Subgroups — #22553589

Order 8384.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₅₂₄	C₂	1	1	yes	maximal, characteristic
2	#16862179	C₂	1	2	yes	maximal, complemented
3	C₄⋊(C₁₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#16862166	C₂	1	2	yes	maximal
5	C₄⋊Q₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₅₂₄	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₁₃₁	—	2	1	no	—
8	C₂×C₁₃₁⋊C₈	C₂²	1	2	yes	—
9	C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
10	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
11	C₂×Q₈×C₁₃₁	—	2	1	no	—
12	C₄⋊C₄×C₁₃₁	—	2	1	no	—
13	C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
14	C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂²×D₁₃₁	—	2	1	no	—
16	D₅₂₄	—	2	1	no	—
17	D₅₂₄	Q₈	1	2	yes	complemented
18	C₂×Dic₁₃₁	—	2	1	no	—
19	C₄×D₁₃₁	—	4	2	no	—
20	C₁₃₁⋊C₈	—	2	2	no	—
21	Q₈×C₁₃₁	—	4	2	no	—
22	C₂×C₄×C₁₃₁	—	2	2	no	—
23	C₂×C₄×C₁₃₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₃₁	D₄	1	1	yes	characteristic
25	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
26	C₂²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
27	C₂×D₁₃₁	—	2	2	no	—
28	C₂×D₁₃₁	—	4	1	no	—
29	C₂×D₁₃₁	—	4	1	no	—
30	C₄×C₁₃₁	—	2	1	no	—
31	C₄×C₁₃₁	—	4	2	no	—
32	C₄×C₁₃₁	D₈	1	2	yes	—
33	C₄×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₁₃₁	#41	1	1	yes	characteristic
35	Dic₁₃₁	—	4	1	no	—
36	C₂×C₁₃₁	C₂²⋊Q₈	1	1	yes	characteristic
37	C₂×C₁₃₁	C₂×D₈	1	1	yes	characteristic
38	C₂×C₁₃₁	#137	1	1	yes	characteristic
39	D₁₃₁	—	4	2	no	—
40	C₁₃₁	#474	1	1	yes	characteristic, complemented, Sylow
41	#474	—	131	1	no	maximal, Sylow
42	C₄⋊Q₈	D₁₃₁	1	1	yes	characteristic, complemented
43	C₄×D₄	—	131	1	no	—
44	#107	—	131	2	no	—
45	C₄⋊C₈	—	131	1	no	—
46	#102	—	131	2	no	—
47	C₂×Q₈	—	2	1	no	—
48	C₂×D₄	—	131	1	no	—
49	C₂²×C₄	—	262	1	no	—
50	C₂×C₈	—	131	2	no	—
51	C₄⋊C₄	—	2	1	no	—
52	C₄⋊C₄	—	131	1	no	—
53	C₄⋊C₄	C₂×D₁₃₁	1	2	yes	—
54	C₂²⋊C₄	—	262	1	no	—
55	C₄²	C₂×D₁₃₁	1	1	yes	characteristic
56	C₂³	—	262	1	no	—
57	Q₈	—	4	2	no	—
58	D₄	—	262	1	no	—
59	D₄	—	131	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	262	1	no	—
62	C₂×C₄	—	524	2	no	—
63	C₂×C₄	C₂²⋊D₁₃₁	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊D₁₃₁	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
66	C₈	—	262	2	no	—
67	C₂²	—	262	2	no	—
68	C₂²	—	524	1	no	—
69	C₂²	—	524	1	no	—
70	C₂²	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	2	no	—
73	C₄	—	524	1	no	—
74	C₄	#14523498	1	2	yes	—
75	C₄	Q₈×D₁₃₁	1	1	yes	characteristic
76	C₄	#14523514	1	1	yes	characteristic
77	C₂	—	524	2	no	—
78	C₂	#16862222	1	1	yes	characteristic
79	C₂	C₂×#14523498	1	1	yes	characteristic
80	C₂	#16862277	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₃₁
commutator	C₂×C₄×C₁₃₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₁
Fitting	C₄⋊Q₈×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂×D₄
hypercenter	C₄⋊Q₈	D₁₃₁
hypocenter	C₁₃₁	#474
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #16862179	C₂	semidirect
3. C₄⋊(C₁₃₁⋊C₈)	C₂	semidirect
5. C₄⋊Q₈×C₁₃₁	C₂	semidirect
17. D₅₂₄	Q₈	semidirect
40. C₁₃₁	#474	semidirect
42. C₄⋊Q₈	D₁₃₁	semidirect