Subgroups — #36712043

Order 22784.

488 conjugacy classes of subgroups.

The subgroup lattice has 278 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	#26022433	C₂	1	2	yes	maximal, complemented
5	#26022410	C₂	1	2	yes	maximal, complemented
6	#26022355	C₂	1	2	yes	maximal, complemented
7	C₄×D₁₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022040	C₂	1	2	yes	maximal, complemented
9	C₈₉×#2169	C₂	1	1	yes	maximal, characteristic, complemented
10	#19152294	—	2	2	no	—
11	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
12	#19152000	C₂²	1	2	yes	—
13	#19151923	—	2	2	no	—
14	#19151917	—	2	2	no	—
15	#19151899	—	2	2	no	—
16	#19151834	C₂²	1	2	yes	—
17	#19151834	C₂²	1	2	yes	—
18	C₄×D₇₁₂	C₂²	1	2	yes	—
19	#19151636	C₂²	1	2	yes	—
20	C₂×#15667946	C₂²	1	2	yes	—
21	#19151426	C₂²	1	2	yes	—
22	#19151426	C₂²	1	2	yes	—
23	#19151424	C₂²	1	2	yes	—
24	#19151424	C₂²	1	1	yes	characteristic
25	#19151424	C₂²	1	1	yes	characteristic
26	#19151424	C₂²	1	1	yes	characteristic
27	#19151400	C₂²	1	2	yes	—
28	#19151400	C₂²	1	2	yes	—
29	#19151400	C₂²	1	2	yes	—
30	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
31	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
32	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
33	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
34	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
35	C₂×D₇₁₂	C₂³	1	2	yes	—
36	#15668007	—	4	2	no	—
37	#15668004	—	2	2	no	—
38	#15667994	—	4	2	no	—
39	#15667986	—	2	2	no	—
40	#15667986	—	2	2	no	—
41	#15667984	—	4	2	no	—
42	C₄×D₃₅₆	—	2	2	no	—
43	#15667972	—	4	2	no	—
44	#15667971	—	2	2	no	—
45	C₂×C₄⋊Dic₈₉	—	2	2	no	—
46	D₁₄₂₄	—	2	2	no	—
47	#15667946	—	2	2	no	—
48	#15667946	C₂³	1	2	yes	—
49	#15667946	C₂³	1	2	yes	—
50	#15667946	C₂³	1	2	yes	—
51	#15667944	—	2	2	no	—
52	#15667944	—	2	2	no	—
53	#15667934	—	2	2	no	—
54	#15667934	—	2	2	no	—
55	#15667934	—	2	2	no	—
56	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
57	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
58	M₃₂×C₈₉	—	2	4	no	—
59	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
60	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
61	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
63	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
64	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
65	C₂×D₃₅₆	—	2	2	no	—
66	C₂×C₂²⋊D₈₉	—	4	2	no	—
67	C₂×C₂²⋊D₈₉	—	4	2	no	—
68	C₂²×Dic₈₉	—	4	2	no	—
69	C₂×C₄×D₈₉	—	4	2	no	—
70	D₇₁₂	—	2	2	no	—
71	D₇₁₂	—	2	2	no	—
72	C₄⋊Dic₈₉	—	2	2	no	—
73	C₄⋊Dic₈₉	—	2	2	no	—
74	C₄⋊Dic₈₉	—	2	2	no	—
75	C₄⋊Dic₈₉	—	4	2	no	—
76	#13083553	—	4	2	no	—
77	#13083553	—	4	2	no	—
78	C₂²⋊Dic₈₉	—	4	2	no	—
79	#13083551	—	4	2	no	—
80	#13083551	—	4	2	no	—
81	#13083551	—	4	2	no	—
82	C₄×Dic₈₉	—	4	2	no	—
83	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₈₉	—	2	2	no	—
86	C₂×C₈×C₈₉	—	2	2	no	—
87	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
