Subgroups — #36542358

Order 22784.

1050 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	#26031172	C₂	1	2	yes	maximal, complemented
2	#26030831	C₂	1	1	yes	maximal, characteristic, complemented
3	#26030704	C₂	1	2	yes	maximal, complemented
4	C₂×#19151975	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019918	C₂	1	4	yes	maximal, complemented
6	#26019867	C₂	1	2	yes	maximal, complemented
7	#26019844	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668040	—	2	4	no	—
10	C₂²×D₇₁₂	—	2	2	no	—
11	#19152148	C₂²	1	2	yes	—
12	C₂×#15668009	C₂²	1	2	yes	—
13	#19152003	C₂²	1	1	yes	characteristic, complemented
14	#19151987	—	2	4	no	—
15	#19151975	—	2	2	no	—
16	#19151975	C₂²	1	2	yes	—
17	#19151975	C₂²	1	2	yes	—
18	#19151975	C₂²	1	1	yes	characteristic, complemented
19	#19151975	C₂²	1	4	yes	complemented
20	#19151975	C₂²	1	4	yes	complemented
21	#19151971	C₂²	1	4	yes	complemented
22	#19151945	C₂²	1	4	yes	complemented
23	#19151945	C₂²	1	2	yes	complemented
24	#19151834	—	2	2	no	—
25	#19151794	—	2	4	no	—
26	#19151734	—	2	4	no	—
27	C₄×D₇₁₂	—	2	2	no	—
28	#431×C₈₉	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
31	#322×C₈₉	C₂²	1	2	yes	complemented
32	C₈²×C₈₉	C₂²	1	2	yes	complemented
33	C₂×#13083574	—	2	2	no	—
34	C₂²×D₃₅₆	—	2	2	no	—
35	C₂²×D₃₅₆	—	2	2	no	—
36	C₂²×D₃₅₆	—	2	2	no	—
37	#15668040	—	4	16	no	—
38	C₂×#13083563	—	2	8	no	—
39	C₂×D₇₁₂	—	2	8	no	—
40	C₂×D₇₁₂	—	2	2	no	—
41	C₂×D₇₁₂	—	2	4	no	—
42	C₂×D₇₁₂	—	2	4	no	—
43	C₂×D₇₁₂	—	2	8	no	—
44	C₂×D₇₁₂	—	4	8	no	—
45	#15668012	C₂³	1	2	yes	—
46	#15668009	—	2	2	no	—
47	#15668009	—	2	2	no	—
48	#15668009	—	2	2	no	—
49	#15668009	C₂³	1	4	yes	—
50	#15668009	C₂³	1	2	yes	—
51	#15668000	—	2	2	no	—
52	#15667986	—	2	2	no	—
53	#15667986	—	2	2	no	—
54	C₄×D₃₅₆	—	2	2	no	—
55	C₄×D₃₅₆	—	2	2	no	—
56	#15667946	—	2	2	no	—
57	#15667934	—	2	4	no	—
58	#15667934	—	2	8	no	—
59	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
60	C₂²×C₈×C₈₉	D₄	1	2	yes	—
61	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
63	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
64	#97×C₈₉	D₄	1	2	yes	—
65	C₄×C₈×C₈₉	D₄	1	2	yes	—
66	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₈₉	C₂³	1	2	yes	—
69	C₄×C₈×C₈₉	C₂³	1	2	yes	—
70	C₂³×D₈₉	—	4	2	no	—
71	#13083574	—	4	8	no	—
72	C₂×Dic₁₇₈	—	2	2	no	—
73	C₂×D₃₅₆	—	2	2	no	—
74	C₂×D₃₅₆	—	2	2	no	—
75	C₂×D₃₅₆	—	2	2	no	—
76	C₂×D₃₅₆	—	2	4	no	—
77	C₂×D₃₅₆	—	2	4	no	—
78	C₂×D₃₅₆	—	2	4	no	—
79	C₂×D₃₅₆	—	4	8	no	—
80	C₂×D₃₅₆	—	4	8	no	—
81	C₂×D₃₅₆	—	4	2	no	—
82	C₂×D₃₅₆	—	4	2	no	—
83	C₂×D₃₅₆	—	4	2	no	—
84	C₂×D₃₅₆	—	8	4	no	—
85	C₂×C₂²⋊D₈₉	—	4	2	no	—
86	C₂×C₄×D₈₉	—	4	2	no	—
87	#13083563	—	4	16	no	—
