Subgroups — #44401610

Order 29056.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₈×Dic₂₂₇	C₂	1	2	yes	maximal, complemented
2	#28408860	C₂	1	1	yes	maximal, characteristic, complemented
3	#28408833	C₂	1	2	yes	maximal, complemented
4	#28408807	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₆×C₂₂₇	C₂	1	1	yes	maximal, characteristic
6	D₄×Dic₂₂₇	—	2	2	no	—
7	C₂×C₂₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	#20300222	C₂²	1	2	yes	—
9	#20300209	—	2	2	no	—
10	C₈×Dic₂₂₇	C₂²	1	1	yes	characteristic
11	C₂×D₈×C₂₂₇	C₂²	1	2	yes	—
12	D₁₆×C₂₂₇	C₄	1	4	yes	complemented
13	C₂×C₁₆×C₂₂₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₂₇	—	4	2	no	—
15	C₂×C₂₂₇⋊C₈	D₄	1	1	yes	characteristic
16	C₄⋊Dic₂₂₇	—	2	2	no	—
17	C₂²⋊Dic₂₂₇	—	4	2	no	—
18	C₄×Dic₂₂₇	D₄	1	1	yes	characteristic
19	C₂₂₇⋊C₁₆	—	2	1	no	—
20	C₂×D₄×C₂₂₇	—	2	2	no	—
21	D₈×C₂₂₇	—	2	2	no	—
22	D₈×C₂₂₇	C₂×C₄	1	4	yes	complemented
23	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
24	C₁₆×C₂₂₇	C₂×C₄	1	2	yes	complemented
25	C₂×Dic₂₂₇	—	4	2	no	—
26	C₂×Dic₂₂₇	—	8	4	no	—
27	C₂×Dic₂₂₇	D₈	1	2	yes	—
28	C₂₂₇⋊C₈	—	2	1	no	—
29	C₂³×C₂₂₇	—	4	2	no	—
30	D₄×C₂₂₇	—	2	4	no	—
31	D₄×C₂₂₇	—	4	2	no	—
32	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
33	C₈×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₂₂₇	#41	1	1	yes	characteristic
35	C₂²×C₂₂₇	—	4	4	no	—
36	C₂²×C₂₂₇	—	8	2	no	—
37	C₂²×C₂₂₇	—	8	2	no	—
38	C₂²×C₂₂₇	C₂×D₈	1	1	yes	characteristic
39	C₄×C₂₂₇	C₄×D₄	1	1	yes	characteristic
40	C₄×C₂₂₇	#135	1	1	yes	characteristic
41	Dic₂₂₇	—	2	2	no	—
42	Dic₂₂₇	—	8	2	no	—
43	C₂×C₂₂₇	—	8	4	no	—
44	C₂×C₂₂₇	C₄×D₈	1	1	yes	characteristic
45	C₂×C₂₂₇	#509	1	2	yes	—
46	C₂₂₇	#2177	1	1	yes	characteristic, complemented, Sylow
47	#2177	—	227	1	no	maximal, Sylow
48	C₂×D₁₆	D₂₂₇	1	1	yes	characteristic
49	C₄×D₈	—	227	2	no	—
50	#367	—	227	1	no	—
51	#357	—	227	2	no	—
52	#346	—	227	1	no	—
53	C₂×D₈	C₂×D₂₂₇	1	2	yes	—
54	C₄×D₄	—	454	2	no	—
55	D₁₆	Dic₂₂₇	1	4	yes	complemented
56	C₂×C₁₆	—	227	1	no	—
57	C₂×C₁₆	C₂×D₂₂₇	1	1	yes	characteristic
58	#107	—	227	2	no	—
59	#102	—	454	2	no	—
60	C₄×C₈	—	227	1	no	—
61	C₂×D₄	—	2	2	no	—
62	C₂²×C₄	—	908	2	no	—
63	D₈	—	2	2	no	—
64	D₈	C₂×Dic₂₂₇	1	4	yes	complemented
65	C₂×C₈	—	227	1	no	—
66	C₂×C₈	C₂²×D₂₂₇	1	1	yes	characteristic
67	C₄⋊C₄	—	454	2	no	—
68	C₂²⋊C₄	—	908	2	no	—
69	C₄²	—	227	1	no	—
70	C₁₆	—	454	1	no	—
71	C₁₆	C₂×Dic₂₂₇	1	2	yes	complemented
72	C₂³	—	4	2	no	—
73	D₄	—	2	4	no	—
74	D₄	—	4	2	no	—
75	C₂×C₄	—	1816	4	no	—
76	C₂×C₄	—	908	2	no	—
77	C₂×C₄	—	227	2	no	—
78	C₂×C₄	D₄×D₂₂₇	1	1	yes	characteristic
79	C₈	—	454	1	no	—
80	C₈	C₂²×Dic₂₂₇	1	1	yes	characteristic
81	C₈	#16813883	1	1	yes	characteristic
82	C₂²	—	4	4	no	—
83	C₂²	—	8	2	no	—
84	C₂²	—	8	2	no	—
85	C₂²	D₈×D₂₂₇	1	1	yes	characteristic
86	C₄	—	1816	2	no	—
87	C₄	—	454	2	no	—
88	C₄	D₄×Dic₂₂₇	1	1	yes	characteristic
89	C₄	#20300307	1	1	yes	characteristic
90	C₂	—	8	4	no	—
91	C₂	D₈×Dic₂₂₇	1	1	yes	characteristic
92	C₂	#28409466	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₂₂₇
commutator	C₈×C₂₂₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₂₇
Fitting, prime radical	C₂×D₁₆×C₂₂₇	C₂
socle	C₂²×C₂₂₇	C₂×D₈
cosocle	C₂×C₈×C₂₂₇	C₂³
hypercenter	C₂×D₁₆	D₂₂₇
hypocenter	C₂₂₇	#2177
LCS	G ▷ C₈×C₂₂₇ ▷ C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

Complements

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

N	H ≅ G/N	type
1. D₈×Dic₂₂₇	C₂	semidirect
2. #28408860	C₂	semidirect
3. #28408833	C₂	semidirect
4. #28408807	C₂	semidirect
12. D₁₆×C₂₂₇	C₄	semidirect
22. D₈×C₂₂₇	C₂×C₄	semidirect
24. C₁₆×C₂₂₇	C₂×C₄	semidirect
46. C₂₂₇	#2177	semidirect
55. D₁₆	Dic₂₂₇	semidirect
64. D₈	C₂×Dic₂₂₇	semidirect
71. C₁₆	C₂×Dic₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
D₈×Dic₂₂₇	2
#28408860	2
#28408833	2
#28408807	2
C₂×D₁₆×C₂₂₇	2
#2177	227