Subgroups — #36712315

Order 22784.

462 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033144	C₂	1	1	yes	maximal, characteristic, complemented
2	#26033120	C₂	1	1	yes	maximal, characteristic
3	#26022534	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022531	C₂	1	2	yes	maximal, complemented
5	#26022417	C₂	1	2	yes	maximal, complemented
6	#26022410	C₂	1	2	yes	maximal
7	#26022405	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022390	C₂	1	1	yes	maximal, characteristic
9	C₄×#15667957	C₂	1	1	yes	maximal, characteristic
10	C₄×#15667957	C₂	1	2	yes	maximal
11	Q₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152326	—	2	1	no	—
13	#19152308	—	2	1	no	—
14	C₂×#15667957	C₂²	1	1	yes	characteristic
15	#19152003	C₂²	1	1	yes	characteristic
16	#19152003	C₂²	1	1	yes	characteristic
17	#19152000	C₂²	1	2	yes	—
18	#19152000	C₂²	1	2	yes	—
19	#19151906	—	2	2	no	—
20	#19151899	—	2	2	no	—
21	#19151889	—	2	1	no	—
22	#19151884	—	2	1	no	—
23	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
24	C₄×Dic₃₅₆	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	1	yes	characteristic
29	#19151426	C₂²	1	2	yes	—
30	#19151424	C₂²	1	2	yes	—
31	#19151424	C₂²	1	2	yes	—
32	#19151403	C₂²	1	2	yes	—
33	#19151403	C₂²	1	1	yes	characteristic
34	#19151400	C₂²	1	1	yes	characteristic
35	#19151400	C₂²	1	2	yes	—
36	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
39	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
42	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
43	#15668012	—	2	1	no	—
44	#15668012	—	2	1	no	—
45	#15668010	—	4	1	no	—
46	#15668004	—	2	2	no	—
47	#15668004	—	2	2	no	—
48	#15668002	—	4	2	no	—
49	#15667990	—	4	2	no	—
50	#15667988	—	4	1	no	—
51	C₄×Dic₁₇₈	—	2	1	no	—
52	C₄×Dic₁₇₈	—	2	2	no	—
53	#15667977	—	4	1	no	—
54	C₄×D₃₅₆	—	2	1	no	—
55	C₄×D₃₅₆	—	2	2	no	—
56	C₄⋊C₄×D₈₉	—	4	1	no	—
57	#15667957	Q₈	1	4	yes	complemented
58	#15667946	—	2	1	no	—
59	#15667946	—	2	1	no	—
60	#15667946	C₂³	1	1	yes	characteristic
61	#15667946	C₂³	1	2	yes	—
62	#15667946	C₂³	1	1	yes	characteristic
63	#15667946	C₂³	1	2	yes	—
64	#15667944	—	2	2	no	—
65	#15667944	—	2	2	no	—
66	#15667937	—	2	2	no	—
67	#15667937	—	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	—	2	1	no	—
79	C₂×Dic₁₇₈	—	4	1	no	—
80	C₂×Dic₁₇₈	—	4	1	no	—
81	C₂×D₃₅₆	—	2	1	no	—
82	C₂×C₄×D₈₉	—	4	1	no	—
83	C₂×C₄×D₈₉	—	4	2	no	—
84	Dic₃₅₆	—	2	1	no	—
85	Dic₃₅₆	C₂×Q₈	1	2	yes	—
86	D₇₁₂	—	2	1	no	—
87	D₇₁₂	C₂×Q₈	1	2	yes	—
88	C₄⋊Dic₈₉	—	2	1	no	—
89	C₄⋊Dic₈₉	—	2	2	no	—
90	C₄⋊Dic₈₉	—	2	1	no	—
91	C₄⋊Dic₈₉	—	2	2	no	—
92	C₄⋊Dic₈₉	—	4	1	no	—
93	C₄⋊Dic₈₉	—	4	1	no	—
94	#13083553	—	4	2	no	—
95	#13083553	—	4	1	no	—
96	#13083553	—	4	2	no	—
97	#13083553	—	4	2	no	—
98	#13083553	—	4	2	no	—
99	#13083553	—	4	2	no	—
100	#13083551	—	4	2	no	—
101	#13083551	—	4	1	no	—
102	C₄×Dic₈₉	—	4	2	no	—
103	C₄×Dic₈₉	—	4	1	no	—
104	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
109	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
110	C₄²×C₈₉	C₂×D₄	1	2	yes	—
111	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₈₉	—	2	2	no	—
113	C₁₆×C₈₉	—	2	1	no	—
114	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
115	C₂²×D₈₉	—	4	1	no	—
