Subgroups — #36712034

Order 22784.

514 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032675	C₂	1	1	yes	maximal, characteristic, complemented
2	#26032600	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022534	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022444	C₂	1	2	yes	maximal, complemented
5	#26022438	C₂	1	2	yes	maximal, complemented
6	#26022400	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022390	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022355	C₂	1	2	yes	maximal, complemented
9	C₄×D₁₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
10	#26022040	C₂	1	2	yes	maximal, complemented
11	C₈₉×#2169	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152313	—	2	1	no	—
13	#19152252	—	2	1	no	—
14	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
15	#19152003	C₂²	1	1	yes	characteristic
16	#19152003	C₂²	1	1	yes	characteristic
17	#19151923	—	2	2	no	—
18	#19151912	—	2	2	no	—
19	#19151889	—	2	1	no	—
20	#19151879	—	2	1	no	—
21	#19151834	C₂²	1	2	yes	—
22	#19151834	C₂²	1	2	yes	—
23	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
24	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
25	#19151636	C₂²	1	2	yes	—
26	C₂×#15667946	C₂²	1	2	yes	—
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	2	yes	—
33	#19151400	C₂²	1	1	yes	characteristic
34	#19151400	C₂²	1	2	yes	—
35	#19151400	C₂²	1	1	yes	characteristic
36	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
37	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#13083573	—	4	1	no	—
42	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
43	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
44	#15668012	—	2	1	no	—
45	#15668012	—	2	1	no	—
46	#15667998	—	4	2	no	—
47	#15667996	—	4	2	no	—
48	#15667992	—	4	1	no	—
49	#15667986	—	2	2	no	—
50	#15667986	—	2	2	no	—
51	#15667983	—	4	1	no	—
52	C₄×D₃₅₆	—	2	1	no	—
53	C₄×D₃₅₆	—	2	1	no	—
54	#15667971	—	2	2	no	—
55	#15667970	—	4	1	no	—
56	C₂×C₄⋊Dic₈₉	—	2	2	no	—
57	D₁₄₂₄	—	2	2	no	—
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	2	yes	—
63	#15667946	C₂³	1	1	yes	characteristic
64	#15667946	C₂³	1	2	yes	—
65	#15667944	—	2	2	no	—
66	#15667934	—	2	2	no	—
67	#15667934	—	2	2	no	—
68	#15667934	—	2	1	no	—
69	#15667934	—	2	1	no	—
70	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
71	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
72	M₃₂×C₈₉	—	2	4	no	—
73	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
74	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
75	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
77	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	#13083573	—	8	4	no	—
80	C₂×Dic₁₇₈	—	4	1	no	—
81	C₂×Dic₁₇₈	—	4	1	no	—
82	C₂×D₃₅₆	—	2	1	no	—
83	C₂×D₃₅₆	—	2	1	no	—
84	C₂×C₂²⋊D₈₉	—	4	2	no	—
85	C₂×C₂²⋊D₈₉	—	4	2	no	—
86	C₂²×Dic₈₉	—	4	2	no	—
87	C₂×C₄×D₈₉	—	4	1	no	—
88	C₂×C₄×D₈₉	—	4	1	no	—
89	D₇₁₂	—	2	1	no	—
90	D₇₁₂	—	2	1	no	—
91	D₇₁₂	—	2	1	no	—
92	D₇₁₂	—	2	1	no	—
93	C₄⋊Dic₈₉	—	2	1	no	—
94	C₄⋊Dic₈₉	—	2	2	no	—
95	C₄⋊Dic₈₉	—	2	1	no	—
96	C₄⋊Dic₈₉	—	2	2	no	—
97	C₄⋊Dic₈₉	—	4	1	no	—
98	C₄⋊Dic₈₉	—	4	2	no	—
99	C₄⋊Dic₈₉	—	4	1	no	—
100	C₂²⋊Dic₈₉	—	4	2	no	—
101	#13083551	—	4	2	no	—
102	#13083551	—	4	1	no	—
103	#13083551	—	4	1	no	—
104	#13083551	—	4	2	no	—
105	C₄×Dic₈₉	—	4	2	no	—
106	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
107	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₈₉	—	2	2	no	—
109	C₂×C₈×C₈₉	—	2	2	no	—
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	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
115	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₁₆×C₈₉	—	2	2	no	—
117	C₁₆×C₈₉	—	2	1	no	—
