Subgroups — #36711216

Order 22784.

646 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033400	C₂	1	2	yes	maximal, complemented
2	#26032039	C₂	1	2	yes	maximal, complemented
3	#26022513	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022382	C₂	1	2	yes	maximal, complemented
5	#26022322	C₂	1	2	yes	maximal, complemented
6	#26022232	C₂	1	2	yes	maximal, complemented
7	#26022211	C₂	1	2	yes	maximal, complemented
8	#26022134	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2170	C₂	1	1	yes	maximal, characteristic, complemented
10	#19152415	C₂²	1	2	yes	—
11	#19152405	—	2	2	no	—
12	#19152403	C₂²	1	2	yes	complemented
13	#19152041	—	2	2	no	—
14	#19152041	C₂²	1	2	yes	complemented
15	#19152036	—	2	2	no	—
16	#19152036	C₂²	1	2	yes	—
17	#19152030	—	2	2	no	—
18	#19152030	C₂²	1	1	yes	characteristic
19	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic, complemented
20	C₂×#15667957	—	2	2	no	—
21	C₂×#15667957	C₂²	1	2	yes	complemented
22	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
23	#19151982	C₂²	1	2	yes	complemented
24	#19151873	C₂²	1	2	yes	complemented
25	#19151868	C₂²	1	2	yes	—
26	#19151862	—	2	2	no	—
27	#19151777	—	2	2	no	—
28	#19151772	—	2	2	no	—
29	#19151716	C₂²	1	2	yes	complemented
30	#19151428	C₂²	1	1	yes	characteristic, complemented
31	#19151415	C₂²	1	2	yes	complemented
32	#19151412	C₂²	1	2	yes	complemented
33	#19151406	C₂²	1	2	yes	complemented
34	#504×C₈₉	C₂²	1	2	yes	complemented
35	#443×C₈₉	C₂²	1	1	yes	characteristic
36	#364×C₈₉	C₂²	1	1	yes	characteristic, complemented
37	#350×C₈₉	C₂²	1	2	yes	complemented
38	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
39	#15668059	—	2	2	no	—
40	#15668058	—	2	2	no	—
41	#15668045	—	2	2	no	—
42	#15668045	—	4	2	no	—
43	#15668044	—	4	2	no	—
44	#15668042	—	2	2	no	—
45	#15668040	—	2	2	no	—
46	#15668040	—	4	2	no	—
47	#15668037	—	2	2	no	—
48	#15668036	—	4	2	no	—
49	#15668034	—	2	2	no	—
50	#15668034	—	2	2	no	—
51	#15668034	C₂³	1	2	yes	—
52	#15668033	—	4	2	no	—
53	C₂×Dic₃₅₆	—	2	2	no	—
54	C₂×Dic₃₅₆	C₂³	1	2	yes	—
55	C₂×#13083563	—	2	2	no	—
56	SD₁₆×D₈₉	—	4	2	no	—
57	C₂×D₇₁₂	—	2	2	no	—
58	C₂×D₇₁₂	C₂³	1	2	yes	—
59	#15668018	—	2	2	no	—
60	#15668000	—	2	2	no	—
61	Dic₇₁₂	—	2	2	no	—
62	Dic₇₁₂	—	4	1	no	—
63	Dic₇₁₂	D₄	1	1	yes	characteristic
64	#15667957	—	2	2	no	—
65	#15667957	—	2	2	no	—
66	#15667957	—	2	2	no	—
67	#15667957	—	2	2	no	—
68	#15667957	D₄	1	2	yes	—
69	D₁₄₂₄	—	2	2	no	—
70	D₁₄₂₄	—	4	1	no	—
71	D₁₄₂₄	D₄	1	1	yes	characteristic
72	#15667948	C₂³	1	2	yes	—
73	#15667940	—	2	2	no	—
74	#15667930	—	2	2	no	—
75	#15667928	—	2	2	no	—
76	#135×C₈₉	D₄	1	1	yes	characteristic
77	#131×C₈₉	C₂³	1	2	yes	—
78	M₃₂×C₈₉	—	2	2	no	—
79	M₃₂×C₈₉	C₂³	1	2	yes	—
80	C₂×C₁₆×C₈₉	—	2	2	no	—
81	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
82	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
83	#108×C₈₉	D₄	1	1	yes	characteristic
84	#104×C₈₉	D₄	1	2	yes	—
