Subgroups — #36712339

Order 22784.

498 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033120	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022538	C₂	1	2	yes	maximal, complemented
4	#26022534	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022410	C₂	1	2	yes	maximal, complemented
6	#26022400	C₂	1	2	yes	maximal, complemented
7	#26022400	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022390	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022129	C₂	1	2	yes	maximal
10	C₄×D₁₄₂₄	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152308	—	2	1	no	—
13	Q₈×D₃₅₆	—	2	1	no	—
14	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
15	#19152003	C₂²	1	1	yes	characteristic
16	#19152003	C₂²	1	2	yes	—
17	#19152003	C₂²	1	1	yes	characteristic
18	#19152000	C₂²	1	2	yes	—
19	#19151899	—	2	2	no	—
20	#19151889	—	2	1	no	—
21	#19151879	—	2	1	no	—
22	#19151879	—	2	2	no	—
23	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
24	C₄×D₇₁₂	C₂²	1	2	yes	—
25	C₄×D₇₁₂	C₂²	1	2	yes	—
26	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
27	#19151426	C₂²	1	2	yes	—
28	#19151426	C₂²	1	2	yes	—
29	#19151426	C₂²	1	2	yes	—
30	#19151424	C₂²	1	2	yes	—
31	#19151424	C₂²	1	1	yes	characteristic
32	#19151400	C₂²	1	1	yes	characteristic
33	#19151400	C₂²	1	2	yes	—
34	#19151400	C₂²	1	2	yes	—
35	#19151400	C₂²	1	1	yes	characteristic
36	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
38	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	#346×C₈₉	C₂²	1	2	yes	—
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×Q₈×D₈₉	—	4	1	no	—
42	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
43	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
44	#15668012	—	2	2	no	—
45	#15668012	—	2	1	no	—
46	#15668012	—	2	1	no	—
47	#15668004	—	2	2	no	—
48	#15667992	—	4	2	no	—
49	#15667992	—	4	1	no	—
50	#15667990	—	4	2	no	—
51	#15667988	—	4	1	no	—
52	C₄×D₃₅₆	—	2	2	no	—
53	C₄×D₃₅₆	—	2	1	no	—
54	C₄×D₃₅₆	—	2	1	no	—
55	C₄×D₃₅₆	—	2	2	no	—
56	C₄⋊C₄×D₈₉	—	4	1	no	—
57	D₁₄₂₄	Q₈	1	4	yes	complemented
58	#15667946	—	2	2	no	—
59	#15667946	—	2	1	no	—
60	#15667946	—	2	1	no	—
61	#15667946	C₂³	1	1	yes	characteristic
62	#15667946	C₂³	1	1	yes	characteristic
63	#15667946	C₂³	1	2	yes	—
64	#15667946	C₂³	1	2	yes	—
65	#15667944	—	2	2	no	—
66	#15667934	—	2	2	no	—
67	#15667934	—	2	1	no	—
68	#15667934	—	2	2	no	—
69	#15667934	—	2	1	no	—
70	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
71	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
72	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
75	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
76	C₄×C₈×C₈₉	C₂³	1	2	yes	—
77	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₂×Dic₁₇₈	—	4	2	no	—
79	C₂×Dic₁₇₈	—	4	1	no	—
80	C₂×Dic₁₇₈	—	4	1	no	—
81	Q₈×D₈₉	—	8	4	no	—
82	C₂×D₃₅₆	—	2	1	no	—
83	C₂×D₃₅₆	—	2	1	no	—
84	C₂×C₄×D₈₉	—	4	1	no	—
85	C₂×C₄×D₈₉	—	4	2	no	—
86	C₂×C₄×D₈₉	—	4	1	no	—
87	C₂×C₄×D₈₉	—	4	2	no	—
88	D₇₁₂	—	2	1	no	—
89	D₇₁₂	—	2	1	no	—
90	D₇₁₂	C₂×Q₈	1	2	yes	—
91	D₇₁₂	C₂×Q₈	1	2	yes	complemented
92	C₄⋊Dic₈₉	—	2	1	no	—
93	C₄⋊Dic₈₉	—	2	2	no	—
94	C₄⋊Dic₈₉	—	2	2	no	—
95	C₄⋊Dic₈₉	—	2	1	no	—
96	C₄⋊Dic₈₉	—	4	1	no	—
97	C₄⋊Dic₈₉	—	4	1	no	—
98	C₄⋊Dic₈₉	—	4	2	no	—
99	#13083553	—	4	2	no	—
100	#13083553	—	4	2	no	—
101	#13083551	—	4	2	no	—
102	#13083551	—	4	1	no	—
103	#13083551	—	4	1	no	—
104	#13083551	—	4	2	no	—
105	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
