Subgroups — #27212915

Order 13504.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16809946	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	#16809946	—	2	2	no	—
4	#16809946	C₄	1	4	yes	complemented
5	C₂×C₂₁₁⋊C₈	—	2	2	no	—
6	C₂²×C₄×C₂₁₁	C₄	1	2	yes	—
7	C₂²×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
8	C₄⋊C₄×C₂₁₁	—	2	2	no	—
9	C₂₁₁⋊C₈	—	2	4	no	—
10	C₂³×C₂₁₁	C₂×C₄	1	1	yes	characteristic
11	C₂×C₄×C₂₁₁	—	2	2	no	—
12	C₂×C₄×C₂₁₁	—	4	2	no	—
13	C₂×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₂₁₁	D₄	1	2	yes	—
15	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
16	C₂×C₄×C₂₁₁	Q₈	1	1	yes	characteristic
17	C₂²×C₂₁₁	—	2	1	no	—
18	C₂²×C₂₁₁	—	2	1	no	—
19	C₂²×C₂₁₁	C₄²	1	1	yes	characteristic
20	C₂²×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
21	C₂²×C₂₁₁	C₄⋊C₄	1	1	yes	characteristic
22	C₄×C₂₁₁	—	4	2	no	—
23	C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
24	C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
25	C₂×C₂₁₁	—	2	1	no	—
26	C₂×C₂₁₁	—	2	1	no	—
27	C₂×C₂₁₁	#95	1	1	yes	characteristic
28	C₂×C₂₁₁	#100	1	1	yes	characteristic
29	C₂×C₂₁₁	#101	1	1	yes	characteristic
30	C₂₁₁	#343	1	1	yes	characteristic, complemented, Sylow
31	#343	—	211	1	no	maximal, Sylow
32	C₂×M₁₆	—	211	2	no	—
33	C₂×C₄⋊C₄	D₂₁₁	1	1	yes	characteristic
34	C₂²×C₄	Dic₂₁₁	1	2	yes	—
35	C₂²×C₄	C₂×D₂₁₁	1	1	yes	characteristic
36	M₁₆	—	422	2	no	—
37	M₁₆	—	211	4	no	—
38	C₂×C₈	—	422	2	no	—
39	C₄⋊C₄	—	2	2	no	—
40	C₂³	C₂×Dic₂₁₁	1	1	yes	characteristic
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	4	2	no	—
43	C₂×C₄	C₄×D₂₁₁	1	2	yes	—
44	C₂×C₄	C₂²⋊D₂₁₁	1	2	yes	—
45	C₂×C₄	D₈₄₄	1	1	yes	characteristic
46	C₂×C₄	Dic₄₂₂	1	1	yes	characteristic
47	C₈	—	422	4	no	—
48	C₂²	—	2	1	no	—
49	C₂²	—	2	1	no	—
50	C₂²	C₄×Dic₂₁₁	1	1	yes	characteristic
51	C₂²	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
52	C₂²	C₄⋊Dic₂₁₁	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
55	C₄	Dic₂₁₁⋊C₄	1	2	yes	—
56	C₂	—	2	1	no	—
57	C₂	—	2	1	no	—
58	C₂	C₂₁₁⋊#95	1	1	yes	characteristic
59	C₂	#20269894	1	1	yes	characteristic
60	C₂	#20269896	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊Dic₂₁₁
commutator	C₂²×C₂₁₁	C₄²
Frattini	C₂²×C₄	C₂×D₂₁₁
Fitting	C₂×C₄⋊C₄×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂²⋊C₄
prime radical	C₂³×C₂₁₁	C₂×C₄
cosocle	C₂²×C₄×C₂₁₁	C₂²
hypercenter	C₂×C₄⋊C₄	D₂₁₁
hypocenter	C₂₁₁	#343
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. #16809946	C₄	semidirect
30. C₂₁₁	#343	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16809946	2
C₂×C₄⋊C₄×C₂₁₁	2
#343	211