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
148	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
149	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
150	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
151	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
152	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
153	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
154	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
155	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
156	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
157	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
158	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
159	#13182008	C₂³	1	2	yes	—
160	#13182008	C₂³	1	2	yes	—
161	#13182008	C₂³	1	1	yes	characteristic
162	#13182008	C₂³	1	1	yes	characteristic
163	#13182008	C₂³	1	2	yes	—
164	#13182008	C₂³	1	2	yes	—
165	#13182008	C₂³	1	2	yes	—
166	#13182008	C₂³	1	2	yes	—
167	#13182008	C₂³	1	2	yes	—
168	#13182008	C₂³	1	1	yes	characteristic
169	#13182008	C₂³	1	2	yes	—
170	#13182008	C₂³	1	2	yes	—
171	#13182008	C₂³	1	1	yes	characteristic
172	#13182008	C₂³	1	2	yes	—
173	#13182008	C₂³	1	1	yes	characteristic
174	#13182008	C₂³	1	2	yes	—
175	#13182008	C₂³	1	1	yes	characteristic
176	C₂²⋊Dic₁₀₉	C₂³	1	2	yes	—
177	C₂²⋊Dic₁₀₉	C₂³	1	2	yes	—
178	C₂²⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
179	C₂²⋊Dic₁₀₉	C₂³	1	2	yes	—
180	C₂²⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
181	#13182006	C₂³	1	2	yes	—
182	#13182006	C₂³	1	2	yes	—
183	#13182006	C₂³	1	1	yes	characteristic
184	#13182006	C₂³	1	1	yes	characteristic
185	#13182006	C₂³	1	2	yes	—
186	#13182006	C₂³	1	2	yes	—
187	#13182006	C₂³	1	2	yes	—
188	#13182006	C₂³	1	2	yes	—
189	#13182006	C₂³	1	2	yes	—
190	#13182006	C₂³	1	2	yes	—
191	#13182006	C₂³	1	1	yes	characteristic
192	#13182006	C₂³	1	1	yes	characteristic
193	#13182006	C₂³	1	2	yes	—
194	#13182006	C₂³	1	2	yes	—
195	#13182006	C₂³	1	1	yes	characteristic
196	#13182006	C₂³	1	2	yes	—
197	#13182006	C₂³	1	1	yes	characteristic
198	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
199	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
200	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
201	C₄×Dic₁₀₉	C₂³	1	2	yes	—
202	C₄×Dic₁₀₉	C₂³	1	2	yes	—
203	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
204	#41×C₁₀₉	—	2	4	no	—
205	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
206	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
207	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
208	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
209	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
210	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
211	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