91	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
92	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₁₆×C₈₉	—	2	2	no	—
94	C₁₆×C₈₉	—	2	1	no	—
95	C₁₆×C₈₉	#41	1	2	yes	—
96	C₂³×C₈₉	C₂×D₈	1	2	yes	—
97	C₂²×D₈₉	—	4	2	no	—
98	D₄×C₈₉	—	4	1	no	—
99	C₂×C₄×C₈₉	—	2	2	no	—
100	C₂×C₄×C₈₉	—	2	2	no	—
101	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
105	C₂²⋊D₈₉	—	8	4	no	—
106	C₂²⋊D₈₉	—	16	2	no	—
107	C₂×Dic₈₉	—	4	2	no	—
108	C₂×Dic₈₉	—	4	2	no	—
109	C₂×Dic₈₉	—	4	2	no	—
110	C₂×Dic₈₉	—	16	2	no	—
111	D₃₅₆	—	4	2	no	—
112	D₃₅₆	—	4	2	no	—
113	C₄×D₈₉	—	16	2	no	—
114	C₈×C₈₉	—	2	2	no	—
115	C₈×C₈₉	—	2	1	no	—
116	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
117	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
118	C₂²×C₈₉	—	2	2	no	—
119	C₂²×C₈₉	—	2	2	no	—
120	C₂²×C₈₉	—	2	2	no	—
121	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
122	C₂×D₈₉	—	8	2	no	—
123	C₂×D₈₉	—	8	2	no	—
124	C₂×D₈₉	—	8	2	no	—
125	C₄×C₈₉	—	2	2	no	—
126	C₄×C₈₉	—	2	1	no	—
127	C₄×C₈₉	—	2	1	no	—
128	C₄×C₈₉	#547	1	1	yes	characteristic
129	C₄×C₈₉	#578	1	1	yes	characteristic
130	Dic₈₉	—	8	2	no	—
131	Dic₈₉	—	8	2	no	—
132	Dic₈₉	—	8	2	no	—
133	#33922	—	89	1	no	maximal, Sylow
134	C₂×C₈₉	—	4	2	no	—
135	C₂×C₈₉	#3334	1	1	yes	characteristic
136	C₂×C₈₉	#3414	1	1	yes	characteristic
137	C₂×C₈₉	#3415	1	1	yes	characteristic
138	D₈₉	—	16	2	no	—
139	#3339	—	89	2	no	—
140	#2253	—	89	1	no	—
141	#2234	—	89	2	no	—
142	#2232	—	89	2	no	—
143	#2228	—	89	2	no	—
144	#2218	—	89	2	no	—
145	C₄×D₁₆	—	89	1	no	—
146	#2169	D₈₉	1	1	yes	characteristic, complemented
147	#2159	—	89	2	no	—
148	C₈₉	#33922	1	1	yes	characteristic, complemented, Sylow
149	#553	—	178	2	no	—
150	C₂×D₁₆	—	89	1	no	—
151	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
152	#499	—	89	2	no	—
153	#482	—	178	2	no	—
154	#481	—	178	2	no	—
155	#478	—	178	2	no	—
156	#466	—	89	2	no	—
157	#466	—	89	2	no	—
158	C₄×D₈	—	89	2	no	—
159	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
160	#427	—	89	2	no	—
161	C₂×#107	—	89	2	no	—
162	#367	—	89	2	no	—
163	#367	—	89	2	no	—
164	#366	—	89	1	no	—
165	#366	—	89	2	no	—
166	#366	—	89	1	no	—
167	#366	—	89	1	no	—
168	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
169	#357	—	89	2	no	—
170	#357	—	89	2	no	—
171	#357	—	89	2	no	—
172	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
173	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
174	C₂×D₈	—	89	2	no	—
175	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
176	#126	—	356	2	no	—
177	#125	—	178	2	no	—
178	#123	—	356	2	no	—
179	#121	—	356	2	no	—
180	#121	—	178	2	no	—
181	#121	—	178	2	no	—
182	C₄×D₄	—	356	2	no	—
183	C₄×D₄	—	178	2	no	—
184	C₄×D₄	D₃₅₆	1	1	yes	characteristic
185	#117	—	178	2	no	—
186	C₂×C₄⋊C₄	—	178	2	no	—
187	D₁₆	—	178	2	no	—
188	M₃₂	—	2	4	no	—
189	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
190	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
