Subgroups — #554×D₁₀₉ #28376559

Order 13952.

1200 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295737	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295736	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295735	C₂	1	2	yes	maximal, complemented
4	#20295734	C₂	1	2	yes	maximal, complemented
5	#20295733	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295732	C₂	1	2	yes	maximal, complemented
7	#20295731	C₂	1	2	yes	maximal, complemented
8	#20295730	C₂	1	4	yes	maximal, complemented
9	C₄⋊Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#125×D₁₀₉	C₂	1	2	yes	maximal, complemented
11	C₂²⋊Q₈×D₁₀₉	C₂	1	4	yes	maximal, complemented
12	C₂²⋊Q₈×D₁₀₉	C₂	1	2	yes	maximal, complemented
13	C₄×Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×D₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
15	C₄×D₄×D₁₀₉	C₂	1	2	yes	maximal
16	C₂×C₄⋊C₄×D₁₀₉	C₂	1	2	yes	maximal, complemented
17	#554×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×Q₈×D₁₀₉	C₂²	1	2	yes	—
20	C₂×D₄×D₁₀₉	C₂²	1	1	yes	characteristic
21	C₂²×C₄×D₁₀₉	C₂²	1	2	yes	—
22	C₂²×C₄×D₁₀₉	C₂²	1	4	yes	—
23	#16812699	C₂²	1	1	yes	characteristic, complemented
24	#16812698	C₂²	1	2	yes	complemented
25	#16812698	C₂²	1	2	yes	complemented
26	#16812697	C₂²	1	2	yes	complemented
27	#16812691	C₂²	1	2	yes	complemented
28	#16812690	C₂²	1	2	yes	complemented
29	#16812690	C₂²	1	2	yes	complemented
30	#16812689	C₂²	1	4	yes	complemented
31	#16812689	C₂²	1	4	yes	complemented
32	#16812679	C₂²	1	2	yes	complemented
33	#16812679	C₂²	1	4	yes	complemented
34	#16812678	C₂²	1	2	yes	complemented
35	#16812678	C₂²	1	4	yes	complemented
36	#16812677	C₂²	1	4	yes	complemented
37	#16812677	C₂²	1	4	yes	complemented
38	#16812677	C₂²	1	4	yes	complemented
39	#16812676	C₂²	1	4	yes	complemented
40	#16812676	C₂²	1	4	yes	complemented
41	#16812676	C₂²	1	4	yes	complemented
42	#16812675	C₂²	1	4	yes	complemented
43	#16812675	C₂²	1	2	yes	complemented
44	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
45	C₄×Dic₂₁₈	C₂²	1	2	yes	complemented
46	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic, complemented
47	#16812664	C₂²	1	2	yes	complemented
48	#16812664	C₂²	1	1	yes	characteristic, complemented
49	C₄×D₄₃₆	C₂²	1	2	yes	—
50	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
51	D₄×Dic₁₀₉	C₂²	1	2	yes	—
52	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
53	#16812661	C₂²	1	1	yes	characteristic
54	#16812661	C₂²	1	2	yes	—
55	C₄×C₂²⋊D₁₀₉	C₂²	1	4	yes	—
56	C₄×C₂²⋊D₁₀₉	C₂²	1	2	yes	—
57	#16812659	C₂²	1	2	yes	—
58	#16812659	C₂²	1	4	yes	—
59	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	complemented
60	C₂×#13182008	C₂²	1	4	yes	complemented
61	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
62	C₄⋊C₄×D₁₀₉	C₂²	1	4	yes	—
63	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	—
64	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
65	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	—
66	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	—
67	C₄⋊C₄×D₁₀₉	C₂²	1	4	yes	—
68	C₄⋊C₄×D₁₀₉	C₂²	1	4	yes	complemented
