Group #44401345 — subgroups

Subgroups — #44401345

Order 29056.

260 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20300228	C₂	1	2	yes	maximal, complemented
2	C₂²⋊(C₂₂₇⋊C₁₆)	C₂	1	4	yes	maximal, complemented
3	C₂³×C₈×C₂₂₇	C₂	1	1	yes	maximal, characteristic
4	#20300228	—	2	4	no	—
5	C₂×C₂₂₇⋊C₁₆	C₂²	1	4	yes	complemented
6	C₂³×C₄×C₂₂₇	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₂₂₇	—	2	4	no	—
8	C₂²×C₈×C₂₂₇	C₄	1	1	yes	characteristic
9	C₂²×C₈×C₂₂₇	C₄	1	2	yes	—
10	C₂²×C₈×C₂₂₇	C₂²	1	2	yes	—
11	C₂²×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
12	C₂₂₇⋊C₁₆	—	2	4	no	—
13	C₂⁴×C₂₂₇	C₈	1	1	yes	characteristic
14	C₂²×C₄×C₂₂₇	—	2	2	no	—
15	C₂²×C₄×C₂₂₇	—	2	2	no	—
16	C₂²×C₄×C₂₂₇	C₈	1	1	yes	characteristic
17	C₂²×C₄×C₂₂₇	C₈	1	2	yes	—
18	C₂²×C₄×C₂₂₇	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₂₂₇	—	2	2	no	—
21	C₂×C₈×C₂₂₇	—	2	4	no	—
22	C₂×C₈×C₂₂₇	—	2	4	no	—
23	C₂×C₈×C₂₂₇	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₂₂₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₂₂₇	D₄	1	4	yes	—
26	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
27	C₂³×C₂₂₇	—	2	2	no	—
28	C₂³×C₂₂₇	—	2	4	no	—
29	C₂³×C₂₂₇	C₂×C₈	1	1	yes	characteristic
30	C₂³×C₂₂₇	C₂×C₈	1	2	yes	—
31	C₂×C₄×C₂₂₇	—	2	2	no	—
32	C₂×C₄×C₂₂₇	—	2	4	no	—
33	C₂×C₄×C₂₂₇	—	2	2	no	—
34	C₂×C₄×C₂₂₇	—	2	2	no	—
35	C₂×C₄×C₂₂₇	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₂₂₇	C₂×C₈	1	2	yes	—
37	C₂×C₄×C₂₂₇	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×C₂₂₇	M₁₆	1	2	yes	—
39	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
40	C₈×C₂₂₇	—	2	2	no	—
41	C₈×C₂₂₇	C₂²⋊C₄	1	2	yes	—
42	C₈×C₂₂₇	C₂×D₄	1	2	yes	—
43	C₂²×C₂₂₇	—	2	8	no	—
44	C₂²×C₂₂₇	—	2	2	no	—
45	C₂²×C₂₂₇	—	2	2	no	—
46	C₂²×C₂₂₇	—	2	2	no	—
47	C₂²×C₂₂₇	C₂²⋊C₈	1	2	yes	—
48	C₂²×C₂₂₇	M₃₂	1	4	yes	complemented
49	C₂²×C₂₂₇	C₂²×C₈	1	1	yes	characteristic
50	C₄×C₂₂₇	—	2	2	no	—
51	C₄×C₂₂₇	C₂²⋊C₈	1	2	yes	—
52	C₄×C₂₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₂₂₇	C₂×M₁₆	1	1	yes	characteristic
