Subgroups — #36712302

Order 22784.

476 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033116	C₂	1	2	yes	maximal, complemented
2	#26022529	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022526	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022408	C₂	1	2	yes	maximal, complemented
5	#26022406	C₂	1	2	yes	maximal, complemented
6	#26022398	C₂	1	2	yes	maximal, complemented
7	C₁₆×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₁₄₂₄	C₂	1	1	yes	maximal, characteristic
9	#26022098	C₂	1	1	yes	maximal, characteristic
10	#26022098	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#2253	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152304	—	2	2	no	—
13	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
14	#19152002	C₂²	1	2	yes	—
15	#19151999	C₂²	1	2	yes	—
16	#19151897	—	2	2	no	—
17	#19151895	—	2	2	no	—
18	#19151887	—	2	2	no	—
19	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
20	C₄×D₇₁₂	C₂²	1	2	yes	—
21	#19151680	C₂²	1	2	yes	—
22	#19151680	C₂²	1	2	yes	—
23	#19151425	C₂²	1	2	yes	—
24	#19151424	C₂²	1	1	yes	characteristic
25	#19151423	C₂²	1	2	yes	—
26	C₄⋊#13083549	C₂²	1	1	yes	characteristic
27	#19151416	C₂²	1	1	yes	characteristic
28	#19151416	C₂²	1	1	yes	characteristic
29	#19151400	C₂²	1	2	yes	—
30	#19151398	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	#499×C₈₉	C₂²	1	2	yes	—
35	#367×C₈₉	C₂²	1	2	yes	—
36	#366×C₈₉	C₂²	1	1	yes	characteristic
37	#366×C₈₉	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×D₇₁₂	C₂³	1	2	yes	—
40	#15668011	—	2	2	no	—
41	#15668003	—	2	2	no	—
42	#15667992	—	4	2	no	—
43	#15667990	—	4	2	no	—
44	#15667990	—	4	2	no	—
45	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
46	C₄×D₃₅₆	—	2	2	no	—
47	#15667974	—	2	2	no	—
48	#15667974	—	2	2	no	—
49	C₄⋊C₄×D₈₉	—	4	2	no	—
50	D₁₄₂₄	Q₈	1	4	yes	complemented
51	C₂×#13083549	C₂³	1	1	yes	characteristic
52	C₂×#13083549	C₂³	1	1	yes	characteristic
53	#15667946	C₂³	1	2	yes	—
54	#15667945	—	2	2	no	—
55	#15667943	—	2	2	no	—
56	C₄⋊#10623494	D₄	1	1	yes	characteristic
57	#15667941	D₄	1	1	yes	characteristic
58	#15667941	D₄	1	1	yes	characteristic
59	#15667934	—	2	2	no	—
60	#15667932	—	2	2	no	—
61	#15667932	—	2	2	no	—
62	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
63	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
64	#125×C₈₉	—	2	2	no	—
65	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
66	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
67	#107×C₈₉	C₂³	1	2	yes	—
68	#107×C₈₉	C₂³	1	2	yes	—
69	#106×C₈₉	—	2	2	no	—
70	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₂×Dic₁₇₈	—	4	2	no	—
72	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
73	C₂×D₃₅₆	—	2	2	no	—
74	C₂×C₄×D₈₉	—	4	2	no	—
75	C₂×C₄×D₈₉	—	4	2	no	—
76	C₂×C₄×D₈₉	—	4	2	no	—
77	D₇₁₂	—	2	2	no	—
78	D₇₁₂	C₂×Q₈	1	4	yes	—
79	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
80	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
81	C₄⋊Dic₈₉	—	2	2	no	—
82	C₄⋊Dic₈₉	—	4	2	no	—
83	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
84	#13083553	—	4	2	no	—
85	#13083553	—	4	2	no	—
86	#13083553	D₈	1	1	yes	characteristic
87	#13083553	D₈	1	1	yes	characteristic
88	#13083551	—	4	2	no	—
89	#13083551	—	4	2	no	—
