Subgroups — #20298509

Order 7232.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14268624	C₂	1	3	yes	maximal
2	#16813529	C₂	1	3	yes	maximal, complemented
3	C₂²×Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₁₁₃	C₂²	1	1	yes	characteristic
5	C₂²×Dic₁₁₃	C₂²	1	3	yes	—
6	#14268624	—	2	6	no	—
7	C₂×Q₈×C₁₁₃	—	2	6	no	—
8	C₂²×C₄×C₁₁₃	C₂²	1	3	yes	—
9	C₂³×C₁₁₃	C₂³	1	1	yes	characteristic
10	C₂²×D₁₁₃	—	2	3	no	—
11	C₂²×D₁₁₃	—	2	3	no	—
12	C₂²×D₁₁₃	Q₈	1	2	yes	complemented
13	Q₈×C₁₁₃	—	2	8	no	—
14	C₂×C₄×C₁₁₃	—	2	6	no	—
15	C₂×C₄×C₁₁₃	D₄	1	6	yes	—
16	C₂×Dic₁₁₃	—	2	3	no	—
17	C₂×Dic₁₁₃	—	2	6	no	—
18	C₂²×C₁₁₃	C₂×D₄	1	3	yes	—
19	C₂²×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
20	C₂²×C₁₁₃	#41	1	3	yes	—
21	C₂×D₁₁₃	—	2	6	no	—
22	C₂×D₁₁₃	—	4	1	no	—
23	C₂×D₁₁₃	—	4	3	no	—
24	C₄×C₁₁₃	—	2	6	no	—
25	Dic₁₁₃	—	4	3	no	—
26	C₂×C₁₁₃	#120	1	1	yes	characteristic
27	C₂×C₁₁₃	C₂²⋊Q₈	1	3	yes	—
28	C₂×C₁₁₃	#124	1	3	yes	—
29	D₁₁₃	—	4	2	no	—
30	C₁₁₃	#393	1	1	yes	characteristic, complemented, Sylow
31	#393	—	113	1	no	maximal, Sylow
32	C₂²×Q₈	D₁₁₃	1	1	yes	characteristic, complemented
33	C₂×C₂²⋊C₄	—	113	3	no	—
34	#95	—	113	3	no	—
35	C₂⁴	—	113	1	no	—
36	C₂×Q₈	—	2	6	no	—
37	C₂²×C₄	—	113	3	no	—
38	C₂²×C₄	C₂×D₁₁₃	1	3	yes	—
39	C₂²⋊C₄	—	226	6	no	—
40	C₂³	—	113	2	no	—
41	C₂³	—	226	3	no	—
42	C₂³	—	226	3	no	—
43	C₂³	C₂²×D₁₁₃	1	1	yes	characteristic
44	Q₈	—	2	8	no	—
45	C₂×C₄	—	2	6	no	—
46	C₂×C₄	—	226	3	no	—
47	C₂×C₄	—	226	6	no	—
48	C₂×C₄	C₂²⋊D₁₁₃	1	6	yes	—
49	C₂²	—	452	1	no	—
50	C₂²	—	452	3	no	—
51	C₂²	—	226	6	no	—
52	C₂²	C₂×C₂²⋊D₁₁₃	1	3	yes	—
53	C₂²	Q₈×D₁₁₃	1	1	yes	characteristic
54	C₂²	#14268648	1	3	yes	—
55	C₄	—	2	6	no	—
56	C₄	—	452	3	no	—
57	C₂	—	452	2	no	—
58	C₂	#16813591	1	1	yes	characteristic
59	C₂	#16813598	1	3	yes	—
60	C₂	#16813611	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₁₃
commutator, socle	C₂³×C₁₁₃	C₂³
Fitting	C₂²×Q₈×C₁₁₃	C₂
hypercenter	C₂²×Q₈	D₁₁₃
hypocenter	C₁₁₃	#393
LCS	G ▷ C₂³×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
2. #16813529	C₂	semidirect
3. C₂²×Q₈×C₁₁₃	C₂	semidirect
12. C₂²×D₁₁₃	Q₈	semidirect
30. C₁₁₃	#393	semidirect
32. C₂²×Q₈	D₁₁₃	semidirect