Subgroups — #20299602

Order 7232.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14268657	C₂	1	2	yes	maximal
2	C₄⋊#11744753	C₂	1	4	yes	maximal, complemented
3	C₄²×D₁₁₃	C₂	1	1	yes	maximal, characteristic
4	#14268657	—	2	4	no	—
5	C₂×#11744753	C₂²	1	4	yes	—
6	C₂×C₄×D₁₁₃	C₄	1	2	yes	—
7	C₂×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
8	C₄×Dic₁₁₃	C₄	1	1	yes	characteristic
9	C₄×Dic₁₁₃	C₂²	1	2	yes	—
10	C₄²×C₁₁₃	C₄	1	1	yes	characteristic
11	C₂²×D₁₁₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
14	C₂×Dic₁₁₃	C₂×C₄	1	2	yes	—
15	C₂×Dic₁₁₃	C₂³	1	1	yes	characteristic
16	C₄×D₁₁₃	—	2	4	no	—
17	C₄×D₁₁₃	D₄	1	2	yes	—
18	C₄×D₁₁₃	Q₈	1	2	yes	—
19	#11744753	—	2	4	no	—
20	C₂²×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
21	C₂×D₁₁₃	—	2	2	no	—
22	C₂×D₁₁₃	C₄⋊C₄	1	2	yes	—
23	C₄×C₁₁₃	C₄⋊C₄	1	2	yes	—
24	C₄×C₁₁₃	M₁₆	1	4	yes	complemented
25	Dic₁₁₃	—	2	2	no	—
26	Dic₁₁₃	C₂×D₄	1	1	yes	characteristic
27	Dic₁₁₃	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₁₁₃	C₂×M₁₆	1	2	yes	—
30	D₁₁₃	—	2	2	no	—
31	C₁₁₃	#423	1	1	yes	characteristic, complemented, Sylow
32	#423	—	113	1	no	maximal, Sylow
33	C₂×M₁₆	—	113	2	no	—
34	C₂×C₄²	—	113	1	no	—
35	C₄⋊C₈	—	113	4	no	—
36	C₂²×C₄	—	113	1	no	—
37	C₂²×C₄	—	113	2	no	—
38	M₁₆	—	226	4	no	—
39	C₂×C₈	—	113	4	no	—
40	C₄²	—	113	2	no	—
41	C₄²	—	113	1	no	—
42	C₄²	#90673	1	1	yes	characteristic
43	C₂³	—	113	1	no	—
44	C₂×C₄	—	113	1	no	—
45	C₂×C₄	—	113	2	no	—
46	C₂×C₄	—	113	2	no	—
47	C₂×C₄	—	113	2	no	—
48	C₂×C₄	—	226	4	no	—
49	C₂×C₄	C₂×#90673	1	1	yes	characteristic
50	C₂×C₄	C₂×#90673	1	2	yes	—
51	C₈	—	226	4	no	—
52	C₂²	—	113	2	no	—
53	C₂²	—	226	2	no	—
54	C₂²	C₂²×#90673	1	1	yes	characteristic
55	C₄	—	113	1	no	—
56	C₄	—	113	1	no	—
57	C₄	—	226	2	no	—
58	C₄	C₄⋊#90673	1	2	yes	—
59	C₄	#14268657	1	4	yes	complemented
60	C₂	—	226	2	no	—
61	C₂	C₂×C₄⋊#90673	1	1	yes	characteristic
62	C₂	C₂×#14268657	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#90673
commutator, socle	C₂²×C₁₁₃	C₂²×C₄
Fitting	C₄²×C₁₁₃	C₄
hypercenter	C₄²	#90673
hypocenter	C₁₁₃	#423
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₄⋊#11744753	C₂	semidirect
24. C₄×C₁₁₃	M₁₆	semidirect
31. C₁₁₃	#423	semidirect
59. C₄	#14268657	semidirect