69	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	complemented
70	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	complemented
71	C₂²⋊C₄×D₁₀₉	C₂²	1	2	yes	—
72	C₂²⋊C₄×D₁₀₉	C₂²	1	4	yes	—
73	C₄²×D₁₀₉	C₂²	1	2	yes	—
74	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
75	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
76	#125×C₁₀₉	C₂²	1	2	yes	complemented
77	C₂²⋊Q₈×C₁₀₉	C₂²	1	4	yes	complemented
78	C₂²⋊Q₈×C₁₀₉	C₂²	1	2	yes	complemented
79	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
80	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
81	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
82	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	complemented
83	C₂³×D₁₀₉	C₂³	1	2	yes	—
84	C₂×Dic₂₁₈	C₂³	1	2	yes	—
85	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
86	C₂×Dic₂₁₈	C₂³	1	2	yes	—
87	C₂×Dic₂₁₈	C₂³	1	4	yes	—
88	Q₈×D₁₀₉	—	2	4	no	—
89	Q₈×D₁₀₉	—	4	4	no	—
90	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
91	C₂×C₂²⋊D₁₀₉	C₂³	1	2	yes	—
92	D₄×D₁₀₉	Q₈	1	8	yes	complemented
93	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
94	C₂²×Dic₁₀₉	C₂³	1	4	yes	—
95	C₂×C₄×D₁₀₉	—	2	8	no	—
96	C₂×C₄×D₁₀₉	—	4	8	no	—
97	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
98	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
99	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
100	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
101	C₂×C₄×D₁₀₉	C₂³	1	4	yes	—
102	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
103	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
104	C₂×C₄×D₁₀₉	C₂³	1	4	yes	—
105	C₂×C₄×D₁₀₉	C₂³	1	4	yes	—
106	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
107	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
108	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
109	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
110	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
111	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
112	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
113	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
114	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
115	#13182008	C₂³	1	2	yes	—
116	#13182008	C₂³	1	4	yes	—
117	#13182008	C₂³	1	4	yes	—
118	#13182008	C₂³	1	4	yes	—
119	#13182008	C₂³	1	2	yes	—
120	#13182008	C₂³	1	4	yes	—
121	#13182008	C₂³	1	2	yes	—
122	#13182008	C₂³	1	2	yes	—
123	#13182008	C₂³	1	4	yes	—
124	#13182008	C₂³	1	4	yes	—
125	C₂²⋊Dic₁₀₉	C₂³	1	4	yes	—
126	C₂²⋊Dic₁₀₉	C₂³	1	2	yes	—
127	#13182006	C₂³	1	2	yes	—
128	#13182006	C₂³	1	4	yes	—
129	#13182006	C₂³	1	4	yes	—
130	#13182006	C₂³	1	2	yes	—
131	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
132	C₄×Dic₁₀₉	C₂³	1	2	yes	—
133	C₄×Dic₁₀₉	C₂³	1	2	yes	—
134	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
135	C₄×Dic₁₀₉	C₂³	1	2	yes	—
136	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
137	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
138	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
139	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
140	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
