Subgroups — #42119499

Order 27584.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₇₂₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#28352135	C₂	1	2	yes	maximal, complemented
3	#28352133	C₂	1	2	yes	maximal, complemented
4	#28352129	C₂	1	1	yes	maximal, characteristic, complemented
5	#135×C₄₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₈₆₂	—	2	1	no	—
7	Dic₁₇₂₄	—	2	1	no	—
8	Dic₁₇₂₄	C₂²	1	2	yes	—
9	C₄₃₁⋊C₁₆	C₂²	1	2	yes	—
10	#41×C₄₃₁	—	2	1	no	—
11	Q₁₆×C₄₃₁	C₂²	1	1	yes	characteristic
12	SD₁₆×C₄₃₁	—	2	1	no	—
13	D₈×C₄₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₄₃₁	C₂²	1	1	yes	characteristic
15	Dic₈₆₂	—	2	2	no	—
16	Dic₈₆₂	—	4	1	no	—
17	C₂×Dic₄₃₁	—	4	1	no	—
18	Q₈×C₄₃₁	—	2	1	no	—
19	D₄×C₄₃₁	—	2	1	no	—
20	D₄×C₄₃₁	—	4	1	no	—
21	C₂×C₄×C₄₃₁	—	4	1	no	—
22	C₂×C₄×C₄₃₁	D₄	1	1	yes	characteristic
23	C₈×C₄₃₁	D₄	1	1	yes	characteristic
24	C₈×C₄₃₁	C₂³	1	1	yes	characteristic
25	C₂²×C₄₃₁	—	4	1	no	—
26	C₂²×C₄₃₁	D₈	1	1	yes	characteristic
27	C₄×C₄₃₁	—	4	1	no	—
28	C₄×C₄₃₁	D₈	1	1	yes	characteristic
29	C₄×C₄₃₁	C₂×D₄	1	1	yes	characteristic
30	Dic₄₃₁	—	4	2	no	—
31	C₂×C₄₃₁	—	2	1	no	—
32	C₂×C₄₃₁	—	8	1	no	—
33	C₂×C₄₃₁	C₂×D₈	1	1	yes	characteristic
34	C₄₃₁	#510	1	1	yes	characteristic, complemented, Sylow
35	#510	—	431	1	no	maximal, Sylow
36	#135	D₄₃₁	1	1	yes	characteristic
37	C₂×Q₁₆	—	431	1	no	—
38	Q₃₂	—	431	2	no	—
39	SD₃₂	—	431	2	no	—
40	M₃₂	—	431	1	no	—
41	#41	—	2	1	no	—
42	C₂×Q₈	—	862	1	no	—
43	Q₁₆	—	862	1	no	—
44	Q₁₆	—	431	2	no	—
45	Q₁₆	C₂×D₄₃₁	1	1	yes	characteristic
46	SD₁₆	—	2	1	no	—
47	D₈	C₂×D₄₃₁	1	1	yes	characteristic
48	C₂×C₈	C₂×D₄₃₁	1	1	yes	characteristic
49	C₁₆	—	431	2	no	—
50	Q₈	—	2	1	no	—
51	Q₈	—	862	2	no	—
52	Q₈	—	1724	1	no	—
53	D₄	—	2	1	no	—
54	D₄	—	4	1	no	—
55	C₂×C₄	—	4	1	no	—
56	C₂×C₄	—	1724	1	no	—
57	C₂×C₄	C₂²⋊D₄₃₁	1	1	yes	characteristic
58	C₈	C₂²⋊D₄₃₁	1	1	yes	characteristic
59	C₈	C₂²×D₄₃₁	1	1	yes	characteristic
60	C₂²	—	4	1	no	—
61	C₂²	#20293785	1	1	yes	characteristic
62	C₄	—	4	1	no	—
63	C₄	—	1724	2	no	—
64	C₄	#20293785	1	1	yes	characteristic
65	C₄	C₂×C₂²⋊D₄₃₁	1	1	yes	characteristic
66	C₂	—	2	1	no	—
67	C₂	—	8	1	no	—
68	C₂	C₂×#20293785	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#20293785
commutator, cosocle	C₈×C₄₃₁	C₂³
Frattini	C₈	C₂²×D₄₃₁
Fitting, prime radical	#135×C₄₃₁	C₂
socle	C₂×C₄₃₁	C₂×D₈
hypercenter	#135	D₄₃₁
hypocenter	C₄₃₁	#510
LCS	G ▷ C₈×C₄₃₁ ▷ C₄×C₄₃₁ ▷ C₂×C₄₃₁ ▷ C₄₃₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₁₇₂₄	C₂	semidirect
2. #28352135	C₂	semidirect
3. #28352133	C₂	semidirect
4. #28352129	C₂	semidirect
34. C₄₃₁	#510	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×Dic₁₇₂₄	2
#28352135	2
#28352133	2
#28352129	2
#135×C₄₃₁	2
#510	431