Subgroups — #20298790

Order 7232.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14268632	C₂	1	2	yes	maximal, complemented
2	C₄×D₄₅₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊#11744751	C₂	1	1	yes	maximal, characteristic, complemented
4	#16813543	C₂	1	2	yes	maximal, complemented
5	#127×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₅₂	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₁₁₃	—	2	1	no	—
8	#14268632	—	2	4	no	—
9	C₂×#11744751	C₂²	1	2	yes	—
10	C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
11	#14268624	—	2	1	no	—
12	C₂×D₄×C₁₁₃	—	2	1	no	—
13	C₂×D₄×C₁₁₃	—	2	1	no	—
14	C₂×D₄×C₁₁₃	C₂²	1	2	yes	—
15	C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
16	C₂³×C₁₁₃	—	2	2	no	—
17	C₂²×D₁₁₃	—	2	1	no	—
18	D₄×C₁₁₃	—	2	2	no	—
19	D₄×C₁₁₃	—	2	2	no	—
20	D₄×C₁₁₃	—	4	2	no	—
21	D₄×C₁₁₃	—	4	2	no	—
22	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
25	C₂×Dic₁₁₃	—	2	1	no	—
26	D₄₅₂	—	2	1	no	—
27	D₄₅₂	D₄	1	2	yes	complemented
28	C₄×D₁₁₃	—	4	2	no	—
29	#11744751	—	2	2	no	—
30	C₂²×C₁₁₃	—	4	2	no	—
31	C₂²×C₁₁₃	—	4	2	no	—
32	C₂²×C₁₁₃	—	4	2	no	—
33	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
34	C₂×D₁₁₃	—	2	2	no	—
35	C₂×D₁₁₃	—	4	1	no	—
36	C₂×D₁₁₃	—	4	1	no	—
37	C₄×C₁₁₃	—	2	1	no	—
38	C₄×C₁₁₃	D₈	1	2	yes	complemented
39	C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
40	C₄×C₁₁₃	#41	1	1	yes	characteristic
41	Dic₁₁₃	—	4	1	no	—
42	C₂×C₁₁₃	—	8	2	no	—
43	C₂×C₁₁₃	#121	1	1	yes	characteristic
44	C₂×C₁₁₃	C₂×D₈	1	1	yes	characteristic
45	C₂×C₁₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
46	D₁₁₃	—	4	2	no	—
47	C₁₁₃	#459	1	1	yes	characteristic, complemented, Sylow
48	#459	—	113	1	no	maximal, Sylow
49	C₂×D₈	—	113	2	no	—
50	#127	D₁₁₃	1	1	yes	characteristic, complemented
51	C₄×D₄	—	113	1	no	—
52	C₄⋊C₈	—	113	1	no	—
53	#102	—	113	2	no	—
54	C₂×D₄	—	2	1	no	—
55	C₂×D₄	—	2	1	no	—
56	C₂×D₄	—	113	1	no	—
57	C₂×D₄	C₂×D₁₁₃	1	2	yes	—
58	C₂²×C₄	—	226	1	no	—
59	D₈	—	226	4	no	—
60	C₂×C₈	—	113	2	no	—
61	C₄⋊C₄	—	113	1	no	—
62	C₂²⋊C₄	—	226	1	no	—
63	C₄²	C₂×D₁₁₃	1	1	yes	characteristic
64	C₂³	—	2	2	no	—
65	C₂³	—	226	1	no	—
66	D₄	—	2	2	no	—
67	D₄	—	2	2	no	—
68	D₄	—	4	2	no	—
69	D₄	—	4	2	no	—
70	D₄	—	113	2	no	—
71	D₄	—	226	1	no	—
72	C₂×C₄	—	452	2	no	—
73	C₂×C₄	—	226	1	no	—
74	C₂×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
75	C₂×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
76	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
77	C₈	—	226	2	no	—
78	C₂²	—	4	2	no	—
79	C₂²	—	4	2	no	—
80	C₂²	—	4	2	no	—
81	C₂²	—	452	1	no	—
82	C₂²	—	452	1	no	—
83	C₂²	—	226	2	no	—
84	C₂²	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	452	1	no	—
87	C₄	#14268632	1	2	yes	complemented
88	C₄	D₄×D₁₁₃	1	1	yes	characteristic
89	C₄	#14268646	1	1	yes	characteristic
90	C₂	—	8	2	no	—
91	C₂	—	452	2	no	—
92	C₂	#16813593	1	1	yes	characteristic
93	C₂	C₂×#14268632	1	1	yes	characteristic
94	C₂	#16813648	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₁₃
commutator	C₂×C₄×C₁₁₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₁₃
Fitting	#127×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×D₄
hypercenter	#127	D₁₁₃
hypocenter	C₁₁₃	#459
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×#14268632	C₂	semidirect
2. C₄×D₄₅₂	C₂	semidirect
3. C₄⋊#11744751	C₂	semidirect
4. #16813543	C₂	semidirect
5. #127×C₁₁₃	C₂	semidirect
27. D₄₅₂	D₄	semidirect
38. C₄×C₁₁₃	D₈	semidirect
47. C₁₁₃	#459	semidirect
50. #127	D₁₁₃	semidirect
87. C₄	#14268632	semidirect