Subgroups — #28376419

Order 13952.

1040 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295650	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295647	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295645	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295636	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295634	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295629	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295627	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295622	C₂	1	2	yes	maximal, complemented
9	#20295617	C₂	1	2	yes	maximal, complemented
10	#20295612	C₂	1	2	yes	maximal, complemented
11	#20295610	C₂	1	2	yes	maximal, complemented
12	#20295581	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295575	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295562	C₂	1	2	yes	maximal, complemented
15	#20295559	C₂	1	2	yes	maximal, complemented
16	#20295555	C₂	1	2	yes	maximal, complemented
17	#20295553	C₂	1	2	yes	maximal, complemented
18	#20295539	C₂	1	1	yes	maximal, characteristic, complemented
19	#20295536	C₂	1	1	yes	maximal, characteristic, complemented
20	C₄×#13182030	C₂	1	1	yes	maximal, characteristic, complemented
21	C₄×#13182028	C₂	1	1	yes	maximal, characteristic, complemented
22	#117×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
23	#542×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#13182030	C₂²	1	1	yes	characteristic
25	C₂×#13182028	C₂²	1	1	yes	characteristic
26	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
27	#16812699	C₂²	1	1	yes	characteristic, complemented
28	#16812697	C₂²	1	1	yes	characteristic, complemented
29	#16812695	C₂²	1	1	yes	characteristic, complemented
30	#16812694	C₂²	1	2	yes	—
31	#16812694	C₂²	1	2	yes	complemented
32	#16812694	C₂²	1	2	yes	complemented
33	#16812693	C₂²	1	2	yes	—
34	#16812692	C₂²	1	2	yes	—
35	#16812692	C₂²	1	2	yes	—
36	#16812692	C₂²	1	2	yes	complemented
37	#16812691	C₂²	1	2	yes	complemented
38	#16812690	C₂²	1	2	yes	complemented
39	#16812690	C₂²	1	1	yes	characteristic, complemented
40	#16812690	C₂²	1	1	yes	characteristic, complemented
41	#16812689	C₂²	1	2	yes	complemented
42	#16812689	C₂²	1	2	yes	complemented
43	#16812688	C₂²	1	2	yes	—
44	#16812687	C₂²	1	1	yes	characteristic, complemented
45	#16812685	C₂²	1	2	yes	—
46	#16812685	C₂²	1	2	yes	—
47	#16812684	C₂²	1	2	yes	—
48	#16812684	C₂²	1	2	yes	—
49	#16812683	C₂²	1	2	yes	—
50	#16812683	C₂²	1	2	yes	complemented
51	#16812682	C₂²	1	2	yes	complemented
52	#16812681	C₂²	1	2	yes	—
53	#16812681	C₂²	1	2	yes	—
54	#16812681	C₂²	1	2	yes	complemented
55	#16812681	C₂²	1	2	yes	complemented
56	#16812679	C₂²	1	1	yes	characteristic
57	#16812679	C₂²	1	2	yes	—
58	#16812678	C₂²	1	1	yes	characteristic
59	#16812677	C₂²	1	2	yes	complemented
60	#16812676	C₂²	1	2	yes	complemented
61	#16812675	C₂²	1	1	yes	characteristic
62	#16812675	C₂²	1	2	yes	—
63	#16812675	C₂²	1	2	yes	complemented
64	#16812674	C₂²	1	1	yes	characteristic
65	#16812673	C₂²	1	1	yes	characteristic