85	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
86	#13083575	—	4	2	no	—
87	#13083574	—	2	2	no	—
88	#13083574	—	4	2	no	—
89	#13083574	—	4	2	no	—
90	#13083573	—	4	2	no	—
91	#13083573	—	8	2	no	—
92	C₂×Dic₁₇₈	—	2	2	no	—
93	C₂×Dic₁₇₈	—	4	2	no	—
94	Q₈×D₈₉	—	4	2	no	—
95	C₂×D₃₅₆	—	2	2	no	—
96	C₂×D₃₅₆	—	4	2	no	—
97	D₄×D₈₉	—	4	2	no	—
98	D₄×D₈₉	—	8	2	no	—
99	Dic₃₅₆	—	2	2	no	—
100	Dic₃₅₆	—	2	2	no	—
101	Dic₃₅₆	—	4	2	no	—
102	Dic₃₅₆	C₂×D₄	1	2	yes	—
103	#13083564	—	4	2	no	—
104	#13083563	—	2	2	no	—
105	#13083563	—	4	2	no	—
106	#13083563	—	4	2	no	—
107	#13083563	—	4	2	no	—
108	#13083563	—	4	2	no	—
109	#13083562	—	4	2	no	—
110	#13083561	—	4	2	no	—
111	D₇₁₂	—	2	2	no	—
112	D₇₁₂	—	2	2	no	—
113	D₇₁₂	—	4	2	no	—
114	D₇₁₂	C₂×D₄	1	2	yes	—
115	#13083559	—	4	2	no	—
116	#13083558	—	2	2	no	—
117	#13083557	—	4	2	no	—
118	C₈×D₈₉	—	4	2	no	—
119	#13083551	—	8	2	no	—
120	#41×C₈₉	C₂×D₄	1	2	yes	—
121	Q₁₆×C₈₉	D₈	1	1	yes	characteristic
122	SD₁₆×C₈₉	D₈	1	2	yes	—
123	D₈×C₈₉	D₈	1	1	yes	characteristic
124	M₁₆×C₈₉	—	2	2	no	—
125	M₁₆×C₈₉	C₂×D₄	1	2	yes	—
126	C₂×C₈×C₈₉	—	2	2	no	—
127	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
129	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₁₆×C₈₉	—	2	2	no	—
131	C₁₆×C₈₉	—	2	1	no	—
132	C₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
133	C₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
134	C₂²×D₈₉	—	4	2	no	—
135	C₂²×D₈₉	—	8	2	no	—
136	Q₈×C₈₉	C₂×D₈	1	2	yes	—
137	D₄×C₈₉	—	2	2	no	—
138	D₄×C₈₉	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₈₉	—	2	2	no	—
140	C₂×C₄×C₈₉	—	2	1	no	—
141	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
142	C₂²⋊D₈₉	—	4	2	no	—
143	C₂²⋊D₈₉	—	8	2	no	—
144	C₂²⋊D₈₉	—	8	2	no	—
145	C₂×Dic₈₉	—	4	2	no	—
146	C₂×Dic₈₉	—	8	2	no	—
147	D₃₅₆	—	2	2	no	—
148	D₃₅₆	—	4	2	no	—
149	D₃₅₆	—	4	2	no	—
150	D₃₅₆	—	8	2	no	—
151	C₄×D₈₉	—	4	2	no	—
152	C₄×D₈₉	—	8	2	no	—
153	C₄×D₈₉	—	8	2	no	—
154	Dic₁₇₈	—	2	2	no	—
155	Dic₁₇₈	—	4	2	no	—
156	Dic₁₇₈	—	4	2	no	—
157	Dic₁₇₈	—	8	2	no	—
158	C₈×C₈₉	—	2	2	no	—
159	C₈×C₈₉	C₂×D₈	1	1	yes	characteristic
160	C₈×C₈₉	C₂×D₈	1	1	yes	characteristic
161	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
162	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
163	#10623494	—	4	2	no	—
164	C₂²×C₈₉	—	2	2	no	—
165	C₂²×C₈₉	#576	1	1	yes	characteristic
166	C₂×D₈₉	—	4	2	no	—
167	C₂×D₈₉	—	8	2	no	—
168	C₂×D₈₉	—	16	2	no	—
169	C₂×D₈₉	—	32	2	no	—
170	C₄×C₈₉	—	2	2	no	—
171	C₄×C₈₉	—	4	1	no	—
172	C₄×C₈₉	D₄²	1	1	yes	characteristic
173	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
174	Dic₈₉	—	4	2	no	—
175	Dic₈₉	—	8	2	no	—
176	#33857	—	89	1	no	maximal, Sylow
177	C₂×C₈₉	—	2	1	no	—
178	C₂×C₈₉	—	4	2	no	—
179	C₂×C₈₉	D₄×D₈	1	1	yes	characteristic
180	D₈₉	—	8	2	no	—
181	D₈₉	—	16	2	no	—
182	#3416	—	89	2	no	—
183	#3290	—	89	2	no	—
184	#2249	—	89	1	no	—
185	#2223	—	89	2	no	—
186	#2212	—	89	2	no	—
187	#2195	—	89	2	no	—
188	#2192	—	89	2	no	—
189	#2178	—	89	1	no	—
