Subgroups — #36712426

Order 22784.

508 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033240	C₂	1	2	yes	maximal, complemented
2	#26022514	C₂	1	1	yes	maximal, characteristic
3	#26022513	C₂	1	1	yes	maximal, characteristic
4	#26022503	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022317	C₂	1	2	yes	maximal, complemented
6	#26022314	C₂	1	2	yes	maximal, complemented
7	#26022296	C₂	1	2	yes	maximal, complemented
8	C₁₆×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022110	C₂	1	1	yes	maximal, characteristic, complemented
10	#26022098	C₂	1	1	yes	maximal, characteristic
11	C₄×Q₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152320	—	2	2	no	—
13	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
14	C₂×#15667958	C₂²	1	2	yes	—
15	C₂×#15667957	C₂²	1	2	yes	—
16	C₂×#15667955	C₂²	1	2	yes	—
17	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
18	#19151983	C₂²	1	2	yes	—
19	#19151982	C₂²	1	2	yes	—
20	#19151976	C₂²	1	2	yes	—
21	#19151823	—	2	2	no	—
22	#19151820	—	2	2	no	—
23	#19151802	—	2	2	no	—
24	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
25	#19151692	C₂²	1	2	yes	—
26	#19151680	C₂²	1	2	yes	—
27	C₄⋊#13083549	C₂²	1	1	yes	characteristic
28	#19151416	C₂²	1	1	yes	characteristic
29	#19151416	C₂²	1	1	yes	characteristic
30	#19151401	C₂²	1	2	yes	—
31	#19151398	C₂²	1	2	yes	—
32	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
34	C₂×Q₃₂×C₈₉	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
36	#366×C₈₉	C₂²	1	1	yes	characteristic
37	#358×C₈₉	C₂²	1	2	yes	—
38	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×Dic₃₅₆	C₂³	1	2	yes	—
40	C₂×#13083564	—	2	2	no	—
41	C₂×#13083563	—	2	2	no	—
42	C₂×#13083561	—	2	2	no	—
43	C₂×D₇₁₂	C₂³	1	2	yes	—
44	#15668010	—	2	2	no	—
45	#15668007	—	4	2	no	—
46	#15668006	—	4	2	no	—
47	#15668001	—	2	2	no	—
48	#15668000	—	2	2	no	—
49	#15667990	—	4	2	no	—
50	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
51	#15667977	—	2	2	no	—
52	#15667974	—	2	2	no	—
53	#15667969	—	4	2	no	—
54	Dic₇₁₂	—	2	2	no	—
55	#15667958	—	4	2	no	—
56	#15667957	—	4	2	no	—
57	#15667955	—	4	2	no	—
58	D₁₄₂₄	—	2	2	no	—
59	C₂×#13083549	C₂³	1	1	yes	characteristic
60	C₂×#13083549	C₂³	1	1	yes	characteristic
61	C₄⋊#10623494	D₄	1	1	yes	characteristic
62	#15667941	D₄	1	1	yes	characteristic
63	#15667941	D₄	1	1	yes	characteristic
64	#15667935	—	2	2	no	—
65	#15667932	—	2	2	no	—
66	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
67	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
68	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
69	C₄×Q₈×C₈₉	—	2	2	no	—
70	Q₃₂×C₈₉	—	2	2	no	—
71	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
72	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
73	#107×C₈₉	C₂³	1	2	yes	—
74	#103×C₈₉	—	2	2	no	—
75	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×Dic₁₇₈	—	2	2	no	—
77	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
78	C₂×D₃₅₆	—	2	2	no	—
79	C₂×C₄×D₈₉	—	4	2	no	—
80	Dic₃₅₆	—	2	2	no	—
81	Dic₃₅₆	—	2	2	no	—
82	#13083564	—	8	2	no	—
83	#13083563	—	8	2	no	—
84	#13083561	—	8	2	no	—
85	D₇₁₂	—	2	2	no	—
86	D₇₁₂	—	2	2	no	—
87	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
88	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
89	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
90	#13083553	—	4	2	no	—
91	#13083553	—	4	2	no	—
92	#13083553	—	4	2	no	—
93	#13083553	D₈	1	1	yes	characteristic