66	#16812673	C₂²	1	2	yes	—
67	#16812672	C₂²	1	2	yes	—
68	#16812672	C₂²	1	2	yes	—
69	#16812672	C₂²	1	2	yes	—
70	#16812671	C₂²	1	2	yes	—
71	#16812670	C₂²	1	1	yes	characteristic
72	#16812670	C₂²	1	2	yes	—
73	#16812669	C₂²	1	2	yes	—
74	#16812667	C₂²	1	2	yes	—
75	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
76	Q₈×Dic₁₀₉	C₂²	1	2	yes	complemented
77	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic, complemented
78	#16812664	C₂²	1	1	yes	characteristic, complemented
79	#16812664	C₂²	1	2	yes	complemented
80	C₄×D₄₃₆	C₂²	1	2	yes	—
81	C₄×D₄₃₆	C₂²	1	1	yes	characteristic, complemented
82	D₄×Dic₁₀₉	C₂²	1	2	yes	—
83	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
84	#16812661	C₂²	1	2	yes	—
85	#16812661	C₂²	1	2	yes	—
86	#16812661	C₂²	1	1	yes	characteristic, complemented
87	C₄×C₂²⋊D₁₀₉	C₂²	1	2	yes	—
88	#16812659	C₂²	1	2	yes	—
89	#16812659	C₂²	1	2	yes	—
90	#16812659	C₂²	1	2	yes	—
91	#16812659	C₂²	1	2	yes	complemented
92	#16812659	C₂²	1	2	yes	complemented
93	#16812658	C₂²	1	1	yes	characteristic
94	#16812657	C₂²	1	1	yes	characteristic
95	#16812657	C₂²	1	1	yes	characteristic
96	#16812657	C₂²	1	1	yes	characteristic
97	#16812657	C₂²	1	1	yes	characteristic
98	#16812657	C₂²	1	2	yes	complemented
99	#16812656	C₂²	1	1	yes	characteristic
100	#16812656	C₂²	1	1	yes	characteristic
101	#16812655	C₂²	1	1	yes	characteristic
102	#16812655	C₂²	1	1	yes	characteristic
103	#16812655	C₂²	1	2	yes	complemented
104	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
105	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
106	C₂²⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
107	C₂²⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
108	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	complemented
109	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
110	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
111	C₄²×D₁₀₉	C₂²	1	2	yes	complemented
112	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
113	#126×C₁₀₉	C₂²	1	2	yes	complemented
114	#124×C₁₀₉	C₂²	1	2	yes	—
115	#124×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
116	#123×C₁₀₉	C₂²	1	2	yes	—
117	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
118	C₂²⋊Q₈×C₁₀₉	C₂²	1	2	yes	complemented
119	#121×C₁₀₉	C₂²	1	1	yes	characteristic
120	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
121	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
122	#117×C₁₀₉	C₂²	1	1	yes	characteristic
123	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
124	#13182030	—	2	4	no	—
125	#13182028	—	2	4	no	—
126	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
127	C₂×Dic₂₁₈	C₂³	1	2	yes	—
128	C₂×D₄₃₆	C₂³	1	2	yes	—
129	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
130	C₂×D₄₃₆	C₂³	1	2	yes	—
131	C₂×C₂²⋊D₁₀₉	C₂³	1	2	yes	—
132	C₂×C₂²⋊D₁₀₉	C₂³	1	2	yes	—
133	C₂×C₂²⋊D₁₀₉	C₂³	1	2	yes	—
134	C₂×C₂²⋊D₁₀₉	C₂³	1	2	yes	—
135	C₂×C₂²⋊D₁₀₉	C₂³	1	2	yes	—
136	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
137	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
