Subgroups — #41011377

Order 27008.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₂₁₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₈×Q₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₁₁⋊C₁₆	—	2	1	no	—
5	C₂×C₂₁₁⋊C₁₆	—	2	1	no	—
6	C₄×Q₈×C₂₁₁	C₄	1	1	yes	characteristic
7	C₄⋊C₈×C₂₁₁	—	2	1	no	—
8	C₄⋊C₈×C₂₁₁	C₄	1	1	yes	characteristic
9	C₄×C₈×C₂₁₁	—	2	1	no	—
10	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₂₁₁⋊C₁₆	—	2	2	no	—
12	C₂₁₁⋊C₁₆	—	4	1	no	—
13	C₂×Q₈×C₂₁₁	C₈	1	1	yes	characteristic
14	C₂×C₈×C₂₁₁	—	2	1	no	—
15	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
16	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
17	C₄⋊C₄×C₂₁₁	—	2	1	no	—
18	C₄⋊C₄×C₂₁₁	C₈	1	1	yes	characteristic
19	C₄²×C₂₁₁	—	2	1	no	—
20	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
21	Q₈×C₂₁₁	—	2	1	no	—
22	Q₈×C₂₁₁	C₁₆	1	2	yes	complemented
23	C₂×C₄×C₂₁₁	—	2	1	no	—
24	C₂×C₄×C₂₁₁	—	2	1	no	—
25	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₂₁₁	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₂₁₁	M₁₆	1	1	yes	characteristic
28	C₈×C₂₁₁	—	2	1	no	—
29	C₈×C₂₁₁	—	4	1	no	—
30	C₈×C₂₁₁	SD₁₆	1	1	yes	characteristic
31	C₈×C₂₁₁	Q₁₆	1	1	yes	characteristic
32	C₂²×C₂₁₁	C₂²⋊C₈	1	1	yes	characteristic
33	C₄×C₂₁₁	—	2	2	no	—
34	C₄×C₂₁₁	—	2	1	no	—
35	C₄×C₂₁₁	—	4	1	no	—
36	C₄×C₂₁₁	#103	1	1	yes	characteristic
37	C₄×C₂₁₁	#104	1	1	yes	characteristic
38	C₄×C₂₁₁	C₂×C₁₆	1	1	yes	characteristic
39	C₄×C₂₁₁	M₃₂	1	1	yes	characteristic
40	C₂×C₂₁₁	Q₈⋊C₈	1	1	yes	characteristic
41	C₂×C₂₁₁	C₂²⋊C₁₆	1	1	yes	characteristic
42	C₂×C₂₁₁	#350	1	1	yes	characteristic
43	C₂₁₁	Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
44	Q₈⋊C₁₆	—	211	1	no	maximal, Sylow
45	C₈×Q₈	D₂₁₁	1	1	yes	characteristic
46	C₄⋊C₁₆	—	211	1	no	—
47	C₄×C₁₆	—	211	1	no	—
48	C₄×Q₈	Dic₂₁₁	1	1	yes	characteristic
49	C₂×C₁₆	—	422	1	no	—
50	C₂×C₁₆	—	422	1	no	—
51	C₄⋊C₈	—	2	1	no	—
52	C₄⋊C₈	Dic₂₁₁	1	1	yes	characteristic
53	C₄×C₈	—	2	1	no	—
54	C₄×C₈	C₂×D₂₁₁	1	1	yes	characteristic
55	C₂×Q₈	C₂₁₁⋊C₈	1	1	yes	characteristic
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
58	C₂×C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	C₂₁₁⋊C₈	1	1	yes	characteristic
61	C₄²	—	2	1	no	—
62	C₄²	C₂×Dic₂₁₁	1	1	yes	characteristic
63	C₁₆	—	844	1	no	—
64	C₁₆	—	422	2	no	—
65	Q₈	—	2	1	no	—
66	Q₈	C₂₁₁⋊C₁₆	1	2	yes	complemented
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
70	C₂×C₄	C₂×C₂₁₁⋊C₈	1	1	yes	characteristic
71	C₂×C₄	#16809946	1	1	yes	characteristic
72	C₈	—	2	1	no	—
73	C₈	—	4	1	no	—
74	C₈	#16809949	1	1	yes	characteristic
75	C₈	#16809952	1	1	yes	characteristic
76	C₂²	C₂²⋊(C₂₁₁⋊C₈)	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	2	1	no	—
79	C₄	—	4	1	no	—
80	C₄	#20269901	1	1	yes	characteristic
81	C₄	#20269904	1	1	yes	characteristic
82	C₄	C₂×C₂₁₁⋊C₁₆	1	1	yes	characteristic
83	C₄	#20269917	1	1	yes	characteristic
84	C₂	Q₈⋊(C₂₁₁⋊C₈)	1	1	yes	characteristic
85	C₂	C₂²⋊(C₂₁₁⋊C₁₆)	1	1	yes	characteristic
86	C₂	#27212930	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊D₂₁₁
commutator	C₄×C₂₁₁	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₂₁₁
Fitting	C₈×Q₈×C₂₁₁	C₂
socle, prime radical	C₂²×C₂₁₁	C₂²⋊C₈
cosocle	C₄×C₈×C₂₁₁	C₂²
hypercenter	C₈×Q₈	D₂₁₁
hypocenter	C₂₁₁	Q₈⋊C₁₆
LCS	G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
22. Q₈×C₂₁₁	C₁₆	semidirect
43. C₂₁₁	Q₈⋊C₁₆	semidirect
66. Q₈	C₂₁₁⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₂₁₁⋊C₁₆)	2
C₄×C₂₁₁⋊C₁₆	2
C₈×Q₈×C₂₁₁	2
Q₈⋊C₁₆	211