191	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
192	#107	—	89	2	no	—
193	#107	—	89	2	no	—
194	#107	—	89	2	no	—
195	#107	—	178	2	no	—
196	#106	—	178	2	no	—
197	#106	—	178	2	no	—
198	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
199	#102	—	178	2	no	—
200	#102	—	178	2	no	—
201	#102	—	178	2	no	—
202	C₂²⋊C₈	D₃₅₆	1	2	yes	—
203	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
204	C₂×D₄	—	356	2	no	—
205	C₂×D₄	—	356	2	no	—
206	C₂×D₄	—	178	2	no	—
207	C₂×D₄	D₇₁₂	1	1	yes	characteristic
208	C₂²×C₄	—	356	2	no	—
209	C₂²×C₄	—	356	2	no	—
210	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
211	D₈	—	178	2	no	—
212	D₈	—	178	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
216	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
217	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
218	C₄⋊C₄	—	356	2	no	—
219	C₄⋊C₄	—	356	2	no	—
220	C₄⋊C₄	—	356	2	no	—
221	C₄⋊C₄	—	178	2	no	—
222	C₄⋊C₄	—	178	2	no	—
223	C₄⋊C₄	—	178	2	no	—
224	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
225	C₂²⋊C₄	—	356	2	no	—
226	C₂²⋊C₄	—	356	2	no	—
227	C₂²⋊C₄	—	356	2	no	—
228	C₂²⋊C₄	—	356	2	no	—
229	C₂²⋊C₄	D₇₁₂	1	2	yes	—
230	C₄²	—	356	2	no	—
231	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
232	C₁₆	—	2	2	no	—
233	C₁₆	—	2	1	no	—
234	C₁₆	#13083573	1	2	yes	—
235	C₂³	—	356	2	no	—
236	C₂³	C₂×D₇₁₂	1	2	yes	—
237	D₄	—	4	1	no	—
238	D₄	—	356	2	no	—
239	D₄	—	356	2	no	—
240	D₄	—	1424	2	no	—
241	D₄	—	712	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	356	2	no	—
245	C₂×C₄	—	356	2	no	—
246	C₂×C₄	—	356	2	no	—
247	C₂×C₄	—	1424	2	no	—
248	C₂×C₄	—	1424	2	no	—
249	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
250	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
251	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
253	C₈	—	2	1	no	—
254	C₈	—	2	2	no	—
255	C₈	C₂×#13083573	1	1	yes	characteristic
256	C₈	#15668059	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	712	2	no	—
261	C₂²	—	712	2	no	—
262	C₂²	—	712	2	no	—
263	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	712	2	no	—
268	C₄	—	712	2	no	—
269	C₄	—	712	2	no	—
270	C₄	#19152252	1	1	yes	characteristic
271	C₄	#19152415	1	1	yes	characteristic
272	C₂	—	4	2	no	—
273	C₂	—	1424	2	no	—
274	C₂	#26032600	1	1	yes	characteristic
275	C₂	#26033388	1	1	yes	characteristic
276	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	C₈₉×#2169	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	#2169	D₈₉
hypocenter	C₈₉	#33922
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #2169

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. #26022433	C₂	semidirect
5. #26022410	C₂	semidirect
6. #26022355	C₂	semidirect
7. C₄×D₁₄₂₄	C₂	semidirect
8. #26022040	C₂	semidirect
9. C₈₉×#2169	C₂	semidirect
146. #2169	D₈₉	semidirect
148. C₈₉	#33922	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
#26022433	2
#26022410	2
#26022355	2
C₄×D₁₄₂₄	2
#26022040	2
C₈₉×#2169	2
#33922	89