190	#2170	D₈₉	1	1	yes	characteristic, complemented
191	C₈₉	#33857	1	1	yes	characteristic, complemented, Sylow
192	#578	—	89	2	no	—
193	#577	—	178	2	no	—
194	#576	—	89	2	no	—
195	#510	—	89	2	no	—
196	#510	—	178	2	no	—
197	#509	—	89	2	no	—
198	#509	—	178	2	no	—
199	#508	—	89	1	no	—
200	#508	—	178	2	no	—
201	C₂×Q₃₂	—	89	1	no	—
202	C₂×SD₃₂	—	89	2	no	—
203	C₂×SD₃₂	—	178	2	no	—
204	C₂×D₁₆	—	89	1	no	—
205	#504	C₂×D₈₉	1	2	yes	complemented
206	#495	—	89	2	no	—
207	#473	—	89	2	no	—
208	#472	—	89	2	no	—
209	#471	—	178	2	no	—
210	#455	—	178	2	no	—
211	#454	—	178	2	no	—
212	#443	—	89	2	no	—
213	#443	C₂×D₈₉	1	1	yes	characteristic
214	#368	—	89	1	no	—
215	#364	C₂×D₈₉	1	1	yes	characteristic, complemented
216	#362	—	89	2	no	—
217	#361	—	89	2	no	—
218	#359	—	89	2	no	—
219	#350	C₂×D₈₉	1	2	yes	complemented
220	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic, complemented
221	2₋¹⁺⁴	—	178	2	no	—
222	H(5,𝔽₂)	—	178	2	no	—
223	#137	—	356	2	no	—
224	#137	—	356	2	no	—
225	#137	—	178	2	no	—
226	#137	—	178	2	no	—
227	AGL(1,ℤ/8)	—	356	2	no	—
228	AGL(1,ℤ/8)	—	356	2	no	—
229	AGL(1,ℤ/8)	—	178	2	no	—
230	AGL(1,ℤ/8)	—	178	2	no	—
231	#135	—	89	2	no	—
232	#135	—	356	2	no	—
233	#135	—	178	2	no	—
234	#135	—	178	2	no	—
235	#135	D₃₅₆	1	1	yes	characteristic
236	C₂×Q₁₆	—	89	2	no	—
237	C₂×Q₁₆	—	178	2	no	—
238	C₂×SD₁₆	—	356	2	no	—
239	C₂×SD₁₆	—	178	2	no	—
240	C₂×D₈	—	89	2	no	—
241	C₂×D₈	—	178	2	no	—
242	#131	—	178	2	no	—
243	#131	C₂²×D₈₉	1	2	yes	—
244	#124	—	178	2	no	—
245	Q₃₂	—	89	1	no	—
246	Q₃₂	—	356	1	no	—
247	Q₃₂	—	178	2	no	—
248	SD₃₂	—	89	2	no	—
249	SD₃₂	—	178	2	no	—
250	SD₃₂	—	178	2	no	—
251	SD₃₂	—	178	2	no	—
252	SD₃₂	—	178	2	no	—
253	D₁₆	—	89	1	no	—
254	D₁₆	—	356	1	no	—
255	D₁₆	—	178	2	no	—
256	M₃₂	—	2	2	no	—
257	M₃₂	C₂²×D₈₉	1	2	yes	—
258	C₂×C₁₆	—	2	2	no	—
259	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
260	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
261	#108	—	89	2	no	—
262	#108	D₃₅₆	1	1	yes	characteristic
263	#104	—	178	2	no	—
264	#104	D₃₅₆	1	2	yes	—
265	#101	—	178	2	no	—
266	#100	—	178	2	no	—
267	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
268	#41	—	356	2	no	—
269	#41	—	356	2	no	—
270	#41	—	356	2	no	—
271	#41	—	356	2	no	—
272	#41	—	178	2	no	—
273	#41	—	712	2	no	—
274	#41	C₂×D₃₅₆	1	2	yes	—
275	C₂×Q₈	—	356	2	no	—
276	C₂×Q₈	—	356	2	no	—
277	C₂×Q₈	—	178	2	no	—
278	C₂×D₄	—	356	2	no	—
279	C₂×D₄	—	356	2	no	—
280	C₂×D₄	—	178	2	no	—
281	C₂×D₄	—	712	2	no	—
282	Q₁₆	—	89	2	no	—
283	Q₁₆	—	356	2	no	—
284	Q₁₆	—	356	2	no	—
285	Q₁₆	—	178	2	no	—
286	Q₁₆	—	178	2	no	—
287	Q₁₆	D₇₁₂	1	1	yes	characteristic
288	SD₁₆	—	356	2	no	—
289	SD₁₆	—	356	2	no	—
290	SD₁₆	—	356	2	no	—
291	SD₁₆	—	356	2	no	—
292	SD₁₆	—	356	2	no	—
293	SD₁₆	—	356	2	no	—
294	SD₁₆	—	178	2	no	—
295	SD₁₆	D₇₁₂	1	2	yes	—
296	D₈	—	89	2	no	—
297	D₈	—	356	2	no	—
298	D₈	—	356	2	no	—
299	D₈	—	178	2	no	—
300	D₈	—	178	2	no	—
301	D₈	D₇₁₂	1	1	yes	characteristic
302	M₁₆	—	2	2	no	—
