Subgroups — #36711841

Order 22784.

494 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032655	C₂	1	2	yes	maximal, complemented
2	#26022531	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022444	C₂	1	2	yes	maximal, complemented
4	#26022438	C₂	1	2	yes	maximal, complemented
5	#26022410	C₂	1	2	yes	maximal, complemented
6	#26022338	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022338	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₁₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022036	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19151424	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152294	—	2	2	no	—
13	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
14	#19152000	C₂²	1	2	yes	—
15	#19151923	—	2	2	no	—
16	#19151917	—	2	2	no	—
17	#19151899	—	2	2	no	—
18	#19151834	C₂²	1	2	yes	—
19	#19151834	C₂²	1	2	yes	—
20	C₄×D₇₁₂	C₂²	1	2	yes	—
21	#19151636	C₂²	1	2	yes	—
22	C₂×#15667946	C₂²	1	2	yes	—
23	#19151426	C₂²	1	2	yes	—
24	#19151426	C₂²	1	2	yes	—
25	#19151424	C₂²	1	2	yes	—
26	#19151424	C₂²	1	1	yes	characteristic
27	#19151424	C₂²	1	1	yes	characteristic
28	#19151424	C₂²	1	1	yes	characteristic
29	#19151400	C₂²	1	2	yes	—
30	#19151400	C₂²	1	2	yes	—
31	#19151400	C₂²	1	2	yes	—
32	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
33	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
34	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
35	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
36	C₂²⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
37	C₂²⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×D₇₁₂	C₂³	1	2	yes	—
40	#15668007	—	4	2	no	—
41	#15668004	—	2	2	no	—
42	#15667994	—	4	2	no	—
43	#15667986	—	2	2	no	—
44	#15667986	—	2	2	no	—
45	#15667984	—	4	2	no	—
46	C₄×D₃₅₆	—	2	2	no	—
47	#15667972	—	4	2	no	—
48	#15667971	—	2	2	no	—
49	C₂×C₄⋊Dic₈₉	—	2	2	no	—
50	D₁₄₂₄	—	2	2	no	—
51	#15667946	—	2	2	no	—
52	#15667946	C₂³	1	2	yes	—
53	#15667946	C₂³	1	2	yes	—
54	#15667946	C₂³	1	2	yes	—
55	#15667944	—	2	2	no	—
56	#15667944	—	2	2	no	—
57	#15667934	—	2	2	no	—
58	#15667934	—	2	2	no	—
59	#15667934	—	2	2	no	—
60	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
62	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
63	C₂×C₁₆×C₈₉	—	2	2	no	—
64	C₂×C₁₆×C₈₉	—	2	2	no	—
65	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
66	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
67	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
70	C₂²⋊C₈×C₈₉	D₄	1	1	yes	characteristic
71	C₂²⋊C₈×C₈₉	D₄	1	1	yes	characteristic
72	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	C₂×D₃₅₆	—	2	2	no	—
74	C₂×C₂²⋊D₈₉	—	4	2	no	—
75	C₂×C₂²⋊D₈₉	—	4	2	no	—
76	C₂²×Dic₈₉	—	4	2	no	—
77	C₂×C₄×D₈₉	—	4	2	no	—
78	D₇₁₂	—	2	2	no	—
79	D₇₁₂	—	2	2	no	—
80	C₄⋊Dic₈₉	—	2	2	no	—
81	C₄⋊Dic₈₉	—	2	2	no	—
82	C₄⋊Dic₈₉	—	2	2	no	—
83	C₄⋊Dic₈₉	—	4	2	no	—
84	#13083553	—	4	2	no	—
85	#13083553	—	4	2	no	—
86	C₂²⋊Dic₈₉	—	4	2	no	—
87	#13083551	—	4	2	no	—
88	#13083551	—	4	2	no	—
89	#13083551	—	4	2	no	—
90	C₄×Dic₈₉	—	4	2	no	—
91	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	—	2	1	no	—
95	C₂×C₈×C₈₉	—	2	1	no	—
96	C₂×C₈×C₈₉	—	2	1	no	—
97	C₂×C₈×C₈₉	—	2	1	no	—
98	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
102	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
103	C₂²⋊C₄×C₈₉	D₈	1	1	yes	characteristic
104	C₂²⋊C₄×C₈₉	D₈	1	1	yes	characteristic
