Subgroups — #42119391

Order 27584.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28352169	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×D₄₃₁	C₂	1	1	yes	maximal, characteristic
3	#28352120	C₂	1	1	yes	maximal, characteristic, complemented
4	#28352114	C₂	1	1	yes	maximal, characteristic, complemented
5	#28352112	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₄₃₁)⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#106×C₄₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₈₆₂	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄₃₁	—	2	1	no	—
10	C₂×C₄×D₄₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄₃₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₄₃₁	—	2	1	no	—
13	C₄⋊Dic₄₃₁	C₂²	1	1	yes	characteristic
14	Dic₄₃₁⋊C₄	—	2	1	no	—
15	(C₂×D₄₃₁)⋊C₄	—	2	1	no	—
16	C₂×C₈×C₄₃₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₄₃₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₄₃₁	C₂²	1	1	yes	characteristic
19	C₂²×D₄₃₁	D₄	1	1	yes	characteristic
20	Dic₈₆₂	—	2	1	no	—
21	Dic₈₆₂	—	2	1	no	—
22	C₂×Dic₄₃₁	—	2	1	no	—
23	C₂×Dic₄₃₁	—	2	1	no	—
24	C₂×Dic₄₃₁	D₄	1	1	yes	characteristic
25	C₄×D₄₃₁	—	2	1	no	—
26	C₄×D₄₃₁	—	2	1	no	—
27	C₄×D₄₃₁	—	4	2	no	—
28	C₄₃₁⋊C₈	—	2	1	no	—
29	C₂×C₄×C₄₃₁	—	2	1	no	—
30	C₂×C₄×C₄₃₁	—	2	1	no	—
31	C₂×C₄×C₄₃₁	C₂³	1	1	yes	characteristic
32	C₈×C₄₃₁	—	2	1	no	—
33	C₂²×C₄₃₁	C₂×D₄	1	1	yes	characteristic
34	C₂×D₄₃₁	—	2	1	no	—
35	C₂×D₄₃₁	—	2	1	no	—
36	C₂×D₄₃₁	SD₁₆	1	2	yes	—
37	C₄×C₄₃₁	—	4	1	no	—
38	C₄×C₄₃₁	—	4	1	no	—
39	C₄×C₄₃₁	#41	1	1	yes	characteristic
40	C₄×C₄₃₁	#41	1	1	yes	characteristic
41	Dic₄₃₁	—	2	1	no	—
42	Dic₄₃₁	—	4	1	no	—
43	Dic₄₃₁	—	4	1	no	—
44	C₂×C₄₃₁	#123	1	1	yes	characteristic
45	C₂×C₄₃₁	C₂×SD₁₆	1	1	yes	characteristic
46	C₂×C₄₃₁	#137	1	1	yes	characteristic
47	D₄₃₁	—	2	2	no	—
48	C₄₃₁	#483	1	1	yes	characteristic, complemented, Sylow
49	#483	—	431	1	no	maximal, Sylow
50	C₂²⋊Q₈	—	431	1	no	—
51	C₂×C₄⋊C₄	—	431	1	no	—
52	#106	—	431	1	no	—
53	#106	D₄₃₁	1	1	yes	characteristic, complemented
54	#103	—	431	1	no	—
55	#103	—	431	1	no	—
56	C₂²⋊C₈	—	431	1	no	—
57	C₂×Q₈	—	431	1	no	—
58	C₂²×C₄	—	862	1	no	—
59	C₂²×C₄	—	431	1	no	—
60	C₂×C₈	—	431	1	no	—
61	C₂×C₈	C₂×D₄₃₁	1	1	yes	characteristic
62	C₄⋊C₄	—	862	1	no	—
63	C₄⋊C₄	—	862	1	no	—
64	C₄⋊C₄	—	431	1	no	—
65	C₄⋊C₄	C₂×D₄₃₁	1	1	yes	characteristic
66	C₄⋊C₄	C₂×D₄₃₁	1	1	yes	characteristic
67	C₂²⋊C₄	—	862	1	no	—
68	C₂³	—	431	1	no	—
69	Q₈	—	862	1	no	—
70	Q₈	—	862	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	862	1	no	—
74	C₂×C₄	—	862	1	no	—
75	C₂×C₄	—	862	1	no	—
76	C₂×C₄	—	862	1	no	—
77	C₂×C₄	—	431	1	no	—
78	C₂×C₄	—	1724	2	no	—
79	C₂×C₄	C₂²×D₄₃₁	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	862	1	no	—
82	C₂²	—	862	1	no	—
83	C₂²	—	862	1	no	—
84	C₂²	—	431	2	no	—
85	C₂²	D₄×D₄₃₁	1	1	yes	characteristic
86	C₄	—	4	1	no	—
87	C₄	—	4	1	no	—
88	C₄	—	862	1	no	—
89	C₄	—	1724	1	no	—
90	C₄	—	1724	1	no	—
91	C₄	#20293800	1	1	yes	characteristic
92	C₄	#20293801	1	1	yes	characteristic
93	C₂	—	862	2	no	—
94	C₂	#28352174	1	1	yes	characteristic
95	C₂	SD₁₆×D₄₃₁	1	1	yes	characteristic
96	C₂	#28352218	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₄₃₁
commutator, cosocle	C₂×C₄×C₄₃₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₃₁
Fitting	#106×C₄₃₁	C₂
socle	C₂²×C₄₃₁	C₂×D₄
prime radical	C₂²×D₄₃₁	D₄
hypercenter	#106	D₄₃₁
hypocenter	C₄₃₁	#483
LCS	G ▷ C₂×C₄×C₄₃₁ ▷ C₂×C₄₃₁ ▷ C₄₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
3. #28352120	C₂	semidirect
4. #28352114	C₂	semidirect
5. #28352112	C₂	semidirect
7. #106×C₄₃₁	C₂	semidirect
48. C₄₃₁	#483	semidirect
53. #106	D₄₃₁	semidirect

Maximal subgroups

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

subgroup	index
#28352169	2
C₄⋊C₄×D₄₃₁	2
#28352120	2
#28352114	2
#28352112	2
(C₂×D₄₃₁)⋊C₈	2
#106×C₄₃₁	2
#483	431