138	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
139	C₂×C₄×D₁₀₉	—	2	4	no	—
140	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
141	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
142	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
143	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
144	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
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	2	yes	—
149	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
150	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
151	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
152	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
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	2	yes	—
157	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
158	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
159	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
160	#13182008	C₂³	1	1	yes	characteristic
161	#13182008	C₂³	1	1	yes	characteristic
162	#13182008	C₂³	1	1	yes	characteristic
163	#13182008	C₂³	1	1	yes	characteristic
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	2	yes	—
169	#13182008	C₂³	1	2	yes	—
170	#13182008	C₂³	1	2	yes	—
171	#13182008	C₂³	1	2	yes	—
172	C₂²⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
173	C₂²⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
174	C₂²⋊Dic₁₀₉	C₂³	1	2	yes	—
175	C₂²⋊Dic₁₀₉	C₂³	1	2	yes	—
176	C₂²⋊Dic₁₀₉	C₂³	1	2	yes	—
177	#13182006	C₂³	1	1	yes	characteristic
178	#13182006	C₂³	1	1	yes	characteristic
179	#13182006	C₂³	1	1	yes	characteristic
180	#13182006	C₂³	1	1	yes	characteristic
181	#13182006	C₂³	1	2	yes	—
182	#13182006	C₂³	1	2	yes	—
183	#13182006	C₂³	1	2	yes	—
184	#13182006	C₂³	1	2	yes	—
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	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
192	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
193	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
194	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
195	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
196	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
197	C₄×Dic₁₀₉	C₂³	1	2	yes	—
198	C₄×Dic₁₀₉	C₂³	1	2	yes	—
199	C₄×Dic₁₀₉	C₂³	1	2	yes	—
200	C₄×Dic₁₀₉	C₂³	1	2	yes	—
201	C₄×Dic₁₀₉	C₂³	1	2	yes	—
202	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
203	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
204	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
205	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
206	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
207	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
208	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
209	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
210	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
211	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
212	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
213	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
214	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
