Subgroups — #36712330

Order 22784.

448 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033144	C₂	1	2	yes	maximal
2	#26022531	C₂	1	2	yes	maximal
3	#26022527	C₂	1	1	yes	maximal, characteristic
4	#26022417	C₂	1	4	yes	maximal
5	#26022395	C₂	1	2	yes	maximal
6	C₄×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic
7	C₄×Dic₇₁₂	C₂	1	2	yes	maximal
8	Q₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
9	#19152326	—	2	2	no	—
10	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
11	#19152003	C₂²	1	2	yes	—
12	#19152000	C₂²	1	4	yes	—
13	#19151906	—	2	4	no	—
14	#19151884	—	2	2	no	—
15	C₄×Dic₃₅₆	C₂²	1	4	yes	—
16	C₄×Dic₃₅₆	C₂²	1	2	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	#19151403	C₂²	1	2	yes	—
22	#19151403	C₂²	1	4	yes	—
23	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
25	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
26	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
27	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
28	C₂×Dic₃₅₆	C₂³	1	2	yes	—
29	#15668012	—	2	2	no	—
30	#15668010	—	4	2	no	—
31	#15668004	—	2	4	no	—
32	#15668002	—	4	4	no	—
33	C₄×Dic₁₇₈	—	2	2	no	—
34	C₄×Dic₁₇₈	—	2	4	no	—
35	#15667977	—	4	2	no	—
36	Dic₇₁₂	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	#15667937	—	2	4	no	—
42	#15667937	—	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	—	2	2	no	—
52	C₂×Dic₁₇₈	—	4	2	no	—
53	Dic₃₅₆	—	2	2	no	—
54	Dic₃₅₆	C₂×Q₈	1	4	yes	—
55	C₄⋊Dic₈₉	—	2	2	no	—
56	C₄⋊Dic₈₉	—	2	4	no	—
57	C₄⋊Dic₈₉	—	4	2	no	—
58	#13083553	—	4	4	no	—
59	#13083553	—	4	2	no	—
60	#13083553	—	4	4	no	—
61	#13083553	—	4	4	no	—
62	C₄×Dic₈₉	—	4	4	no	—
63	C₄×Dic₈₉	—	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	1	yes	characteristic
69	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
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	Q₈×C₈₉	—	2	2	no	—
76	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
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	1	yes	characteristic
80	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
81	C₂×Dic₈₉	—	4	2	no	—
82	C₂×Dic₈₉	—	4	2	no	—
83	C₂×Dic₈₉	—	4	4	no	—
84	Dic₁₇₈	—	2	4	no	—
85	Dic₁₇₈	—	4	2	no	—
86	Dic₁₇₈	—	8	4	no	—
87	C₈×C₈₉	—	2	2	no	—
88	C₈×C₈₉	—	2	1	no	—
89	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
90	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
91	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
92	C₄×C₈₉	—	2	2	no	—
93	C₄×C₈₉	—	2	2	no	—
94	C₄×C₈₉	—	2	1	no	—
95	C₄×C₈₉	#508	1	2	yes	—
96	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
97	C₄×C₈₉	#578	1	1	yes	characteristic
98	Dic₈₉	—	4	4	no	—
99	Dic₈₉	—	8	4	no	—
100	Dic₈₉	—	8	2	no	—
101	#33950	—	89	1	no	maximal, Sylow
102	C₂×C₈₉	Q₈×D₈	1	1	yes	characteristic
103	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
104	C₂×C₈₉	#3417	1	1	yes	characteristic
105	#3383	—	89	2	no	—
106	#2253	—	89	2	no	—
107	#2252	—	89	1	no	—
108	#2229	—	89	4	no	—
109	#2225	—	89	2	no	—
110	Q₈×C₁₆	D₈₉	1	1	yes	characteristic
111	C₄×Q₃₂	—	89	1	no	—
112	C₄×Q₃₂	—	89	2	no	—
113	C₈₉	#33950	1	1	yes	characteristic, complemented, Sylow
114	#557	—	178	2	no	—
115	C₂×Q₃₂	—	89	1	no	—
116	#500	—	89	2	no	—
117	#499	—	89	4	no	—
118	#479	—	178	4	no	—
119	#475	—	178	2	no	—
120	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
