Subgroups — #20299270

Order 7232.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14268648	C₂	1	2	yes	maximal, complemented
2	#16813619	C₂	1	3	yes	maximal, complemented
3	#16813597	C₂	1	6	yes	maximal, complemented
4	C₄×D₄₅₂	C₂	1	3	yes	maximal, complemented
5	C₄×Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#14268648	—	2	8	no	—
7	C₂×D₄₅₂	C₂²	1	3	yes	—
8	C₂×D₄₅₂	C₂²	1	6	yes	—
9	C₂×D₄₅₂	C₂²	1	6	yes	—
10	C₂×C₄×D₁₁₃	C₂²	1	6	yes	—
11	C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
12	#14268624	C₂²	1	6	yes	—
13	C₂×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₁₁₃	C₂²	1	3	yes	—
15	C₄²×C₁₁₃	C₂²	1	3	yes	—
16	C₂²×D₁₁₃	C₂³	1	6	yes	—
17	Q₈×C₁₁₃	D₄	1	4	yes	complemented
18	C₂×C₄×C₁₁₃	C₂³	1	3	yes	—
19	C₂×C₄×C₁₁₃	C₂³	1	3	yes	—
20	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
21	C₂×Dic₁₁₃	C₂³	1	2	yes	—
22	D₄₅₂	—	2	12	no	—
23	D₄₅₂	—	2	6	no	—
24	D₄₅₂	—	4	6	no	—
25	C₄×D₁₁₃	—	2	12	no	—
26	C₂²×C₁₁₃	C₂⁴	1	1	yes	characteristic
27	C₂×D₁₁₃	—	2	6	no	—
28	C₂×D₁₁₃	—	2	6	no	—
29	C₂×D₁₁₃	—	2	6	no	—
30	C₄×C₁₁₃	—	2	3	no	—
31	C₄×C₁₁₃	C₂×D₄	1	6	yes	—
32	C₄×C₁₁₃	#41	1	2	yes	complemented
33	Dic₁₁₃	—	2	2	no	—
34	C₂×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
35	C₂×C₁₁₃	C₂×#41	1	1	yes	characteristic
36	C₂×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
37	D₁₁₃	—	4	6	no	—
38	C₁₁₃	#550	1	1	yes	characteristic, complemented, Sylow
39	#550	—	113	1	no	maximal, Sylow
40	C₂×#41	—	113	2	no	—
41	#127	—	113	3	no	—
42	#121	—	113	6	no	—
43	C₄×Q₈	D₁₁₃	1	1	yes	characteristic, complemented
44	C₄×D₄	—	113	3	no	—
45	#41	—	226	8	no	—
46	C₂×Q₈	C₂×D₁₁₃	1	1	yes	characteristic
47	C₂×D₄	—	113	6	no	—
48	C₂×D₄	—	113	3	no	—
49	C₂×D₄	—	113	6	no	—
50	C₂²×C₄	—	113	6	no	—
51	C₄⋊C₄	—	113	1	no	—
52	C₄⋊C₄	C₂×D₁₁₃	1	3	yes	—
53	C₂²⋊C₄	—	113	6	no	—
54	C₄²	C₂×D₁₁₃	1	3	yes	—
55	C₂³	—	113	6	no	—
56	Q₈	D₄₅₂	1	4	yes	complemented
57	D₄	—	452	6	no	—
58	D₄	—	226	12	no	—
59	D₄	—	226	6	no	—
60	C₂×C₄	—	113	2	no	—
61	C₂×C₄	—	226	12	no	—
62	C₂×C₄	C₂²×D₁₁₃	1	3	yes	—
63	C₂×C₄	C₂²×D₁₁₃	1	3	yes	—
64	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
65	C₂²	—	226	6	no	—
66	C₂²	—	226	6	no	—
67	C₂²	—	226	6	no	—
68	C₂²	C₂³×D₁₁₃	1	1	yes	characteristic
69	C₄	—	2	3	no	—
70	C₄	—	226	2	no	—
71	C₄	C₂×D₄₅₂	1	6	yes	—
72	C₄	#14268648	1	2	yes	complemented
73	C₂	—	452	6	no	—
74	C₂	C₂²×D₄₅₂	1	1	yes	characteristic
75	C₂	C₂×#14268648	1	1	yes	characteristic
76	C₂	#16813668	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₁₃
commutator, socle	C₂²×C₁₁₃	C₂⁴
Fitting	C₄×Q₈×C₁₁₃	C₂
hypercenter	C₄×Q₈	D₁₁₃
hypocenter	C₁₁₃	#550
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#14268648	C₂	semidirect
2. #16813619	C₂	semidirect
3. #16813597	C₂	semidirect
4. C₄×D₄₅₂	C₂	semidirect
5. C₄×Q₈×C₁₁₃	C₂	semidirect
17. Q₈×C₁₁₃	D₄	semidirect
32. C₄×C₁₁₃	#41	semidirect
38. C₁₁₃	#550	semidirect
43. C₄×Q₈	D₁₁₃	semidirect
56. Q₈	D₄₅₂	semidirect
72. C₄	#14268648	semidirect