215	C₂²⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
216	C₂²⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
217	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
218	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
219	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
220	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
221	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
222	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
223	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
224	C₄²×C₁₀₉	C₂³	1	2	yes	—
225	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
226	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
227	C₂²×D₁₀₉	—	2	2	no	—
228	C₂²×D₁₀₉	—	4	1	no	—
229	C₂²×D₁₀₉	—	4	1	no	—
230	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
231	C₂²×D₁₀₉	C₂⁴	1	2	yes	—
232	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
233	Q₈×C₁₀₉	—	2	2	no	—
234	Q₈×C₁₀₉	—	4	2	no	—
235	D₄×C₁₀₉	—	2	2	no	—
236	D₄×C₁₀₉	—	4	2	no	—
237	C₂×C₄×C₁₀₉	—	2	2	no	—
238	C₂×C₄×C₁₀₉	—	4	2	no	—
239	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
240	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
241	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
242	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
243	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
244	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
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	C₂²⋊D₁₀₉	—	2	4	no	—
249	C₂²⋊D₁₀₉	—	4	2	no	—
250	C₂²⋊D₁₀₉	—	4	2	no	—
251	C₂²⋊D₁₀₉	—	4	2	no	—
252	C₂²⋊D₁₀₉	—	4	2	no	—
253	C₂×Dic₁₀₉	—	2	2	no	—
254	C₂×Dic₁₀₉	—	2	4	no	—
255	C₂×Dic₁₀₉	—	4	2	no	—
256	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
257	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
258	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
259	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
260	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
261	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
262	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
263	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
264	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
265	D₄₃₆	—	2	4	no	—
266	D₄₃₆	—	2	2	no	—
267	D₄₃₆	—	4	2	no	—
268	C₄×D₁₀₉	—	2	2	no	—
269	C₄×D₁₀₉	—	2	2	no	—
270	C₄×D₁₀₉	—	2	2	no	—
271	C₄×D₁₀₉	—	2	4	no	—
272	C₄×D₁₀₉	—	4	2	no	—
273	C₄×D₁₀₉	—	4	1	no	—
274	C₄×D₁₀₉	—	4	2	no	—
275	C₄×D₁₀₉	—	4	1	no	—
276	C₄×D₁₀₉	—	4	1	no	—
277	C₄×D₁₀₉	—	4	1	no	—
278	C₄×D₁₀₉	#41	1	4	yes	—
279	Dic₂₁₈	—	2	2	no	—
280	Dic₂₁₈	—	4	2	no	—
281	C₂²×C₁₀₉	—	2	2	no	—
282	C₂²×C₁₀₉	—	2	1	no	—
283	C₂²×C₁₀₉	—	2	2	no	—
284	C₂²×C₁₀₉	—	2	1	no	—
285	C₂²×C₁₀₉	—	2	2	no	—
286	C₂²×C₁₀₉	—	2	1	no	—
287	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
288	C₂×D₁₀₉	—	2	1	no	—
289	C₂×D₁₀₉	—	2	1	no	—
290	C₂×D₁₀₉	—	2	1	no	—
291	C₂×D₁₀₉	—	2	2	no	—
292	C₂×D₁₀₉	—	2	2	no	—
293	C₂×D₁₀₉	—	2	2	no	—
294	C₂×D₁₀₉	—	2	1	no	—
295	C₂×D₁₀₉	—	2	1	no	—
296	C₂×D₁₀₉	—	4	4	no	—
297	C₂×D₁₀₉	C₂×#41	1	2	yes	—
