Subgroups — #42119378

Order 27584.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28352163	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₄₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#28352121	C₂	1	2	yes	maximal, complemented
4	#28352111	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₄₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₇₂₄	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₄₃₁	—	2	1	no	—
8	C₂²×Dic₄₃₁	—	2	1	no	—
9	C₄⋊Dic₄₃₁	—	2	1	no	—
10	C₄⋊Dic₄₃₁	C₂²	1	2	yes	—
11	C₄⋊Dic₄₃₁	C₂²	1	1	yes	characteristic
12	(C₂×D₄₃₁)⋊C₄	—	2	1	no	—
13	C₂²×C₄×C₄₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₄₃₁	C₂²	1	2	yes	—
15	C₂²×D₄₃₁	—	2	1	no	—
16	D₁₇₂₄	—	2	2	no	—
17	C₂²⋊D₄₃₁	—	4	2	no	—
18	C₂×Dic₄₃₁	—	2	1	no	—
19	C₂×Dic₄₃₁	—	2	2	no	—
20	C₂×Dic₄₃₁	—	4	2	no	—
21	C₂³×C₄₃₁	D₄	1	1	yes	characteristic
22	C₂×C₄×C₄₃₁	—	2	2	no	—
23	C₂×C₄×C₄₃₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₄₃₁	C₂³	1	1	yes	characteristic
25	C₈×C₄₃₁	—	2	2	no	—
26	C₂²×C₄₃₁	—	2	1	no	—
27	C₂²×C₄₃₁	—	2	1	no	—
28	C₂²×C₄₃₁	SD₁₆	1	2	yes	—
29	C₂²×C₄₃₁	C₂×D₄	1	1	yes	characteristic
30	C₂×D₄₃₁	—	4	1	no	—
31	C₂×D₄₃₁	—	4	1	no	—
32	C₂×D₄₃₁	—	4	1	no	—
33	C₄×C₄₃₁	—	2	1	no	—
34	C₄×C₄₃₁	#41	1	2	yes	—
35	Dic₄₃₁	—	4	1	no	—
36	Dic₄₃₁	—	4	2	no	—
37	C₂×C₄₃₁	—	2	2	no	—
38	C₂×C₄₃₁	#123	1	1	yes	characteristic
39	C₂×C₄₃₁	C₂×SD₁₆	1	1	yes	characteristic
40	C₂×C₄₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
41	D₄₃₁	—	8	1	no	—
42	C₄₃₁	#481	1	1	yes	characteristic, complemented, Sylow
43	#481	—	431	1	no	maximal, Sylow
44	#121	—	431	1	no	—
45	C₂×C₄⋊C₄	—	431	1	no	—
46	#106	—	431	2	no	—
47	#102	—	431	2	no	—
48	C₂²⋊C₈	D₄₃₁	1	1	yes	characteristic, complemented
49	C₂×D₄	—	862	1	no	—
50	C₂×D₄	—	431	1	no	—
51	C₂²×C₄	—	862	1	no	—
52	C₂²×C₄	C₂×D₄₃₁	1	1	yes	characteristic
53	C₂×C₈	C₂×D₄₃₁	1	2	yes	—
54	C₄⋊C₄	—	862	1	no	—
55	C₄⋊C₄	—	431	2	no	—
56	C₄⋊C₄	—	431	1	no	—
57	C₂²⋊C₄	—	862	1	no	—
58	C₂³	—	862	1	no	—
59	C₂³	D₁₇₂₄	1	1	yes	characteristic
60	D₄	—	862	2	no	—
61	D₄	—	1724	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	862	1	no	—
64	C₂×C₄	—	862	2	no	—
65	C₂×C₄	—	1724	2	no	—
66	C₂×C₄	D₁₇₂₄	1	1	yes	characteristic
67	C₂×C₄	C₂²×D₄₃₁	1	1	yes	characteristic
68	C₈	—	2	2	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	1724	1	no	—
72	C₂²	—	1724	1	no	—
73	C₂²	—	1724	1	no	—
74	C₂²	#20293789	1	2	yes	—
75	C₂²	C₂×D₁₇₂₄	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	1724	1	no	—
78	C₄	—	1724	2	no	—
79	C₄	#20293799	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	—	3448	1	no	—
82	C₂	#28352173	1	1	yes	characteristic
83	C₂	C₂×#20293789	1	1	yes	characteristic
84	C₂	#28352217	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₇₂₄
commutator, cosocle	C₂×C₄×C₄₃₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₃₁
Fitting	C₂²⋊C₈×C₄₃₁	C₂
socle	C₂²×C₄₃₁	C₂×D₄
prime radical	#28352163	C₂
hypercenter	C₂²⋊C₈	D₄₃₁
hypocenter	C₄₃₁	#481
LCS	G ▷ C₂×C₄×C₄₃₁ ▷ C₂×C₄₃₁ ▷ C₄₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊Dic₄₃₁	C₂	semidirect
3. #28352121	C₂	semidirect
4. #28352111	C₂	semidirect
5. C₂²⋊C₈×C₄₃₁	C₂	semidirect
42. C₄₃₁	#481	semidirect
48. C₂²⋊C₈	D₄₃₁	semidirect

Maximal subgroups

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

subgroup	index
#28352163	2
C₂×C₄⋊Dic₄₃₁	2
#28352121	2
#28352111	2
C₂²⋊C₈×C₄₃₁	2
#481	431