Subgroups — #20298939

Order 7232.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813601	C₂	1	1	yes	maximal, characteristic
2	#16813555	C₂	1	1	yes	maximal, characteristic, complemented
3	#16813553	C₂	1	1	yes	maximal, characteristic, complemented
4	#16813545	C₂	1	1	yes	maximal, characteristic, complemented
5	#16813545	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²⋊#11744751	C₂	1	1	yes	maximal, characteristic
7	#117×C₁₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₂₆	C₂²	1	1	yes	characteristic
9	C₂×#11744751	C₂²	1	1	yes	characteristic
10	C₂×#11744751	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
12	#14268626	—	2	1	no	—
13	C₂²⋊Dic₁₁₃	—	2	1	no	—
14	C₂²×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
17	C₂²⋊C₄×C₁₁₃	—	2	1	no	—
18	C₄²×C₁₁₃	—	2	1	no	—
19	C₂³×C₁₁₃	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₁₃	—	2	1	no	—
21	C₂×C₄×C₁₁₃	—	2	1	no	—
22	C₂×C₄×C₁₁₃	—	2	1	no	—
23	C₂×C₄×C₁₁₃	—	2	1	no	—
24	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
25	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₁₃	—	2	1	no	—
27	C₂×Dic₁₁₃	—	2	1	no	—
28	Dic₂₂₆	—	2	1	no	—
29	Dic₂₂₆	—	2	1	no	—
30	#11744751	—	2	1	no	—
31	#11744751	—	2	1	no	—
32	C₂²×C₁₁₃	—	2	1	no	—
33	C₂²×C₁₁₃	—	2	1	no	—
34	C₂²×C₁₁₃	—	2	1	no	—
35	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
36	C₄×C₁₁₃	—	2	1	no	—
37	C₄×C₁₁₃	—	4	1	no	—
38	C₄×C₁₁₃	—	4	1	no	—
39	C₄×C₁₁₃	#41	1	1	yes	characteristic
40	C₄×C₁₁₃	#41	1	1	yes	characteristic
41	Dic₁₁₃	—	4	1	no	—
42	Dic₁₁₃	—	4	1	no	—
43	C₂×C₁₁₃	—	4	1	no	—
44	C₂×C₁₁₃	#123	1	1	yes	characteristic
45	C₂×C₁₁₃	#135	1	1	yes	characteristic
46	C₂×C₁₁₃	#137	1	1	yes	characteristic
47	C₁₁₃	#485	1	1	yes	characteristic, complemented, Sylow
48	#485	—	113	1	no	maximal, Sylow
49	C₂²⋊Q₈	—	113	1	no	—
50	#117	D₁₁₃	1	1	yes	characteristic
51	#107	—	113	1	no	—
52	#106	—	113	1	no	—
53	#103	—	113	1	no	—
54	#103	—	113	1	no	—
55	C₂²⋊C₈	—	113	1	no	—
56	C₂×Q₈	—	113	1	no	—
57	C₂²×C₄	C₂×D₁₁₃	1	1	yes	characteristic
58	C₂×C₈	—	113	1	no	—
59	C₂×C₈	—	113	1	no	—
60	C₄⋊C₄	—	113	1	no	—
61	C₄⋊C₄	—	226	1	no	—
62	C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
63	C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
64	C₂²⋊C₄	—	2	1	no	—
65	C₂²⋊C₄	—	226	1	no	—
66	C₄²	—	2	1	no	—
67	C₂³	C₂²⋊D₁₁₃	1	1	yes	characteristic
68	Q₈	—	226	1	no	—
69	Q₈	—	226	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	226	1	no	—
75	C₂×C₄	—	226	1	no	—
76	C₂×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
77	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
78	C₈	—	226	1	no	—
79	C₈	—	226	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	4	1	no	—
87	C₄	—	452	1	no	—
88	C₄	—	452	1	no	—
89	C₄	#14268647	1	1	yes	characteristic
90	C₄	#14268647	1	1	yes	characteristic
91	C₂	—	4	1	no	—
92	C₂	#16813605	1	1	yes	characteristic
93	C₂	#16813642	1	1	yes	characteristic
94	C₂	#16813652	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₁₃
commutator	C₂×C₄×C₁₁₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₁₃
Fitting	#117×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×D₄
hypercenter	#117	D₁₁₃
hypocenter	C₁₁₃	#485
LCS	G ▷ C₂×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
2. #16813555	C₂	semidirect
3. #16813553	C₂	semidirect
4. #16813545	C₂	semidirect
5. #16813545	C₂	semidirect
47. C₁₁₃	#485	semidirect