118	C₁₆×C₈₉	#41	1	2	yes	complemented
119	C₂³×C₈₉	C₂×D₈	1	2	yes	—
120	C₂²×D₈₉	—	4	1	no	—
121	C₂²×D₈₉	—	4	1	no	—
122	D₄×C₈₉	—	2	2	no	—
123	C₂×C₄×C₈₉	—	2	2	no	—
124	C₂×C₄×C₈₉	—	2	2	no	—
125	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
127	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
129	C₂²⋊D₈₉	—	8	4	no	—
130	C₂²⋊D₈₉	—	16	2	no	—
131	C₂×Dic₈₉	—	4	1	no	—
132	C₂×Dic₈₉	—	4	2	no	—
133	C₂×Dic₈₉	—	4	1	no	—
134	C₂×Dic₈₉	—	4	2	no	—
135	C₂×Dic₈₉	—	8	4	no	—
136	D₃₅₆	—	4	1	no	—
137	D₃₅₆	—	4	1	no	—
138	D₃₅₆	—	4	1	no	—
139	D₃₅₆	—	4	1	no	—
140	C₄×D₈₉	—	8	2	no	—
141	C₄×D₈₉	—	8	2	no	—
142	Dic₁₇₈	—	8	2	no	—
143	Dic₁₇₈	—	16	1	no	—
144	C₈×C₈₉	—	2	2	no	—
145	C₈×C₈₉	—	2	1	no	—
146	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
147	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
148	C₂²×C₈₉	—	2	2	no	—
149	C₂²×C₈₉	—	2	2	no	—
150	C₂²×C₈₉	—	2	2	no	—
151	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
152	C₂×D₈₉	—	8	1	no	—
153	C₂×D₈₉	—	8	1	no	—
154	C₂×D₈₉	—	8	1	no	—
155	C₂×D₈₉	—	8	1	no	—
156	C₂×D₈₉	—	8	1	no	—
157	C₂×D₈₉	—	8	1	no	—
158	C₄×C₈₉	—	2	2	no	—
159	C₄×C₈₉	—	2	1	no	—
160	C₄×C₈₉	#509	1	2	yes	—
161	C₄×C₈₉	#547	1	1	yes	characteristic
162	C₄×C₈₉	#578	1	1	yes	characteristic
163	Dic₈₉	—	8	2	no	—
164	Dic₈₉	—	8	1	no	—
165	Dic₈₉	—	8	1	no	—
166	Dic₈₉	—	8	2	no	—
167	#33921	—	89	1	no	maximal, Sylow
168	C₂×C₈₉	—	4	2	no	—
169	C₂×C₈₉	#3334	1	1	yes	characteristic
170	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
171	C₂×C₈₉	#3416	1	1	yes	characteristic
172	D₈₉	—	16	1	no	—
173	D₈₉	—	16	1	no	—
174	#3341	—	89	1	no	—
175	#3334	—	89	1	no	—
176	#2254	—	89	1	no	—
177	#2234	—	89	2	no	—
178	#2233	—	89	2	no	—
179	#2226	—	89	1	no	—
180	#2224	—	89	1	no	—
181	#2218	—	89	2	no	—
182	C₄×D₁₆	—	89	1	no	—
183	#2169	D₈₉	1	1	yes	characteristic, complemented
184	#2159	—	89	2	no	—
185	C₈₉	#33921	1	1	yes	characteristic, complemented, Sylow
186	#555	—	178	1	no	—
187	#547	—	178	1	no	—
188	C₂×D₁₆	—	89	1	no	—
189	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
190	#500	—	89	1	no	—
191	#500	—	89	1	no	—
192	#482	—	178	2	no	—
193	#480	—	178	2	no	—
194	#476	—	178	1	no	—
195	#474	—	178	1	no	—
196	#466	—	89	2	no	—
197	#466	—	89	2	no	—
198	C₄×D₈	—	89	1	no	—
199	C₄×D₈	—	89	1	no	—
200	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
201	#427	—	89	2	no	—
202	C₂×#107	—	89	2	no	—
203	#367	—	89	2	no	—
204	#367	—	89	2	no	—
205	#367	—	89	2	no	—
206	#366	—	89	1	no	—
207	#366	—	89	2	no	—
208	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
209	#357	—	89	1	no	—
210	#357	—	89	1	no	—
211	#357	—	89	2	no	—
212	#357	—	89	2	no	—
213	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
214	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
215	C₂×#41	—	356	1	no	—
216	C₂×D₈	—	89	1	no	—
217	C₂×D₈	—	89	1	no	—
218	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
219	C₄⋊Q₈	—	178	1	no	—
220	C₄⋊Q₈	—	178	1	no	—
221	#124	—	356	2	no	—
222	#123	—	356	2	no	—
223	C₂²⋊Q₈	—	356	1	no	—
224	#121	—	356	1	no	—
225	#121	—	178	2	no	—
226	#121	—	178	2	no	—
227	C₄×D₄	—	178	1	no	—
228	C₄×D₄	—	178	1	no	—
229	C₄×D₄	D₃₅₆	1	1	yes	characteristic
230	#117	—	356	1	no	—
231	#117	—	178	2	no	—
232	C₂×C₄⋊C₄	—	178	2	no	—
233	D₁₆	—	178	2	no	—
234	M₃₂	—	2	4	no	—
235	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
236	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
237	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
238	#107	—	89	1	no	—
239	#107	—	89	1	no	—
240	#107	—	89	2	no	—
241	#107	—	89	2	no	—
242	#107	—	178	2	no	—
243	#107	—	178	1	no	—
244	#107	—	178	1	no	—
245	#106	—	178	2	no	—
246	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