298	C₄×C₁₀₉	—	2	1	no	—
299	C₄×C₁₀₉	—	2	2	no	—
300	C₄×C₁₀₉	—	2	1	no	—
301	C₄×C₁₀₉	—	2	1	no	—
302	C₄×C₁₀₉	—	2	1	no	—
303	C₄×C₁₀₉	—	2	2	no	—
304	C₄×C₁₀₉	—	2	1	no	—
305	C₄×C₁₀₉	—	2	2	no	—
306	C₄×C₁₀₉	C₂×#41	1	2	yes	—
307	Dic₁₀₉	—	2	2	no	—
308	Dic₁₀₉	—	2	1	no	—
309	Dic₁₀₉	—	2	2	no	—
310	Dic₁₀₉	—	2	2	no	—
311	Dic₁₀₉	—	2	1	no	—
312	Dic₁₀₉	—	2	1	no	—
313	Dic₁₀₉	—	2	1	no	—
314	Dic₁₀₉	—	2	1	no	—
315	Dic₁₀₉	C₂×#41	1	2	yes	—
316	C₂×C₁₀₉	—	4	1	no	—
317	C₂×C₁₀₉	—	4	2	no	—
318	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
319	C₂×C₁₀₉	#585	1	1	yes	characteristic
320	C₂×C₁₀₉	#585	1	1	yes	characteristic
321	D₁₀₉	—	2	2	no	—
322	D₁₀₉	—	4	2	no	—
323	D₁₀₉	—	4	1	no	—
324	#3455	—	109	1	no	maximal, Sylow
325	C₁₀₉	#3455	1	1	yes	characteristic, complemented, Sylow
326	#542	—	109	1	no	—
327	#542	—	109	1	no	—
328	#542	—	109	1	no	—
329	#542	D₁₀₉	1	1	yes	characteristic, complemented
330	#541	—	109	1	no	—
331	#541	—	109	1	no	—
332	#540	—	109	1	no	—
333	#540	—	109	1	no	—
334	#539	—	109	2	no	—
335	#539	—	109	2	no	—
336	#538	—	109	2	no	—
337	#538	—	109	2	no	—
338	#533	—	109	1	no	—
339	#532	—	109	1	no	—
340	#529	—	109	2	no	—
341	#529	—	109	2	no	—
342	#529	—	109	2	no	—
343	#529	—	109	2	no	—
344	#525	—	109	1	no	—
345	#525	—	109	1	no	—
346	C₄×#41	—	109	1	no	—
347	C₄×#41	—	109	1	no	—
348	C₂×#117	—	109	1	no	—
349	C₂×#41	—	109	1	no	—
350	C₂×#41	—	109	1	no	—
351	C₂³×C₄	—	109	1	no	—
352	C₄⋊Q₈	—	109	1	no	—
353	C₄⋊Q₈	—	109	1	no	—
354	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
355	#127	—	109	1	no	—
356	#126	—	109	2	no	—
357	#126	—	109	2	no	—
358	#126	—	109	2	no	—
359	#126	—	109	2	no	—
360	#126	—	109	2	no	—
361	#126	—	109	2	no	—
362	#126	—	109	2	no	—
363	#126	C₂×D₁₀₉	1	2	yes	complemented
364	#125	—	109	2	no	—
365	#125	—	109	2	no	—
366	#125	—	109	2	no	—
367	#125	—	109	1	no	—
368	#125	—	109	1	no	—
369	#125	—	109	2	no	—
370	#124	—	109	2	no	—
371	#124	—	109	2	no	—
372	#124	—	109	2	no	—
373	#124	—	109	1	no	—
374	#124	C₂×D₁₀₉	1	2	yes	—
375	#124	C₂×D₁₀₉	1	1	yes	characteristic, complemented
376	#123	—	109	2	no	—
377	#123	—	109	2	no	—
378	#123	—	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	2	no	—
385	#123	C₂×D₁₀₉	1	2	yes	—
386	C₂²⋊Q₈	—	109	1	no	—
387	C₂²⋊Q₈	—	109	2	no	—
388	C₂²⋊Q₈	—	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	2	no	—
394	C₂²⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
395	C₂²⋊Q₈	C₂×D₁₀₉	1	2	yes	complemented
396	#121	—	109	1	no	—
397	#121	—	109	2	no	—
398	#121	—	109	2	no	—
399	#121	—	109	1	no	—
400	#121	—	109	2	no	—
401	#121	—	109	1	no	—
402	#121	—	109	2	no	—
403	#121	—	109	2	no	—
404	#121	—	109	2	no	—
405	#121	C₂×D₁₀₉	1	1	yes	characteristic
406	#120	—	109	2	no	—
407	#120	—	109	2	no	—
408	C₄×Q₈	—	109	1	no	—
409	C₄×Q₈	—	109	2	no	—
410	C₄×Q₈	—	109	1	no	—
411	C₄×Q₈	—	109	2	no	—
412	C₄×Q₈	—	109	1	no	—
413	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
414	C₄×D₄	—	109	1	no	—
415	C₄×D₄	—	109	2	no	—
416	C₄×D₄	—	109	2	no	—
417	C₄×D₄	—	109	2	no	—
418	C₄×D₄	—	109	1	no	—
419	C₄×D₄	—	109	2	no	—
