Subgroups — #36712297

Order 22784.

494 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₆×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#26033104	C₂	1	1	yes	maximal, characteristic
3	#26022533	C₂	1	2	yes	maximal, complemented
4	#26022532	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022400	C₂	1	1	yes	maximal, characteristic
6	#26022398	C₂	1	2	yes	maximal
7	#26022392	C₂	1	2	yes	maximal, complemented
8	#26022154	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#15667955	C₂	1	1	yes	maximal, characteristic
10	#26022101	C₂	1	2	yes	maximal
11	C₈₉×#2225	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈×Dic₁₇₈	—	2	1	no	—
13	#19152263	—	2	1	no	—
14	C₂×#15667955	C₂²	1	1	yes	characteristic
15	#19152003	C₂²	1	1	yes	characteristic
16	#19152002	C₂²	1	2	yes	—
17	#19152002	C₂²	1	2	yes	—
18	#19151893	—	2	1	no	—
19	#19151891	—	2	2	no	—
20	#19151877	—	2	2	no	—
21	#19151875	—	2	1	no	—
22	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
23	Q₁₆×Dic₈₉	C₂²	1	2	yes	—
24	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
25	#19151680	C₂²	1	2	yes	—
26	#19151426	C₂²	1	2	yes	—
27	#19151425	C₂²	1	2	yes	—
28	#19151425	C₂²	1	2	yes	—
29	#19151425	C₂²	1	1	yes	characteristic
30	#19151416	C₂²	1	2	yes	—
31	#19151402	C₂²	1	1	yes	characteristic
32	#19151400	C₂²	1	2	yes	—
33	#19151398	C₂²	1	1	yes	characteristic
34	C₄×#13083549	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₈₉	C₂²	1	2	yes	—
36	#500×C₈₉	C₂²	1	1	yes	characteristic
37	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
38	#367×C₈₉	C₂²	1	2	yes	—
39	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	#358×C₈₉	C₂²	1	2	yes	—
41	C₂×Q₈×D₈₉	—	4	1	no	—
42	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
43	#15668012	—	2	1	no	—
44	#15668012	—	4	1	no	—
45	#15668011	—	2	2	no	—
46	#15668011	—	2	2	no	—
47	#15668011	—	4	2	no	—
48	#15667990	—	4	2	no	—
49	#15667988	—	4	1	no	—
50	Q₈×Dic₈₉	—	2	2	no	—
51	C₄×Dic₁₇₈	—	4	1	no	—
52	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
53	C₄×D₃₅₆	—	2	1	no	—
54	#15667974	—	2	2	no	—
55	#15667955	Q₈	1	4	yes	complemented
56	C₂×#13083549	C₂³	1	1	yes	characteristic
57	C₂×#13083549	C₂³	1	1	yes	characteristic
58	#15667946	—	2	1	no	—
59	#15667946	C₂³	1	2	yes	—
60	#15667946	C₂³	1	1	yes	characteristic
61	#15667945	—	2	2	no	—
62	#15667945	—	2	2	no	—
63	C₄⋊#10623494	D₄	1	1	yes	characteristic
64	#15667941	D₄	1	2	yes	—
65	#15667936	—	2	2	no	—
66	#15667934	—	2	1	no	—
67	#15667932	—	2	2	no	—
68	C₈×Dic₈₉	C₂³	1	2	yes	—
69	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