88	D₇₁₂	—	4	8	no	—
89	D₇₁₂	—	4	16	no	—
90	D₇₁₂	—	4	4	no	—
91	D₇₁₂	—	4	8	no	—
92	D₇₁₂	—	8	8	no	—
93	C₄⋊Dic₈₉	—	2	2	no	—
94	C₄⋊Dic₈₉	—	2	2	no	—
95	#13083551	—	4	2	no	—
96	#13083551	—	4	2	no	—
97	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
100	M₁₆×C₈₉	D₈	1	8	yes	complemented
101	C₂×C₈×C₈₉	—	2	4	no	—
102	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
104	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
105	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
106	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
110	C₂³×C₈₉	#127	1	1	yes	characteristic
111	C₂²×D₈₉	—	4	2	no	—
112	C₂²×D₈₉	—	4	4	no	—
113	C₂²×D₈₉	—	8	4	no	—
114	C₂²×D₈₉	—	8	4	no	—
115	C₂²×D₈₉	—	8	4	no	—
116	C₂×C₄×C₈₉	—	2	2	no	—
117	C₂×C₄×C₈₉	—	2	2	no	—
118	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
119	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
120	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂×D₈	1	4	yes	—
122	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
125	C₂²⋊D₈₉	—	8	4	no	—
126	C₂×Dic₈₉	—	4	2	no	—
127	D₃₅₆	—	4	4	no	—
128	D₃₅₆	—	4	8	no	—
129	D₃₅₆	—	4	8	no	—
130	D₃₅₆	—	4	4	no	—
131	D₃₅₆	—	4	4	no	—
132	D₃₅₆	—	4	8	no	—
133	D₃₅₆	—	8	4	no	—
134	D₃₅₆	—	8	4	no	—
135	D₃₅₆	—	8	4	no	—
136	C₄×D₈₉	—	8	4	no	—
137	Dic₁₇₈	—	4	4	no	—
138	C₈×C₈₉	—	2	2	no	—
139	C₈×C₈₉	—	2	4	no	—
140	C₈×C₈₉	C₂×D₈	1	8	yes	complemented
141	C₈×C₈₉	AGL(1,ℤ/8)	1	4	yes	complemented
142	C₂²×C₈₉	—	2	1	no	—
143	C₂²×C₈₉	—	2	1	no	—
144	C₂²×C₈₉	#493	1	2	yes	—
145	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
146	C₂×D₈₉	—	8	2	no	—
147	C₂×D₈₉	—	8	4	no	—
148	C₂×D₈₉	—	8	4	no	—
149	C₂×D₈₉	—	8	2	no	—
150	C₂×D₈₉	—	8	4	no	—
151	C₂×D₈₉	—	8	2	no	—
152	C₂×D₈₉	—	16	8	no	—
153	C₄×C₈₉	—	2	1	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	#493	1	2	yes	—
156	C₄×C₈₉	C₂²×D₈	1	2	yes	—
157	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
158	C₄×C₈₉	#576	1	2	yes	—
159	Dic₈₉	—	8	2	no	—
160	#19629	—	89	1	no	maximal, Sylow
161	C₂×C₈₉	—	2	2	no	—
162	C₂×C₈₉	C₂×#493	1	1	yes	characteristic
163	C₂×C₈₉	#3151	1	1	yes	characteristic
164	C₂×C₈₉	#3155	1	1	yes	characteristic
165	D₈₉	—	16	2	no	—
166	D₈₉	—	16	4	no	—
167	#3205	—	89	2	no	—
168	#3169	—	89	1	no	—
169	#3151	—	89	2	no	—
170	C₂×#493	—	89	1	no	—
171	#1721	—	89	4	no	—
172	#1712	—	89	2	no	—
173	#1706	—	89	2	no	—
174	C₈×M₁₆	D₈₉	1	1	yes	characteristic, complemented
175	C₈₉	#19629	1	1	yes	characteristic, complemented, Sylow
176	C₂×AGL(1,ℤ/8)	—	178	4	no	—
177	C₂²×D₈	—	178	2	no	—
178	#532	—	89	2	no	—
179	C₂×#127	—	89	2	no	—
180	#500	—	89	1	no	—
181	#496	—	178	4	no	—
182	#493	—	89	2	no	—
183	#493	—	89	2	no	—