105	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₁₆×C₈₉	—	2	1	no	—
107	C₁₆×C₈₉	—	2	1	no	—
108	C₁₆×C₈₉	—	2	1	no	—
109	C₁₆×C₈₉	#41	1	2	yes	—
110	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
111	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
112	C₂²×D₈₉	—	4	2	no	—
113	D₄×C₈₉	—	2	2	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
123	C₂²⋊D₈₉	—	8	4	no	—
124	C₂²⋊D₈₉	—	16	2	no	—
125	C₂×Dic₈₉	—	4	2	no	—
126	C₂×Dic₈₉	—	4	2	no	—
127	C₂×Dic₈₉	—	4	2	no	—
128	C₂×Dic₈₉	—	16	2	no	—
129	D₃₅₆	—	4	2	no	—
130	D₃₅₆	—	4	2	no	—
131	C₄×D₈₉	—	16	2	no	—
132	C₈×C₈₉	—	2	1	no	—
133	C₈×C₈₉	—	2	1	no	—
134	C₈×C₈₉	—	2	1	no	—
135	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
136	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
137	C₂²×C₈₉	—	2	1	no	—
138	C₂²×C₈₉	—	2	1	no	—
139	C₂²×C₈₉	—	2	1	no	—
140	C₂²×C₈₉	—	2	1	no	—
141	C₂²×C₈₉	—	2	1	no	—
142	C₂²×C₈₉	#508	1	2	yes	—
143	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
144	C₂×D₈₉	—	8	2	no	—
145	C₂×D₈₉	—	8	2	no	—
146	C₂×D₈₉	—	8	2	no	—
147	C₄×C₈₉	—	2	1	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	—	2	1	no	—
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	#547	1	1	yes	characteristic
152	C₄×C₈₉	#578	1	1	yes	characteristic
153	Dic₈₉	—	8	2	no	—
154	Dic₈₉	—	8	2	no	—
155	Dic₈₉	—	8	2	no	—
156	#33906	—	89	1	no	maximal, Sylow
157	C₂×C₈₉	—	2	2	no	—
158	C₂×C₈₉	—	4	1	no	—
159	C₂×C₈₉	#3334	1	1	yes	characteristic
160	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
161	C₂×C₈₉	#3415	1	1	yes	characteristic
162	D₈₉	—	16	2	no	—
163	#3339	—	89	2	no	—
164	#2253	—	89	1	no	—
165	#2234	—	89	2	no	—
166	#2233	—	89	2	no	—
167	#2228	—	89	2	no	—
168	#2215	—	89	1	no	—
169	#2215	—	89	1	no	—
170	C₄×D₁₆	—	89	1	no	—
171	D₄×C₁₆	D₈₉	1	1	yes	characteristic, complemented
172	#2158	—	89	1	no	—
173	C₂×#366	—	89	1	no	—
174	C₈₉	#33906	1	1	yes	characteristic, complemented, Sylow
175	#553	—	178	2	no	—
176	C₂×D₁₆	—	89	1	no	—
177	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
178	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
179	#499	—	89	2	no	—
180	#482	—	178	2	no	—
181	#481	—	178	2	no	—
182	#478	—	178	2	no	—
183	#466	—	89	2	no	—
184	#466	—	89	2	no	—
185	C₄×D₈	—	89	2	no	—
186	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
187	#427	—	89	2	no	—
188	C₂×#107	—	89	2	no	—
189	#367	—	89	2	no	—
190	#367	—	89	2	no	—
191	#366	—	89	1	no	—
192	#366	—	89	1	no	—
193	#366	—	89	2	no	—
194	#366	—	89	1	no	—
195	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
196	#357	—	89	2	no	—
197	#357	—	89	2	no	—
198	#357	—	89	2	no	—
199	C₂²⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
200	C₂²⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
201	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
202	C₂×D₈	—	89	2	no	—
203	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
205	#126	—	356	2	no	—
206	#125	—	178	2	no	—
207	#123	—	356	2	no	—
208	#121	—	356	2	no	—
209	#121	—	178	2	no	—
210	#121	—	178	2	no	—
211	C₄×D₄	—	356	2	no	—
212	C₄×D₄	—	178	2	no	—
213	C₄×D₄	D₃₅₆	1	1	yes	characteristic
214	#117	—	178	2	no	—
215	C₂×C₄⋊C₄	—	178	2	no	—
216	D₁₆	—	178	2	no	—
217	C₂×C₁₆	—	2	2	no	—
218	C₂×C₁₆	—	2	2	no	—
219	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
220	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
221	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
