Subgroups — #41011575

Order 27008.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₃₇₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₁₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#366×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₆₈₈	—	2	1	no	—
5	D₃₃₇₆	—	2	1	no	—
6	D₃₃₇₆	C₄	1	2	yes	complemented
7	C₂₁₁⋊C₃₂	—	2	1	no	—
8	C₂×C₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
9	#107×C₂₁₁	—	2	1	no	—
10	C₂×D₈₄₄	—	4	1	no	—
11	D₁₆₈₈	—	2	2	no	—
12	D₁₆₈₈	—	4	1	no	—
13	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
14	C₄⋊C₄×C₂₁₁	—	4	1	no	—
15	C₁₆×C₂₁₁	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₂₁₁	D₄	1	1	yes	characteristic
17	C₂²×D₂₁₁	—	8	1	no	—
18	D₈₄₄	—	4	2	no	—
19	D₈₄₄	—	8	1	no	—
20	C₂×C₄×C₂₁₁	—	8	1	no	—
21	C₂×C₄×C₂₁₁	D₈	1	1	yes	characteristic
22	C₈×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₂₁₁	SD₁₆	1	1	yes	characteristic
24	C₂²×C₂₁₁	D₁₆	1	1	yes	characteristic
25	C₂×D₂₁₁	—	8	2	no	—
26	C₂×D₂₁₁	—	16	1	no	—
27	C₂×D₂₁₁	—	16	1	no	—
28	C₄×C₂₁₁	—	16	1	no	—
29	C₄×C₂₁₁	#102	1	1	yes	characteristic
30	C₄×C₂₁₁	SD₃₂	1	1	yes	characteristic
31	C₂×C₂₁₁	#357	1	1	yes	characteristic
32	C₂×C₂₁₁	D₃₂	1	1	yes	characteristic
33	C₂×C₂₁₁	SD₆₄	1	1	yes	characteristic
34	D₂₁₁	—	16	2	no	—
35	C₂₁₁	#1415	1	1	yes	characteristic, complemented, Sylow
36	#1415	—	211	1	no	maximal, Sylow
37	C₂×D₁₆	—	211	1	no	—
38	C₂×C₃₂	—	211	1	no	—
39	#366	D₂₁₁	1	1	yes	characteristic, complemented
40	C₂×D₈	—	422	1	no	—
41	D₁₆	—	422	1	no	—
42	D₁₆	—	211	2	no	—
43	C₂×C₁₆	C₂×D₂₁₁	1	1	yes	characteristic
44	#107	—	2	1	no	—
45	C₃₂	—	422	1	no	—
46	C₂×D₄	—	844	1	no	—
47	D₈	—	844	1	no	—
48	D₈	—	422	2	no	—
49	C₂×C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
50	C₄⋊C₄	—	4	1	no	—
51	C₁₆	C₄×D₂₁₁	1	1	yes	characteristic
52	C₁₆	D₈₄₄	1	1	yes	characteristic
53	C₂³	—	1688	1	no	—
54	D₄	—	844	2	no	—
55	D₄	—	1688	1	no	—
56	C₂×C₄	—	8	1	no	—
57	C₂×C₄	#16809947	1	1	yes	characteristic
58	C₈	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
59	C₈	#16809949	1	1	yes	characteristic
60	C₂²	—	3376	1	no	—
61	C₂²	—	3376	1	no	—
62	C₂²	—	1688	2	no	—
63	C₂²	#20269918	1	1	yes	characteristic
64	C₄	—	16	1	no	—
65	C₄	#20269897	1	1	yes	characteristic
66	C₄	#20269920	1	1	yes	characteristic
67	C₂	—	3376	2	no	—
68	C₂	#27212946	1	1	yes	characteristic
69	C₂	#27212981	1	1	yes	characteristic
70	C₂	#27212983	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20269918
commutator	C₁₆×C₂₁₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₁₁
Fitting	#366×C₂₁₁	C₂
socle	C₂²×C₂₁₁	D₁₆
prime radical	C₂×D₃₃₇₆	C₂
cosocle	C₂×C₁₆×C₂₁₁	C₂²
hypercenter	#366	D₂₁₁
hypocenter	C₂₁₁	#1415
LCS	G ▷ C₁₆×C₂₁₁ ▷ C₈×C₂₁₁ ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #366

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₁₁⋊C₃₂	C₂	semidirect
3. #366×C₂₁₁	C₂	semidirect
6. D₃₃₇₆	C₄	semidirect
35. C₂₁₁	#1415	semidirect
39. #366	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃₃₇₆	2
C₂×C₂₁₁⋊C₃₂	2
#366×C₂₁₁	2
#1415	211