Subgroups — #36712254

Order 22784.

486 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033265	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022489	C₂	1	2	yes	maximal, complemented
4	#26022476	C₂	1	1	yes	maximal, characteristic
5	#26022405	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022390	C₂	1	2	yes	maximal, complemented
7	#26022315	C₂	1	2	yes	maximal, complemented
8	#26022293	C₂	1	1	yes	maximal, characteristic
9	#26022118	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152333	—	2	1	no	—
13	Q₈×Dic₁₇₈	—	2	1	no	—
14	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
15	C₂×#15667957	C₂²	1	2	yes	—
16	#19152003	C₂²	1	1	yes	characteristic
17	#19152003	C₂²	1	2	yes	—
18	#19151982	C₂²	1	2	yes	—
19	#19151975	C₂²	1	1	yes	characteristic
20	#19151894	—	2	1	no	—
21	#19151894	—	2	2	no	—
22	#19151814	—	2	2	no	—
23	#19151799	—	2	1	no	—
24	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
25	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
26	C₄×Dic₃₅₆	C₂²	1	2	yes	—
27	C₄×D₇₁₂	C₂²	1	2	yes	—
28	#19151424	C₂²	1	2	yes	—
29	#19151424	C₂²	1	1	yes	characteristic
30	#19151403	C₂²	1	2	yes	—
31	#19151403	C₂²	1	1	yes	characteristic
32	#19151403	C₂²	1	1	yes	characteristic
33	#19151400	C₂²	1	2	yes	—
34	#19151400	C₂²	1	2	yes	—
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	#346×C₈₉	C₂²	1	2	yes	—
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
42	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
43	C₂×#13083563	—	2	2	no	—
44	C₂×D₇₁₂	—	2	1	no	—
45	C₂×D₇₁₂	C₂³	1	2	yes	—
46	#15668012	—	2	2	no	—
47	#15668012	—	2	1	no	—
48	#15668011	—	4	1	no	—
49	#15668011	—	4	2	no	—
50	#15668009	—	2	1	no	—
51	#15668001	—	4	1	no	—
52	#15668000	—	2	2	no	—
53	#15667997	—	4	2	no	—
54	Q₈×Dic₈₉	—	4	1	no	—
55	C₄×Dic₁₇₈	—	2	2	no	—
56	C₄×Dic₁₇₈	—	2	1	no	—
57	C₄×Dic₁₇₈	—	2	1	no	—
58	C₄×D₃₅₆	—	2	2	no	—
59	#15667974	—	4	1	no	—
60	Dic₇₁₂	—	2	2	no	—
61	#15667957	—	2	4	no	—
62	#15667946	—	2	1	no	—
63	#15667946	—	2	2	no	—
64	#15667946	C₂³	1	2	yes	—
65	#15667946	C₂³	1	1	yes	characteristic
66	#15667946	C₂³	1	1	yes	characteristic
67	#15667937	—	2	1	no	—
68	#15667937	—	2	2	no	—
69	#15667937	—	2	1	no	—
70	#15667934	—	2	2	no	—
71	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
72	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
75	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
76	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
77	C₄×C₈×C₈₉	C₂³	1	2	yes	—
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₇₈	—	2	1	no	—
80	C₂×Dic₁₇₈	—	2	1	no	—
81	C₂×Dic₁₇₈	—	4	2	no	—
82	C₂×Dic₁₇₈	—	4	1	no	—
83	C₂×D₃₅₆	—	2	2	no	—
84	C₂×D₃₅₆	—	4	1	no	—
85	C₂×D₃₅₆	—	4	1	no	—
86	C₂×C₄×D₈₉	—	4	2	no	—
87	Dic₃₅₆	—	2	1	no	—
88	Dic₃₅₆	—	2	1	no	—
89	Dic₃₅₆	—	2	1	no	—
90	Dic₃₅₆	#41	1	2	yes	—
91	#13083563	—	8	2	no	—
92	D₇₁₂	—	2	2	no	—
93	D₇₁₂	—	2	2	no	—
94	D₇₁₂	—	8	1	no	—
95	C₄⋊Dic₈₉	—	2	1	no	—
96	C₄⋊Dic₈₉	—	2	2	no	—
97	C₄⋊Dic₈₉	—	2	1	no	—
98	C₄⋊Dic₈₉	—	4	2	no	—
99	C₄⋊Dic₈₉	—	4	1	no	—
100	#13083553	—	4	1	no	—
101	#13083553	—	4	2	no	—
102	#13083553	—	4	1	no	—
103	#13083551	—	4	2	no	—
104	#13083551	—	4	2	no	—
105	#13083551	—	4	2	no	—
106	C₄×Dic₈₉	—	4	1	no	—
107	C₄×Dic₈₉	—	4	2	no	—
108	C₄×Dic₈₉	—	4	1	no	—
109	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
114	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