141	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
142	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
143	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
144	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
145	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
146	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
147	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
148	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
149	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
150	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
151	C₄²×C₁₀₉	C₂³	1	2	yes	—
152	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
153	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
154	C₂²×D₁₀₉	—	2	4	no	—
155	C₂²×D₁₀₉	—	2	4	no	—
156	C₂²×D₁₀₉	C₂×Q₈	1	8	yes	—
157	C₂²×D₁₀₉	C₂⁴	1	2	yes	—
158	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
159	Q₈×C₁₀₉	—	2	2	no	—
160	Q₈×C₁₀₉	—	4	2	no	—
161	D₄×C₁₀₉	C₂×Q₈	1	4	yes	complemented
162	C₂×C₄×C₁₀₉	—	2	4	no	—
163	C₂×C₄×C₁₀₉	—	4	4	no	—
164	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
165	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
166	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
167	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
168	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
169	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
170	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
171	C₂²⋊D₁₀₉	C₂×Q₈	1	8	yes	complemented
172	C₂×Dic₁₀₉	—	2	4	no	—
173	C₂×Dic₁₀₉	—	4	4	no	—
174	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
175	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
176	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
177	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
178	C₂×Dic₁₀₉	C₂⁴	1	4	yes	—
179	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
180	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
181	D₄₃₆	C₂×Q₈	1	4	yes	complemented
182	C₄×D₁₀₉	—	2	4	no	—
183	C₄×D₁₀₉	—	2	4	no	—
184	C₄×D₁₀₉	—	2	4	no	—
185	C₄×D₁₀₉	—	2	8	no	—
186	C₄×D₁₀₉	—	2	2	no	—
187	C₄×D₁₀₉	—	2	4	no	—
188	C₄×D₁₀₉	—	2	4	no	—
189	C₄×D₁₀₉	—	4	4	no	—
190	C₄×D₁₀₉	—	4	4	no	—
191	C₄×D₁₀₉	C₂×Q₈	1	4	yes	—
192	C₄×D₁₀₉	#41	1	4	yes	—
193	Dic₂₁₈	—	2	2	no	—
194	Dic₂₁₈	—	2	4	no	—
195	Dic₂₁₈	—	4	4	no	—
196	Dic₂₁₈	—	4	2	no	—
197	C₂²×C₁₀₉	—	2	2	no	—
198	C₂²×C₁₀₉	—	2	2	no	—
199	C₂²×C₁₀₉	C₂²×Q₈	1	4	yes	—
200	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
201	C₂×D₁₀₉	—	2	2	no	—
202	C₂×D₁₀₉	—	2	2	no	—
203	C₂×D₁₀₉	—	2	16	no	—
204	C₂×D₁₀₉	C₂²×Q₈	1	4	yes	—
205	C₂×D₁₀₉	C₂²×Q₈	1	2	yes	—
206	C₂×D₁₀₉	C₂×#41	1	2	yes	—
207	C₂×D₁₀₉	H(5,𝔽₂)	1	2	yes	—
208	C₄×C₁₀₉	—	2	2	no	—
209	C₄×C₁₀₉	—	2	2	no	—
210	C₄×C₁₀₉	—	2	1	no	—
211	C₄×C₁₀₉	—	2	4	no	—
212	C₄×C₁₀₉	—	2	2	no	—
213	C₄×C₁₀₉	C₂²×Q₈	1	2	yes	—
214	C₄×C₁₀₉	C₂×#41	1	2	yes	—
215	Dic₁₀₉	—	2	2	no	—
216	Dic₁₀₉	—	2	2	no	—
217	Dic₁₀₉	—	2	1	no	—
218	Dic₁₀₉	—	2	4	no	—
219	Dic₁₀₉	—	2	2	no	—
220	Dic₁₀₉	C₂²×Q₈	1	2	yes	—
221	Dic₁₀₉	C₂×#41	1	2	yes	—
222	C₂×C₁₀₉	—	2	4	no	—
223	C₂×C₁₀₉	C₂³×Q₈	1	1	yes	characteristic
