Subgroups — #20299209

Order 7232.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813617	C₂	1	4	yes	maximal
2	#16813614	C₂	1	1	yes	maximal, characteristic, complemented
3	#16813612	C₂	1	4	yes	maximal, complemented
4	#16813598	C₂	1	2	yes	maximal
5	Q₈×Dic₁₁₃	C₂	1	2	yes	maximal, complemented
6	#16813579	C₂	1	1	yes	maximal, characteristic
7	C₄⋊Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
10	C₄⋊Dic₁₁₃	C₂²	1	4	yes	—
11	#14268626	C₂²	1	2	yes	—
12	#14268626	C₂²	1	4	yes	—
13	#14268626	C₂²	1	4	yes	—
14	#14268624	C₂²	1	2	yes	—
15	#14268624	C₂²	1	4	yes	—
16	C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
17	C₄×Dic₁₁₃	C₂²	1	4	yes	—
18	C₂×Q₈×C₁₁₃	C₂²	1	2	yes	—
19	C₄⋊C₄×C₁₁₃	C₂²	1	4	yes	—
20	C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
21	C₂²×D₁₁₃	C₂³	1	1	yes	characteristic
22	Q₈×C₁₁₃	—	2	4	no	—
23	C₂×C₄×C₁₁₃	C₂³	1	2	yes	—
24	C₂×C₄×C₁₁₃	C₂³	1	4	yes	—
25	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₁₃	C₂³	1	2	yes	—
27	C₂×Dic₁₁₃	C₂³	1	4	yes	—
28	C₂×Dic₁₁₃	C₂³	1	1	yes	characteristic
29	C₄×D₁₁₃	—	2	2	no	—
30	C₂²×C₁₁₃	C₂⁴	1	1	yes	characteristic
31	C₂×D₁₁₃	—	2	1	no	—
32	C₂×D₁₁₃	—	2	2	no	—
33	C₄×C₁₁₃	—	2	4	no	—
34	C₄×C₁₁₃	—	2	2	no	—
35	C₄×C₁₁₃	#41	1	2	yes	—
36	Dic₁₁₃	—	2	1	no	—
37	Dic₁₁₃	—	2	2	no	—
38	Dic₁₁₃	—	2	4	no	—
39	C₂×C₁₁₃	C₂×#41	1	1	yes	characteristic
40	C₂×C₁₁₃	2₋¹⁺⁴	1	2	yes	—
41	D₁₁₃	—	4	1	no	—
42	C₁₁₃	#541	1	1	yes	characteristic, complemented, Sylow
43	#541	—	113	1	no	maximal, Sylow
44	C₄⋊Q₈	D₁₁₃	1	1	yes	characteristic, complemented
45	#126	—	113	4	no	—
46	#125	—	113	4	no	—
47	#125	—	113	1	no	—
48	C₂²⋊Q₈	—	113	2	no	—
49	C₄×Q₈	—	113	2	no	—
50	#117	—	113	1	no	—
51	C₂×Q₈	C₂×D₁₁₃	1	2	yes	—
52	C₂²×C₄	—	113	1	no	—
53	C₄⋊C₄	—	113	2	no	—
54	C₄⋊C₄	—	113	2	no	—
55	C₄⋊C₄	—	113	4	no	—
56	C₄⋊C₄	—	113	4	no	—
57	C₄⋊C₄	—	113	4	no	—
58	C₄⋊C₄	C₂×D₁₁₃	1	4	yes	—
59	C₂²⋊C₄	—	113	2	no	—
60	C₂²⋊C₄	—	113	4	no	—
61	C₄²	—	113	1	no	—
62	C₄²	—	113	4	no	—
63	C₄²	C₂×D₁₁₃	1	1	yes	characteristic
64	C₂³	—	113	1	no	—
65	Q₈	—	2	4	no	—
66	C₂×C₄	—	113	2	no	—
67	C₂×C₄	—	113	4	no	—
68	C₂×C₄	—	113	1	no	—
69	C₂×C₄	—	226	2	no	—
70	C₂×C₄	C₂²×D₁₁₃	1	4	yes	—
71	C₂×C₄	C₂²×D₁₁₃	1	2	yes	—
72	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
73	C₂²	—	226	1	no	—
74	C₂²	—	226	2	no	—
75	C₂²	C₂³×D₁₁₃	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	2	2	no	—
78	C₄	—	226	1	no	—
79	C₄	—	226	2	no	—
80	C₄	—	226	4	no	—
81	C₄	#14268646	1	2	yes	—
82	C₂	—	452	1	no	—
83	C₂	C₂×#14268646	1	1	yes	characteristic
84	C₂	#16813670	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₁₃
commutator, socle	C₂²×C₁₁₃	C₂⁴
Fitting	C₄⋊Q₈×C₁₁₃	C₂
hypercenter	C₄⋊Q₈	D₁₁₃
hypocenter	C₁₁₃	#541
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
2. #16813614	C₂	semidirect
3. #16813612	C₂	semidirect
5. Q₈×Dic₁₁₃	C₂	semidirect
7. C₄⋊Q₈×C₁₁₃	C₂	semidirect
42. C₁₁₃	#541	semidirect
44. C₄⋊Q₈	D₁₁₃	semidirect