115	C₄²×C₈₉	C₂×D₄	1	2	yes	—
116	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₈₉	—	2	2	no	—
118	C₁₆×C₈₉	—	2	1	no	—
119	C₁₆×C₈₉	—	2	1	no	—
120	C₂²×D₈₉	—	4	2	no	—
121	Q₈×C₈₉	—	2	2	no	—
122	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₈₉	—	4	1	no	—
128	C₂×Dic₈₉	—	4	1	no	—
129	C₂×Dic₈₉	—	4	1	no	—
130	C₂×Dic₈₉	—	4	2	no	—
131	C₂×Dic₈₉	—	4	1	no	—
132	D₃₅₆	—	4	2	no	—
133	D₃₅₆	—	4	2	no	—
134	D₃₅₆	—	8	2	no	—
135	D₃₅₆	—	16	1	no	—
136	C₄×D₈₉	—	16	2	no	—
137	Dic₁₇₈	—	2	2	no	—
138	Dic₁₇₈	—	4	1	no	—
139	Dic₁₇₈	—	4	1	no	—
140	Dic₁₇₈	—	4	1	no	—
141	Dic₁₇₈	—	16	2	no	—
142	Dic₁₇₈	—	16	1	no	—
143	C₈×C₈₉	—	2	1	no	—
144	C₈×C₈₉	—	2	2	no	—
145	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
146	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
147	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
148	C₂×D₈₉	—	8	2	no	—
149	C₂×D₈₉	—	8	2	no	—
150	C₂×D₈₉	—	8	2	no	—
151	C₄×C₈₉	—	2	2	no	—
152	C₄×C₈₉	—	2	2	no	—
153	C₄×C₈₉	—	2	1	no	—
154	C₄×C₈₉	#510	1	2	yes	—
155	C₄×C₈₉	#548	1	1	yes	characteristic
156	C₄×C₈₉	#577	1	1	yes	characteristic
157	Dic₈₉	—	4	2	no	—
158	Dic₈₉	—	8	1	no	—
159	Dic₈₉	—	8	1	no	—
160	Dic₈₉	—	8	2	no	—
161	Dic₈₉	—	8	1	no	—
162	#33942	—	89	1	no	maximal, Sylow
163	C₂×C₈₉	#3358	1	1	yes	characteristic
164	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
165	C₂×C₈₉	#3415	1	1	yes	characteristic
166	D₈₉	—	16	2	no	—
167	#3396	—	89	1	no	—
168	Q₈×Q₁₆	—	89	1	no	—
169	#2243	—	89	2	no	—
170	#2240	—	89	1	no	—
171	#2227	—	89	1	no	—
172	#2224	—	89	2	no	—
173	#2211	—	89	2	no	—
174	#2207	—	89	1	no	—
175	#2183	D₈₉	1	1	yes	characteristic, complemented
176	#2175	—	89	2	no	—
177	C₄×Q₃₂	—	89	1	no	—
178	C₈₉	#33942	1	1	yes	characteristic, complemented, Sylow
179	#559	—	178	1	no	—
180	Q₈²	—	178	1	no	—
181	C₂×Q₃₂	—	89	1	no	—
182	C₂×SD₃₂	—	89	2	no	—
183	#500	—	89	2	no	—
184	#500	—	89	1	no	—
185	#495	—	89	2	no	—
186	#493	—	89	1	no	—
187	#477	—	178	1	no	—
188	#477	—	178	2	no	—
189	#463	—	178	2	no	—
190	#460	—	178	1	no	—
191	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
192	C₄×Q₁₆	—	89	1	no	—
193	C₄×Q₁₆	—	89	2	no	—
194	C₄×Q₁₆	—	89	1	no	—
195	C₄×D₈	—	89	2	no	—
196	#366	—	89	1	no	—
197	#366	—	89	2	no	—
198	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
199	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
200	#358	—	89	1	no	—
201	#358	—	89	1	no	—
202	#358	—	89	2	no	—
203	#357	—	89	2	no	—
204	#357	—	89	2	no	—
205	#357	—	89	2	no	—
206	#346	C₂×D₈₉	1	2	yes	—
207	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
208	C₂×Q₁₆	—	89	1	no	—
209	C₂×Q₁₆	—	89	1	no	—
210	C₂×SD₁₆	—	178	2	no	—
211	C₂×D₈	—	89	2	no	—
212	C₂×D₈	—	178	1	no	—
213	C₄⋊Q₈	—	356	1	no	—
214	C₄⋊Q₈	—	356	2	no	—
215	C₄⋊Q₈	—	178	2	no	—
216	C₄⋊Q₈	—	178	1	no	—
217	#127	—	178	1	no	—
218	#124	—	356	1	no	—
219	#124	—	178	2	no	—
220	#123	—	356	2	no	—
221	C₄×Q₈	—	356	1	no	—
222	C₄×Q₈	—	178	2	no	—
223	C₄×Q₈	—	178	1	no	—
224	C₄×Q₈	—	178	1	no	—
225	C₄×Q₈	D₃₅₆	1	1	yes	characteristic
226	C₄×D₄	—	356	1	no	—
227	C₄×D₄	—	178	2	no	—
228	Q₃₂	—	178	2	no	—
229	SD₃₂	—	178	4	no	—
230	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
231	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
232	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
233	#107	—	89	2	no	—
234	#107	—	89	1	no	—
235	#107	—	89	1	no	—
236	#107	—	178	2	no	—
237	#107	—	178	1	no	—