94	#13083553	D₈	1	1	yes	characteristic
95	#13083551	—	4	2	no	—
96	#13083551	—	4	2	no	—
97	#13083551	—	4	2	no	—
98	#13083551	—	4	2	no	—
99	#13083551	—	4	2	no	—
100	C₄×Dic₈₉	—	4	2	no	—
101	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
102	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
103	#13083549	—	2	1	no	—
104	#13083549	—	2	1	no	—
105	C₂×Q₈×C₈₉	—	2	2	no	—
106	Q₁₆×C₈₉	—	2	2	no	—
107	Q₁₆×C₈₉	—	2	2	no	—
108	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
110	C₄⋊C₄×C₈₉	—	2	2	no	—
111	C₄⋊C₄×C₈₉	—	4	2	no	—
112	C₄²×C₈₉	—	4	2	no	—
113	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₁₆×C₈₉	—	2	1	no	—
115	C₁₆×C₈₉	#41	1	2	yes	—
116	C₂²×D₈₉	—	4	2	no	—
117	Q₈×C₈₉	—	4	2	no	—
118	Q₈×C₈₉	—	4	2	no	—
119	C₂×C₄×C₈₉	—	4	2	no	—
120	C₂×C₄×C₈₉	—	4	2	no	—
121	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
124	C₂×Dic₈₉	—	4	2	no	—
125	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
126	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
127	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
128	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
129	D₃₅₆	—	4	2	no	—
130	D₃₅₆	—	4	2	no	—
131	C₄×D₈₉	—	8	4	no	—
132	Dic₁₇₈	—	2	2	no	—
133	Dic₁₇₈	—	4	2	no	—
134	Dic₁₇₈	—	4	2	no	—
135	C₈×C₈₉	—	2	1	no	—
136	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
137	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
138	#10623494	—	2	1	no	—
139	#10623494	—	2	1	no	—
140	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
141	C₂×D₈₉	—	8	2	no	—
142	C₂×D₈₉	—	8	2	no	—
143	C₂×D₈₉	—	8	2	no	—
144	C₄×C₈₉	—	2	1	no	—
145	C₄×C₈₉	—	2	1	no	—
146	C₄×C₈₉	—	8	2	no	—
147	C₄×C₈₉	—	8	2	no	—
148	C₄×C₈₉	#550	1	1	yes	characteristic
149	C₄×C₈₉	#576	1	1	yes	characteristic
150	Dic₈₉	—	2	1	no	—
151	Dic₈₉	—	2	1	no	—
152	Dic₈₉	—	2	1	no	—
153	Dic₈₉	—	8	2	no	—
154	Dic₈₉	#508	1	2	yes	—
155	#33960	—	89	1	no	maximal, Sylow
156	C₂×C₈₉	#3391	1	1	yes	characteristic
157	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
158	C₂×C₈₉	#3417	1	1	yes	characteristic
159	D₈₉	—	16	2	no	—
160	#3393	—	89	2	no	—
161	#2249	—	89	1	no	—
162	#2249	—	89	1	no	—
163	#2247	—	89	1	no	—
164	#2211	—	89	2	no	—
165	#2211	—	89	2	no	—
166	#2208	—	89	2	no	—
167	Q₈×C₁₆	—	89	1	no	—
168	C₄×Q₃₂	—	89	1	no	—
169	C₄×Q₃₂	D₈₉	1	1	yes	characteristic, complemented
170	C₄×D₁₆	—	89	1	no	—
171	C₈₉	#33960	1	1	yes	characteristic, complemented, Sylow
172	#556	—	178	2	no	—
173	C₂×Q₃₂	—	89	2	no	—
174	C₂×Q₃₂	—	89	1	no	—
175	C₂×Q₃₂	C₂×D₈₉	1	1	yes	characteristic
176	C₂×SD₃₂	—	89	2	no	—
177	C₂×SD₃₂	—	89	2	no	—
178	C₂×D₁₆	—	89	1	no	—
179	#495	—	89	2	no	—
180	#495	—	89	2	no	—
181	#494	—	89	2	no	—
182	#464	—	178	2	no	—
183	#464	—	178	2	no	—
184	#461	—	178	2	no	—
185	C₈×Q₈	—	89	1	no	—
186	C₄×Q₁₆	—	89	2	no	—
187	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
188	C₄×D₈	—	89	2	no	—
189	#366	C₂×D₈₉	1	1	yes	characteristic
190	C₄⋊C₁₆	—	89	1	no	—
191	C₄⋊C₁₆	—	89	1	no	—
192	C₄⋊C₁₆	—	89	1	no	—
193	#358	—	89	2	no	—
194	#358	C₂×D₈₉	1	2	yes	—
195	#357	—	89	2	no	—
196	C₄×C₁₆	—	89	1	no	—
197	C₄×C₁₆	—	89	1	no	—
198	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
199	C₂×Q₁₆	—	89	2	no	—
200	C₂×Q₁₆	—	178	2	no	—
201	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
202	C₂×SD₁₆	—	178	2	no	—
203	C₂×SD₁₆	—	178	2	no	—
204	C₂×D₈	—	89	2	no	—
205	C₄⋊Q₈	—	178	2	no	—
206	#126	—	356	2	no	—
207	#126	—	356	2	no	—