212	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
213	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
214	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
215	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
216	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
217	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
218	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
219	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
220	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
221	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
222	C₂²⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
223	C₂²⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
224	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
225	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
226	C₄²×C₁₀₉	C₂³	1	2	yes	—
227	C₄²×C₁₀₉	C₂³	1	2	yes	—
228	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
229	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
230	C₂²×D₁₀₉	—	2	4	no	—
231	C₂²×D₁₀₉	—	2	4	no	—
232	C₂²×D₁₀₉	—	4	2	no	—
233	C₂²×D₁₀₉	C₂⁴	1	2	yes	—
234	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
235	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
236	Q₈×C₁₀₉	—	2	2	no	—
237	D₄×C₁₀₉	—	2	4	no	—
238	D₄×C₁₀₉	#41	1	4	yes	complemented
239	C₂×C₄×C₁₀₉	—	2	4	no	—
240	C₂×C₄×C₁₀₉	—	2	2	no	—
241	C₂×C₄×C₁₀₉	—	4	2	no	—
242	C₂×C₄×C₁₀₉	—	4	2	no	—
243	C₂×C₄×C₁₀₉	—	4	2	no	—
244	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
245	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
246	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
247	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
248	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
249	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
250	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
251	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
252	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
253	C₂²⋊D₁₀₉	—	2	4	no	—
254	C₂²⋊D₁₀₉	—	2	4	no	—
255	C₂²⋊D₁₀₉	—	2	2	no	—
256	C₂²⋊D₁₀₉	—	2	2	no	—
257	C₂²⋊D₁₀₉	—	4	2	no	—
258	C₂×Dic₁₀₉	—	2	4	no	—
259	C₂×Dic₁₀₉	—	2	4	no	—
260	C₂×Dic₁₀₉	—	4	2	no	—
261	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
262	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
263	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
264	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
265	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
266	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
267	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
268	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
269	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
270	D₄₃₆	—	2	2	no	—
271	D₄₃₆	—	2	2	no	—
272	D₄₃₆	—	4	2	no	—
273	C₄×D₁₀₉	—	2	2	no	—
274	C₄×D₁₀₉	—	2	2	no	—