238	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
239	C₄⋊C₈	D₃₅₆	1	2	yes	—
240	#103	—	178	1	no	—
241	#103	—	178	2	no	—
242	#103	—	178	1	no	—
243	#102	—	178	2	no	—
244	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₈₉	1	2	yes	—
246	C₂×Q₈	—	356	2	no	—
247	C₂×Q₈	—	356	1	no	—
248	C₂×Q₈	—	178	1	no	—
249	C₂×Q₈	—	178	1	no	—
250	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
251	C₂×D₄	—	356	1	no	—
252	C₂×D₄	—	356	1	no	—
253	C₂×D₄	—	178	2	no	—
254	C₂²×C₄	—	356	2	no	—
255	Q₁₆	—	89	2	no	—
256	Q₁₆	—	178	1	no	—
257	Q₁₆	—	178	1	no	—
258	Q₁₆	—	178	1	no	—
259	SD₁₆	—	712	2	no	—
260	D₈	—	178	2	no	—
261	D₈	—	178	2	no	—
262	D₈	—	712	1	no	—
263	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
264	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
265	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
266	C₄⋊C₄	—	356	2	no	—
267	C₄⋊C₄	—	356	1	no	—
268	C₄⋊C₄	—	356	1	no	—
269	C₄⋊C₄	—	356	2	no	—
270	C₄⋊C₄	—	356	1	no	—
271	C₄⋊C₄	—	178	1	no	—
272	C₄⋊C₄	—	178	2	no	—
273	C₄⋊C₄	—	178	1	no	—
274	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
275	C₄⋊C₄	D₇₁₂	1	2	yes	—
276	C₂²⋊C₄	—	356	2	no	—
277	C₂²⋊C₄	—	356	2	no	—
278	C₂²⋊C₄	—	356	2	no	—
279	C₄²	—	356	1	no	—
280	C₄²	—	356	2	no	—
281	C₄²	—	356	1	no	—
282	C₄²	C₂×D₃₅₆	1	2	yes	—
283	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
284	C₁₆	—	2	2	no	—
285	C₁₆	—	2	1	no	—
286	C₁₆	—	2	1	no	—
287	C₂³	—	356	2	no	—
288	Q₈	—	2	2	no	—
289	Q₈	—	356	1	no	—
290	Q₈	—	356	1	no	—
291	Q₈	—	356	1	no	—
292	Q₈	—	1424	2	no	—
293	Q₈	—	1424	1	no	—
294	Q₈	—	178	2	no	—
295	D₄	—	356	2	no	—
296	D₄	—	356	2	no	—
297	D₄	—	1424	1	no	—
298	D₄	—	712	2	no	—
299	C₂×C₄	—	356	1	no	—
300	C₂×C₄	—	356	1	no	—
301	C₂×C₄	—	356	1	no	—
302	C₂×C₄	—	356	2	no	—
303	C₂×C₄	—	356	1	no	—
304	C₂×C₄	—	1424	2	no	—
305	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
306	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
307	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
308	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
309	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	2	no	—
312	C₈	#41×D₈₉	1	1	yes	characteristic
313	C₈	#15668060	1	1	yes	characteristic
314	C₂²	—	712	2	no	—
315	C₂²	—	712	2	no	—
316	C₂²	—	712	2	no	—
317	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	1	no	—
321	C₄	—	356	2	no	—
322	C₄	—	712	1	no	—
323	C₄	—	712	1	no	—
324	C₄	—	712	2	no	—
325	C₄	—	712	1	no	—
326	C₄	#19152041	1	2	yes	—
327	C₄	#19152260	1	1	yes	characteristic
328	C₄	#19152410	1	1	yes	characteristic
329	C₂	—	1424	2	no	—
330	C₂	#26032888	1	1	yes	characteristic
331	C₂	C₂×#19152041	1	1	yes	characteristic
332	C₂	#26033393	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	#19151975	C₂²
hypercenter	#2183	D₈₉
hypocenter	C₈₉	#33942
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
2. Q₈×Dic₃₅₆	C₂	semidirect
3. #26022489	C₂	semidirect
5. #26022405	C₂	semidirect
6. #26022390	C₂	semidirect
7. #26022315	C₂	semidirect
9. #26022118	C₂	semidirect
10. C₄×Dic₇₁₂	C₂	semidirect
11. C₈₉×#2183	C₂	semidirect
175. #2183	D₈₉	semidirect
178. C₈₉	#33942	semidirect

Maximal subgroups

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

subgroup	index
#26033265	2
Q₈×Dic₃₅₆	2
#26022489	2
#26022476	2
#26022405	2
#26022390	2
#26022315	2
#26022293	2
#26022118	2
C₄×Dic₇₁₂	2
C₈₉×#2183	2
#33942	89