70	C₄⋊Q₈×C₈₉	—	2	1	no	—
71	C₄×Q₈×C₈₉	—	2	1	no	—
72	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
73	#107×C₈₉	—	2	1	no	—
74	#107×C₈₉	C₂³	1	1	yes	characteristic
75	#107×C₈₉	C₂³	1	2	yes	—
76	#103×C₈₉	—	2	1	no	—
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	2	no	—
81	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
82	Q₈×D₈₉	—	8	4	no	—
83	C₂×D₃₅₆	—	2	1	no	—
84	C₂×C₄×D₈₉	—	4	1	no	—
85	C₂×C₄×D₈₉	—	4	2	no	—
86	D₇₁₂	—	2	1	no	—
87	D₇₁₂	C₂×Q₈	1	2	yes	—
88	C₂×#10623494	C₂×D₄	1	2	yes	—
89	C₄⋊Dic₈₉	—	2	1	no	—
90	C₄⋊Dic₈₉	—	2	2	no	—
91	C₄⋊Dic₈₉	—	4	1	no	—
92	C₄⋊Dic₈₉	—	4	2	no	—
93	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
94	#13083553	—	4	2	no	—
95	#13083553	—	4	2	no	—
96	#13083553	D₈	1	2	yes	—
97	#13083551	—	4	1	no	—
98	#13083551	—	4	2	no	—
99	C₄×Dic₈₉	—	4	2	no	—
100	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
101	#13083549	—	2	1	no	—
102	#13083549	C₂×Q₈	1	2	yes	complemented
103	C₂×Q₈×C₈₉	—	2	1	no	—
104	C₂×Q₈×C₈₉	—	4	1	no	—
105	Q₁₆×C₈₉	—	2	1	no	—
106	Q₁₆×C₈₉	C₂×Q₈	1	2	yes	complemented
107	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₈₉	—	2	2	no	—
110	C₄⋊C₄×C₈₉	—	2	1	no	—
111	C₄⋊C₄×C₈₉	—	4	1	no	—
112	C₄⋊C₄×C₈₉	—	4	1	no	—
113	C₄²×C₈₉	—	4	1	no	—
114	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₈₉	—	2	2	no	—
116	C₂²×D₈₉	—	4	1	no	—
117	Q₈×C₈₉	—	2	2	no	—
118	Q₈×C₈₉	—	4	1	no	—
119	Q₈×C₈₉	—	8	2	no	—
120	C₂×C₄×C₈₉	—	4	1	no	—
121	C₂×C₄×C₈₉	—	4	1	no	—
122	C₂×C₄×C₈₉	—	4	2	no	—
123	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
126	C₂×Dic₈₉	—	4	2	no	—
127	C₂×Dic₈₉	—	4	1	no	—
128	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
129	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
130	D₃₅₆	—	2	2	no	—
131	D₃₅₆	—	4	1	no	—
132	C₄×D₈₉	—	8	2	no	—
133	C₄×D₈₉	—	8	4	no	—
134	Dic₁₇₈	—	8	4	no	—
135	Dic₁₇₈	—	16	2	no	—
136	Dic₁₇₈	—	16	1	no	—
137	Dic₁₇₈	SD₃₂	1	4	yes	complemented
138	C₈×C₈₉	—	2	1	no	—
139	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
140	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
141	#10623494	—	2	2	no	—
142	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
143	C₂×D₈₉	—	4	2	no	—
144	C₂×D₈₉	—	8	1	no	—
145	C₂×D₈₉	—	8	1	no	—
146	C₄×C₈₉	—	2	1	no	—
147	C₄×C₈₉	—	4	2	no	—
148	C₄×C₈₉	—	8	2	no	—
149	C₄×C₈₉	—	8	1	no	—
150	C₄×C₈₉	C₂×SD₃₂	1	2	yes	—
151	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
152	C₄×C₈₉	#578	1	1	yes	characteristic
153	Dic₈₉	—	2	2	no	—
154	Dic₈₉	—	8	1	no	—
155	Dic₈₉	—	8	2	no	—
156	Dic₈₉	C₂×SD₃₂	1	4	yes	—
157	Q₈×SD₃₂	—	89	1	no	maximal, Sylow
158	C₂×C₈₉	Q₈×D₈	1	1	yes	characteristic
