Subgroups — #20298606

Order 7232.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₂₂₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#14268631	C₂	1	1	yes	maximal, characteristic
3	#16813545	C₂	1	4	yes	maximal, complemented
4	#117×C₁₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₂₂₆	—	2	2	no	—
6	C₂×Dic₂₂₆	—	2	1	no	—
7	C₂×Dic₂₂₆	C₄	1	4	yes	complemented
8	C₂×Dic₂₂₆	C₂²	1	2	yes	—
9	C₂²×Dic₁₁₃	—	2	1	no	—
10	#14268631	—	2	2	no	—
11	C₂×#11744751	C₂²	1	2	yes	—
12	C₂²×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
14	C₂²⋊C₄×C₁₁₃	—	2	1	no	—
15	C₄²×C₁₁₃	—	2	1	no	—
16	C₂³×C₁₁₃	D₄	1	1	yes	characteristic
17	C₂×C₄×C₁₁₃	—	2	2	no	—
18	C₂×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₁₁₃	D₄	1	2	yes	—
20	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₁₃	—	2	4	no	—
23	C₂×Dic₁₁₃	—	2	2	no	—
24	Dic₂₂₆	—	2	4	no	—
25	Dic₂₂₆	—	2	2	no	—
26	Dic₂₂₆	C₂×C₄	1	4	yes	—
27	#11744751	—	2	2	no	—
28	C₂²×C₁₁₃	—	2	2	no	—
29	C₂²×C₁₁₃	C₂²⋊C₄	1	2	yes	—
30	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
31	C₄×C₁₁₃	—	4	2	no	—
32	C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
33	C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
35	Dic₁₁₃	—	2	4	no	—
36	C₂×C₁₁₃	—	2	2	no	—
37	C₂×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₁₁₃	#137	1	2	yes	—
39	C₁₁₃	#419	1	1	yes	characteristic, complemented, Sylow
40	#419	—	113	1	no	maximal, Sylow
41	C₂²×Q₈	—	113	1	no	—
42	C₂×M₁₆	—	113	1	no	—
43	#117	D₁₁₃	1	1	yes	characteristic
44	#103	—	113	4	no	—
45	C₂×Q₈	—	113	4	no	—
46	C₂×Q₈	—	113	2	no	—
47	C₂×Q₈	—	226	2	no	—
48	C₂×Q₈	—	226	1	no	—
49	C₂²×C₄	—	226	1	no	—
50	C₂²×C₄	C₂×D₁₁₃	1	1	yes	characteristic
51	M₁₆	—	226	2	no	—
52	C₂×C₈	—	113	2	no	—
53	C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
54	C₂²⋊C₄	—	2	1	no	—
55	C₄²	—	2	1	no	—
56	C₂³	C₂²⋊D₁₁₃	1	1	yes	characteristic
57	Q₈	—	113	4	no	—
58	Q₈	—	226	4	no	—
59	Q₈	—	226	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	226	4	no	—
62	C₂×C₄	—	226	2	no	—
63	C₂×C₄	C₄×D₁₁₃	1	2	yes	—
64	C₂×C₄	D₄₅₂	1	2	yes	—
65	C₂×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
66	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
67	C₈	—	226	2	no	—
68	C₂²	—	2	2	no	—
69	C₂²	#14268624	1	2	yes	—
70	C₂²	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
71	C₄	—	4	2	no	—
72	C₄	—	226	4	no	—
73	C₄	#14268624	1	2	yes	—
74	C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
75	C₄	C₂×D₄₅₂	1	1	yes	characteristic
76	C₂	—	2	2	no	—
77	C₂	C₂×#14268624	1	1	yes	characteristic
78	C₂	#16813652	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₁₃
commutator	C₄×C₁₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₁₃
Fitting	#117×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×D₄
hypercenter	#117	D₁₁₃
hypocenter	C₁₁₃	#419
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. #16813545	C₂	semidirect
7. C₂×Dic₂₂₆	C₄	semidirect
39. C₁₁₃	#419	semidirect