Subgroups — #36712136

Order 22784.

484 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033265	C₂	1	1	yes	maximal, characteristic
2	#26033144	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022484	C₂	1	2	yes	maximal, complemented
4	#26022476	C₂	1	1	yes	maximal, characteristic
5	#26022410	C₂	1	2	yes	maximal, complemented
6	#26022395	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022315	C₂	1	2	yes	maximal, complemented
8	#26022293	C₂	1	1	yes	maximal, characteristic
9	C₄×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#15667957	C₂	1	2	yes	maximal, complemented
11	Q₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152333	—	2	1	no	—
13	#19152326	—	2	1	no	—
14	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
15	C₂×#15667957	C₂²	1	2	yes	—
16	#19152003	C₂²	1	1	yes	characteristic
17	#19152000	C₂²	1	2	yes	—
18	#19151982	C₂²	1	2	yes	—
19	#19151975	C₂²	1	1	yes	characteristic
20	#19151906	—	2	2	no	—
21	#19151884	—	2	1	no	—
22	#19151814	—	2	2	no	—
23	#19151799	—	2	1	no	—
24	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
25	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
26	C₄×Dic₃₅₆	C₂²	1	2	yes	—
27	C₄×D₇₁₂	C₂²	1	2	yes	—
28	#19151426	C₂²	1	2	yes	—
29	#19151424	C₂²	1	1	yes	characteristic
30	#19151403	C₂²	1	1	yes	characteristic
31	#19151403	C₂²	1	2	yes	—
32	#19151403	C₂²	1	1	yes	characteristic
33	#19151400	C₂²	1	2	yes	—
34	#19151400	C₂²	1	2	yes	—
35	#19151400	C₂²	1	2	yes	—
36	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
38	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
41	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
42	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
43	C₂×#13083563	—	2	2	no	—
44	C₂×D₇₁₂	—	2	1	no	—
45	C₂×D₇₁₂	C₂³	1	2	yes	—
46	#15668012	—	2	1	no	—
47	#15668010	—	4	1	no	—
48	#15668009	—	2	1	no	—
49	#15668004	—	2	2	no	—
50	#15668002	—	4	2	no	—
51	#15668001	—	4	1	no	—
52	#15668000	—	2	2	no	—
53	#15667997	—	4	2	no	—
54	C₄×Dic₁₇₈	—	2	1	no	—
55	C₄×Dic₁₇₈	—	2	2	no	—
56	C₄×Dic₁₇₈	—	2	1	no	—
57	#15667977	—	4	1	no	—
58	C₄×D₃₅₆	—	2	2	no	—
59	#15667974	—	4	1	no	—
60	Dic₇₁₂	—	2	2	no	—
61	#15667957	—	2	4	no	—
62	#15667946	—	2	1	no	—
63	#15667946	C₂³	1	1	yes	characteristic
64	#15667946	C₂³	1	2	yes	—
65	#15667946	C₂³	1	1	yes	characteristic
66	#15667944	—	2	2	no	—
67	#15667937	—	2	2	no	—
68	#15667937	—	2	1	no	—
69	#15667937	—	2	1	no	—
70	#15667934	—	2	2	no	—
71	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
72	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
75	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
76	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
77	C₄×C₈×C₈₉	C₂³	1	2	yes	—
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₇₈	—	2	1	no	—
80	C₂×Dic₁₇₈	—	2	1	no	—
81	C₂×Dic₁₇₈	—	4	1	no	—
82	C₂×D₃₅₆	—	2	2	no	—
83	C₂×D₃₅₆	—	4	1	no	—
84	C₂×D₃₅₆	—	4	1	no	—
85	C₂×C₄×D₈₉	—	4	2	no	—
86	Dic₃₅₆	—	2	1	no	—
87	Dic₃₅₆	—	2	1	no	—
88	Dic₃₅₆	—	2	1	no	—
89	Dic₃₅₆	#41	1	2	yes	—
90	#13083563	—	8	2	no	—
91	D₇₁₂	—	2	2	no	—
92	D₇₁₂	—	2	2	no	—
93	D₇₁₂	—	8	1	no	—
94	C₄⋊Dic₈₉	—	2	1	no	—
95	C₄⋊Dic₈₉	—	2	1	no	—
96	C₄⋊Dic₈₉	—	2	2	no	—
97	C₄⋊Dic₈₉	—	4	1	no	—
98	#13083553	—	4	2	no	—
99	#13083553	—	4	1	no	—
100	#13083553	—	4	1	no	—
101	#13083553	—	4	2	no	—
102	#13083553	—	4	2	no	—
103	#13083551	—	4	2	no	—
104	#13083551	—	4	2	no	—
105	#13083551	—	4	2	no	—
106	C₄×Dic₈₉	—	4	1	no	—