159	C₂×C₈₉	C₂²×SD₃₂	1	1	yes	characteristic
160	C₂×C₈₉	#3416	1	1	yes	characteristic
161	D₈₉	—	8	2	no	—
162	Q₈×Q₁₆	—	89	1	no	—
163	Q₈×D₈	—	89	1	no	—
164	#2254	—	89	1	no	—
165	#2254	—	89	2	no	—
166	#2226	—	89	2	no	—
167	#2226	—	89	1	no	—
168	#2225	—	89	2	no	—
169	#2225	D₈₉	1	1	yes	characteristic, complemented
170	Q₈×C₁₆	—	89	1	no	—
171	C₄×SD₃₂	—	89	1	no	—
172	C₄×SD₃₂	—	89	2	no	—
173	C₈₉	Q₈×SD₃₂	1	1	yes	characteristic, complemented, Sylow
174	Q₈²	—	178	1	no	—
175	D₄×Q₈	—	178	1	no	—
176	C₂×SD₃₂	—	89	1	no	—
177	#500	—	89	2	no	—
178	#500	—	89	2	no	—
179	#500	—	89	1	no	—
180	#500	C₂×D₈₉	1	1	yes	characteristic
181	#477	—	178	2	no	—
182	#477	—	178	1	no	—
183	#474	—	178	2	no	—
184	#474	—	178	1	no	—
185	C₈×Q₈	—	89	1	no	—
186	C₄×Q₁₆	—	89	2	no	—
187	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
188	C₄×D₈	—	89	2	no	—
189	C₄×D₈	—	89	1	no	—
190	#367	—	89	2	no	—
191	#367	—	89	2	no	—
192	#367	—	89	1	no	—
193	#367	—	89	2	no	—
194	#367	C₂×D₈₉	1	2	yes	—
195	C₄⋊C₁₆	—	89	2	no	—
196	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
197	#358	—	89	1	no	—
198	#358	C₂×D₈₉	1	2	yes	—
199	#357	—	89	2	no	—
200	#357	—	89	1	no	—
201	C₄×C₁₆	—	89	2	no	—
202	C₄×C₁₆	—	89	1	no	—
203	C₂²×Q₈	—	356	1	no	—
204	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
205	C₂×D₈	—	89	1	no	—
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	—	356	1	no	—
208	C₄⋊Q₈	—	356	2	no	—
209	C₄⋊Q₈	—	178	2	no	—
210	C₄⋊Q₈	—	178	2	no	—
211	C₄⋊Q₈	—	178	1	no	—
212	C₂²⋊Q₈	—	356	2	no	—
213	C₂²⋊Q₈	—	356	1	no	—
214	C₄×Q₈	—	2	1	no	—
215	C₄×Q₈	—	89	1	no	—
216	C₄×Q₈	—	356	1	no	—
217	C₄×Q₈	—	178	2	no	—
218	C₄×D₄	—	178	2	no	—
219	C₄×D₄	—	178	1	no	—
220	SD₃₂	—	89	4	no	—
221	C₂×C₁₆	—	89	1	no	—
222	C₂×C₁₆	—	89	1	no	—
223	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
224	#107	—	2	1	no	—
225	#107	—	89	2	no	—
226	#107	—	89	1	no	—
227	#107	—	178	2	no	—
228	#107	—	178	2	no	—
229	#107	—	178	1	no	—
230	#107	C₂²×D₈₉	1	1	yes	characteristic
231	#107	C₂²×D₈₉	1	2	yes	—
232	C₄⋊C₈	—	89	2	no	—
233	C₄⋊C₈	—	89	1	no	—
234	#103	—	2	1	no	—
235	#103	—	178	2	no	—
236	#102	—	178	2	no	—
237	#102	—	178	1	no	—
238	C₄×C₈	—	89	2	no	—
239	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
240	C₂×Q₈	—	2	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	89	1	no	—
243	C₂×Q₈	—	356	2	no	—
244	C₂×Q₈	—	356	2	no	—
245	C₂×Q₈	—	356	1	no	—
246	C₂×Q₈	—	712	4	no	—
247	C₂×D₄	—	178	1	no	—
248	C₂²×C₄	—	356	2	no	—
249	C₂²×C₄	—	356	1	no	—
250	Q₁₆	—	2	1	no	—
251	Q₁₆	Q₈×D₈₉	1	2	yes	complemented
252	D₈	—	89	2	no	—
253	D₈	—	178	1	no	—
254	C₂×C₈	—	89	2	no	—
