Subgroups — #20299613

Order 7232.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14268657	C₂	1	1	yes	maximal, characteristic
2	D₁₁₃⋊M₁₆	C₂	1	1	yes	maximal, characteristic
3	#16813693	C₂	1	2	yes	maximal, complemented
4	#16813692	C₂	1	2	yes	maximal, complemented
5	M₁₆×D₁₁₃	C₂	1	1	yes	maximal, characteristic
6	#14268657	—	2	1	no	—
7	#14268657	C₂²	1	1	yes	characteristic
8	#14268657	C₂²	1	2	yes	—
9	#14268657	C₂²	1	1	yes	characteristic
10	#14268656	—	2	1	no	—
11	#14268655	—	2	1	no	—
12	C₂×#11744753	—	2	1	no	—
13	C₂×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
14	#14268631	C₄	1	1	yes	characteristic
15	#14268630	C₄	1	2	yes	—
16	C₈×D₁₁₃	C₂²	1	2	yes	—
17	M₁₆×C₁₁₃	C₄	1	1	yes	characteristic
18	C₂²×D₁₁₃	Q₈	1	1	yes	characteristic
19	C₂×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₁₁₃	D₄	1	1	yes	characteristic
21	C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
22	C₄×D₁₁₃	D₄	1	1	yes	characteristic
23	C₄×D₁₁₃	Q₈	1	1	yes	characteristic
24	C₄×D₁₁₃	C₂³	1	1	yes	characteristic
25	#11744753	—	2	2	no	—
26	#11744753	—	2	1	no	—
27	#11744753	—	2	1	no	—
28	C₈×C₁₁₃	C₂×C₄	1	2	yes	—
29	#11744751	C₂×C₄	1	2	yes	—
30	C₂²×C₁₁₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×D₁₁₃	—	4	1	no	—
32	C₂×D₁₁₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×D₁₁₃	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₁₁₃	C₄⋊C₄	1	1	yes	characteristic
35	C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
36	Dic₁₁₃	C₄⋊C₄	1	1	yes	characteristic
37	Dic₁₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₁₁₃	—	2	1	no	—
39	C₂×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
40	D₁₁₃	—	2	1	no	—
41	D₁₁₃	—	2	1	no	—
42	C₁₁₃	#430	1	1	yes	characteristic, complemented, Sylow
43	#430	—	113	1	no	maximal, Sylow
44	C₂×M₁₆	—	113	1	no	—
45	C₂×M₁₆	—	113	1	no	—
46	C₂×M₁₆	—	113	1	no	—
47	#108	—	113	2	no	—
48	#108	—	113	2	no	—
49	C₂²×C₄	—	113	1	no	—
50	M₁₆	—	113	1	no	—
51	M₁₆	—	113	2	no	—
52	M₁₆	—	113	1	no	—
53	M₁₆	—	113	2	no	—
54	M₁₆	—	113	1	no	—
55	M₁₆	—	226	1	no	—
56	M₁₆	—	226	1	no	—
57	M₁₆	#90673	1	1	yes	characteristic
58	C₂×C₈	—	113	2	no	—
59	C₂×C₈	—	226	1	no	—
60	C₂×C₈	—	226	1	no	—
61	C₂³	—	113	1	no	—
62	C₂×C₄	—	113	1	no	—
63	C₂×C₄	—	113	1	no	—
64	C₂×C₄	—	113	1	no	—
65	C₂×C₄	—	113	1	no	—
66	C₂×C₄	—	113	1	no	—
67	C₂×C₄	C₂×#90673	1	1	yes	characteristic
68	C₈	—	113	2	no	—
69	C₈	—	226	2	no	—
70	C₈	—	226	1	no	—
71	C₈	—	226	1	no	—
72	C₈	C₂×#90673	1	2	yes	—
73	C₂²	—	113	1	no	—
74	C₂²	—	113	1	no	—
75	C₂²	—	452	1	no	—
76	C₂²	C₄⋊#90673	1	1	yes	characteristic
77	C₄	—	113	1	no	—
78	C₄	—	113	1	no	—
79	C₄	C₄⋊#90673	1	1	yes	characteristic
80	C₄	C₂²×#90673	1	1	yes	characteristic
81	C₂	—	2	1	no	—
82	C₂	—	226	1	no	—
83	C₂	—	226	1	no	—
84	C₂	C₂×C₄⋊#90673	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₄⋊#90673
commutator	C₄×C₁₁₃	C₂²×C₄
Frattini	C₄	C₂²×#90673
Fitting	M₁₆×C₁₁₃	C₄
socle	C₂×C₁₁₃	C₂×C₄⋊C₄
hypercenter	M₁₆	#90673
hypocenter	C₁₁₃	#430
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
3. #16813693	C₂	semidirect
4. #16813692	C₂	semidirect
42. C₁₁₃	#430	semidirect