247	#102	—	178	1	no	—
248	#102	—	178	1	no	—
249	#102	—	178	2	no	—
250	#102	—	178	2	no	—
251	C₂²⋊C₈	D₃₅₆	1	2	yes	—
252	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
253	#41	—	712	4	no	—
254	C₂×Q₈	—	356	1	no	—
255	C₂×Q₈	—	356	1	no	—
256	C₂×D₄	—	356	2	no	—
257	C₂×D₄	—	356	2	no	—
258	C₂×D₄	—	178	1	no	—
259	C₂×D₄	—	178	1	no	—
260	C₂×D₄	D₇₁₂	1	1	yes	characteristic
261	C₂²×C₄	—	356	2	no	—
262	C₂²×C₄	—	356	1	no	—
263	C₂²×C₄	—	356	1	no	—
264	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
265	D₈	—	178	1	no	—
266	D₈	—	178	1	no	—
267	D₈	—	178	1	no	—
268	D₈	—	178	1	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
272	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
273	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
274	C₄⋊C₄	—	356	2	no	—
275	C₄⋊C₄	—	356	1	no	—
276	C₄⋊C₄	—	356	1	no	—
277	C₄⋊C₄	—	178	1	no	—
278	C₄⋊C₄	—	178	2	no	—
279	C₄⋊C₄	—	178	1	no	—
280	C₄⋊C₄	—	178	2	no	—
281	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
282	C₂²⋊C₄	—	356	2	no	—
283	C₂²⋊C₄	—	356	2	no	—
284	C₂²⋊C₄	—	356	1	no	—
285	C₂²⋊C₄	—	356	1	no	—
286	C₂²⋊C₄	—	356	2	no	—
287	C₂²⋊C₄	D₇₁₂	1	2	yes	—
288	C₄²	—	356	2	no	—
289	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
290	C₁₆	—	2	2	no	—
291	C₁₆	—	2	1	no	—
292	C₁₆	#13083573	1	2	yes	complemented
293	C₂³	—	356	1	no	—
294	C₂³	—	356	1	no	—
295	C₂³	C₂×D₇₁₂	1	2	yes	—
296	Q₈	—	1424	1	no	—
297	Q₈	—	712	2	no	—
298	D₄	—	2	2	no	—
299	D₄	—	356	1	no	—
300	D₄	—	356	1	no	—
301	D₄	—	356	1	no	—
302	D₄	—	356	1	no	—
303	D₄	—	1424	2	no	—
304	D₄	—	712	4	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	356	1	no	—
308	C₂×C₄	—	356	2	no	—
309	C₂×C₄	—	356	1	no	—
310	C₂×C₄	—	356	2	no	—
311	C₂×C₄	—	712	4	no	—
312	C₂×C₄	—	712	2	no	—
313	C₂×C₄	—	712	2	no	—
314	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
315	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
316	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
317	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	2	no	—
320	C₈	C₂×#13083573	1	1	yes	characteristic
321	C₈	#15668059	1	1	yes	characteristic
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	712	1	no	—
326	C₂²	—	712	1	no	—
327	C₂²	—	712	1	no	—
328	C₂²	—	712	1	no	—
329	C₂²	—	712	1	no	—
330	C₂²	—	712	1	no	—
331	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	1	no	—
334	C₄	—	712	1	no	—
335	C₄	—	712	2	no	—
336	C₄	—	712	1	no	—
337	C₄	—	712	2	no	—
338	C₄	#19152036	1	2	yes	—
339	C₄	#19152252	1	1	yes	characteristic
340	C₄	#19152415	1	1	yes	characteristic
341	C₂	—	4	2	no	—
342	C₂	—	1424	1	no	—
343	C₂	—	1424	1	no	—
344	C₂	#26032600	1	1	yes	characteristic
345	C₂	C₂×#19152036	1	1	yes	characteristic
346	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×#2169	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	#2169	D₈₉
hypocenter	C₈₉	#33921
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #2169

Complements

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

N	H ≅ G/N	type
1. #26032675	C₂	semidirect
2. #26032600	C₂	semidirect
3. #26022534	C₂	semidirect
4. #26022444	C₂	semidirect
5. #26022438	C₂	semidirect
6. #26022400	C₂	semidirect
7. #26022390	C₂	semidirect
8. #26022355	C₂	semidirect
9. C₄×D₁₄₂₄	C₂	semidirect
10. #26022040	C₂	semidirect
11. C₈₉×#2169	C₂	semidirect
118. C₁₆×C₈₉	#41	semidirect
183. #2169	D₈₉	semidirect
185. C₈₉	#33921	semidirect
292. C₁₆	#13083573	semidirect

Maximal subgroups

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

subgroup	index
#26032675	2
#26032600	2
#26022534	2
#26022444	2
#26022438	2
#26022400	2
#26022390	2
#26022355	2
C₄×D₁₄₂₄	2
#26022040	2
C₈₉×#2169	2
#33921	89