Subgroups — #42119448

Order 27584.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20293789	C₂	1	2	yes	maximal, complemented
2	C₂×D₃₄₄₈	C₂	1	2	yes	maximal, complemented
3	#28352186	C₂	1	1	yes	maximal, characteristic, complemented
4	#28352177	C₂	1	1	yes	maximal, characteristic, complemented
5	#97×C₄₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₈₆₂	C₂²	1	1	yes	characteristic
7	C₂×D₁₇₂₄	—	2	2	no	—
8	C₂×D₁₇₂₄	C₂²	1	1	yes	characteristic
9	C₂×D₁₇₂₄	C₂²	1	2	yes	—
10	#20293789	—	2	4	no	—
11	D₃₄₄₈	—	2	4	no	—
12	(C₂×D₄₃₁)⋊C₄	—	2	2	no	—
13	C₂×C₈×C₄₃₁	C₂²	1	2	yes	—
14	C₄²×C₄₃₁	C₂²	1	1	yes	characteristic
15	C₂²×D₄₃₁	—	2	1	no	—
16	C₂²×D₄₃₁	—	2	2	no	—
17	Dic₈₆₂	—	2	2	no	—
18	D₁₇₂₄	—	2	2	no	—
19	D₁₇₂₄	—	2	2	no	—
20	D₁₇₂₄	—	2	2	no	—
21	D₁₇₂₄	—	4	4	no	—
22	C₂×Dic₄₃₁	—	2	1	no	—
23	C₂×C₄×C₄₃₁	D₄	1	2	yes	—
24	C₂×C₄×C₄₃₁	C₂³	1	1	yes	characteristic
25	C₈×C₄₃₁	D₄	1	4	yes	complemented
26	C₂²×C₄₃₁	C₂×D₄	1	1	yes	characteristic
27	C₂×D₄₃₁	—	4	1	no	—
28	C₂×D₄₃₁	—	4	2	no	—
29	C₂×D₄₃₁	—	4	4	no	—
30	C₂×D₄₃₁	—	4	2	no	—
31	C₄×C₄₃₁	—	2	2	no	—
32	C₄×C₄₃₁	C₂×D₄	1	2	yes	—
33	Dic₄₃₁	—	4	1	no	—
34	C₂×C₄₃₁	#127	1	1	yes	characteristic
35	C₂×C₄₃₁	AGL(1,ℤ/8)	1	2	yes	—
36	D₄₃₁	—	8	2	no	—
37	D₄₃₁	—	8	1	no	—
38	C₄₃₁	#496	1	1	yes	characteristic, complemented, Sylow
39	#496	—	431	1	no	maximal, Sylow
40	C₂×SD₁₆	—	431	2	no	—
41	C₂×D₈	—	431	2	no	—
42	#127	—	431	1	no	—
43	#124	—	431	1	no	—
44	#97	D₄₃₁	1	1	yes	characteristic, complemented
45	C₂×Q₈	—	431	1	no	—
46	C₂×D₄	—	862	2	no	—
47	C₂×D₄	—	431	1	no	—
48	C₂×D₄	—	431	2	no	—
49	SD₁₆	—	862	4	no	—
50	D₈	—	862	4	no	—
51	C₂×C₈	C₂×D₄₃₁	1	2	yes	—
52	C₂²⋊C₄	—	862	2	no	—
53	C₄²	C₂×D₄₃₁	1	1	yes	characteristic
54	C₂³	—	862	1	no	—
55	C₂³	—	862	2	no	—
56	Q₈	—	862	2	no	—
57	D₄	—	862	2	no	—
58	D₄	—	862	2	no	—
59	D₄	—	862	2	no	—
60	D₄	—	1724	4	no	—
61	C₂×C₄	—	862	1	no	—
62	C₂×C₄	D₁₇₂₄	1	2	yes	—
63	C₂×C₄	C₂²×D₄₃₁	1	1	yes	characteristic
64	C₈	D₁₇₂₄	1	4	yes	complemented
65	C₂²	—	1724	1	no	—
66	C₂²	—	1724	2	no	—
67	C₂²	—	1724	4	no	—
68	C₂²	—	1724	2	no	—
69	C₂²	C₂×D₁₇₂₄	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	1724	1	no	—
72	C₄	C₂×D₁₇₂₄	1	2	yes	—
73	C₂	—	3448	2	no	—
74	C₂	—	3448	1	no	—
75	C₂	#28352186	1	1	yes	characteristic
76	C₂	#28352217	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₇₂₄
commutator, cosocle	C₂×C₄×C₄₃₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₃₁
Fitting	#97×C₄₃₁	C₂
socle	C₂²×C₄₃₁	C₂×D₄
hypercenter	#97	D₄₃₁
hypocenter	C₄₃₁	#496
LCS	G ▷ C₂×C₄×C₄₃₁ ▷ C₂×C₄₃₁ ▷ C₄₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
1. C₂×#20293789	C₂	semidirect
2. C₂×D₃₄₄₈	C₂	semidirect
3. #28352186	C₂	semidirect
4. #28352177	C₂	semidirect
5. #97×C₄₃₁	C₂	semidirect
25. C₈×C₄₃₁	D₄	semidirect
38. C₄₃₁	#496	semidirect
44. #97	D₄₃₁	semidirect
64. C₈	D₁₇₂₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20293789	2
C₂×D₃₄₄₈	2
#28352186	2
#28352177	2
#97×C₄₃₁	2
#496	431