224	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
225	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
226	D₁₀₉	—	2	4	no	—
227	D₁₀₉	#554	1	4	yes	direct product factor
228	C₂×#554	—	109	1	no	maximal, Sylow
229	C₁₀₉	C₂×#554	1	1	yes	characteristic, complemented, Sylow
230	#554	—	109	1	no	—
231	#554	—	109	2	no	—
232	#554	—	109	4	no	—
233	#554	—	109	2	no	—
234	#554	—	109	1	no	—
235	#554	—	109	1	no	—
236	#554	—	109	2	no	—
237	#554	—	109	2	no	—
238	#554	D₁₀₉	1	1	yes	characteristic, direct product factor
239	C₂×C₄⋊Q₈	—	109	1	no	—
240	C₂×#125	—	109	2	no	—
241	C₂×C₂²⋊Q₈	—	109	4	no	—
242	C₂×C₂²⋊Q₈	—	109	2	no	—
243	C₂×C₄×Q₈	—	109	1	no	—
244	C₂×C₄×D₄	—	109	2	no	—
245	C₂×C₄×D₄	—	109	1	no	—
246	C₂²×C₄⋊C₄	—	109	2	no	—
247	C₂²×Q₈	—	109	2	no	—
248	C₂²×Q₈	—	109	1	no	—
249	C₂²×D₄	—	109	1	no	—
250	C₂³×C₄	—	109	4	no	—
251	C₂³×C₄	—	109	2	no	—
252	C₄⋊Q₈	—	109	2	no	—
253	C₄⋊Q₈	—	109	2	no	—
254	C₄⋊Q₈	—	109	2	no	—
255	C₄⋊Q₈	—	109	1	no	—
256	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
257	#125	—	109	4	no	—
258	#125	—	109	4	no	—
259	#125	—	109	2	no	—
260	#125	—	109	2	no	—
261	#125	—	109	2	no	—
262	#125	C₂×D₁₀₉	1	2	yes	complemented
263	C₂²⋊Q₈	—	109	4	no	—
264	C₂²⋊Q₈	—	109	2	no	—
265	C₂²⋊Q₈	—	109	4	no	—
266	C₂²⋊Q₈	—	109	2	no	—
267	C₂²⋊Q₈	—	109	4	no	—
268	C₂²⋊Q₈	—	109	4	no	—
269	C₂²⋊Q₈	—	109	4	no	—
270	C₂²⋊Q₈	—	109	4	no	—
271	C₂²⋊Q₈	—	109	2	no	—
272	C₂²⋊Q₈	—	109	4	no	—
273	C₂²⋊Q₈	—	109	4	no	—
274	C₂²⋊Q₈	—	109	4	no	—
275	C₂²⋊Q₈	C₂×D₁₀₉	1	4	yes	complemented
276	C₂²⋊Q₈	C₂×D₁₀₉	1	2	yes	complemented
277	C₄×Q₈	—	109	1	no	—
278	C₄×Q₈	—	109	2	no	—
279	C₄×Q₈	—	109	1	no	—
280	C₄×Q₈	—	109	1	no	—
281	C₄×Q₈	—	109	2	no	—
282	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
283	C₄×D₄	—	109	2	no	—
284	C₄×D₄	—	109	1	no	—
285	C₄×D₄	—	109	4	no	—
286	C₄×D₄	—	109	2	no	—
287	C₄×D₄	—	109	2	no	—
288	C₄×D₄	—	109	1	no	—
289	C₄×D₄	—	109	4	no	—
290	C₄×D₄	—	109	2	no	—
291	C₄×D₄	—	109	2	no	—
292	C₄×D₄	—	109	1	no	—
293	C₄×D₄	C₂×D₁₀₉	1	2	yes	—
294	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
295	C₂×C₄⋊C₄	—	109	4	no	—
296	C₂×C₄⋊C₄	—	109	2	no	—
297	C₂×C₄⋊C₄	—	109	1	no	—
298	C₂×C₄⋊C₄	—	109	4	no	—
299	C₂×C₄⋊C₄	—	109	2	no	—
300	C₂×C₄⋊C₄	—	109	4	no	—
301	C₂×C₄⋊C₄	—	109	2	no	—
302	C₂×C₄⋊C₄	—	109	2	no	—
303	C₂×C₄⋊C₄	—	109	1	no	—
304	C₂×C₄⋊C₄	—	109	2	no	—
305	C₂×C₄⋊C₄	—	109	2	no	—
306	C₂×C₄⋊C₄	—	109	4	no	—
307	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	complemented
308	C₂×C₂²⋊C₄	—	109	4	no	—
309	C₂×C₂²⋊C₄	—	109	2	no	—
310	C₂×C₄²	—	109	2	no	—
311	C₂×C₄²	—	109	1	no	—
312	C₂⁴	—	109	2	no	—
313	C₂×Q₈	—	109	4	no	—
314	C₂×Q₈	—	109	2	no	—
315	C₂×Q₈	—	109	2	no	—
316	C₂×Q₈	—	109	1	no	—
317	C₂×Q₈	—	436	4	no	—
318	C₂×Q₈	—	218	4	no	—
319	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
320	C₂×Q₈	C₂²×D₁₀₉	1	2	yes	—
321	C₂×D₄	—	109	2	no	—
322	C₂×D₄	—	109	8	no	—
323	C₂×D₄	—	109	1	no	—
324	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
325	C₂²×C₄	—	109	1	no	—
326	C₂²×C₄	—	109	1	no	—
327	C₂²×C₄	—	109	4	no	—
328	C₂²×C₄	—	109	2	no	—
329	C₂²×C₄	—	109	4	no	—
330	C₂²×C₄	—	109	4	no	—
331	C₂²×C₄	—	109	2	no	—