420	C₄×D₄	—	109	2	no	—
421	C₄×D₄	—	109	2	no	—
422	C₄×D₄	—	109	2	no	—
423	C₄×D₄	—	109	2	no	—
424	C₄×D₄	—	109	2	no	—
425	C₄×D₄	—	109	2	no	—
426	C₄×D₄	—	109	1	no	—
427	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic, complemented
428	#117	—	109	1	no	—
429	#117	—	109	1	no	—
430	#117	—	109	1	no	—
431	#117	—	109	1	no	—
432	#117	—	109	1	no	—
433	#117	—	109	2	no	—
434	#117	—	109	1	no	—
435	#117	—	109	1	no	—
436	#117	—	109	2	no	—
437	#117	—	109	1	no	—
438	#117	—	109	1	no	—
439	#117	C₂×D₁₀₉	1	1	yes	characteristic
440	C₂×C₄⋊C₄	—	109	1	no	—
441	C₂×C₄⋊C₄	—	109	1	no	—
442	C₂×C₂²⋊C₄	—	109	1	no	—
443	C₂×C₂²⋊C₄	—	109	1	no	—
444	C₂×C₄²	—	109	2	no	—
445	C₂×C₄²	—	109	1	no	—
446	C₂×C₄²	—	109	1	no	—
447	C₂×C₄²	—	109	2	no	—
448	C₂⁴	—	109	1	no	—
449	#41	—	218	4	no	—
450	#41	—	218	4	no	—
451	C₂×Q₈	—	109	2	no	—
452	C₂×Q₈	—	109	1	no	—
453	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
454	C₂×Q₈	C₂²×D₁₀₉	1	2	yes	—
455	C₂×D₄	—	109	2	no	—
456	C₂×D₄	—	109	2	no	—
457	C₂×D₄	—	109	2	no	—
458	C₂×D₄	—	109	2	no	—
459	C₂×D₄	—	109	2	no	—
460	C₂×D₄	—	109	2	no	—
461	C₂×D₄	—	109	2	no	—
462	C₂×D₄	—	109	1	no	—
463	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
464	C₂×D₄	C₂²×D₁₀₉	1	2	yes	—
465	C₂²×C₄	—	109	1	no	—
466	C₂²×C₄	—	109	2	no	—
467	C₂²×C₄	—	109	2	no	—
468	C₂²×C₄	—	109	1	no	—
469	C₂²×C₄	—	109	1	no	—
470	C₂²×C₄	—	109	1	no	—
471	C₂²×C₄	—	109	1	no	—
472	C₂²×C₄	—	109	1	no	—
473	C₂²×C₄	—	109	2	no	—
474	C₂²×C₄	—	109	2	no	—
475	C₂²×C₄	—	109	2	no	—
476	C₂²×C₄	—	109	1	no	—
477	C₂²×C₄	—	109	1	no	—
478	C₂²×C₄	—	109	1	no	—
479	C₂²×C₄	—	218	4	no	—
480	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
481	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
482	C₄⋊C₄	—	109	1	no	—
483	C₄⋊C₄	—	109	1	no	—
484	C₄⋊C₄	—	109	1	no	—
485	C₄⋊C₄	—	109	1	no	—
486	C₄⋊C₄	—	109	2	no	—
487	C₄⋊C₄	—	109	2	no	—
488	C₄⋊C₄	—	109	2	no	—
489	C₄⋊C₄	—	109	2	no	—
490	C₄⋊C₄	—	109	2	no	—
491	C₄⋊C₄	—	109	2	no	—
492	C₄⋊C₄	—	109	2	no	—
493	C₄⋊C₄	—	109	2	no	—
494	C₄⋊C₄	—	109	1	no	—
495	C₄⋊C₄	—	109	1	no	—
496	C₄⋊C₄	—	109	1	no	—
497	C₄⋊C₄	—	109	1	no	—
498	C₄⋊C₄	—	109	2	no	—
499	C₄⋊C₄	—	109	2	no	—
500	C₄⋊C₄	—	109	2	no	—
501	C₄⋊C₄	—	109	2	no	—
502	C₄⋊C₄	—	109	2	no	—
503	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
504	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
505	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
506	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
507	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
508	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
509	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
510	C₂²⋊C₄	—	109	1	no	—
511	C₂²⋊C₄	—	109	1	no	—
512	C₂²⋊C₄	—	109	2	no	—
513	C₂²⋊C₄	—	109	2	no	—
514	C₂²⋊C₄	—	109	2	no	—
515	C₂²⋊C₄	—	109	1	no	—
516	C₂²⋊C₄	—	109	1	no	—
517	C₂²⋊C₄	—	109	1	no	—
518	C₂²⋊C₄	—	109	1	no	—
519	C₂²⋊C₄	—	109	2	no	—
520	C₂²⋊C₄	—	109	2	no	—
521	C₂²⋊C₄	—	109	2	no	—
522	C₂²⋊C₄	—	109	2	no	—
523	C₂²⋊C₄	—	109	2	no	—
524	C₂²⋊C₄	—	109	2	no	—
525	C₂²⋊C₄	—	109	2	no	—
526	C₂²⋊C₄	—	109	2	no	—