110	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
111	C₄²×C₈₉	C₂×D₄	1	2	yes	—
112	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₁₆×C₈₉	—	2	2	no	—
114	C₁₆×C₈₉	—	2	1	no	—
115	C₁₆×C₈₉	C₂×Q₈	1	2	yes	complemented
116	C₂²×D₈₉	—	4	1	no	—
117	C₂²×D₈₉	—	4	1	no	—
118	Q₈×C₈₉	—	2	2	no	—
119	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
124	C₂×Dic₈₉	—	4	1	no	—
125	C₂×Dic₈₉	—	4	2	no	—
126	C₂×Dic₈₉	—	4	1	no	—
127	C₂×Dic₈₉	—	4	2	no	—
128	D₃₅₆	—	2	2	no	—
129	D₃₅₆	—	2	2	no	—
130	D₃₅₆	—	4	1	no	—
131	D₃₅₆	—	4	1	no	—
132	C₄×D₈₉	—	8	4	no	—
133	C₄×D₈₉	—	8	2	no	—
134	C₄×D₈₉	—	8	2	no	—
135	C₄×D₈₉	—	16	2	no	—
136	Dic₁₇₈	—	8	4	no	—
137	Dic₁₇₈	—	8	2	no	—
138	Dic₁₇₈	—	16	1	no	—
139	C₈×C₈₉	—	2	2	no	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
142	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
144	C₂×D₈₉	—	4	2	no	—
145	C₂×D₈₉	—	4	2	no	—
146	C₂×D₈₉	—	8	1	no	—
147	C₂×D₈₉	—	8	1	no	—
148	C₂×D₈₉	—	8	1	no	—
149	C₂×D₈₉	—	8	1	no	—
150	C₄×C₈₉	—	2	2	no	—
151	C₄×C₈₉	—	2	1	no	—
152	C₄×C₈₉	—	2	2	no	—
153	C₄×C₈₉	#509	1	2	yes	—
154	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
155	C₄×C₈₉	#578	1	1	yes	characteristic
156	Dic₈₉	—	8	1	no	—
157	Dic₈₉	—	8	2	no	—
158	Dic₈₉	—	8	1	no	—
159	Dic₈₉	—	8	2	no	—
160	#33951	—	89	1	no	maximal, Sylow
161	C₂×C₈₉	Q₈×D₈	1	1	yes	characteristic
162	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
163	C₂×C₈₉	#3416	1	1	yes	characteristic
164	D₈₉	—	8	2	no	—
165	D₈₉	—	8	2	no	—
166	#3380	—	89	1	no	—
167	Q₈×D₈	—	89	1	no	—
168	#2255	—	89	2	no	—
169	#2254	—	89	1	no	—
170	#2228	—	89	2	no	—
171	#2226	—	89	2	no	—
172	#2226	—	89	1	no	—
173	#2224	—	89	1	no	—
174	#2183	D₈₉	1	1	yes	characteristic, complemented
175	#2177	—	89	2	no	—
176	C₄×D₁₆	—	89	1	no	—
177	C₈₉	#33951	1	1	yes	characteristic, complemented, Sylow
178	#554	—	178	1	no	—
179	D₄×Q₈	—	178	1	no	—
180	C₂×D₁₆	—	89	1	no	—
181	#500	—	89	1	no	—
182	#500	—	89	2	no	—
183	#500	—	89	1	no	—
184	#499	—	89	2	no	—
185	#478	—	178	2	no	—
186	#476	—	178	1	no	—
187	#474	—	178	1	no	—
188	#474	—	178	2	no	—
189	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
190	C₄×D₈	—	89	1	no	—
191	C₄×D₈	—	89	2	no	—
192	C₄×D₈	—	89	1	no	—
193	C₄×D₈	—	89	2	no	—
194	#367	—	89	2	no	—
195	#367	—	89	2	no	—
196	#367	—	89	2	no	—
197	#366	—	89	1	no	—
198	#366	—	89	2	no	—
199	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
200	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
201	#357	—	89	1	no	—
202	#357	—	89	1	no	—
203	#357	—	89	2	no	—
204	#357	—	89	2	no	—
205	#346	C₂×D₈₉	1	2	yes	—
206	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
207	C₂²×Q₈	—	356	1	no	—
208	C₂×D₈	—	89	1	no	—
209	C₂×D₈	—	89	1	no	—
210	C₄⋊Q₈	—	178	2	no	—
211	C₄⋊Q₈	—	178	1	no	—
212	C₄⋊Q₈	—	178	1	no	—
213	#125	—	178	2	no	—
214	C₂²⋊Q₈	—	356	1	no	—
215	C₂²⋊Q₈	—	356	2	no	—
216	C₂²⋊Q₈	—	356	1	no	—
217	C₂²⋊Q₈	—	356	2	no	—
218	C₄×Q₈	D₃₅₆	1	1	yes	characteristic
219	C₄×D₄	—	178	2	no	—
220	C₄×D₄	—	178	1	no	—
221	C₄×D₄	—	178	1	no	—
222	C₄×D₄	—	178	2	no	—
223	C₂×C₄⋊C₄	—	356	1	no	—
224	D₁₆	—	89	4	no	—
225	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
226	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
227	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
228	#107	—	89	1	no	—
229	#107	—	89	1	no	—
230	#107	—	89	2	no	—
231	#107	—	89	2	no	—
232	#107	—	178	2	no	—
233	#107	—	178	1	no	—
234	#107	—	178	1	no	—
235	#106	—	178	2	no	—