332	C₂²×C₄	—	109	1	no	—
333	C₂²×C₄	—	109	2	no	—
334	C₂²×C₄	—	109	2	no	—
335	C₂²×C₄	—	109	4	no	—
336	C₂²×C₄	—	109	2	no	—
337	C₂²×C₄	—	109	2	no	—
338	C₂²×C₄	—	436	8	no	—
339	C₂²×C₄	—	218	8	no	—
340	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
341	C₂²×C₄	C₂²×D₁₀₉	1	4	yes	—
342	C₄⋊C₄	—	109	2	no	—
343	C₄⋊C₄	—	109	4	no	—
344	C₄⋊C₄	—	109	4	no	—
345	C₄⋊C₄	—	109	4	no	—
346	C₄⋊C₄	—	109	2	no	—
347	C₄⋊C₄	—	109	4	no	—
348	C₄⋊C₄	—	109	2	no	—
349	C₄⋊C₄	—	109	2	no	—
350	C₄⋊C₄	—	109	4	no	—
351	C₄⋊C₄	—	109	4	no	—
352	C₄⋊C₄	—	109	1	no	—
353	C₄⋊C₄	—	109	4	no	—
354	C₄⋊C₄	—	109	2	no	—
355	C₄⋊C₄	—	109	1	no	—
356	C₄⋊C₄	—	109	2	no	—
357	C₄⋊C₄	—	109	2	no	—
358	C₄⋊C₄	—	109	4	no	—
359	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
360	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
361	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
362	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
363	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
364	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
365	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
366	C₂²⋊C₄	—	109	4	no	—
367	C₂²⋊C₄	—	109	2	no	—
368	C₂²⋊C₄	—	109	4	no	—
369	C₂²⋊C₄	—	109	2	no	—
370	C₂²⋊C₄	—	109	2	no	—
371	C₂²⋊C₄	—	109	4	no	—
372	C₂²⋊C₄	C₂²×D₁₀₉	1	4	yes	—
373	C₂²⋊C₄	C₂²×D₁₀₉	1	2	yes	—
374	C₄²	—	109	2	no	—
375	C₄²	—	109	2	no	—
376	C₄²	—	109	2	no	—
377	C₄²	—	109	1	no	—
378	C₄²	—	109	1	no	—
379	C₄²	—	109	1	no	—
380	C₄²	C₂²×D₁₀₉	1	2	yes	—
381	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
382	C₂³	—	109	1	no	—
383	C₂³	—	109	8	no	—
384	C₂³	—	109	2	no	—
385	C₂³	—	218	4	no	—
386	C₂³	—	218	4	no	—
387	C₂³	C₂³×D₁₀₉	1	2	yes	—
388	Q₈	—	2	2	no	—
389	Q₈	—	4	2	no	—
390	Q₈	—	436	4	no	—
391	Q₈	—	436	2	no	—
392	Q₈	—	218	4	no	—
393	Q₈	—	218	2	no	—
394	D₄	—	109	8	no	—
395	D₄	—	109	4	no	—
396	D₄	Q₈×D₁₀₉	1	4	yes	complemented
397	C₂×C₄	—	2	4	no	—
398	C₂×C₄	—	4	4	no	—
399	C₂×C₄	—	109	1	no	—
400	C₂×C₄	—	109	1	no	—
401	C₂×C₄	—	109	4	no	—
402	C₂×C₄	—	109	4	no	—
403	C₂×C₄	—	109	2	no	—
404	C₂×C₄	—	109	1	no	—
405	C₂×C₄	—	109	2	no	—
406	C₂×C₄	—	109	2	no	—
407	C₂×C₄	—	109	4	no	—
408	C₂×C₄	—	436	4	no	—
409	C₂×C₄	—	436	4	no	—
410	C₂×C₄	—	436	4	no	—
411	C₂×C₄	—	218	4	no	—
412	C₂×C₄	—	218	4	no	—
413	C₂×C₄	—	218	8	no	—
414	C₂×C₄	—	218	4	no	—
415	C₂×C₄	—	218	2	no	—
416	C₂×C₄	—	218	4	no	—
417	C₂×C₄	—	218	4	no	—
418	C₂×C₄	—	218	4	no	—
419	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
420	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
421	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
422	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
423	C₂×C₄	C₂³×D₁₀₉	1	4	yes	—
424	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
425	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
426	C₂²	—	2	2	no	—
427	C₂²	—	2	2	no	—
428	C₂²	—	109	2	no	—
429	C₂²	—	109	2	no	—
430	C₂²	—	109	2	no	—
431	C₂²	—	109	4	no	—
432	C₂²	—	218	16	no	—
433	C₂²	—	218	2	no	—
434	C₂²	—	218	2	no	—
435	C₂²	C₂×Q₈×D₁₀₉	1	4	yes	—
436	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
437	C₄	—	2	2	no	—
438	C₄	—	2	2	no	—
439	C₄	—	2	2	no	—
440	C₄	—	2	4	no	—