90	#13083551	—	4	2	no	—
91	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
92	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
93	#13083549	—	2	1	no	—
94	#13083549	—	2	1	no	—
95	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₈₉	—	2	2	no	—
98	C₄⋊C₄×C₈₉	—	2	2	no	—
99	C₄⋊C₄×C₈₉	—	4	2	no	—
100	C₄⋊C₄×C₈₉	—	4	2	no	—
101	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₁₆×C₈₉	—	2	1	no	—
103	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
104	C₂²×D₈₉	—	4	2	no	—
105	C₂×C₄×C₈₉	—	4	2	no	—
106	C₂×C₄×C₈₉	—	4	2	no	—
107	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
110	C₂×Dic₈₉	—	4	2	no	—
111	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
112	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
113	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
114	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
115	D₃₅₆	—	2	4	no	—
116	D₃₅₆	—	4	2	no	—
117	C₄×D₈₉	—	8	4	no	—
118	C₄×D₈₉	—	16	2	no	—
119	C₄×D₈₉	—	16	2	no	—
120	Dic₁₇₈	—	2	2	no	—
121	Dic₁₇₈	—	16	2	no	—
122	C₈×C₈₉	—	2	1	no	—
123	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
124	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
125	#10623494	—	2	1	no	—
126	#10623494	—	2	1	no	—
127	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
128	C₂×D₈₉	—	4	4	no	—
129	C₂×D₈₉	—	8	2	no	—
130	C₂×D₈₉	—	8	2	no	—
131	C₄×C₈₉	—	2	1	no	—
132	C₄×C₈₉	—	2	1	no	—
133	C₄×C₈₉	—	8	2	no	—
134	C₄×C₈₉	—	8	2	no	—
135	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
136	C₄×C₈₉	#578	1	1	yes	characteristic
137	Dic₈₉	—	2	1	no	—
138	Dic₈₉	—	2	1	no	—
139	Dic₈₉	—	2	1	no	—
140	Dic₈₉	—	8	2	no	—
141	Dic₈₉	#508	1	2	yes	—
142	#33947	—	89	1	no	maximal, Sylow
143	C₂×C₈₉	Q₈×D₈	1	1	yes	characteristic
144	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
145	C₂×C₈₉	#3415	1	1	yes	characteristic
146	D₈₉	—	8	4	no	—
147	#3380	—	89	2	no	—
148	#2253	—	89	1	no	—
149	#2253	D₈₉	1	1	yes	characteristic, complemented
150	#2252	—	89	1	no	—
151	#2228	—	89	2	no	—
152	#2228	—	89	2	no	—
153	#2226	—	89	2	no	—
154	Q₈×C₁₆	—	89	1	no	—
155	C₄×D₁₆	—	89	1	no	—
156	C₄×D₁₆	—	89	1	no	—
157	C₄×D₁₆	—	89	1	no	—
158	C₈₉	#33947	1	1	yes	characteristic, complemented, Sylow
159	#554	—	178	2	no	—
160	C₂×D₁₆	—	89	1	no	—
161	#500	—	89	2	no	—
162	#499	—	89	2	no	—
163	#499	C₂×D₈₉	1	2	yes	—
164	#478	—	178	2	no	—
165	#478	—	178	2	no	—
166	#476	—	178	2	no	—
167	C₈×Q₈	—	89	1	no	—
168	C₄×D₈	—	89	2	no	—
169	C₄×D₈	—	89	2	no	—
170	C₄×D₈	—	89	2	no	—
171	#367	—	89	2	no	—
172	#367	C₂×D₈₉	1	2	yes	—
173	#366	—	89	1	no	—
174	#366	—	89	2	no	—
175	#366	C₂×D₈₉	1	1	yes	characteristic
176	#366	C₂×D₈₉	1	1	yes	characteristic
177	C₄⋊C₁₆	—	89	1	no	—
178	C₄⋊C₁₆	—	89	1	no	—
179	C₄⋊C₁₆	—	89	1	no	—
180	#357	—	89	2	no	—
181	#357	—	89	2	no	—
182	#357	—	89	2	no	—
183	C₄×C₁₆	—	89	1	no	—
184	C₄×C₁₆	—	89	1	no	—
185	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
186	C₂×D₈	—	89	2	no	—
187	C₄⋊Q₈	—	178	2	no	—
188	#125	—	2	2	no	—
189	#125	—	178	2	no	—
190	C₂²⋊Q₈	—	356	2	no	—
191	C₂²⋊Q₈	—	356	2	no	—
192	C₂²⋊Q₈	—	356	2	no	—
193	C₄×Q₈	—	89	1	no	—
194	C₄×D₄	—	178	2	no	—
195	C₄×D₄	—	178	2	no	—
196	C₄×D₄	—	178	2	no	—
197	C₂×C₄⋊C₄	—	356	2	no	—
198	D₁₆	—	89	4	no	—
199	C₂×C₁₆	—	89	1	no	—