255	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
256	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	89	2	no	—
262	C₄⋊C₄	—	89	1	no	—
263	C₄⋊C₄	—	356	2	no	—
264	C₄⋊C₄	—	356	2	no	—
265	C₄⋊C₄	—	356	2	no	—
266	C₄⋊C₄	—	356	1	no	—
267	C₄⋊C₄	—	178	1	no	—
268	C₄⋊C₄	—	178	2	no	—
269	C₂²⋊C₄	—	356	2	no	—
270	C₂²⋊C₄	—	356	1	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	89	2	no	—
273	C₄²	—	356	2	no	—
274	C₄²	D₄×D₈₉	1	1	yes	characteristic
275	C₁₆	—	2	2	no	—
276	C₁₆	—	89	2	no	—
277	C₁₆	—	178	1	no	—
278	C₂³	—	356	1	no	—
279	Q₈	—	2	2	no	—
280	Q₈	—	4	1	no	—
281	Q₈	—	8	2	no	—
282	Q₈	—	89	4	no	—
283	Q₈	—	1424	1	no	—
284	Q₈	—	1424	2	no	—
285	Q₈	—	712	4	no	—
286	D₄	—	356	1	no	—
287	D₄	—	178	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	89	2	no	—
292	C₂×C₄	—	89	2	no	—
293	C₂×C₄	—	356	2	no	—
294	C₂×C₄	—	356	1	no	—
295	C₂×C₄	—	712	4	no	—
296	C₂×C₄	—	712	2	no	—
297	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
298	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
299	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
300	C₈	—	2	1	no	—
301	C₈	—	178	2	no	—
302	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
303	C₈	#15668059	1	1	yes	characteristic
304	C₂²	—	356	2	no	—
305	C₂²	—	712	1	no	—
306	C₂²	—	712	1	no	—
307	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	4	2	no	—
310	C₄	—	8	2	no	—
311	C₄	—	8	1	no	—
312	C₄	—	89	4	no	—
313	C₄	—	178	2	no	—
314	C₄	—	712	1	no	—
315	C₄	—	712	2	no	—
316	C₄	SD₃₂×D₈₉	1	2	yes	—
317	C₄	#19152261	1	1	yes	characteristic
318	C₄	#19152414	1	1	yes	characteristic
319	C₂	—	712	2	no	—
320	C₂	#26033102	1	1	yes	characteristic
321	C₂	C₂×SD₃₂×D₈₉	1	1	yes	characteristic
322	C₂	#26033397	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2225	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₇₁₂	C₂³
hypercenter	#2225	D₈₉
hypocenter	C₈₉	Q₈×SD₃₂
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2225

Complements

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

N	H ≅ G/N	type
1. Q₁₆×Dic₁₇₈	C₂	semidirect
3. #26022533	C₂	semidirect
4. #26022532	C₂	semidirect
7. #26022392	C₂	semidirect
8. #26022154	C₂	semidirect
11. C₈₉×#2225	C₂	semidirect
55. #15667955	Q₈	semidirect
102. #13083549	C₂×Q₈	semidirect
106. Q₁₆×C₈₉	C₂×Q₈	semidirect
137. Dic₁₇₈	SD₃₂	semidirect
169. #2225	D₈₉	semidirect
173. C₈₉	Q₈×SD₃₂	semidirect
251. Q₁₆	Q₈×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
Q₁₆×Dic₁₇₈	2
#26033104	2
#26022533	2
#26022532	2
#26022400	2
#26022398	2
#26022392	2
#26022154	2
C₄×#15667955	2
#26022101	2
C₈₉×#2225	2
Q₈×SD₃₂	89