275	C₄×D₁₀₉	—	2	2	no	—
276	C₄×D₁₀₉	—	2	2	no	—
277	C₄×D₁₀₉	—	2	4	no	—
278	C₄×D₁₀₉	—	4	2	no	—
279	C₄×D₁₀₉	—	4	2	no	—
280	C₄×D₁₀₉	—	4	2	no	—
281	Dic₂₁₈	—	2	2	no	—
282	Dic₂₁₈	—	2	2	no	—
283	Dic₂₁₈	—	2	4	no	—
284	Dic₂₁₈	—	4	2	no	—
285	Dic₂₁₈	—	4	2	no	—
286	C₂²×C₁₀₉	—	2	2	no	—
287	C₂²×C₁₀₉	—	2	2	no	—
288	C₂²×C₁₀₉	—	2	1	no	—
289	C₂²×C₁₀₉	—	2	1	no	—
290	C₂²×C₁₀₉	—	2	1	no	—
291	C₂²×C₁₀₉	C₂×#41	1	4	yes	—
292	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
293	C₂×D₁₀₉	—	2	1	no	—
294	C₂×D₁₀₉	—	2	1	no	—
295	C₂×D₁₀₉	—	2	1	no	—
296	C₂×D₁₀₉	—	2	2	no	—
297	C₂×D₁₀₉	—	2	2	no	—
298	C₂×D₁₀₉	—	2	1	no	—
299	C₂×D₁₀₉	—	2	1	no	—
300	C₂×D₁₀₉	—	2	2	no	—
301	C₂×D₁₀₉	—	4	8	no	—
302	C₂×D₁₀₉	2₋¹⁺⁴	1	2	yes	—
303	C₄×C₁₀₉	—	2	1	no	—
304	C₄×C₁₀₉	—	2	2	no	—
305	C₄×C₁₀₉	—	2	1	no	—
306	C₄×C₁₀₉	—	2	2	no	—
307	C₄×C₁₀₉	—	2	1	no	—
308	C₄×C₁₀₉	—	2	2	no	—
309	C₄×C₁₀₉	—	2	1	no	—
310	C₄×C₁₀₉	—	2	1	no	—
311	C₄×C₁₀₉	C₂×#41	1	2	yes	—
312	Dic₁₀₉	—	2	2	no	—
313	Dic₁₀₉	—	2	1	no	—
314	Dic₁₀₉	—	2	1	no	—
315	Dic₁₀₉	—	2	1	no	—
316	Dic₁₀₉	—	2	2	no	—
317	Dic₁₀₉	—	2	2	no	—
318	Dic₁₀₉	—	2	1	no	—
319	Dic₁₀₉	—	2	1	no	—
320	Dic₁₀₉	H(5,𝔽₂)	1	2	yes	—
321	C₂×C₁₀₉	—	2	4	no	—
322	C₂×C₁₀₉	—	4	1	no	—
323	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
324	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
325	C₂×C₁₀₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
326	D₁₀₉	—	2	2	no	—
327	D₁₀₉	—	4	1	no	—
328	D₁₀₉	—	4	2	no	—
329	#3520	—	109	1	no	maximal, Sylow
330	C₁₀₉	#3520	1	1	yes	characteristic, complemented, Sylow
331	#556	—	109	1	no	—
332	#554	—	109	1	no	—
333	#554	—	109	1	no	—
334	#552	—	109	2	no	—
335	#551	—	109	2	no	—
336	#551	—	109	2	no	—
337	D₄×Q₈	—	109	1	no	—
338	#548	—	109	2	no	—
339	#547	—	109	1	no	—
340	#547	—	109	1	no	—
341	#546	—	109	1	no	—
342	#546	—	109	1	no	—
343	#542	—	109	1	no	—
344	#542	—	109	1	no	—
345	#539	—	109	2	no	—
346	#539	—	109	2	no	—
347	#533	—	109	1	no	—
348	#529	—	109	1	no	—
349	#525	—	109	2	no	—
350	C₂×C₂²⋊Q₈	—	109	2	no	—
351	#520	—	109	1	no	—
352	#520	D₁₀₉	1	1	yes	characteristic, complemented
353	C₂×C₄×D₄	—	109	1	no	—
354	C₂×#41	—	109	1	no	—
355	C₂×#41	—	109	2	no	—
356	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
357	C₂²×Q₈	—	109	2	no	—
358	C₂²×D₄	—	109	1	no	—
359	C₂³×C₄	—	109	2	no	—
360	C₄⋊Q₈	—	109	1	no	—
361	C₄⋊Q₈	—	109	1	no	—
362	C₄⋊Q₈	—	109	1	no	—
363	C₄⋊Q₈	—	109	1	no	—
364	#126	—	109	1	no	—
365	#126	—	109	2	no	—
366	#126	—	109	2	no	—
367	#126	—	109	1	no	—
368	#126	—	109	2	no	—
369	#126	—	109	2	no	—
370	#125	—	109	1	no	—
371	#125	—	109	2	no	—
372	#125	—	109	2	no	—
373	#125	—	109	2	no	—
374	#124	—	109	1	no	—
375	#124	—	109	1	no	—
376	#124	—	109	2	no	—
377	#124	—	109	1	no	—
378	#124	—	109	2	no	—
379	#123	—	109	2	no	—
380	#123	—	109	2	no	—
381	#123	—	109	2	no	—
382	#123	—	109	2	no	—
383	#123	—	109	2	no	—