116	Q₈×C₈₉	—	2	2	no	—
117	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
122	C₂×Dic₈₉	—	4	1	no	—
123	C₂×Dic₈₉	—	4	1	no	—
124	C₂×Dic₈₉	—	4	2	no	—
125	C₂×Dic₈₉	—	4	1	no	—
126	C₂×Dic₈₉	—	4	2	no	—
127	D₃₅₆	—	2	2	no	—
128	D₃₅₆	—	4	1	no	—
129	C₄×D₈₉	—	8	2	no	—
130	C₄×D₈₉	—	16	2	no	—
131	Dic₁₇₈	—	2	2	no	—
132	Dic₁₇₈	—	4	1	no	—
133	Dic₁₇₈	—	8	2	no	—
134	Dic₁₇₈	—	16	1	no	—
135	C₈×C₈₉	—	2	2	no	—
136	C₈×C₈₉	—	2	1	no	—
137	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
138	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
139	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
140	C₂×D₈₉	—	4	2	no	—
141	C₂×D₈₉	—	8	1	no	—
142	C₂×D₈₉	—	8	1	no	—
143	C₄×C₈₉	—	2	2	no	—
144	C₄×C₈₉	—	2	1	no	—
145	C₄×C₈₉	—	2	2	no	—
146	C₄×C₈₉	#508	1	2	yes	—
147	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
148	C₄×C₈₉	#578	1	1	yes	characteristic
149	Dic₈₉	—	4	2	no	—
150	Dic₈₉	—	8	2	no	—
151	Dic₈₉	—	8	1	no	—
152	Dic₈₉	—	8	2	no	—
153	Dic₈₉	—	8	1	no	—
154	#33948	—	89	1	no	maximal, Sylow
155	C₂×C₈₉	Q₈×D₈	1	1	yes	characteristic
156	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
157	C₂×C₈₉	#3416	1	1	yes	characteristic
158	D₈₉	—	8	2	no	—
159	#3383	—	89	1	no	—
160	#3380	—	89	1	no	—
161	#2254	—	89	1	no	—
162	#2253	—	89	2	no	—
163	#2229	—	89	2	no	—
164	#2228	—	89	2	no	—
165	#2227	—	89	1	no	—
166	#2224	—	89	1	no	—
167	Q₈×C₁₆	D₈₉	1	1	yes	characteristic, complemented
168	C₄×SD₃₂	—	89	2	no	—
169	C₄×SD₃₂	—	89	1	no	—
170	C₈₉	#33948	1	1	yes	characteristic, complemented, Sylow
171	#557	—	178	1	no	—
172	#554	—	178	1	no	—
173	C₂×SD₃₂	—	89	1	no	—
174	#500	—	89	1	no	—
175	#500	—	89	1	no	—
176	#499	—	89	2	no	—
177	#499	—	89	2	no	—
178	#479	—	178	2	no	—
179	#478	—	178	2	no	—
180	#476	—	178	1	no	—
181	#475	—	178	1	no	—
182	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
183	C₄×Q₁₆	—	89	1	no	—
184	C₄×Q₁₆	—	89	2	no	—
185	C₄×D₈	—	89	1	no	—
186	C₄×D₈	—	89	2	no	—
187	#367	—	89	1	no	—
188	#367	—	89	2	no	—
189	#367	—	89	2	no	—
190	#366	—	89	2	no	—
191	#366	—	89	2	no	—
192	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
193	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
194	#358	—	89	2	no	—
195	#358	—	89	1	no	—
196	#357	—	89	2	no	—
197	#357	—	89	1	no	—
198	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
199	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
200	C₂×Q₁₆	—	89	1	no	—
201	C₂×D₈	—	89	1	no	—
202	C₄⋊Q₈	—	356	1	no	—
203	C₄⋊Q₈	—	178	1	no	—
204	C₄⋊Q₈	—	178	1	no	—
205	#125	—	356	2	no	—
206	#125	—	178	2	no	—
207	#125	—	178	2	no	—
208	C₂²⋊Q₈	—	356	1	no	—
209	C₂²⋊Q₈	—	356	2	no	—
210	C₄×Q₈	—	356	1	no	—
211	C₄×Q₈	—	178	1	no	—
212	C₄×Q₈	—	178	2	no	—
213	C₄×Q₈	D₃₅₆	1	1	yes	characteristic
214	C₄×D₄	—	178	1	no	—
215	C₄×D₄	—	178	2	no	—
216	C₂×C₄⋊C₄	—	356	1	no	—
217	SD₃₂	—	89	4	no	—
218	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
219	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
220	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
221	#107	—	89	1	no	—
222	#107	—	89	2	no	—
223	#107	—	89	1	no	—
224	#107	—	89	2	no	—
225	#107	—	178	1	no	—
226	#107	—	178	1	no	—
227	#106	—	178	2	no	—
228	#106	—	178	2	no	—
