Subgroups — #20298885

Order 7232.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₄₅₂	C₂	1	1	yes	maximal, characteristic
2	#16813553	C₂	1	2	yes	maximal, complemented
3	C₄⋊#11744751	C₂	1	1	yes	maximal, characteristic, complemented
4	#16813542	C₂	1	2	yes	maximal
5	C₄⋊Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₅₂	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₁₁₃	—	2	1	no	—
8	C₂×#11744751	C₂²	1	2	yes	—
9	C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
10	#14268624	—	2	1	no	—
11	C₂×Q₈×C₁₁₃	—	2	1	no	—
12	C₄⋊C₄×C₁₁₃	—	2	1	no	—
13	C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
14	C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
15	C₂²×D₁₁₃	—	2	1	no	—
16	Q₈×C₁₁₃	—	4	2	no	—
17	C₂×C₄×C₁₁₃	—	2	2	no	—
18	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
21	C₂×Dic₁₁₃	—	2	1	no	—
22	D₄₅₂	—	2	1	no	—
23	D₄₅₂	Q₈	1	2	yes	complemented
24	C₄×D₁₁₃	—	4	2	no	—
25	#11744751	—	2	2	no	—
26	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
27	C₂×D₁₁₃	—	2	2	no	—
28	C₂×D₁₁₃	—	4	1	no	—
29	C₂×D₁₁₃	—	4	1	no	—
30	C₄×C₁₁₃	—	2	1	no	—
31	C₄×C₁₁₃	—	4	2	no	—
32	C₄×C₁₁₃	SD₁₆	1	2	yes	—
33	C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₁₁₃	#41	1	1	yes	characteristic
35	Dic₁₁₃	—	4	1	no	—
36	C₂×C₁₁₃	C₂²⋊Q₈	1	1	yes	characteristic
37	C₂×C₁₁₃	C₂×SD₁₆	1	1	yes	characteristic
38	C₂×C₁₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
39	D₁₁₃	—	4	2	no	—
40	C₁₁₃	#476	1	1	yes	characteristic, complemented, Sylow
41	#476	—	113	1	no	maximal, Sylow
42	C₄⋊Q₈	D₁₁₃	1	1	yes	characteristic, complemented
43	C₄×D₄	—	113	1	no	—
44	#106	—	113	2	no	—
45	C₄⋊C₈	—	113	1	no	—
46	#102	—	113	2	no	—
47	C₂×Q₈	—	2	1	no	—
48	C₂×D₄	—	113	1	no	—
49	C₂²×C₄	—	226	1	no	—
50	C₂×C₈	—	113	2	no	—
51	C₄⋊C₄	—	2	1	no	—
52	C₄⋊C₄	—	113	1	no	—
53	C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
54	C₂²⋊C₄	—	226	1	no	—
55	C₄²	C₂×D₁₁₃	1	1	yes	characteristic
56	C₂³	—	226	1	no	—
57	Q₈	—	4	2	no	—
58	D₄	—	113	2	no	—
59	D₄	—	226	1	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	452	2	no	—
62	C₂×C₄	—	226	1	no	—
63	C₂×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
66	C₈	—	226	2	no	—
67	C₂²	—	452	1	no	—
68	C₂²	—	452	1	no	—
69	C₂²	—	226	2	no	—
70	C₂²	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	2	no	—
73	C₄	—	452	1	no	—
74	C₄	#14268634	1	2	yes	—
75	C₄	Q₈×D₁₁₃	1	1	yes	characteristic
76	C₄	#14268648	1	1	yes	characteristic
77	C₂	—	452	2	no	—
78	C₂	#16813598	1	1	yes	characteristic
79	C₂	C₂×#14268634	1	1	yes	characteristic
80	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	C₄⋊Q₈×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×D₄
hypercenter	C₄⋊Q₈	D₁₁₃
hypocenter	C₁₁₃	#476
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #16813553	C₂	semidirect
3. C₄⋊#11744751	C₂	semidirect
5. C₄⋊Q₈×C₁₁₃	C₂	semidirect
23. D₄₅₂	Q₈	semidirect
40. C₁₁₃	#476	semidirect
42. C₄⋊Q₈	D₁₁₃	semidirect