384	#123	—	109	1	no	—
385	#123	—	109	2	no	—
386	#123	—	109	2	no	—
387	#123	—	109	1	no	—
388	#123	—	109	2	no	—
389	C₂²⋊Q₈	—	109	1	no	—
390	C₂²⋊Q₈	—	109	1	no	—
391	C₂²⋊Q₈	—	109	2	no	—
392	C₂²⋊Q₈	—	109	2	no	—
393	C₂²⋊Q₈	—	109	1	no	—
394	C₂²⋊Q₈	—	109	1	no	—
395	C₂²⋊Q₈	—	109	2	no	—
396	C₂²⋊Q₈	—	109	2	no	—
397	C₂²⋊Q₈	—	109	2	no	—
398	C₂²⋊Q₈	—	109	2	no	—
399	C₂²⋊Q₈	—	109	2	no	—
400	C₂²⋊Q₈	—	109	2	no	—
401	C₂²⋊Q₈	—	109	2	no	—
402	C₂²⋊Q₈	—	109	1	no	—
403	C₂²⋊Q₈	—	109	2	no	—
404	C₂²⋊Q₈	—	109	2	no	—
405	C₂²⋊Q₈	—	109	2	no	—
406	#121	—	109	2	no	—
407	#121	—	109	2	no	—
408	#121	—	109	2	no	—
409	#121	—	109	1	no	—
410	#121	—	109	1	no	—
411	#121	—	109	1	no	—
412	#120	—	109	2	no	—
413	#120	—	109	2	no	—
414	C₄×Q₈	—	109	1	no	—
415	C₄×Q₈	—	109	1	no	—
416	C₄×Q₈	—	109	1	no	—
417	C₄×Q₈	—	109	2	no	—
418	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
419	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
420	C₄×D₄	—	109	1	no	—
421	C₄×D₄	—	109	1	no	—
422	C₄×D₄	—	109	2	no	—
423	C₄×D₄	—	109	2	no	—
424	C₄×D₄	—	109	2	no	—
425	C₄×D₄	—	109	1	no	—
426	C₄×D₄	—	109	1	no	—
427	C₄×D₄	—	109	1	no	—
428	C₄×D₄	—	109	2	no	—
429	C₄×D₄	—	109	2	no	—
430	C₄×D₄	—	109	1	no	—
431	C₄×D₄	—	109	2	no	—
432	C₄×D₄	—	109	1	no	—
433	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
434	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
435	C₄×D₄	C₂×D₁₀₉	1	2	yes	complemented
436	C₄×D₄	C₂×D₁₀₉	1	2	yes	complemented
437	#117	—	109	2	no	—
438	#117	—	109	1	no	—
439	#117	—	109	2	no	—
440	#117	—	109	2	no	—
441	#117	C₂×D₁₀₉	1	1	yes	characteristic, complemented
442	#117	C₂×D₁₀₉	1	2	yes	complemented
443	C₂×C₄⋊C₄	—	109	2	no	—
444	C₂×C₄⋊C₄	—	109	2	no	—
445	C₂×C₄⋊C₄	—	109	1	no	—
446	C₂×C₄⋊C₄	—	109	1	no	—
447	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	complemented
448	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic, complemented
449	C₂×C₂²⋊C₄	—	109	2	no	—
450	C₂×C₂²⋊C₄	—	109	2	no	—
451	C₂×C₂²⋊C₄	—	109	2	no	—
452	C₂×C₄²	—	109	1	no	—
453	C₂⁴	—	109	2	no	—
454	#41	—	2	4	no	—
455	#41	—	436	4	no	—
456	#41	—	218	4	no	—
457	C₂×Q₈	—	109	2	no	—
458	C₂×Q₈	—	109	2	no	—
459	C₂×Q₈	—	109	1	no	—
460	C₂×Q₈	—	109	1	no	—
461	C₂×Q₈	—	109	2	no	—
462	C₂×Q₈	—	436	4	no	—
463	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
464	C₂×D₄	—	109	2	no	—
465	C₂×D₄	—	109	1	no	—
466	C₂×D₄	—	109	2	no	—
467	C₂×D₄	—	109	2	no	—
468	C₂×D₄	—	109	1	no	—
469	C₂×D₄	—	109	2	no	—
470	C₂×D₄	—	109	1	no	—
471	C₂×D₄	—	109	1	no	—
472	C₂×D₄	—	218	4	no	—
473	C₂×D₄	C₂²×D₁₀₉	1	2	yes	—
474	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
475	C₂²×C₄	—	109	2	no	—
476	C₂²×C₄	—	109	2	no	—
477	C₂²×C₄	—	109	2	no	—
478	C₂²×C₄	—	109	2	no	—
479	C₂²×C₄	—	109	2	no	—
480	C₂²×C₄	—	109	2	no	—
481	C₂²×C₄	—	109	1	no	—
482	C₂²×C₄	—	109	1	no	—
483	C₂²×C₄	—	109	1	no	—
484	C₂²×C₄	—	109	1	no	—