229	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
230	C₄⋊C₈	D₃₅₆	1	2	yes	—
231	#103	—	178	2	no	—
232	#103	—	178	1	no	—
233	#102	—	178	2	no	—
234	#102	—	178	1	no	—
235	C₄×C₈	C₂²×D₈₉	1	2	yes	—
236	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
237	C₂×Q₈	—	356	1	no	—
238	C₂×Q₈	—	356	1	no	—
239	C₂×Q₈	—	178	1	no	—
240	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
241	C₂×D₄	—	178	1	no	—
242	C₂²×C₄	—	356	2	no	—
243	C₂²×C₄	—	356	1	no	—
244	Q₁₆	—	89	2	no	—
245	Q₁₆	—	178	1	no	—
246	D₈	—	89	2	no	—
247	D₈	—	178	1	no	—
248	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
249	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
250	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
251	C₄⋊C₄	—	356	2	no	—
252	C₄⋊C₄	—	356	1	no	—
253	C₄⋊C₄	—	356	2	no	—
254	C₄⋊C₄	—	356	1	no	—
255	C₄⋊C₄	—	356	2	no	—
256	C₄⋊C₄	—	356	2	no	—
257	C₄⋊C₄	—	356	1	no	—
258	C₄⋊C₄	—	356	2	no	—
259	C₄⋊C₄	—	178	1	no	—
260	C₄⋊C₄	—	178	2	no	—
261	C₄⋊C₄	—	178	1	no	—
262	C₄⋊C₄	—	178	2	no	—
263	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
264	C₄⋊C₄	D₇₁₂	1	2	yes	—
265	C₂²⋊C₄	—	356	1	no	—
266	C₂²⋊C₄	—	356	2	no	—
267	C₄²	—	356	2	no	—
268	C₄²	—	356	1	no	—
269	C₄²	C₂×D₃₅₆	1	2	yes	—
270	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
271	C₁₆	—	2	2	no	—
272	C₁₆	—	2	1	no	—
273	C₁₆	Q₈×D₈₉	1	2	yes	—
274	C₂³	—	356	1	no	—
275	Q₈	—	2	2	no	—
276	Q₈	—	356	1	no	—
277	Q₈	—	1424	1	no	—
278	Q₈	—	178	2	no	—
279	Q₈	—	712	2	no	—
280	D₄	—	356	1	no	—
281	D₄	—	178	2	no	—
282	C₂×C₄	—	356	1	no	—
283	C₂×C₄	—	356	1	no	—
284	C₂×C₄	—	356	2	no	—
285	C₂×C₄	—	356	1	no	—
286	C₂×C₄	—	356	2	no	—
287	C₂×C₄	—	1424	2	no	—
288	C₂×C₄	—	712	2	no	—
289	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
290	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
291	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
292	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	—	2	1	no	—
296	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
297	C₈	#15668059	1	1	yes	characteristic
298	C₂²	—	356	2	no	—
299	C₂²	—	712	1	no	—
300	C₂²	—	712	1	no	—
301	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	1	no	—
305	C₄	—	356	2	no	—
306	C₄	—	712	1	no	—
307	C₄	—	712	2	no	—
308	C₄	—	712	2	no	—
309	C₄	—	712	1	no	—
310	C₄	#19152030	1	2	yes	—
311	C₄	Q₈×D₃₅₆	1	1	yes	characteristic
312	C₄	#19152415	1	1	yes	characteristic
313	C₂	—	712	2	no	—
314	C₂	Q₈×D₇₁₂	1	1	yes	characteristic
315	C₂	C₂×#19152030	1	1	yes	characteristic
316	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	Q₈×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₇₁₂	C₂³
hypercenter	Q₈×C₁₆	D₈₉
hypocenter	C₈₉	#33948
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
1. #26033144	C₂	semidirect
3. #26022534	C₂	semidirect
4. #26022531	C₂	semidirect
5. #26022417	C₂	semidirect
7. #26022405	C₂	semidirect
11. Q₈×C₁₆×C₈₉	C₂	semidirect
57. #15667957	Q₈	semidirect
167. Q₈×C₁₆	D₈₉	semidirect
170. C₈₉	#33948	semidirect

Maximal subgroups

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

subgroup	index
#26033144	2
#26033120	2
#26022534	2
#26022531	2
#26022417	2
#26022410	2
#26022405	2
#26022390	2
C₄×#15667957	2
Q₈×C₁₆×C₈₉	2
#33948	89