441	C₄	—	2	1	no	—
442	C₄	—	109	2	no	—
443	C₄	—	109	2	no	—
444	C₄	—	218	2	no	—
445	C₄	—	218	2	no	—
446	C₄	—	218	2	no	—
447	C₄	—	218	4	no	—
448	C₄	—	218	1	no	—
449	C₄	C₂×Q₈×D₁₀₉	1	2	yes	—
450	C₄	#41×D₁₀₉	1	2	yes	—
451	C₂	—	2	4	no	—
452	C₂	—	109	4	no	—
453	C₂	—	218	4	no	—
454	C₂	C₂²×Q₈×D₁₀₉	1	1	yes	characteristic
455	C₂	C₂×#41×D₁₀₉	1	1	yes	characteristic
456	C₂	H(5,𝔽₂)×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₉
commutator, socle, cosocle	C₂²×C₁₀₉	C₂⁵
Fitting	#554×C₁₀₉	C₂
prime radical	C₂×D₄×D₁₀₉	C₂²
hypercenter	#554	D₁₀₉
hypocenter	C₁₀₉	C₂×#554
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #554

Complements

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

N	H ≅ G/N	type
1. #20295737	C₂	semidirect
2. #20295736	C₂	semidirect
3. #20295735	C₂	semidirect
4. #20295734	C₂	semidirect
5. #20295733	C₂	semidirect
6. #20295732	C₂	semidirect
7. #20295731	C₂	semidirect
8. #20295730	C₂	semidirect
9. C₄⋊Q₈×D₁₀₉	C₂	semidirect
10. #125×D₁₀₉	C₂	semidirect
11. C₂²⋊Q₈×D₁₀₉	C₂	semidirect
12. C₂²⋊Q₈×D₁₀₉	C₂	semidirect
13. C₄×Q₈×D₁₀₉	C₂	semidirect
16. C₂×C₄⋊C₄×D₁₀₉	C₂	semidirect
17. #554×C₁₀₉	C₂	semidirect
23. #16812699	C₂²	semidirect
24. #16812698	C₂²	semidirect
25. #16812698	C₂²	semidirect
26. #16812697	C₂²	semidirect
27. #16812691	C₂²	semidirect
28. #16812690	C₂²	semidirect
29. #16812690	C₂²	semidirect
30. #16812689	C₂²	semidirect
31. #16812689	C₂²	semidirect
32. #16812679	C₂²	semidirect
33. #16812679	C₂²	semidirect
34. #16812678	C₂²	semidirect
35. #16812678	C₂²	semidirect
36. #16812677	C₂²	semidirect
37. #16812677	C₂²	semidirect
38. #16812677	C₂²	semidirect
39. #16812676	C₂²	semidirect
40. #16812676	C₂²	semidirect
41. #16812676	C₂²	semidirect
42. #16812675	C₂²	semidirect
43. #16812675	C₂²	semidirect
44. Q₈×Dic₁₀₉	C₂²	semidirect
45. C₄×Dic₂₁₈	C₂²	semidirect
46. C₄×Dic₂₁₈	C₂²	semidirect
47. #16812664	C₂²	semidirect
48. #16812664	C₂²	semidirect
59. C₂×C₄⋊Dic₁₀₉	C₂²	semidirect
60. C₂×#13182008	C₂²	semidirect
68. C₄⋊C₄×D₁₀₉	C₂²	semidirect
69. C₄⋊C₄×D₁₀₉	C₂²	semidirect
70. C₄⋊C₄×D₁₀₉	C₂²	semidirect
75. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
76. #125×C₁₀₉	C₂²	semidirect
77. C₂²⋊Q₈×C₁₀₉	C₂²	semidirect
78. C₂²⋊Q₈×C₁₀₉	C₂²	semidirect
79. C₄×Q₈×C₁₀₉	C₂²	semidirect
82. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
92. D₄×D₁₀₉	Q₈	semidirect
161. D₄×C₁₀₉	C₂×Q₈	semidirect
171. C₂²⋊D₁₀₉	C₂×Q₈	semidirect
181. D₄₃₆	C₂×Q₈	semidirect
227. D₁₀₉	#554	direct (N × H)
229. C₁₀₉	C₂×#554	semidirect
238. #554	D₁₀₉	direct (N × H)
256. C₄⋊Q₈	C₂×D₁₀₉	semidirect
262. #125	C₂×D₁₀₉	semidirect
275. C₂²⋊Q₈	C₂×D₁₀₉	semidirect
276. C₂²⋊Q₈	C₂×D₁₀₉	semidirect
282. C₄×Q₈	C₂×D₁₀₉	semidirect
307. C₂×C₄⋊C₄	C₂×D₁₀₉	semidirect
396. D₄	Q₈×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295737	2
#20295736	2
#20295735	2
#20295734	2
#20295733	2
#20295732	2
#20295731	2
#20295730	2
C₄⋊Q₈×D₁₀₉	2
#125×D₁₀₉	2
C₂²⋊Q₈×D₁₀₉	2
C₄×Q₈×D₁₀₉	2
C₄×D₄×D₁₀₉	2
C₂×C₄⋊C₄×D₁₀₉	2
#554×C₁₀₉	2
C₂×#554	109

Subgroups — #554×D109 #28376559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #554×D₁₀₉ #28376559