54	C₂×C₂₂₇	—	2	2	no	—
55	C₂×C₂₂₇	—	2	4	no	—
56	C₂×C₂₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
57	C₂×C₂₂₇	C₂×M₃₂	1	2	yes	—
58	C₂₂₇	#2112	1	1	yes	characteristic, complemented, Sylow
59	#2112	—	227	1	no	maximal, Sylow
60	C₂³×C₈	D₂₂₇	1	1	yes	characteristic
61	C₂×M₃₂	—	227	2	no	—
62	C₂²⋊C₁₆	—	227	4	no	—
63	C₂³×C₄	Dic₂₂₇	1	1	yes	characteristic
64	C₂²×C₈	—	2	4	no	—
65	C₂²×C₈	Dic₂₂₇	1	1	yes	characteristic
66	C₂²×C₈	Dic₂₂₇	1	2	yes	—
67	C₂²×C₈	C₂×D₂₂₇	1	2	yes	—
68	C₂²×C₈	C₂×D₂₂₇	1	1	yes	characteristic
69	M₃₂	—	454	4	no	—
70	C₂×C₁₆	—	227	4	no	—
71	C₂⁴	C₂₂₇⋊C₈	1	1	yes	characteristic
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	C₂₂₇⋊C₈	1	1	yes	characteristic
75	C₂²×C₄	C₂₂₇⋊C₈	1	2	yes	—
76	C₂²×C₄	C₂×Dic₂₂₇	1	2	yes	—
77	C₂²×C₄	C₂×Dic₂₂₇	1	1	yes	characteristic
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	4	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	C₂×Dic₂₂₇	1	2	yes	—
82	C₂×C₈	C₂×Dic₂₂₇	1	1	yes	characteristic
83	C₂×C₈	C₂²⋊D₂₂₇	1	4	yes	—
84	C₂×C₈	C₂²×D₂₂₇	1	1	yes	characteristic
85	C₁₆	—	454	4	no	—
86	C₂³	—	2	2	no	—
87	C₂³	—	2	4	no	—
88	C₂³	C₂×C₂₂₇⋊C₈	1	1	yes	characteristic
89	C₂³	C₂×C₂₂₇⋊C₈	1	2	yes	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	C₂²⋊Dic₂₂₇	1	2	yes	—
95	C₂×C₄	C₂×C₂₂₇⋊C₈	1	2	yes	—
96	C₂×C₄	C₂×C₂₂₇⋊C₈	1	1	yes	characteristic
97	C₂×C₄	#16813868	1	2	yes	—
98	C₂×C₄	C₂²×Dic₂₂₇	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	C₂²⋊Dic₂₂₇	1	2	yes	—
101	C₈	C₂×C₂²⋊D₂₂₇	1	2	yes	—
102	C₂²	—	2	8	no	—
103	C₂²	—	2	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	—	2	2	no	—
106	C₂²	C₂²⋊(C₂₂₇⋊C₈)	1	2	yes	—
107	C₂²	#20300228	1	4	yes	complemented
108	C₂²	C₂²×C₂₂₇⋊C₈	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	C₂²⋊(C₂₂₇⋊C₈)	1	2	yes	—
111	C₄	C₂×C₂²⋊Dic₂₂₇	1	1	yes	characteristic
112	C₄	C₂×#16813868	1	1	yes	characteristic
113	C₂	—	2	2	no	—
114	C₂	—	2	4	no	—
115	C₂	C₂×C₂²⋊(C₂₂₇⋊C₈)	1	1	yes	characteristic
116	C₂	C₂×#20300228	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20300228