485	C₂²×C₄	—	109	2	no	—
486	C₂²×C₄	—	436	4	no	—
487	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
488	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
489	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
490	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
491	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
492	C₄⋊C₄	—	109	1	no	—
493	C₄⋊C₄	—	109	2	no	—
494	C₄⋊C₄	—	109	1	no	—
495	C₄⋊C₄	—	109	2	no	—
496	C₄⋊C₄	—	109	2	no	—
497	C₄⋊C₄	—	109	2	no	—
498	C₄⋊C₄	—	109	2	no	—
499	C₄⋊C₄	—	109	2	no	—
500	C₄⋊C₄	—	109	2	no	—
501	C₄⋊C₄	—	109	1	no	—
502	C₄⋊C₄	—	109	1	no	—
503	C₄⋊C₄	—	109	2	no	—
504	C₄⋊C₄	—	109	2	no	—
505	C₄⋊C₄	—	109	2	no	—
506	C₄⋊C₄	—	109	1	no	—
507	C₄⋊C₄	—	109	2	no	—
508	C₄⋊C₄	—	109	1	no	—
509	C₄⋊C₄	—	109	1	no	—
510	C₄⋊C₄	—	109	1	no	—
511	C₄⋊C₄	—	109	2	no	—
512	C₄⋊C₄	—	109	1	no	—
513	C₄⋊C₄	—	109	2	no	—
514	C₄⋊C₄	—	109	1	no	—
515	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
516	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
517	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
518	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
519	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
520	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
521	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
522	C₂²⋊C₄	—	109	2	no	—
523	C₂²⋊C₄	—	109	2	no	—
524	C₂²⋊C₄	—	109	1	no	—
525	C₂²⋊C₄	—	109	2	no	—
526	C₂²⋊C₄	—	109	1	no	—
527	C₂²⋊C₄	—	109	1	no	—
528	C₂²⋊C₄	—	109	1	no	—
529	C₂²⋊C₄	—	109	2	no	—
530	C₂²⋊C₄	—	109	2	no	—
531	C₂²⋊C₄	—	109	2	no	—
532	C₂²⋊C₄	—	109	2	no	—
533	C₂²⋊C₄	—	109	2	no	—
534	C₂²⋊C₄	—	109	2	no	—
535	C₂²⋊C₄	—	109	2	no	—
536	C₂²⋊C₄	—	109	2	no	—
537	C₂²⋊C₄	—	109	1	no	—
538	C₂²⋊C₄	—	109	1	no	—
539	C₂²⋊C₄	—	109	2	no	—
540	C₂²⋊C₄	—	109	2	no	—
541	C₂²⋊C₄	—	109	1	no	—
542	C₂²⋊C₄	—	109	2	no	—
543	C₂²⋊C₄	—	109	1	no	—
544	C₂²⋊C₄	C₂²×D₁₀₉	1	2	yes	—
545	C₂²⋊C₄	C₂²×D₁₀₉	1	2	yes	—
546	C₂²⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
547	C₂²⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
548	C₄²	—	109	1	no	—
549	C₄²	—	109	1	no	—
550	C₄²	—	109	1	no	—
551	C₄²	—	109	2	no	—
552	C₄²	—	109	2	no	—
553	C₄²	—	109	1	no	—
554	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
555	C₄²	C₂²×D₁₀₉	1	2	yes	—
556	C₄²	C₂²×D₁₀₉	1	2	yes	—
557	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
558	C₂³	—	109	1	no	—
559	C₂³	—	109	2	no	—
560	C₂³	—	109	1	no	—
561	C₂³	—	436	2	no	—
562	C₂³	—	218	4	no	—
563	C₂³	—	218	4	no	—
564	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
565	C₂³	C₂³×D₁₀₉	1	2	yes	—
566	Q₈	—	2	2	no	—
567	Q₈	—	436	2	no	—
568	Q₈	—	436	2	no	—
569	Q₈	—	218	2	no	—
570	Q₈	—	218	4	no	—
571	Q₈	—	218	2	no	—
572	D₄	—	2	4	no	—
573	D₄	—	436	2	no	—
574	D₄	—	436	2	no	—
575	D₄	—	218	2	no	—
576	D₄	—	218	2	no	—
577	D₄	—	218	2	no	—
578	D₄	—	218	4	no	—
