Subgroups — #27212916

Order 13504.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16809946	C₂	1	1	yes	maximal, characteristic
2	C₂×#16809946	C₂	1	1	yes	maximal, characteristic
3	#117×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	#16809946	—	2	1	no	—
5	#16809946	—	2	1	no	—
6	#16809946	—	2	1	no	—
7	#16809946	C₄	1	2	yes	complemented
8	C₂×C₂₁₁⋊C₈	—	2	1	no	—
9	C₂×C₂₁₁⋊C₈	C₄	1	2	yes	complemented
10	C₂²×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×C₂₁₁	—	2	1	no	—
12	C₂²⋊C₄×C₂₁₁	C₄	1	2	yes	—
13	C₄²×C₂₁₁	—	2	1	no	—
14	C₂₁₁⋊C₈	—	2	2	no	—
15	C₂₁₁⋊C₈	—	2	2	no	—
16	C₂³×C₂₁₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₂₁₁	—	2	2	no	—
18	C₂×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₁₁	Q₈	1	1	yes	characteristic
24	C₂²×C₂₁₁	—	4	1	no	—
25	C₂²×C₂₁₁	C₄²	1	1	yes	characteristic
26	C₂²×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₂₁₁	C₄⋊C₄	1	1	yes	characteristic
28	C₄×C₂₁₁	—	4	2	no	—
29	C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
30	C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
31	C₄×C₂₁₁	C₄⋊C₄	1	1	yes	characteristic
32	C₄×C₂₁₁	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₂₁₁	—	2	1	no	—
34	C₂×C₂₁₁	—	2	1	no	—
35	C₂×C₂₁₁	—	2	1	no	—
36	C₂×C₂₁₁	#95	1	1	yes	characteristic
37	C₂₁₁	#344	1	1	yes	characteristic, complemented, Sylow
38	#344	—	211	1	no	maximal, Sylow
39	C₂×M₁₆	—	211	1	no	—
40	C₂×M₁₆	—	211	1	no	—
41	#117	D₂₁₁	1	1	yes	characteristic
42	C₂²×C₄	C₂×D₂₁₁	1	1	yes	characteristic
43	M₁₆	—	422	1	no	—
44	M₁₆	—	422	1	no	—
45	M₁₆	—	422	1	no	—
46	M₁₆	—	211	2	no	—
47	C₂×C₈	—	422	1	no	—
48	C₂×C₈	—	211	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₂²⋊C₄	Dic₂₁₁	1	2	yes	—
51	C₄²	—	2	1	no	—
52	C₂³	C₂×Dic₂₁₁	1	1	yes	characteristic
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	C₄×D₂₁₁	1	1	yes	characteristic
55	C₂×C₄	C₄×D₂₁₁	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
57	C₂×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
58	C₂×C₄	D₈₄₄	1	1	yes	characteristic
59	C₂×C₄	Dic₄₂₂	1	1	yes	characteristic
60	C₈	—	422	2	no	—
61	C₈	—	422	2	no	—
62	C₂²	—	4	1	no	—
63	C₂²	C₄×Dic₂₁₁	1	1	yes	characteristic
64	C₂²	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
65	C₂²	C₄⋊Dic₂₁₁	1	1	yes	characteristic
66	C₄	—	4	2	no	—
67	C₄	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
68	C₄	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
69	C₄	Dic₂₁₁⋊C₄	1	1	yes	characteristic
70	C₄	Dic₂₁₁⋊C₄	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	—	2	1	no	—
73	C₂	—	2	1	no	—
74	C₂	C₂₁₁⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄	(C₂×D₂₁₁)⋊C₄
commutator	C₂²×C₂₁₁	C₄²
Frattini	C₂²×C₄	C₂×D₂₁₁
Fitting	#117×C₂₁₁	C₂
socle	C₂×C₂₁₁	#95
prime radical	C₂³×C₂₁₁	C₂×C₄
cosocle	C₂²×C₄×C₂₁₁	C₂²
hypercenter	#117	D₂₁₁
hypocenter	C₂₁₁	#344
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
7. #16809946	C₄	semidirect
9. C₂×C₂₁₁⋊C₈	C₄	semidirect
37. C₂₁₁	#344	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
#117×C₂₁₁	2
#344	211