107	C₄×Dic₈₉	—	4	2	no	—
108	C₄×Dic₈₉	—	4	1	no	—
109	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
114	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
115	C₄²×C₈₉	C₂×D₄	1	2	yes	—
116	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₈₉	—	2	2	no	—
118	C₁₆×C₈₉	—	2	1	no	—
119	C₁₆×C₈₉	—	2	1	no	—
120	C₂²×D₈₉	—	4	2	no	—
121	Q₈×C₈₉	—	2	2	no	—
122	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₈₉	—	4	1	no	—
128	C₂×Dic₈₉	—	4	1	no	—
129	C₂×Dic₈₉	—	4	1	no	—
130	C₂×Dic₈₉	—	4	1	no	—
131	C₂×Dic₈₉	—	4	2	no	—
132	D₃₅₆	—	4	2	no	—
133	D₃₅₆	—	4	2	no	—
134	D₃₅₆	—	8	2	no	—
135	D₃₅₆	—	16	1	no	—
136	C₄×D₈₉	—	16	2	no	—
137	Dic₁₇₈	—	2	2	no	—
138	Dic₁₇₈	—	4	1	no	—
139	Dic₁₇₈	—	4	1	no	—
140	Dic₁₇₈	—	4	1	no	—
141	Dic₁₇₈	—	8	2	no	—
142	C₈×C₈₉	—	2	1	no	—
143	C₈×C₈₉	—	2	2	no	—
144	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
145	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
146	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₈₉	—	8	2	no	—
148	C₂×D₈₉	—	8	2	no	—
149	C₂×D₈₉	—	8	2	no	—
150	C₄×C₈₉	—	2	2	no	—
151	C₄×C₈₉	—	2	2	no	—
152	C₄×C₈₉	—	2	1	no	—
153	C₄×C₈₉	#508	1	2	yes	—
154	C₄×C₈₉	#548	1	1	yes	characteristic
155	C₄×C₈₉	#577	1	1	yes	characteristic
156	Dic₈₉	—	4	2	no	—
157	Dic₈₉	—	8	1	no	—
158	Dic₈₉	—	8	2	no	—
159	Dic₈₉	—	8	1	no	—
160	Dic₈₉	—	8	1	no	—
161	#33932	—	89	1	no	maximal, Sylow
162	C₂×C₈₉	#3358	1	1	yes	characteristic
163	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
164	C₂×C₈₉	#3416	1	1	yes	characteristic
165	D₈₉	—	16	2	no	—
166	#3396	—	89	1	no	—
167	#3383	—	89	1	no	—
168	#2242	—	89	2	no	—
169	#2240	—	89	1	no	—
170	#2228	—	89	2	no	—
171	#2225	—	89	1	no	—
172	#2211	—	89	2	no	—
173	#2207	—	89	1	no	—
174	Q₈×C₁₆	D₈₉	1	1	yes	characteristic, complemented
175	C₄×Q₃₂	—	89	1	no	—
176	C₄×SD₃₂	—	89	2	no	—
177	C₈₉	#33932	1	1	yes	characteristic, complemented, Sylow
178	#559	—	178	1	no	—
179	#557	—	178	1	no	—
180	C₂×Q₃₂	—	89	1	no	—
181	C₂×SD₃₂	—	89	2	no	—
182	#500	—	89	1	no	—
183	#499	—	89	2	no	—
184	#495	—	89	2	no	—
185	#493	—	89	1	no	—
186	#479	—	178	2	no	—
187	#475	—	178	1	no	—
188	#463	—	178	2	no	—
189	#460	—	178	1	no	—
190	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
191	C₄×Q₁₆	—	89	2	no	—
192	C₄×Q₁₆	—	89	1	no	—
193	C₄×Q₁₆	—	89	1	no	—
194	C₄×D₈	—	89	2	no	—
195	#367	—	89	2	no	—
196	#366	—	89	1	no	—
197	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
198	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
199	#358	—	89	1	no	—
200	#358	—	89	1	no	—
201	#358	—	89	2	no	—
202	#357	—	89	2	no	—
203	#357	—	89	2	no	—
204	#357	—	89	2	no	—
205	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
206	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
207	C₂×Q₁₆	—	89	1	no	—
208	C₂×Q₁₆	—	89	1	no	—
209	C₂×SD₁₆	—	178	2	no	—
210	C₂×D₈	—	89	2	no	—
211	C₂×D₈	—	178	1	no	—
212	C₄⋊Q₈	—	356	1	no	—
213	C₄⋊Q₈	—	178	1	no	—
214	#127	—	178	1	no	—
215	#125	—	356	2	no	—
216	#125	—	178	2	no	—
217	#124	—	356	1	no	—
218	#124	—	178	2	no	—
219	#123	—	356	2	no	—
220	C₄×Q₈	—	356	1	no	—
221	C₄×Q₈	—	178	1	no	—
222	C₄×Q₈	—	178	2	no	—
223	C₄×Q₈	—	178	1	no	—
224	C₄×Q₈	D₃₅₆	1	1	yes	characteristic
225	C₄×D₄	—	356	1	no	—
226	C₄×D₄	—	178	2	no	—
227	Q₃₂	—	178	2	no	—
228	SD₃₂	—	178	4	no	—
229	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
230	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
231	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
