Subgroups — #36712036

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	#26032676	C₂	1	1	yes	maximal, characteristic, complemented
2	#26032599	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022532	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022446	C₂	1	2	yes	maximal, complemented
5	#26022436	C₂	1	2	yes	maximal, complemented
6	#26022400	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022354	C₂	1	2	yes	maximal, complemented
8	C₄×#15667953	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022083	C₂	1	1	yes	maximal, characteristic, complemented
10	#26022037	C₂	1	2	yes	maximal, complemented
11	C₈₉×#2224	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152313	—	2	1	no	—
13	#19152251	—	2	1	no	—
14	C₂×#15667953	C₂²	1	1	yes	characteristic
15	C₂×#15667951	C₂²	1	2	yes	—
16	#19152003	C₂²	1	1	yes	characteristic
17	#19151920	—	2	2	no	—
18	#19151913	—	2	2	no	—
19	#19151888	—	2	1	no	—
20	#19151875	—	2	1	no	—
21	#19151835	C₂²	1	2	yes	—
22	#19151831	C₂²	1	2	yes	—
23	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
24	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
25	#19151635	C₂²	1	2	yes	—
26	#107×D₈₉	C₂²	1	2	yes	—
27	#19151426	C₂²	1	2	yes	—
28	#19151425	C₂²	1	2	yes	—
29	#19151425	C₂²	1	2	yes	—
30	#19151423	C₂²	1	1	yes	characteristic
31	#19151400	C₂²	1	2	yes	—
32	#19151399	C₂²	1	1	yes	characteristic
33	#19151398	C₂²	1	1	yes	characteristic
34	#19151377	C₂²	1	2	yes	—
35	C₄×#13083549	C₂²	1	1	yes	characteristic
36	#500×C₈₉	C₂²	1	1	yes	characteristic
37	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
38	#366×C₈₉	C₂²	1	2	yes	—
39	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	#357×C₈₉	C₂²	1	2	yes	—
41	C₂×#13083575	—	4	1	no	—
42	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
43	C₂×C₈×D₈₉	C₂³	1	2	yes	—
44	#15668012	—	2	1	no	—
45	#15667998	—	4	2	no	—
46	#15667997	—	4	2	no	—
47	#15667988	—	4	1	no	—
48	#15667987	—	2	2	no	—
49	#15667986	—	4	1	no	—
50	#15667983	—	2	2	no	—
51	C₄×D₃₅₆	—	2	1	no	—
52	C₄×D₃₅₆	D₄	1	1	yes	characteristic
53	#15667971	—	4	1	no	—
54	#15667970	—	2	2	no	—
55	C₄⋊C₄×D₈₉	—	2	2	no	—
56	#15667953	—	2	2	no	—
57	#15667951	—	2	4	no	—
58	C₂×#13083549	C₂³	1	1	yes	characteristic
59	C₂×#13083549	C₂³	1	1	yes	characteristic
60	#15667946	—	2	1	no	—
61	#15667946	C₂³	1	1	yes	characteristic
62	#15667946	C₂³	1	2	yes	—
63	#15667945	—	2	2	no	—
64	#15667943	—	2	2	no	—
65	C₄⋊#10623494	D₄	1	1	yes	characteristic
66	#15667934	—	2	1	no	—
67	#15667933	—	2	2	no	—
68	#15667932	—	2	2	no	—
69	#15667924	D₄	1	2	yes	—
70	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₈₉	—	2	1	no	—
72	C₄×D₄×C₈₉	—	2	1	no	—
73	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
74	#107×C₈₉	—	2	1	no	—
75	#107×C₈₉	C₂³	1	2	yes	—
76	#107×C₈₉	C₂³	1	1	yes	characteristic
77	#102×C₈₉	—	2	1	no	—
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	#13083575	—	8	4	no	—
80	C₂×Dic₁₇₈	—	4	1	no	—
81	C₂×D₃₅₆	—	2	1	no	—
82	C₂×D₃₅₆	—	4	2	no	—
83	C₂×D₃₅₆	D₈	1	1	yes	characteristic
84	C₂×C₂²⋊D₈₉	—	4	2	no	—
85	C₂×C₄×D₈₉	—	4	2	no	—
86	C₂×C₄×D₈₉	—	4	1	no	—
87	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
88	D₇₁₂	—	2	1	no	—
89	D₇₁₂	—	2	1	no	—
90	C₂×#10623494	C₂×D₄	1	2	yes	—
91	C₈×D₈₉	—	2	2	no	—
92	C₈×D₈₉	—	2	2	no	—
93	C₄⋊Dic₈₉	—	2	2	no	—
94	C₄⋊Dic₈₉	—	2	1	no	—
95	C₄⋊Dic₈₉	—	4	1	no	—
96	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