208	#124	—	178	2	no	—
209	#124	—	178	2	no	—
210	C₂²⋊Q₈	—	356	2	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	89	1	no	—
213	C₄×Q₈	—	178	2	no	—
214	C₄×D₄	—	178	2	no	—
215	#117	—	356	2	no	—
216	Q₃₂	—	2	2	no	—
217	Q₃₂	—	356	2	no	—
218	Q₃₂	—	178	2	no	—
219	SD₃₂	—	356	2	no	—
220	SD₃₂	—	356	2	no	—
221	D₁₆	—	178	2	no	—
222	C₂×C₁₆	—	89	1	no	—
223	C₂×C₁₆	—	89	1	no	—
224	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
225	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
226	#107	C₂²×D₈₉	1	2	yes	—
227	C₄⋊C₈	—	89	1	no	—
228	C₄⋊C₈	—	89	1	no	—
229	C₄⋊C₈	—	89	1	no	—
230	#103	—	2	2	no	—
231	#103	—	178	2	no	—
232	#102	—	178	2	no	—
233	C₄×C₈	—	89	1	no	—
234	C₄×C₈	—	89	1	no	—
235	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	89	1	no	—
238	C₂×Q₈	—	178	2	no	—
239	C₂×D₄	—	178	2	no	—
240	C₂²×C₄	—	356	2	no	—
241	Q₁₆	—	2	2	no	—
242	Q₁₆	—	2	2	no	—
243	Q₁₆	—	178	2	no	—
244	Q₁₆	—	178	2	no	—
245	Q₁₆	—	712	2	no	—
246	SD₁₆	—	712	2	no	—
247	SD₁₆	—	712	2	no	—
248	D₈	—	178	2	no	—
249	D₈	—	178	2	no	—
250	C₂×C₈	—	89	1	no	—
251	C₂×C₈	—	89	1	no	—
252	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	89	1	no	—
257	C₄⋊C₄	—	89	1	no	—
258	C₄⋊C₄	—	89	1	no	—
259	C₄⋊C₄	—	356	2	no	—
260	C₄⋊C₄	—	356	2	no	—
261	C₄⋊C₄	—	356	2	no	—
262	C₂²⋊C₄	—	356	2	no	—
263	C₂²⋊C₄	—	356	2	no	—
264	C₂²⋊C₄	—	356	2	no	—
265	C₂²⋊C₄	—	356	2	no	—
266	C₂²⋊C₄	—	356	2	no	—
267	C₄²	—	4	2	no	—
268	C₄²	—	89	1	no	—
269	C₄²	—	89	1	no	—
270	C₄²	—	356	2	no	—
271	C₄²	D₄×D₈₉	1	1	yes	characteristic
272	C₁₆	—	2	1	no	—
273	C₁₆	—	178	1	no	—
274	C₁₆	—	178	1	no	—
275	C₁₆	#13083574	1	2	yes	—
276	C₂³	—	356	2	no	—
277	Q₈	—	4	2	no	—
278	Q₈	—	4	2	no	—
279	Q₈	—	356	2	no	—
280	Q₈	—	356	2	no	—
281	Q₈	—	178	2	no	—
282	D₄	—	356	2	no	—
283	D₄	—	356	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	89	1	no	—
287	C₂×C₄	—	89	1	no	—
288	C₂×C₄	—	89	1	no	—
289	C₂×C₄	—	89	1	no	—
290	C₂×C₄	—	356	2	no	—
291	C₂×C₄	—	712	4	no	—
292	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
293	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
294	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	178	1	no	—
297	C₈	—	178	1	no	—
298	C₈	C₂×#13083574	1	1	yes	characteristic
299	C₈	#15668058	1	1	yes	characteristic
300	C₂²	—	712	2	no	—
301	C₂²	—	712	2	no	—
302	C₂²	—	712	2	no	—
303	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	89	2	no	—
309	C₄	—	178	1	no	—
310	C₄	—	178	1	no	—
311	C₄	—	178	1	no	—
312	C₄	—	712	2	no	—
313	C₄	#19152269	1	1	yes	characteristic
314	C₄	#19152400	1	1	yes	characteristic
315	C₂	—	1424	2	no	—
316	C₂	#26033226	1	1	yes	characteristic
317	C₂	#508×D₈₉	1	1	yes	characteristic
318	C₂	#26033401	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₄×Q₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₁₄₂₄	C₂²
hypercenter	C₄×Q₃₂	D₈₉
hypocenter	C₈₉	#33960
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
1. #26033240	C₂	semidirect
4. #26022503	C₂	semidirect
5. #26022317	C₂	semidirect
6. #26022314	C₂	semidirect
7. #26022296	C₂	semidirect
8. C₁₆×Dic₁₇₈	C₂	semidirect
9. #26022110	C₂	semidirect
11. C₄×Q₃₂×C₈₉	C₂	semidirect
169. C₄×Q₃₂	D₈₉	semidirect
171. C₈₉	#33960	semidirect

Maximal subgroups

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

subgroup	index
#26033240	2
#26022514	2
#26022513	2
#26022503	2
#26022317	2
#26022314	2
#26022296	2
C₁₆×Dic₁₇₈	2
#26022110	2
#26022098	2
C₄×Q₃₂×C₈₉	2
#33960	89