184	#493	—	89	4	no	—
185	#493	—	89	1	no	—
186	#493	—	89	4	no	—
187	#493	—	178	2	no	—
188	#492	—	89	4	no	—
189	#486	—	89	2	no	—
190	#486	—	89	4	no	—
191	#466	—	178	2	no	—
192	#459	—	178	4	no	—
193	#447	—	178	4	no	—
194	C₄×D₈	—	178	2	no	—
195	#431	C₂×D₈₉	1	1	yes	characteristic
196	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
197	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
198	#322	C₂×D₈₉	1	2	yes	complemented
199	C₈²	C₂×D₈₉	1	2	yes	complemented
200	C₂×#41	—	178	2	no	—
201	C₂²×D₄	—	178	2	no	—
202	C₂²×D₄	—	178	2	no	—
203	C₂²×D₄	—	178	2	no	—
204	AGL(1,ℤ/8)	—	356	16	no	—
205	C₂×SD₁₆	—	178	8	no	—
206	C₂×D₈	—	356	8	no	—
207	C₂×D₈	—	178	4	no	—
208	C₂×D₈	—	178	4	no	—
209	C₂×D₈	—	178	8	no	—
210	C₂×D₈	—	178	2	no	—
211	C₂×D₈	—	178	8	no	—
212	C₂×M₁₆	D₃₅₆	1	2	yes	—
213	C₂²×C₈	D₃₅₆	1	2	yes	—
214	C₄⋊Q₈	—	89	2	no	—
215	#127	—	89	4	no	—
216	#127	—	89	2	no	—
217	#127	—	178	2	no	—
218	#127	—	178	2	no	—
219	#127	—	178	2	no	—
220	#124	—	178	2	no	—
221	#121	—	178	2	no	—
222	#121	—	178	2	no	—
223	C₄×D₄	—	178	2	no	—
224	C₄×D₄	—	178	2	no	—
225	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
226	#107	—	178	2	no	—
227	C₄⋊C₈	D₃₅₆	1	2	yes	—
228	#102	—	178	4	no	—
229	#102	—	178	8	no	—
230	C₂²⋊C₈	D₃₅₆	1	2	yes	—
231	#97	D₃₅₆	1	2	yes	—
232	C₄×C₈	D₃₅₆	1	2	yes	—
233	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
234	C₄×C₈	C₂²×D₈₉	1	2	yes	—
235	C₄×C₈	C₂²×D₈₉	1	2	yes	—
236	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
237	C₂⁴	—	356	2	no	—
238	#41	—	356	8	no	—
239	C₂×Q₈	—	178	2	no	—
240	C₂×D₄	—	356	8	no	—
241	C₂×D₄	—	356	2	no	—
242	C₂×D₄	—	356	2	no	—
243	C₂×D₄	—	356	2	no	—
244	C₂×D₄	—	356	2	no	—
245	C₂×D₄	—	356	8	no	—
246	C₂×D₄	—	178	2	no	—
247	C₂×D₄	—	178	4	no	—
248	C₂×D₄	—	178	2	no	—
249	C₂×D₄	—	178	2	no	—
250	C₂×D₄	—	178	4	no	—
251	C₂×D₄	—	178	4	no	—
252	C₂×D₄	—	712	4	no	—
253	C₂²×C₄	—	356	2	no	—
254	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
257	SD₁₆	—	356	16	no	—
258	D₈	—	356	8	no	—
259	D₈	—	356	16	no	—
260	D₈	—	356	8	no	—
261	D₈	—	356	4	no	—
262	D₈	—	712	8	no	—
263	M₁₆	D₇₁₂	1	8	yes	complemented
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
266	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
267	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
268	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
269	C₄⋊C₄	—	178	2	no	—
270	C₄⋊C₄	—	178	2	no	—
271	C₂²⋊C₄	—	356	2	no	—
272	C₂²⋊C₄	—	356	2	no	—
273	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
274	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
275	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
276	C₄²	C₂³×D₈₉	1	1	yes	characteristic