97	#13083553	—	4	2	no	—
98	C₂²⋊Dic₈₉	—	4	2	no	—
99	#13083551	—	4	2	no	—
100	#13083551	—	4	1	no	—
101	#13083551	—	4	2	no	—
102	#13083551	D₈	1	2	yes	—
103	C₄×Dic₈₉	—	4	2	no	—
104	#13083549	—	2	1	no	—
105	#13083549	#41	1	2	yes	complemented
106	C₂×Q₈×C₈₉	—	4	1	no	—
107	C₂×D₄×C₈₉	—	2	1	no	—
108	C₂²×C₄×C₈₉	—	4	1	no	—
109	D₈×C₈₉	—	2	1	no	—
110	D₈×C₈₉	—	2	1	no	—
111	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₈₉	—	2	1	no	—
114	C₄⋊C₄×C₈₉	—	2	2	no	—
115	C₄⋊C₄×C₈₉	—	4	1	no	—
116	C₂²⋊C₄×C₈₉	—	4	1	no	—
117	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₁₆×C₈₉	—	2	2	no	—
119	C₂³×C₈₉	—	4	1	no	—
120	C₂²×D₈₉	—	4	1	no	—
121	C₂²×D₈₉	C₂×D₈	1	2	yes	—
122	Q₈×C₈₉	—	8	2	no	—
123	D₄×C₈₉	—	4	1	no	—
124	D₄×C₈₉	—	4	1	no	—
125	C₂×C₄×C₈₉	—	4	1	no	—
126	C₂×C₄×C₈₉	—	4	2	no	—
127	C₂×C₄×C₈₉	—	8	2	no	—
128	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
131	C₂²⋊D₈₉	—	16	2	no	—
132	C₂×Dic₈₉	—	4	2	no	—
133	C₂×Dic₈₉	—	4	1	no	—
134	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
135	D₃₅₆	—	2	2	no	—
136	D₃₅₆	—	4	1	no	—
137	D₃₅₆	—	4	1	no	—
138	D₃₅₆	—	8	4	no	—
139	C₄×D₈₉	—	2	2	no	—
140	C₄×D₈₉	—	2	2	no	—
141	C₄×D₈₉	—	8	4	no	—
142	C₄×D₈₉	—	8	2	no	—
143	Dic₁₇₈	—	16	1	no	—
144	C₈×C₈₉	—	2	1	no	—
145	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
146	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
147	#10623494	—	2	2	no	—
148	C₂²×C₈₉	—	8	1	no	—
149	C₂²×C₈₉	—	8	1	no	—
150	C₂²×C₈₉	—	8	1	no	—
151	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
152	C₂×D₈₉	—	2	2	no	—
153	C₂×D₈₉	—	2	2	no	—
154	C₂×D₈₉	—	2	2	no	—
155	C₂×D₈₉	—	8	1	no	—
156	C₂×D₈₉	—	8	1	no	—
157	C₂×D₈₉	—	8	1	no	—
158	C₄×C₈₉	—	2	1	no	—
159	C₄×C₈₉	—	8	2	no	—
160	C₄×C₈₉	—	8	1	no	—
161	C₄×C₈₉	#509	1	2	yes	—
162	C₄×C₈₉	#547	1	1	yes	characteristic
163	C₄×C₈₉	#578	1	1	yes	characteristic
164	Dic₈₉	—	2	2	no	—
165	Dic₈₉	—	8	1	no	—
166	Dic₈₉	—	8	2	no	—
167	#33921	—	89	1	no	maximal, Sylow
168	C₂×C₈₉	—	16	1	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	—	4	2	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	D₈₉	1	1	yes	characteristic, complemented
181	#2218	—	89	2	no	—
182	C₄×D₁₆	—	89	1	no	—
183	#2169	—	89	1	no	—
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	—	89	2	no	—
190	#500	—	89	1	no	—
191	#500	C₂×D₈₉	1	1	yes	characteristic
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	C₂×D₈₉	1	1	yes	characteristic
200	C₈×D₄	—	89	1	no	—
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	C₂×D₈₉	1	2	yes	—
208	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
209	#357	—	89	1	no	—
210	#357	—	89	2	no	—
211	#357	—	89	1	no	—
212	#357	C₂×D₈₉	1	2	yes	—
213	C₂²⋊C₁₆	—	89	2	no	—
214	C₄×C₁₆	—	89	1	no	—
215	C₂×#41	—	356	1	no	—
216	C₂×D₈	—	89	1	no	—
217	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
218	C₂²×C₈	—	89	2	no	—
219	C₄⋊Q₈	—	2	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	—	2	1	no	—
228	C₄×D₄	—	89	1	no	—
229	C₄×D₄	—	178	1	no	—
230	#117	—	356	1	no	—
231	#117	—	178	2	no	—
232	C₂×C₄⋊C₄	—	178	2	no	—
233	D₁₆	—	178	2	no	—
234	M₃₂	—	178	4	no	—
235	C₂×C₁₆	—	89	1	no	—
236	C₂×C₁₆	—	89	1	no	—
237	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
238	#107	—	2	1	no	—
239	#107	—	89	1	no	—
240	#107	—	89	2	no	—
241	#107	—	178	2	no	—
242	#107	—	178	1	no	—
243	#107	C₂²×D₈₉	1	1	yes	characteristic
244	#107	C₂²×D₈₉	1	2	yes	—
245	#106	—	178	2	no	—
246	C₄⋊C₈	—	89	1	no	—