527	C₂²⋊C₄	—	109	2	no	—
528	C₂²⋊C₄	—	109	2	no	—
529	C₂²⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
530	C₂²⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
531	C₂²⋊C₄	C₂²×D₁₀₉	1	2	yes	—
532	C₂²⋊C₄	C₂²×D₁₀₉	1	2	yes	—
533	C₂²⋊C₄	C₂²×D₁₀₉	1	2	yes	—
534	C₂²⋊C₄	C₂²×D₁₀₉	1	2	yes	—
535	C₂²⋊C₄	C₂²×D₁₀₉	1	2	yes	—
536	C₄²	—	109	1	no	—
537	C₄²	—	109	1	no	—
538	C₄²	—	109	2	no	—
539	C₄²	—	109	2	no	—
540	C₄²	—	109	1	no	—
541	C₄²	—	109	1	no	—
542	C₄²	—	109	1	no	—
543	C₄²	—	109	1	no	—
544	C₄²	—	109	2	no	—
545	C₄²	—	109	2	no	—
546	C₄²	—	109	2	no	—
547	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
548	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
549	C₄²	C₂²×D₁₀₉	1	2	yes	—
550	C₂³	—	109	1	no	—
551	C₂³	—	109	1	no	—
552	C₂³	—	109	2	no	—
553	C₂³	—	436	1	no	—
554	C₂³	—	436	1	no	—
555	C₂³	—	218	2	no	—
556	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
557	C₂³	C₂³×D₁₀₉	1	2	yes	—
558	Q₈	—	2	2	no	—
559	Q₈	—	4	2	no	—
560	Q₈	—	436	2	no	—
561	Q₈	—	218	2	no	—
562	D₄	—	2	2	no	—
563	D₄	—	4	2	no	—
564	D₄	—	436	2	no	—
565	D₄	—	436	2	no	—
566	D₄	—	436	2	no	—
567	D₄	—	436	2	no	—
568	D₄	—	436	2	no	—
569	D₄	—	218	4	no	—
570	D₄	—	218	4	no	—
571	D₄	—	218	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	4	2	no	—
574	C₂×C₄	—	109	1	no	—
575	C₂×C₄	—	109	4	no	—
576	C₂×C₄	—	109	1	no	—
577	C₂×C₄	—	109	1	no	—
578	C₂×C₄	—	109	2	no	—
579	C₂×C₄	—	109	2	no	—
580	C₂×C₄	—	109	2	no	—
581	C₂×C₄	—	109	1	no	—
582	C₂×C₄	—	109	1	no	—
583	C₂×C₄	—	109	1	no	—
584	C₂×C₄	—	436	1	no	—
585	C₂×C₄	—	436	2	no	—
586	C₂×C₄	—	436	1	no	—
587	C₂×C₄	—	436	1	no	—
588	C₂×C₄	—	436	2	no	—
589	C₂×C₄	—	436	1	no	—
590	C₂×C₄	—	436	2	no	—
591	C₂×C₄	—	218	2	no	—
592	C₂×C₄	—	218	4	no	—
593	C₂×C₄	—	218	2	no	—
594	C₂×C₄	—	218	2	no	—
595	C₂×C₄	—	218	4	no	—
596	C₂×C₄	—	218	2	no	—
597	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
598	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
599	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
600	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
601	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
602	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
603	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
604	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
605	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
606	C₂²	—	2	1	no	—
607	C₂²	—	2	2	no	—
608	C₂²	—	2	2	no	—
609	C₂²	—	2	1	no	—
610	C₂²	—	2	2	no	—
611	C₂²	—	2	1	no	—
612	C₂²	—	109	2	no	—
613	C₂²	—	436	4	no	—
614	C₂²	—	218	1	no	—
615	C₂²	—	218	2	no	—
616	C₂²	—	218	2	no	—
617	C₂²	—	218	1	no	—
618	C₂²	—	218	2	no	—
619	C₂²	—	218	1	no	—
620	C₂²	—	218	1	no	—
621	C₂²	—	218	1	no	—
622	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
623	C₄	—	2	1	no	—
624	C₄	—	2	2	no	—
625	C₄	—	2	1	no	—
626	C₄	—	2	2	no	—
627	C₄	—	2	2	no	—
628	C₄	—	2	1	no	—
629	C₄	—	2	1	no	—
630	C₄	—	2	1	no	—
631	C₄	—	109	2	no	—
632	C₄	—	218	1	no	—
633	C₄	—	218	2	no	—
634	C₄	—	218	1	no	—
635	C₄	—	218	1	no	—
636	C₄	—	218	1	no	—
637	C₄	—	218	1	no	—
