Subgroups — #36712306

Order 22784.

476 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033120	C₂	1	2	yes	maximal, complemented
2	#26022531	C₂	1	2	yes	maximal, complemented
3	#26022527	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022410	C₂	1	4	yes	maximal, complemented
5	#26022400	C₂	1	2	yes	maximal, complemented
6	C₄×D₁₄₂₄	C₂	1	2	yes	maximal
7	C₄×D₁₄₂₄	C₂	1	1	yes	maximal, characteristic
8	Q₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#19152308	—	2	2	no	—
10	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
11	#19152003	C₂²	1	2	yes	—
12	#19152000	C₂²	1	4	yes	—
13	#19151899	—	2	4	no	—
14	#19151889	—	2	2	no	—
15	C₄×D₇₁₂	C₂²	1	2	yes	—
16	C₄×D₇₁₂	C₂²	1	4	yes	—
17	#19151426	C₂²	1	4	yes	—
18	#19151424	C₂²	1	2	yes	—
19	#19151424	C₂²	1	2	yes	—
20	#19151424	C₂²	1	1	yes	characteristic
21	#19151400	C₂²	1	4	yes	—
22	#19151400	C₂²	1	2	yes	—
23	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
25	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
26	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
27	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
28	C₂×D₇₁₂	C₂³	1	2	yes	—
29	#15668012	—	2	2	no	—
30	#15668004	—	2	4	no	—
31	#15667990	—	4	4	no	—
32	#15667988	—	4	2	no	—
33	C₄×D₃₅₆	—	2	2	no	—
34	C₄×D₃₅₆	—	2	4	no	—
35	C₄⋊C₄×D₈₉	—	4	2	no	—
36	D₁₄₂₄	Q₈	1	4	yes	complemented
37	#15667946	—	2	2	no	—
38	#15667946	C₂³	1	4	yes	—
39	#15667946	C₂³	1	2	yes	—
40	#15667944	—	2	4	no	—
41	#15667934	—	2	4	no	—
42	#15667934	—	2	2	no	—
43	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
44	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
45	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
46	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
48	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
49	C₄×C₈×C₈₉	C₂³	1	2	yes	—
50	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
51	C₂×Dic₁₇₈	—	4	2	no	—
52	C₂×D₃₅₆	—	2	2	no	—
53	C₂×C₄×D₈₉	—	4	2	no	—
54	C₂×C₄×D₈₉	—	4	4	no	—
55	D₇₁₂	—	2	2	no	—
56	D₇₁₂	C₂×Q₈	1	4	yes	—
57	C₄⋊Dic₈₉	—	2	2	no	—
58	C₄⋊Dic₈₉	—	2	4	no	—
59	C₄⋊Dic₈₉	—	4	2	no	—
60	#13083553	—	4	4	no	—
61	#13083553	—	4	4	no	—
62	#13083551	—	4	4	no	—
63	#13083551	—	4	2	no	—
64	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
65	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
68	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
69	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
70	C₄²×C₈₉	C₂×D₄	1	2	yes	—
71	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₁₆×C₈₉	—	2	1	no	—
73	C₁₆×C₈₉	—	2	2	no	—
74	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
75	C₂²×D₈₉	—	4	2	no	—
76	Q₈×C₈₉	—	2	2	no	—
77	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
79	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
80	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
82	C₂×Dic₈₉	—	4	2	no	—
83	C₂×Dic₈₉	—	4	4	no	—
84	D₃₅₆	—	2	4	no	—
85	D₃₅₆	—	4	2	no	—
86	C₄×D₈₉	—	8	4	no	—
87	C₄×D₈₉	—	16	4	no	—
88	Dic₁₇₈	—	16	2	no	—
89	C₈×C₈₉	—	2	2	no	—
90	C₈×C₈₉	—	2	1	no	—
91	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
92	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
93	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
94	C₂×D₈₉	—	4	4	no	—
95	C₂×D₈₉	—	8	2	no	—
96	C₂×D₈₉	—	8	2	no	—
97	C₄×C₈₉	—	2	2	no	—
98	C₄×C₈₉	—	2	1	no	—
99	C₄×C₈₉	—	2	2	no	—
100	C₄×C₈₉	#508	1	2	yes	—
101	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
102	C₄×C₈₉	#578	1	1	yes	characteristic
103	Dic₈₉	—	8	2	no	—
104	Dic₈₉	—	8	4	no	—
105	#33947	—	89	1	no	maximal, Sylow
106	C₂×C₈₉	Q₈×D₈	1	1	yes	characteristic
107	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
108	C₂×C₈₉	#3415	1	1	yes	characteristic
109	D₈₉	—	8	4	no	—
110	#3380	—	89	2	no	—
111	#2253	—	89	2	no	—
112	#2252	—	89	1	no	—
113	#2228	—	89	4	no	—
114	#2226	—	89	2	no	—
115	Q₈×C₁₆	D₈₉	1	1	yes	characteristic, complemented
116	C₄×D₁₆	—	89	1	no	—