247	#102	—	2	1	no	—
248	#102	—	178	2	no	—
249	#102	—	178	2	no	—
250	#102	—	178	1	no	—
251	C₂²⋊C₈	—	89	2	no	—
252	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
253	#41	—	712	4	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	—	356	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	89	1	no	—
258	C₂×D₄	—	356	2	no	—
259	C₂×D₄	—	356	2	no	—
260	C₂×D₄	—	178	1	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	—	89	2	no	—
263	C₂²×C₄	—	356	2	no	—
264	C₂²×C₄	—	356	1	no	—
265	D₈	—	2	1	no	—
266	D₈	—	2	1	no	—
267	D₈	—	178	1	no	—
268	D₈	—	178	1	no	—
269	C₂×C₈	—	89	2	no	—
270	C₂×C₈	—	178	2	no	—
271	C₂×C₈	—	178	2	no	—
272	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
273	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	89	1	no	—
278	C₄⋊C₄	—	356	2	no	—
279	C₄⋊C₄	—	356	1	no	—
280	C₄⋊C₄	—	178	2	no	—
281	C₄⋊C₄	—	178	1	no	—
282	C₂²⋊C₄	—	4	1	no	—
283	C₂²⋊C₄	—	89	2	no	—
284	C₂²⋊C₄	—	356	2	no	—
285	C₂²⋊C₄	—	356	2	no	—
286	C₂²⋊C₄	—	356	2	no	—
287	C₂²⋊C₄	—	356	1	no	—
288	C₄²	—	356	2	no	—
289	C₄²	D₄×D₈₉	1	1	yes	characteristic
290	C₁₆	—	2	2	no	—
291	C₁₆	—	89	2	no	—
292	C₁₆	—	178	1	no	—
293	C₂³	—	4	1	no	—
294	C₂³	—	89	2	no	—
295	C₂³	—	356	1	no	—
296	Q₈	—	8	2	no	—
297	Q₈	—	1424	1	no	—
298	D₄	—	4	1	no	—
299	D₄	—	4	1	no	—
300	D₄	—	356	1	no	—
301	D₄	—	356	1	no	—
302	D₄	—	1424	2	no	—
303	D₄	—	178	2	no	—
304	D₄	—	712	4	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	8	2	no	—
308	C₂×C₄	—	89	2	no	—
309	C₂×C₄	—	356	2	no	—
310	C₂×C₄	—	356	1	no	—
311	C₂×C₄	—	178	2	no	—
312	C₂×C₄	—	178	2	no	—
313	C₂×C₄	—	712	4	no	—
314	C₂×C₄	—	712	2	no	—
315	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
316	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
317	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	178	2	no	—
320	C₈	#41×D₈₉	1	1	yes	characteristic
321	C₈	#15668060	1	1	yes	characteristic
322	C₂²	—	8	1	no	—
323	C₂²	—	8	1	no	—
324	C₂²	—	8	1	no	—
325	C₂²	—	178	2	no	—
326	C₂²	—	178	2	no	—
327	C₂²	—	178	2	no	—
328	C₂²	—	712	1	no	—
329	C₂²	—	712	1	no	—
330	C₂²	—	712	1	no	—
331	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	2	no	—
335	C₄	—	178	2	no	—
336	C₄	—	712	1	no	—
337	C₄	—	712	2	no	—
338	C₄	#19152032	1	2	yes	—
339	C₄	#19152250	1	1	yes	characteristic
340	C₄	#19152414	1	1	yes	characteristic
341	C₂	—	16	1	no	—
342	C₂	—	356	2	no	—
343	C₂	—	1424	1	no	—
344	C₂	#26032598	1	1	yes	characteristic
345	C₂	C₂×#19152032	1	1	yes	characteristic
346	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×#2224	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	#2224	D₈₉
hypocenter	C₈₉	#33921
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2224

Complements

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

N	H ≅ G/N	type
1. #26032676	C₂	semidirect
2. #26032599	C₂	semidirect
3. #26022532	C₂	semidirect
4. #26022446	C₂	semidirect
5. #26022436	C₂	semidirect
6. #26022400	C₂	semidirect
7. #26022354	C₂	semidirect
8. C₄×#15667953	C₂	semidirect
9. #26022083	C₂	semidirect
10. #26022037	C₂	semidirect
11. C₈₉×#2224	C₂	semidirect
105. #13083549	#41	semidirect
180. #2224	D₈₉	semidirect
185. C₈₉	#33921	semidirect

Maximal subgroups

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

subgroup	index
#26032676	2
#26032599	2
#26022532	2
#26022446	2
#26022436	2
#26022400	2
#26022354	2
C₄×#15667953	2
#26022083	2
#26022037	2
C₈₉×#2224	2
#33921	89