303	M₁₆	—	356	2	no	—
304	M₁₆	—	178	2	no	—
305	M₁₆	C₂×D₃₅₆	1	2	yes	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	356	2	no	—
308	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
309	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
310	C₂²⋊C₄	—	712	2	no	—
311	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
312	C₁₆	—	2	2	no	—
313	C₁₆	—	2	1	no	—
314	C₁₆	D₄×D₈₉	1	1	yes	characteristic
315	C₁₆	D₄×D₈₉	1	1	yes	characteristic
316	C₂³	—	356	2	no	—
317	C₂³	—	712	2	no	—
318	Q₈	—	356	2	no	—
319	Q₈	—	356	2	no	—
320	Q₈	—	178	2	no	—
321	Q₈	—	712	2	no	—
322	Q₈	C₂×D₇₁₂	1	2	yes	—
323	D₄	—	2	2	no	—
324	D₄	—	356	2	no	—
325	D₄	—	356	2	no	—
326	D₄	—	356	2	no	—
327	D₄	—	178	2	no	—
328	D₄	—	712	2	no	—
329	D₄	—	712	2	no	—
330	D₄	—	712	2	no	—
331	D₄	C₂×D₇₁₂	1	2	yes	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	356	2	no	—
335	C₂×C₄	—	356	2	no	—
336	C₂×C₄	—	712	2	no	—
337	C₂×C₄	—	712	2	no	—
338	C₂×C₄	—	712	2	no	—
339	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
340	C₈	—	2	2	no	—
341	C₈	—	356	2	no	—
342	C₈	C₂×D₇₁₂	1	1	yes	characteristic
343	C₈	C₂×D₇₁₂	1	1	yes	characteristic
344	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
345	C₈	#15668058	1	1	yes	characteristic
346	C₂²	—	2	2	no	—
347	C₂²	—	2848	2	no	—
348	C₂²	—	356	2	no	—
349	C₂²	—	1424	2	no	—
350	C₂²	—	712	2	no	—
351	C₂²	#19152403	1	1	yes	characteristic
352	C₄	—	2	2	no	—
353	C₄	—	4	1	no	—
354	C₄	—	356	2	no	—
355	C₄	—	712	2	no	—
356	C₄	D₄×D₃₅₆	1	1	yes	characteristic
357	C₄	C₂²×D₇₁₂	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	4	2	no	—
360	C₂	—	1424	2	no	—
361	C₂	—	712	2	no	—
362	C₂	D₄×D₇₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2170	C₂
socle	C₂×C₈₉	D₄×D₈
hypercenter	#2170	D₈₉
hypocenter	C₈₉	#33857
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443 ◁ #2170

Complements

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

N	H ≅ G/N	type
1. #26033400	C₂	semidirect
2. #26032039	C₂	semidirect
3. #26022513	C₂	semidirect
4. #26022382	C₂	semidirect
5. #26022322	C₂	semidirect
6. #26022232	C₂	semidirect
7. #26022211	C₂	semidirect
8. #26022134	C₂	semidirect
9. C₈₉×#2170	C₂	semidirect
12. #19152403	C₂²	semidirect
14. #19152041	C₂²	semidirect
19. C₂×Dic₇₁₂	C₂²	semidirect
21. C₂×#15667957	C₂²	semidirect
23. #19151982	C₂²	semidirect
24. #19151873	C₂²	semidirect
29. #19151716	C₂²	semidirect
30. #19151428	C₂²	semidirect
31. #19151415	C₂²	semidirect
32. #19151412	C₂²	semidirect
33. #19151406	C₂²	semidirect
34. #504×C₈₉	C₂²	semidirect
36. #364×C₈₉	C₂²	semidirect
37. #350×C₈₉	C₂²	semidirect
38. C₄×C₁₆×C₈₉	C₂²	semidirect
190. #2170	D₈₉	semidirect
191. C₈₉	#33857	semidirect
205. #504	C₂×D₈₉	semidirect
215. #364	C₂×D₈₉	semidirect
219. #350	C₂×D₈₉	semidirect
220. C₄×C₁₆	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26033400	2
#26032039	2
#26022513	2
#26022382	2
#26022322	2
#26022232	2
#26022211	2
#26022134	2
C₈₉×#2170	2
#33857	89