Subgroups — #41011453

Order 27008.

86 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	#27212899	C₂	1	1	yes	maximal, characteristic
3	#500×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₁₁⋊C₁₆	—	2	1	no	—
5	C₄⋊(C₂₁₁⋊C₈)	—	2	1	no	—
6	C₄⋊Q₈×C₂₁₁	—	2	1	no	—
7	#107×C₂₁₁	—	2	1	no	—
8	#107×C₂₁₁	C₄	1	2	yes	—
9	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
10	C₂×C₂₁₁⋊C₈	—	4	1	no	—
11	C₂₁₁⋊C₁₆	—	4	1	no	—
12	C₂×Q₈×C₂₁₁	—	4	1	no	—
13	C₂×C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
15	C₄⋊C₄×C₂₁₁	—	2	2	no	—
16	C₄⋊C₄×C₂₁₁	—	4	1	no	—
17	C₄²×C₂₁₁	D₄	1	1	yes	characteristic
18	C₂₁₁⋊C₈	—	8	1	no	—
19	Q₈×C₂₁₁	—	8	1	no	—
20	Q₈×C₂₁₁	—	8	1	no	—
21	C₂×C₄×C₂₁₁	—	4	2	no	—
22	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₁₁	D₈	1	1	yes	characteristic
24	C₂×C₄×C₂₁₁	SD₁₆	1	1	yes	characteristic
25	C₈×C₂₁₁	—	2	1	no	—
26	C₈×C₂₁₁	SD₁₆	1	1	yes	characteristic
27	C₈×C₂₁₁	Q₁₆	1	1	yes	characteristic
28	C₂²×C₂₁₁	#102	1	1	yes	characteristic
29	C₄×C₂₁₁	—	2	1	no	—
30	C₄×C₂₁₁	—	8	2	no	—
31	C₄×C₂₁₁	#101	1	1	yes	characteristic
32	C₄×C₂₁₁	#103	1	1	yes	characteristic
33	C₄×C₂₁₁	D₁₆	1	1	yes	characteristic
34	C₄×C₂₁₁	SD₃₂	1	1	yes	characteristic
35	C₂×C₂₁₁	#332	1	1	yes	characteristic
36	C₂×C₂₁₁	#357	1	1	yes	characteristic
37	C₂×C₂₁₁	#362	1	1	yes	characteristic
38	C₂₁₁	#1364	1	1	yes	characteristic, complemented, Sylow
39	#1364	—	211	1	no	maximal, Sylow
40	#500	D₂₁₁	1	1	yes	characteristic
41	C₄⋊C₁₆	—	211	1	no	—
42	#335	—	211	1	no	—
43	C₄⋊Q₈	—	2	1	no	—
44	C₂×C₁₆	—	422	1	no	—
45	#107	—	2	1	no	—
46	#107	Dic₂₁₁	1	2	yes	—
47	C₄⋊C₈	—	422	1	no	—
48	C₄×C₈	C₂×D₂₁₁	1	1	yes	characteristic
49	C₂×Q₈	—	4	1	no	—
50	C₂×C₈	—	844	1	no	—
51	C₂×C₈	C₂×Dic₂₁₁	1	1	yes	characteristic
52	C₂×C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
53	C₄⋊C₄	—	2	2	no	—
54	C₄⋊C₄	—	4	1	no	—
55	C₄²	C₂²⋊D₂₁₁	1	1	yes	characteristic
56	C₁₆	—	844	1	no	—
57	Q₈	—	8	1	no	—
58	Q₈	—	8	1	no	—
59	C₂×C₄	—	4	2	no	—
60	C₂×C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
61	C₂×C₄	#16809947	1	1	yes	characteristic
62	C₂×C₄	#16809950	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	1688	1	no	—
65	C₈	#16809949	1	1	yes	characteristic
66	C₈	#16809952	1	1	yes	characteristic
67	C₂²	#20269898	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	8	2	no	—
70	C₄	#20269896	1	1	yes	characteristic
71	C₄	#20269901	1	1	yes	characteristic
72	C₄	#20269918	1	1	yes	characteristic
73	C₄	#20269921	1	1	yes	characteristic
74	C₂	#27212892	1	1	yes	characteristic
75	C₂	#27212947	1	1	yes	characteristic
76	C₂	#27212962	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20269898
commutator	C₂×C₈×C₂₁₁	C₂×C₄
Frattini	C₄×C₈	C₂×D₂₁₁
Fitting	#500×C₂₁₁	C₂
socle, prime radical	C₂²×C₂₁₁	#102
cosocle	C₄×C₈×C₂₁₁	C₂²
hypercenter	#500	D₂₁₁
hypocenter	C₂₁₁	#1364
LCS	G ▷ C₂×C₈×C₂₁₁ ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
38. C₂₁₁	#1364	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
#27212899	2
#500×C₂₁₁	2
#1364	211