232	#107	—	89	1	no	—
233	#107	—	89	2	no	—
234	#107	—	89	1	no	—
235	#107	—	178	1	no	—
236	#106	—	178	2	no	—
237	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
238	C₄⋊C₈	D₃₅₆	1	2	yes	—
239	#103	—	178	1	no	—
240	#103	—	178	2	no	—
241	#103	—	178	1	no	—
242	#102	—	178	2	no	—
243	C₄×C₈	C₂²×D₈₉	1	2	yes	—
244	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
245	C₂×Q₈	—	356	1	no	—
246	C₂×Q₈	—	178	1	no	—
247	C₂×Q₈	—	178	1	no	—
248	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
249	C₂×D₄	—	356	1	no	—
250	C₂×D₄	—	356	1	no	—
251	C₂×D₄	—	178	2	no	—
252	C₂²×C₄	—	356	2	no	—
253	Q₁₆	—	89	2	no	—
254	Q₁₆	—	178	1	no	—
255	Q₁₆	—	178	1	no	—
256	Q₁₆	—	178	1	no	—
257	SD₁₆	—	712	2	no	—
258	D₈	—	178	2	no	—
259	D₈	—	178	2	no	—
260	D₈	—	712	1	no	—
261	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
262	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
264	C₄⋊C₄	—	356	2	no	—
265	C₄⋊C₄	—	356	1	no	—
266	C₄⋊C₄	—	356	1	no	—
267	C₄⋊C₄	—	356	2	no	—
268	C₄⋊C₄	—	356	1	no	—
269	C₄⋊C₄	—	356	2	no	—
270	C₄⋊C₄	—	178	1	no	—
271	C₄⋊C₄	—	178	1	no	—
272	C₄⋊C₄	—	178	2	no	—
273	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
274	C₄⋊C₄	D₇₁₂	1	2	yes	—
275	C₂²⋊C₄	—	356	2	no	—
276	C₂²⋊C₄	—	356	2	no	—
277	C₂²⋊C₄	—	356	2	no	—
278	C₄²	—	356	1	no	—
279	C₄²	—	356	2	no	—
280	C₄²	—	356	1	no	—
281	C₄²	C₂×D₃₅₆	1	2	yes	—
282	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
283	C₁₆	—	2	2	no	—
284	C₁₆	—	2	1	no	—
285	C₁₆	—	2	1	no	—
286	C₂³	—	356	2	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	356	1	no	—
289	Q₈	—	356	1	no	—
290	Q₈	—	356	1	no	—
291	Q₈	—	178	2	no	—
292	Q₈	—	712	2	no	—
293	D₄	—	356	2	no	—
294	D₄	—	356	2	no	—
295	D₄	—	1424	1	no	—
296	D₄	—	712	2	no	—
297	C₂×C₄	—	356	1	no	—
298	C₂×C₄	—	356	1	no	—
299	C₂×C₄	—	356	1	no	—
300	C₂×C₄	—	356	1	no	—
301	C₂×C₄	—	356	2	no	—
302	C₂×C₄	—	1424	2	no	—
303	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
304	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
305	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
306	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	2	no	—
310	C₈	#41×D₈₉	1	1	yes	characteristic
311	C₈	#15668060	1	1	yes	characteristic
312	C₂²	—	712	2	no	—
313	C₂²	—	712	2	no	—
314	C₂²	—	712	2	no	—
315	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	1	no	—
319	C₄	—	356	2	no	—
320	C₄	—	712	2	no	—
321	C₄	—	712	1	no	—
322	C₄	—	712	1	no	—
323	C₄	—	712	1	no	—
324	C₄	#19152030	1	2	yes	—
325	C₄	#19152260	1	1	yes	characteristic
326	C₄	#19152410	1	1	yes	characteristic
327	C₂	—	1424	2	no	—
328	C₂	#26032888	1	1	yes	characteristic
329	C₂	C₂×#19152030	1	1	yes	characteristic
330	C₂	#26033400	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	Q₈×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#19151975	C₂²
hypercenter	Q₈×C₁₆	D₈₉
hypocenter	C₈₉	#33932
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. #26033144	C₂	semidirect
3. #26022484	C₂	semidirect
5. #26022410	C₂	semidirect
6. #26022395	C₂	semidirect
7. #26022315	C₂	semidirect
9. C₄×Dic₇₁₂	C₂	semidirect
10. C₄×#15667957	C₂	semidirect
11. Q₈×C₁₆×C₈₉	C₂	semidirect
174. Q₈×C₁₆	D₈₉	semidirect
177. C₈₉	#33932	semidirect

Maximal subgroups

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

subgroup	index
#26033265	2
#26033144	2
#26022484	2
#26022476	2
#26022410	2
#26022395	2
#26022315	2
#26022293	2
C₄×Dic₇₁₂	2
C₄×#15667957	2
Q₈×C₁₆×C₈₉	2
#33932	89