117	C₄×D₁₆	—	89	2	no	—
118	C₈₉	#33947	1	1	yes	characteristic, complemented, Sylow
119	#554	—	178	2	no	—
120	C₂×D₁₆	—	89	1	no	—
121	#500	—	89	2	no	—
122	#499	—	89	4	no	—
123	#478	—	178	4	no	—
124	#476	—	178	2	no	—
125	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
126	C₄×D₈	—	89	2	no	—
127	C₄×D₈	—	89	4	no	—
128	#367	—	89	4	no	—
129	#366	—	89	2	no	—
130	#366	—	89	1	no	—
131	#366	—	89	2	no	—
132	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
133	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
134	#357	—	89	4	no	—
135	#357	—	89	2	no	—
136	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
137	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
138	C₂×D₈	—	89	2	no	—
139	C₄⋊Q₈	—	178	2	no	—
140	#125	—	178	4	no	—
141	C₂²⋊Q₈	—	356	2	no	—
142	C₂²⋊Q₈	—	356	4	no	—
143	C₄×Q₈	D₃₅₆	1	1	yes	characteristic
144	C₄×D₄	—	178	2	no	—
145	C₄×D₄	—	178	4	no	—
146	C₂×C₄⋊C₄	—	356	2	no	—
147	D₁₆	—	89	4	no	—
148	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
149	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
150	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
151	#107	—	89	4	no	—
152	#107	—	89	2	no	—
153	#107	—	178	2	no	—
154	#106	—	178	4	no	—
155	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
156	C₄⋊C₈	D₃₅₆	1	2	yes	—
157	#102	—	178	4	no	—
158	#102	—	178	2	no	—
159	C₄×C₈	C₂²×D₈₉	1	2	yes	—
160	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
161	C₂×Q₈	—	356	2	no	—
162	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
163	C₂×D₄	—	178	2	no	—
164	C₂²×C₄	—	356	4	no	—
165	C₂²×C₄	—	356	2	no	—
166	D₈	—	89	4	no	—
167	D₈	—	178	2	no	—
168	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
169	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
170	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
171	C₄⋊C₄	—	356	4	no	—
172	C₄⋊C₄	—	356	2	no	—
173	C₄⋊C₄	—	356	4	no	—
174	C₄⋊C₄	—	178	2	no	—
175	C₄⋊C₄	—	178	4	no	—
176	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
177	C₄⋊C₄	D₇₁₂	1	2	yes	—
178	C₂²⋊C₄	—	356	2	no	—
179	C₂²⋊C₄	—	356	4	no	—
180	C₄²	C₂×D₃₅₆	1	2	yes	—
181	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
182	C₁₆	—	2	1	no	—
183	C₁₆	—	2	2	no	—
184	C₁₆	Q₈×D₈₉	1	2	yes	—
185	C₂³	—	356	2	no	—
186	Q₈	—	2	2	no	—
187	Q₈	—	1424	2	no	—
188	D₄	—	356	2	no	—
189	D₄	—	178	4	no	—
190	C₂×C₄	—	356	2	no	—
191	C₂×C₄	—	356	4	no	—
192	C₂×C₄	—	1424	4	no	—
193	C₂×C₄	—	712	4	no	—
194	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
195	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
196	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
197	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
198	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
199	C₈	—	2	2	no	—
200	C₈	—	2	1	no	—
201	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
202	C₈	#15668059	1	1	yes	characteristic
203	C₂²	—	356	4	no	—
204	C₂²	—	712	2	no	—
205	C₂²	—	712	2	no	—
206	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	1	no	—
210	C₄	—	712	2	no	—
211	C₄	—	712	4	no	—
212	C₄	#19152030	1	2	yes	—
213	C₄	Q₈×D₃₅₆	1	1	yes	characteristic
214	C₄	#19152415	1	1	yes	characteristic
215	C₂	—	712	4	no	—
216	C₂	Q₈×D₇₁₂	1	1	yes	characteristic
217	C₂	C₂×#19152030	1	1	yes	characteristic
218	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	Q₈×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₁₄₂₄	C₂²
hypercenter	Q₈×C₁₆	D₈₉
hypocenter	C₈₉	#33947
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
1. #26033120	C₂	semidirect
2. #26022531	C₂	semidirect
3. #26022527	C₂	semidirect
4. #26022410	C₂	semidirect
5. #26022400	C₂	semidirect
8. Q₈×C₁₆×C₈₉	C₂	semidirect
36. D₁₄₂₄	Q₈	semidirect
115. Q₈×C₁₆	D₈₉	semidirect
118. C₈₉	#33947	semidirect

Maximal subgroups

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

subgroup	index
#26033120	2
#26022531	2
#26022527	2
#26022410	2
#26022400	2
C₄×D₁₄₂₄	2
Q₈×C₁₆×C₈₉	2
#33947	89