638	C₄	—	218	2	no	—
639	C₄	—	218	2	no	—
640	C₄	#41×D₁₀₉	1	2	yes	—
641	C₂	—	4	1	no	—
642	C₂	—	4	2	no	—
643	C₂	—	436	1	no	—
644	C₂	—	436	2	no	—
645	C₂	—	218	2	no	—
646	C₂	C₂×#41×D₁₀₉	1	1	yes	characteristic
647	C₂	#20295861	1	1	yes	characteristic
648	C₂	#20295863	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₉
commutator, socle, cosocle	C₂²×C₁₀₉	C₂⁵
Fitting	#542×C₁₀₉	C₂
hypercenter	#542	D₁₀₉
hypocenter	C₁₀₉	#3455
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #542

Complements

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

N	H ≅ G/N	type
1. #20295650	C₂	semidirect
2. #20295647	C₂	semidirect
3. #20295645	C₂	semidirect
4. #20295636	C₂	semidirect
5. #20295634	C₂	semidirect
6. #20295629	C₂	semidirect
7. #20295627	C₂	semidirect
8. #20295622	C₂	semidirect
9. #20295617	C₂	semidirect
10. #20295612	C₂	semidirect
11. #20295610	C₂	semidirect
12. #20295581	C₂	semidirect
13. #20295575	C₂	semidirect
14. #20295562	C₂	semidirect
15. #20295559	C₂	semidirect
16. #20295555	C₂	semidirect
17. #20295553	C₂	semidirect
18. #20295539	C₂	semidirect
19. #20295536	C₂	semidirect
20. C₄×#13182030	C₂	semidirect
21. C₄×#13182028	C₂	semidirect
22. #117×D₁₀₉	C₂	semidirect
23. #542×C₁₀₉	C₂	semidirect
27. #16812699	C₂²	semidirect
28. #16812697	C₂²	semidirect
29. #16812695	C₂²	semidirect
31. #16812694	C₂²	semidirect
32. #16812694	C₂²	semidirect
36. #16812692	C₂²	semidirect
37. #16812691	C₂²	semidirect
38. #16812690	C₂²	semidirect
39. #16812690	C₂²	semidirect
40. #16812690	C₂²	semidirect
41. #16812689	C₂²	semidirect
42. #16812689	C₂²	semidirect
44. #16812687	C₂²	semidirect
50. #16812683	C₂²	semidirect
51. #16812682	C₂²	semidirect
54. #16812681	C₂²	semidirect
55. #16812681	C₂²	semidirect
59. #16812677	C₂²	semidirect
60. #16812676	C₂²	semidirect
63. #16812675	C₂²	semidirect
75. Q₈×Dic₁₀₉	C₂²	semidirect
76. Q₈×Dic₁₀₉	C₂²	semidirect
77. C₄×Dic₂₁₈	C₂²	semidirect
78. #16812664	C₂²	semidirect
79. #16812664	C₂²	semidirect
81. C₄×D₄₃₆	C₂²	semidirect
83. D₄×Dic₁₀₉	C₂²	semidirect
86. #16812661	C₂²	semidirect
91. #16812659	C₂²	semidirect
92. #16812659	C₂²	semidirect
98. #16812657	C₂²	semidirect
103. #16812655	C₂²	semidirect
108. C₂×C₄×Dic₁₀₉	C₂²	semidirect
111. C₄²×D₁₀₉	C₂²	semidirect
112. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
113. #126×C₁₀₉	C₂²	semidirect
115. #124×C₁₀₉	C₂²	semidirect
118. C₂²⋊Q₈×C₁₀₉	C₂²	semidirect
120. C₄×Q₈×C₁₀₉	C₂²	semidirect
121. C₄×D₄×C₁₀₉	C₂²	semidirect
325. C₁₀₉	#3455	semidirect
329. #542	D₁₀₉	semidirect
354. C₄⋊Q₈	C₂×D₁₀₉	semidirect
363. #126	C₂×D₁₀₉	semidirect
375. #124	C₂×D₁₀₉	semidirect
395. C₂²⋊Q₈	C₂×D₁₀₉	semidirect
413. C₄×Q₈	C₂×D₁₀₉	semidirect
427. C₄×D₄	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295650	2
#20295647	2
#20295645	2
#20295636	2
#20295634	2
#20295629	2
#20295627	2
#20295622	2
#20295617	2
#20295612	2
#20295610	2
#20295581	2
#20295575	2
#20295562	2
#20295559	2
#20295555	2
#20295553	2
#20295539	2
#20295536	2
C₄×#13182030	2
C₄×#13182028	2
#117×D₁₀₉	2
#542×C₁₀₉	2
#3455	109