236	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
237	C₄⋊C₈	D₃₅₆	1	2	yes	—
238	#102	—	178	2	no	—
239	#102	—	178	1	no	—
240	#102	—	178	2	no	—
241	#102	—	178	1	no	—
242	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₈₉	1	2	yes	—
244	C₂×Q₈	—	356	1	no	—
245	C₂×Q₈	—	356	2	no	—
246	C₂×Q₈	—	356	1	no	—
247	C₂×Q₈	—	712	4	no	—
248	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
249	C₂×D₄	—	178	1	no	—
250	C₂×D₄	—	178	1	no	—
251	C₂²×C₄	—	356	2	no	—
252	C₂²×C₄	—	356	1	no	—
253	C₂²×C₄	—	356	2	no	—
254	C₂²×C₄	—	356	1	no	—
255	D₈	—	89	2	no	—
256	D₈	—	89	2	no	—
257	D₈	—	178	1	no	—
258	D₈	—	178	1	no	—
259	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
260	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
261	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
262	C₄⋊C₄	—	356	2	no	—
263	C₄⋊C₄	—	356	1	no	—
264	C₄⋊C₄	—	356	2	no	—
265	C₄⋊C₄	—	356	1	no	—
266	C₄⋊C₄	—	356	2	no	—
267	C₄⋊C₄	—	178	1	no	—
268	C₄⋊C₄	—	178	2	no	—
269	C₄⋊C₄	—	178	2	no	—
270	C₄⋊C₄	—	178	1	no	—
271	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
272	C₄⋊C₄	D₇₁₂	1	2	yes	—
273	C₂²⋊C₄	—	356	1	no	—
274	C₂²⋊C₄	—	356	2	no	—
275	C₂²⋊C₄	—	356	1	no	—
276	C₂²⋊C₄	—	356	2	no	—
277	C₄²	C₂×D₃₅₆	1	2	yes	—
278	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
279	C₁₆	—	2	1	no	—
280	C₁₆	—	2	2	no	—
281	C₁₆	Q₈×D₈₉	1	2	yes	complemented
282	C₂³	—	356	1	no	—
283	C₂³	—	356	1	no	—
284	Q₈	—	2	2	no	—
285	Q₈	—	1424	1	no	—
286	Q₈	—	712	4	no	—
287	Q₈	—	712	2	no	—
288	D₄	—	356	1	no	—
289	D₄	—	356	1	no	—
290	D₄	—	178	2	no	—
291	D₄	—	178	2	no	—
292	C₂×C₄	—	356	1	no	—
293	C₂×C₄	—	356	2	no	—
294	C₂×C₄	—	356	2	no	—
295	C₂×C₄	—	356	1	no	—
296	C₂×C₄	—	1424	2	no	—
297	C₂×C₄	—	712	4	no	—
298	C₂×C₄	—	712	2	no	—
299	C₂×C₄	—	712	2	no	—
300	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
301	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
302	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
303	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	2	1	no	—
307	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
308	C₈	#15668059	1	1	yes	characteristic
309	C₂²	—	356	2	no	—
310	C₂²	—	356	2	no	—
311	C₂²	—	712	1	no	—
312	C₂²	—	712	1	no	—
313	C₂²	—	712	1	no	—
314	C₂²	—	712	1	no	—
315	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	1	no	—
319	C₄	—	712	2	no	—
320	C₄	—	712	1	no	—
321	C₄	—	712	1	no	—
322	C₄	—	712	2	no	—
323	C₄	#19152036	1	2	yes	—
324	C₄	Q₈×D₃₅₆	1	1	yes	characteristic
325	C₄	#19152415	1	1	yes	characteristic
326	C₂	—	712	2	no	—
327	C₂	—	712	2	no	—
328	C₂	Q₈×D₇₁₂	1	1	yes	characteristic
329	C₂	C₂×#19152036	1	1	yes	characteristic
330	C₂	#26033400	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2183	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₁₄₂₄	C₂²
hypercenter	#2183	D₈₉
hypocenter	C₈₉	#33951
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
1. #26033120	C₂	semidirect
2. Q₈×D₇₁₂	C₂	semidirect
3. #26022538	C₂	semidirect
4. #26022534	C₂	semidirect
5. #26022410	C₂	semidirect
6. #26022400	C₂	semidirect
7. #26022400	C₂	semidirect
8. #26022390	C₂	semidirect
11. C₈₉×#2183	C₂	semidirect
57. D₁₄₂₄	Q₈	semidirect
91. D₇₁₂	C₂×Q₈	semidirect
115. C₁₆×C₈₉	C₂×Q₈	semidirect
174. #2183	D₈₉	semidirect
177. C₈₉	#33951	semidirect
281. C₁₆	Q₈×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26033120	2
Q₈×D₇₁₂	2
#26022538	2
#26022534	2
#26022410	2
#26022400	2
#26022390	2
#26022129	2
C₄×D₁₄₂₄	2
C₈₉×#2183	2
#33951	89