579	D₄	—	218	4	no	—
580	D₄	—	218	2	no	—
581	D₄	#13182029	1	4	yes	complemented
582	C₂×C₄	—	2	4	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	4	2	no	—
585	C₂×C₄	—	4	2	no	—
586	C₂×C₄	—	4	2	no	—
587	C₂×C₄	—	109	1	no	—
588	C₂×C₄	—	109	1	no	—
589	C₂×C₄	—	109	1	no	—
590	C₂×C₄	—	109	2	no	—
591	C₂×C₄	—	109	1	no	—
592	C₂×C₄	—	109	1	no	—
593	C₂×C₄	—	109	2	no	—
594	C₂×C₄	—	109	1	no	—
595	C₂×C₄	—	109	2	no	—
596	C₂×C₄	—	436	2	no	—
597	C₂×C₄	—	436	2	no	—
598	C₂×C₄	—	436	2	no	—
599	C₂×C₄	—	436	2	no	—
600	C₂×C₄	—	218	2	no	—
601	C₂×C₄	—	218	4	no	—
602	C₂×C₄	—	218	2	no	—
603	C₂×C₄	—	218	4	no	—
604	C₂×C₄	—	218	4	no	—
605	C₂×C₄	—	218	2	no	—
606	C₂×C₄	—	218	2	no	—
607	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
608	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
609	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
610	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
611	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
612	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
613	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
614	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
615	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
616	C₂²	—	2	2	no	—
617	C₂²	—	2	2	no	—
618	C₂²	—	2	1	no	—
619	C₂²	—	2	1	no	—
620	C₂²	—	2	1	no	—
621	C₂²	—	109	2	no	—
622	C₂²	—	436	8	no	—
623	C₂²	—	218	1	no	—
624	C₂²	—	218	2	no	—
625	C₂²	—	218	1	no	—
626	C₂²	—	218	2	no	—
627	C₂²	—	218	1	no	—
628	C₂²	—	218	2	no	—
629	C₂²	—	218	1	no	—
630	C₂²	—	218	1	no	—
631	C₂²	C₂×#13182029	1	4	yes	—
632	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
633	C₄	—	2	1	no	—
634	C₄	—	2	2	no	—
635	C₄	—	2	1	no	—
636	C₄	—	2	2	no	—
637	C₄	—	2	2	no	—
638	C₄	—	2	1	no	—
639	C₄	—	2	1	no	—
640	C₄	—	2	1	no	—
641	C₄	—	109	2	no	—
642	C₄	—	218	1	no	—
643	C₄	—	218	1	no	—
644	C₄	—	218	2	no	—
645	C₄	—	218	2	no	—
646	C₄	—	218	2	no	—
647	C₄	—	218	1	no	—
648	C₄	—	218	1	no	—
649	C₄	—	218	1	no	—
650	C₄	C₂×#13182029	1	2	yes	—
651	C₂	—	2	4	no	—
652	C₂	—	4	1	no	—
653	C₂	—	436	2	no	—
654	C₂	—	436	1	no	—
655	C₂	—	218	2	no	—
656	C₂	C₂²×#13182029	1	1	yes	characteristic
657	C₂	H(5,𝔽₂)×D₁₀₉	1	1	yes	characteristic
658	C₂	2₋¹⁺⁴×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₉
commutator, socle, cosocle	C₂²×C₁₀₉	C₂⁵
Fitting	#520×C₁₀₉	C₂
hypercenter	#520	D₁₀₉
hypocenter	C₁₀₉	#3520
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #520

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #20295748	C₂	semidirect
2. #20295736	C₂	semidirect
3. #20295731	C₂	semidirect
4. #20295709	C₂	semidirect
5. #20295705	C₂	semidirect
6. #20295700	C₂	semidirect
7. D₄×Dic₂₁₈	C₂	semidirect
8. #20295685	C₂	semidirect
9. #20295677	C₂	semidirect
10. #20295675	C₂	semidirect
11. #20295671	C₂	semidirect
12. #20295664	C₂	semidirect
13. #20295648	C₂	semidirect