200	C₂×C₁₆	—	89	1	no	—
201	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
202	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
203	#107	—	89	2	no	—
204	#107	—	178	2	no	—
205	#107	C₂²×D₈₉	1	2	yes	—
206	#107	C₂²×D₈₉	1	2	yes	—
207	#106	—	2	2	no	—
208	#106	—	178	2	no	—
209	C₄⋊C₈	—	89	1	no	—
210	C₄⋊C₈	—	89	1	no	—
211	C₄⋊C₈	—	89	1	no	—
212	#102	—	178	2	no	—
213	#102	—	178	2	no	—
214	#102	—	178	2	no	—
215	C₄×C₈	—	89	1	no	—
216	C₄×C₈	—	89	1	no	—
217	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
218	C₂×Q₈	—	89	1	no	—
219	C₂×Q₈	—	356	2	no	—
220	C₂×D₄	—	178	2	no	—
221	C₂²×C₄	—	356	2	no	—
222	C₂²×C₄	—	356	2	no	—
223	C₂²×C₄	—	356	2	no	—
224	D₈	—	89	4	no	—
225	D₈	—	178	2	no	—
226	C₂×C₈	—	89	1	no	—
227	C₂×C₈	—	89	1	no	—
228	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
229	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	89	1	no	—
235	C₄⋊C₄	—	89	1	no	—
236	C₄⋊C₄	—	89	1	no	—
237	C₄⋊C₄	—	356	2	no	—
238	C₄⋊C₄	—	356	2	no	—
239	C₄⋊C₄	—	356	2	no	—
240	C₄⋊C₄	—	178	2	no	—
241	C₂²⋊C₄	—	356	2	no	—
242	C₂²⋊C₄	—	356	2	no	—
243	C₂²⋊C₄	—	356	2	no	—
244	C₄²	—	89	1	no	—
245	C₄²	—	89	1	no	—
246	C₄²	D₄×D₈₉	1	1	yes	characteristic
247	C₁₆	—	2	1	no	—
248	C₁₆	—	178	1	no	—
249	C₁₆	—	178	1	no	—
250	C₁₆	Q₈×D₈₉	1	2	yes	—
251	C₂³	—	356	2	no	—
252	Q₈	—	1424	2	no	—
253	Q₈	—	178	2	no	—
254	D₄	—	356	2	no	—
255	D₄	—	178	4	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	89	1	no	—
259	C₂×C₄	—	89	1	no	—
260	C₂×C₄	—	89	1	no	—
261	C₂×C₄	—	89	1	no	—
262	C₂×C₄	—	356	2	no	—
263	C₂×C₄	—	1424	2	no	—
264	C₂×C₄	—	1424	2	no	—
265	C₂×C₄	—	712	4	no	—
266	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
267	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
268	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
269	C₈	—	2	1	no	—
270	C₈	—	178	1	no	—
271	C₈	—	178	1	no	—
272	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
273	C₈	#15668060	1	1	yes	characteristic
274	C₂²	—	356	4	no	—
275	C₂²	—	712	2	no	—
276	C₂²	—	712	2	no	—
277	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	89	2	no	—
283	C₄	—	178	1	no	—
284	C₄	—	178	1	no	—
285	C₄	—	178	1	no	—
286	C₄	—	712	2	no	—
287	C₄	#19152263	1	1	yes	characteristic
288	C₄	#19152411	1	1	yes	characteristic
289	C₂	—	712	4	no	—
290	C₂	#26033104	1	1	yes	characteristic
291	C₂	#508×D₈₉	1	1	yes	characteristic
292	C₂	#26033390	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×#2253	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₁₄₂₄	C₂²
hypercenter	#2253	D₈₉
hypocenter	C₈₉	#33947
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2253

Complements

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

N	H ≅ G/N	type
1. #26033116	C₂	semidirect
2. #26022529	C₂	semidirect
3. #26022526	C₂	semidirect
4. #26022408	C₂	semidirect
5. #26022406	C₂	semidirect
6. #26022398	C₂	semidirect
7. C₁₆×Dic₁₇₈	C₂	semidirect
11. C₈₉×#2253	C₂	semidirect
50. D₁₄₂₄	Q₈	semidirect
149. #2253	D₈₉	semidirect
158. C₈₉	#33947	semidirect

Maximal subgroups

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

subgroup	index
#26033116	2
#26022529	2
#26022526	2
#26022408	2
#26022406	2
#26022398	2
C₁₆×Dic₁₇₈	2
C₄×D₁₄₂₄	2
#26022098	2
C₈₉×#2253	2
#33947	89