277	C₂³	—	356	2	no	—
278	C₂³	—	356	4	no	—
279	C₂³	—	712	4	no	—
280	C₂³	—	712	4	no	—
281	C₂³	—	712	4	no	—
282	C₂³	#15668009	1	1	yes	characteristic
283	Q₈	—	356	4	no	—
284	D₄	—	356	8	no	—
285	D₄	—	356	4	no	—
286	D₄	—	356	4	no	—
287	D₄	—	356	8	no	—
288	D₄	—	356	4	no	—
289	D₄	—	356	8	no	—
290	D₄	—	712	4	no	—
291	D₄	—	712	4	no	—
292	D₄	—	712	4	no	—
293	D₄	—	712	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	356	2	no	—
297	C₂×C₄	—	712	4	no	—
298	C₂×C₄	#15668009	1	1	yes	characteristic
299	C₂×C₄	#15668009	1	1	yes	characteristic
300	C₂×C₄	#15668009	1	1	yes	characteristic
301	C₂×C₄	C₂×D₇₁₂	1	4	yes	—
302	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	2	4	no	—
307	C₈	C₂×D₇₁₂	1	8	yes	complemented
308	C₈	#15668040	1	4	yes	complemented
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	1424	8	no	—
312	C₂²	—	712	2	no	—
313	C₂²	—	712	4	no	—
314	C₂²	—	712	4	no	—
315	C₂²	—	712	2	no	—
316	C₂²	—	712	4	no	—
317	C₂²	—	712	2	no	—
318	C₂²	#19151975	1	2	yes	—
319	C₂²	C₂×#15668009	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	712	2	no	—
323	C₄	#19151975	1	2	yes	—
324	C₄	C₂²×D₇₁₂	1	2	yes	—
325	C₄	C₂×#15668040	1	2	yes	—
326	C₄	#19152403	1	2	yes	—
327	C₂	—	2	2	no	—
328	C₂	—	1424	4	no	—
329	C₂	—	1424	2	no	—
330	C₂	C₂×#19151975	1	1	yes	characteristic
331	C₂	#26030704	1	1	yes	characteristic
332	C₂	#26030738	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15668009
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈×M₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#127
hypercenter	C₈×M₁₆	D₈₉
hypocenter	C₈₉	#19629
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
1. #26031172	C₂	semidirect
2. #26030831	C₂	semidirect
3. #26030704	C₂	semidirect
4. C₂×#19151975	C₂	semidirect
5. #26019918	C₂	semidirect
6. #26019867	C₂	semidirect
7. #26019844	C₂	semidirect
8. C₈×M₁₆×C₈₉	C₂	semidirect
13. #19152003	C₂²	semidirect
18. #19151975	C₂²	semidirect
19. #19151975	C₂²	semidirect
20. #19151975	C₂²	semidirect
21. #19151971	C₂²	semidirect
22. #19151945	C₂²	semidirect
23. #19151945	C₂²	semidirect
31. #322×C₈₉	C₂²	semidirect
32. C₈²×C₈₉	C₂²	semidirect
100. M₁₆×C₈₉	D₈	semidirect
140. C₈×C₈₉	C₂×D₈	semidirect
141. C₈×C₈₉	AGL(1,ℤ/8)	semidirect
174. C₈×M₁₆	D₈₉	semidirect
175. C₈₉	#19629	semidirect
198. #322	C₂×D₈₉	semidirect
199. C₈²	C₂×D₈₉	semidirect
263. M₁₆	D₇₁₂	semidirect
307. C₈	C₂×D₇₁₂	semidirect
308. C₈	#15668040	semidirect

Maximal subgroups

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

subgroup	index
#26031172	2
#26030831	2
#26030704	2
C₂×#19151975	2
#26019918	2
#26019867	2
#26019844	2
C₈×M₁₆×C₈₉	2
#19629	89