14. #20295647	C₂	semidirect
15. #20295623	C₂	semidirect
16. #20295618	C₂	semidirect
17. #20295580	C₂	semidirect
18. #20295554	C₂	semidirect
19. #20295535	C₂	semidirect
20. C₂×#16812677	C₂	semidirect
21. #20295505	C₂	semidirect
22. C₄×D₄×D₁₀₉	C₂	semidirect
23. #520×C₁₀₉	C₂	semidirect
24. C₂×#13182029	C₂²	semidirect
26. C₂²×Dic₂₁₈	C₂²	semidirect
29. #16812699	C₂²	semidirect
30. #16812698	C₂²	semidirect
31. #16812698	C₂²	semidirect
32. #16812697	C₂²	semidirect
33. #16812694	C₂²	semidirect
34. #16812694	C₂²	semidirect
35. #16812694	C₂²	semidirect
36. #16812693	C₂²	semidirect
37. #16812693	C₂²	semidirect
38. #16812692	C₂²	semidirect
39. #16812691	C₂²	semidirect
40. #16812689	C₂²	semidirect
41. #16812689	C₂²	semidirect
42. #16812689	C₂²	semidirect
45. #16812687	C₂²	semidirect
46. #16812685	C₂²	semidirect
47. #16812685	C₂²	semidirect
49. #16812684	C₂²	semidirect
50. #16812684	C₂²	semidirect
53. #16812682	C₂²	semidirect
54. #16812682	C₂²	semidirect
56. #16812681	C₂²	semidirect
57. #16812680	C₂²	semidirect
59. #16812678	C₂²	semidirect
60. #16812678	C₂²	semidirect
61. #16812678	C₂²	semidirect
66. #16812677	C₂²	semidirect
67. #16812677	C₂²	semidirect
68. #16812677	C₂²	semidirect
69. #16812677	C₂²	semidirect
70. #16812677	C₂²	semidirect
71. #16812676	C₂²	semidirect
72. #16812676	C₂²	semidirect
74. #16812675	C₂²	semidirect
80. #16812671	C₂²	semidirect
83. C₄×Dic₂₁₈	C₂²	semidirect
84. C₄×Dic₂₁₈	C₂²	semidirect
85. C₄×Dic₂₁₈	C₂²	semidirect
86. #16812664	C₂²	semidirect
96. C₄×C₂²⋊D₁₀₉	C₂²	semidirect
97. C₄×C₂²⋊D₁₀₉	C₂²	semidirect
99. #16812659	C₂²	semidirect
101. #16812656	C₂²	semidirect
102. #16812656	C₂²	semidirect
103. #16812655	C₂²	semidirect
104. C₂×#13182008	C₂²	semidirect
105. C₂×#13182008	C₂²	semidirect
107. C₄⋊C₄×D₁₀₉	C₂²	semidirect
113. C₄×Q₈×C₁₀₉	C₂²	semidirect
114. C₄×Q₈×C₁₀₉	C₂²	semidirect
117. C₄×D₄×C₁₀₉	C₂²	semidirect
118. C₄×D₄×C₁₀₉	C₂²	semidirect
119. #117×C₁₀₉	C₂²	semidirect
120. #117×C₁₀₉	C₂²	semidirect
121. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
122. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
238. D₄×C₁₀₉	#41	semidirect
330. C₁₀₉	#3520	semidirect
352. #520	D₁₀₉	semidirect
418. C₄×Q₈	C₂×D₁₀₉	semidirect
419. C₄×Q₈	C₂×D₁₀₉	semidirect
435. C₄×D₄	C₂×D₁₀₉	semidirect
436. C₄×D₄	C₂×D₁₀₉	semidirect
441. #117	C₂×D₁₀₉	semidirect
442. #117	C₂×D₁₀₉	semidirect
447. C₂×C₄⋊C₄	C₂×D₁₀₉	semidirect
448. C₂×C₄⋊C₄	C₂×D₁₀₉	semidirect
581. D₄	#13182029	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20295748	2
#20295736	2
#20295731	2
#20295709	2
#20295705	2
#20295700	2
D₄×Dic₂₁₈	2
#20295685	2
#20295677	2
#20295675	2
#20295671	2
#20295664	2
#20295648	2
#20295647	2
#20295623	2
#20295618	2
#20295580	2
#20295554	2
#20295535	2
C₂×#16812677	2
#20295505	2
C₄×D₄×D₁₀₉	2
#520×C₁₀₉	2
#3520	109