222	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
223	#107	—	89	2	no	—
224	#107	—	89	2	no	—
225	#107	—	89	2	no	—
226	#107	—	178	2	no	—
227	#106	—	178	2	no	—
228	#106	—	178	2	no	—
229	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
230	#102	—	178	2	no	—
231	#102	—	178	2	no	—
232	#102	—	178	2	no	—
233	C₂²⋊C₈	D₃₅₆	1	1	yes	characteristic
234	C₂²⋊C₈	D₃₅₆	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
236	C₂×D₄	—	356	2	no	—
237	C₂×D₄	—	356	2	no	—
238	C₂×D₄	—	178	2	no	—
239	C₂×D₄	D₇₁₂	1	1	yes	characteristic
240	C₂²×C₄	—	356	2	no	—
241	C₂²×C₄	—	356	2	no	—
242	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
244	D₈	—	178	2	no	—
245	D₈	—	178	2	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
251	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
252	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
254	C₄⋊C₄	—	356	2	no	—
255	C₄⋊C₄	—	356	2	no	—
256	C₄⋊C₄	—	356	2	no	—
257	C₄⋊C₄	—	178	2	no	—
258	C₄⋊C₄	—	178	2	no	—
259	C₄⋊C₄	—	178	2	no	—
260	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
261	C₂²⋊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	D₇₁₂	1	1	yes	characteristic
266	C₂²⋊C₄	D₇₁₂	1	1	yes	characteristic
267	C₄²	—	356	2	no	—
268	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
269	C₁₆	—	2	1	no	—
270	C₁₆	—	2	1	no	—
271	C₁₆	—	2	1	no	—
272	C₁₆	#13083573	1	2	yes	—
273	C₂³	—	356	2	no	—
274	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
275	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
276	D₄	—	2	2	no	—
277	D₄	—	356	2	no	—
278	D₄	—	356	2	no	—
279	D₄	—	1424	2	no	—
280	D₄	—	712	4	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	356	2	no	—
286	C₂×C₄	—	356	2	no	—
287	C₂×C₄	—	356	2	no	—
288	C₂×C₄	—	1424	2	no	—
289	C₂×C₄	—	1424	2	no	—
290	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
291	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
292	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
293	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	C₂×#13083573	1	1	yes	characteristic
299	C₈	#15668059	1	1	yes	characteristic
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	712	2	no	—
306	C₂²	—	712	2	no	—
307	C₂²	—	712	2	no	—
308	C₂²	#19152030	1	2	yes	—
309	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	712	2	no	—
315	C₄	—	712	2	no	—
316	C₄	—	712	2	no	—
317	C₄	#19152252	1	1	yes	characteristic
318	C₄	#19152415	1	1	yes	characteristic
319	C₂	—	2	2	no	—
320	C₂	—	4	1	no	—
321	C₂	—	1424	2	no	—
322	C₂	#26032600	1	1	yes	characteristic
323	C₂	C₂×#19152030	1	1	yes	characteristic
324	C₂	#26033393	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	D₄×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	D₄×C₁₆	D₈₉
hypocenter	C₈₉	#33906
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. #26032655	C₂	semidirect
2. #26022531	C₂	semidirect
3. #26022444	C₂	semidirect
4. #26022438	C₂	semidirect
5. #26022410	C₂	semidirect
6. #26022338	C₂	semidirect
7. #26022338	C₂	semidirect
8. C₄×D₁₄₂₄	C₂	semidirect
9. #26022036	C₂	semidirect
10. C₂×#19151424	C₂	semidirect
11. D₄×C₁₆×C₈₉	C₂	semidirect
171. D₄×C₁₆	D₈₉	semidirect
174. C₈₉	#33906	semidirect

Maximal subgroups

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

subgroup	index
#26032655	2
#26022531	2
#26022444	2
#26022438	2
#26022410	2
#26022338	2
C₄×D₁₄₂₄	2
#26022036	2
C₂×#19151424	2
D₄×C₁₆×C₈₉	2
#33906	89