Subgroups — #41011570

Order 27008.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27213514	C₂	1	1	yes	maximal, characteristic
2	#27212945	C₂	1	1	yes	maximal, characteristic
3	#356×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	#20269969	C₄	1	1	yes	characteristic
5	#20269901	—	2	1	no	—
6	#20269901	—	2	1	no	—
7	#20269896	—	2	1	no	—
8	#20269888	C₄	1	1	yes	characteristic
9	C₄⋊Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
10	#101×C₂₁₁	—	2	1	no	—
11	#16809946	—	4	1	no	—
12	C₂×C₂₁₁⋊C₈	—	2	1	no	—
13	C₄⋊Dic₂₁₁	—	2	1	no	—
14	Dic₂₁₁⋊C₄	—	4	1	no	—
15	C₂×Q₈×C₂₁₁	D₄	1	1	yes	characteristic
16	C₂×Q₈×C₂₁₁	D₄	1	1	yes	characteristic
17	M₁₆×C₂₁₁	—	4	1	no	—
18	C₄⋊C₄×C₂₁₁	—	4	1	no	—
19	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₂₁₁	—	4	1	no	—
21	C₂₁₁⋊C₈	—	4	1	no	—
22	C₂₁₁⋊C₈	—	8	1	no	—
23	Q₈×C₂₁₁	—	4	1	no	—
24	Q₈×C₂₁₁	—	4	1	no	—
25	C₂×C₄×C₂₁₁	—	2	1	no	—
26	C₂×C₄×C₂₁₁	—	2	1	no	—
27	C₂×C₄×C₂₁₁	—	2	1	no	—
28	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
29	C₈×C₂₁₁	—	8	1	no	—
30	C₂²×C₂₁₁	#99	1	1	yes	characteristic
31	C₄×C₂₁₁	—	2	1	no	—
32	C₄×C₂₁₁	—	4	1	no	—
33	C₄×C₂₁₁	—	4	1	no	—
34	C₄×C₂₁₁	—	4	1	no	—
35	Dic₂₁₁	—	8	1	no	—
36	C₂×C₂₁₁	—	2	1	no	—
37	C₂×C₂₁₁	#354	1	1	yes	characteristic
38	C₂₁₁	#1413	1	1	yes	characteristic, complemented, Sylow
39	#1413	—	211	1	no	maximal, Sylow
40	#491	—	211	1	no	—
41	#356	—	211	1	no	—
42	#356	D₂₁₁	1	1	yes	characteristic
43	C₄⋊Q₈	C₂×D₂₁₁	1	1	yes	characteristic
44	#125	—	211	1	no	—
45	#103	—	422	1	no	—
46	#103	—	422	1	no	—
47	#101	—	2	1	no	—
48	#101	—	422	1	no	—
49	#97	—	211	1	no	—
50	C₂×Q₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
51	C₂×Q₈	D₈₄₄	1	1	yes	characteristic
52	M₁₆	—	4	1	no	—
53	M₁₆	—	844	1	no	—
54	C₂×C₈	—	422	1	no	—
55	C₄⋊C₄	—	4	1	no	—
56	C₄⋊C₄	—	844	1	no	—
57	C₄⋊C₄	—	422	1	no	—
58	C₄²	C₄×D₂₁₁	1	1	yes	characteristic
59	Q₈	—	4	1	no	—
60	Q₈	—	4	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	844	1	no	—
65	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
66	C₈	—	8	1	no	—
67	C₈	—	844	1	no	—
68	C₈	—	1688	1	no	—
69	C₂²	#20269891	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	4	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	1688	1	no	—
75	C₂	—	2	1	no	—
76	C₂	#27212940	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27212940
commutator	C₄²×C₂₁₁	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₂₁₁
Fitting	#356×C₂₁₁	C₂
socle	C₂×C₂₁₁	#354
prime radical	C₂²×C₂₁₁	#99
cosocle	C₄⋊Q₈×C₂₁₁	C₂²
hypercenter	#356	D₂₁₁
hypocenter	C₂₁₁	#1413
LCS	G ▷ C₄²×C₂₁₁ ▷ C₂×C₄×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #356

Complements

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

N	H ≅ G/N	type
38. C₂₁₁	#1413	semidirect

Maximal subgroups

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

subgroup	index
#27213514	2
#27212945	2
#356×C₂₁₁	2
#1413	211