121	C₄×Q₁₆	—	89	2	no	—
122	C₄×Q₁₆	—	89	4	no	—
123	#367	—	89	4	no	—
124	#366	—	89	2	no	—
125	#366	—	89	1	no	—
126	#366	—	89	2	no	—
127	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
128	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
129	#358	—	89	4	no	—
130	#358	—	89	2	no	—
131	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
132	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
133	C₂×Q₁₆	—	89	2	no	—
134	C₄⋊Q₈	—	356	2	no	—
135	C₄⋊Q₈	—	178	2	no	—
136	#125	—	356	4	no	—
137	#125	—	178	4	no	—
138	C₄×Q₈	—	356	2	no	—
139	C₄×Q₈	—	178	2	no	—
140	C₄×Q₈	—	178	4	no	—
141	C₄×Q₈	D₃₅₆	1	1	yes	characteristic
142	Q₃₂	—	89	4	no	—
143	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
144	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
145	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
146	#107	—	89	4	no	—
147	#107	—	89	2	no	—
148	#107	—	178	2	no	—
149	#106	—	178	4	no	—
150	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
151	C₄⋊C₈	D₃₅₆	1	2	yes	—
152	#103	—	178	4	no	—
153	#103	—	178	2	no	—
154	C₄×C₈	C₂²×D₈₉	1	2	yes	—
155	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
156	C₂×Q₈	—	356	2	no	—
157	C₂×Q₈	—	178	2	no	—
158	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
159	Q₁₆	—	89	4	no	—
160	Q₁₆	—	178	2	no	—
161	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
162	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
163	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
164	C₄⋊C₄	—	356	4	no	—
165	C₄⋊C₄	—	356	2	no	—
166	C₄⋊C₄	—	356	4	no	—
167	C₄⋊C₄	—	356	2	no	—
168	C₄⋊C₄	—	356	4	no	—
169	C₄⋊C₄	—	178	2	no	—
170	C₄⋊C₄	—	178	4	no	—
171	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
172	C₄⋊C₄	D₇₁₂	1	2	yes	—
173	C₄²	—	356	4	no	—
174	C₄²	—	356	2	no	—
175	C₄²	C₂×D₃₅₆	1	2	yes	—
176	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
177	C₁₆	—	2	1	no	—
178	C₁₆	—	2	2	no	—
179	C₁₆	Q₈×D₈₉	1	2	yes	—
180	Q₈	—	2	2	no	—
181	Q₈	—	356	2	no	—
182	Q₈	—	178	4	no	—
183	Q₈	—	712	4	no	—
184	C₂×C₄	—	356	2	no	—
185	C₂×C₄	—	356	2	no	—
186	C₂×C₄	—	356	4	no	—
187	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
188	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
189	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
190	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
191	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	—	2	1	no	—
194	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
195	C₈	#15668059	1	1	yes	characteristic
196	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	1	no	—
200	C₄	—	356	4	no	—
201	C₄	—	712	4	no	—
202	C₄	—	712	2	no	—
203	C₄	#19152030	1	2	yes	—
204	C₄	Q₈×D₃₅₆	1	1	yes	characteristic
205	C₄	#19152415	1	1	yes	characteristic
206	C₂	Q₈×D₇₁₂	1	1	yes	characteristic
207	C₂	C₂×#19152030	1	1	yes	characteristic
208	C₂	#26033405	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, prime radical	C₂²×C₈₉	C₂²×D₈
hypercenter	Q₈×C₁₆	D₈₉
hypocenter	C₈₉	#33950
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
36. Dic₇₁₂	Q₈	semidirect
113. C₈₉	#33950	semidirect

Maximal subgroups

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

subgroup	index
#26033144	2
#26022531	2
#26022527	2
#26022417	2
#26022395	2
C₄×Dic₇₁₂	2
Q₈×C₁₆×C₈₉	2
#33950	89