C₂

yes

maximal, complemented

C₂²⋊(C₂₂₇⋊C₁₆)

C₂

yes

maximal, complemented

C₂³×C₈×C₂₂₇

C₂

yes

maximal, characteristic

#20300228

—

C₂×C₂₂₇⋊C₁₆

C₂²

yes

complemented

C₂³×C₄×C₂₂₇

C₄

yes

characteristic

C₂²×C₈×C₂₂₇

—

C₂²×C₈×C₂₂₇

C₄

yes

characteristic

C₂²×C₈×C₂₂₇

C₄

yes

—

C₂²×C₈×C₂₂₇

C₂²

yes

—

C₂²×C₈×C₂₂₇

C₂²

yes

characteristic

C₂₂₇⋊C₁₆

—

C₂⁴×C₂₂₇

C₈

yes

characteristic

C₂²×C₄×C₂₂₇

—

C₂²×C₄×C₂₂₇

—

C₂²×C₄×C₂₂₇

C₈

yes

characteristic

C₂²×C₄×C₂₂₇

C₈

yes

—

C₂²×C₄×C₂₂₇

C₂×C₄

yes

—

C₂²×C₄×C₂₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₂₂₇

—

C₂×C₈×C₂₂₇

—

C₂×C₈×C₂₂₇

—

C₂×C₈×C₂₂₇

C₂×C₄

yes

—

C₂×C₈×C₂₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₂₂₇

D₄

yes

—

C₂×C₈×C₂₂₇

C₂³

yes

characteristic

C₂³×C₂₂₇

—

C₂³×C₂₂₇

—

C₂³×C₂₂₇

C₂×C₈

yes

characteristic

C₂³×C₂₂₇

C₂×C₈

yes

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₂₇

C₂×C₈

yes

—

C₂×C₄×C₂₂₇

C₂×C₈

yes

characteristic

C₂×C₄×C₂₂₇

M₁₆

yes

—

C₂×C₄×C₂₂₇

C₂²×C₄

yes

characteristic

C₈×C₂₂₇

—

C₈×C₂₂₇

C₂²⋊C₄

yes

—

C₈×C₂₂₇

C₂×D₄

yes

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

C₂²⋊C₈

yes

—

C₂²×C₂₂₇

M₃₂

yes

complemented

C₂²×C₂₂₇

C₂²×C₈

yes

characteristic

C₄×C₂₂₇

—

C₄×C₂₂₇

C₂²⋊C₈

yes

—

C₄×C₂₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₂₇

C₂×M₁₆

yes

characteristic

C₂×C₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₂₂₇

C₂×M₃₂

yes

—

C₂₂₇

#2112

yes

characteristic, complemented, Sylow

#2112

—

227

maximal, Sylow

C₂³×C₈

D₂₂₇

yes

characteristic

C₂×M₃₂

—

227

—

C₂²⋊C₁₆

—

227

—

C₂³×C₄

Dic₂₂₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

Dic₂₂₇

yes

characteristic

C₂²×C₈

Dic₂₂₇

yes

—

C₂²×C₈

C₂×D₂₂₇

yes

—

C₂²×C₈

C₂×D₂₂₇

yes

characteristic

M₃₂

—

454

—

C₂×C₁₆

—

227

—

C₂⁴

C₂₂₇⋊C₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂₂₇⋊C₈

yes

characteristic

C₂²×C₄

C₂₂₇⋊C₈

yes

—

C₂²×C₄

C₂×Dic₂₂₇

yes

—

C₂²×C₄

C₂×Dic₂₂₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₂₂₇

yes

—

C₂×C₈

C₂×Dic₂₂₇

yes

characteristic

C₂×C₈

C₂²⋊D₂₂₇

yes

—

C₂×C₈

C₂²×D₂₂₇

yes

characteristic

C₁₆

—

454

—

C₂³

—

C₂³

—

C₂³

C₂×C₂₂₇⋊C₈

yes

characteristic

C₂³

C₂×C₂₂₇⋊C₈

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₂₂₇

yes

—

C₂×C₄

C₂×C₂₂₇⋊C₈

yes

—

C₂×C₄

C₂×C₂₂₇⋊C₈

yes

characteristic

C₂×C₄

#16813868

yes

—

C₂×C₄

C₂²×Dic₂₂₇

yes

characteristic

C₈

—

100

C₈

C₂²⋊Dic₂₂₇

yes

—

101

C₈

C₂×C₂²⋊D₂₂₇

yes

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

C₂²⋊(C₂₂₇⋊C₈)

yes

—

107

C₂²

#20300228

yes

complemented

108

C₂²

C₂²×C₂₂₇⋊C₈

yes

characteristic

109

C₄

—

110

C₄

C₂²⋊(C₂₂₇⋊C₈)

yes

—

111

C₄

C₂×C₂²⋊Dic₂₂₇

yes

characteristic

112

C₄

C₂×#16813868

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

C₂×C₂²⋊(C₂₂₇⋊C₈)

yes

characteristic

116

C₂

C₂×#20300228

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₂₂₇
commutator, socle	C₂²×C₂₂₇	C₂²×C₈
Fitting	C₂³×C₈×C₂₂₇	C₂
prime radical	C₂⁴×C₂₂₇	C₈
cosocle	C₂×C₈×C₂₂₇	C₂³
hypercenter	C₂³×C₈	D₂₂₇
hypocenter	C₂₂₇	#2112
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₂₂₇

commutator, socle

C₂²×C₂₂₇

C₂²×C₈

Fitting

C₂³×C₈×C₂₂₇

C₂

prime radical

C₂⁴×C₂₂₇

C₈

cosocle

C₂×C₈×C₂₂₇

C₂³

hypercenter

C₂³×C₈

D₂₂₇

C₂₂₇

G ▷ C₂²×C₂₂₇ ▷ C₂₂₇

C₁ ◁ C₂×C₈ ◁ C₂³×C₈

N	H ≅ G/N	type
1. C₂×#20300228	C₂	semidirect
2. C₂²⋊(C₂₂₇⋊C₁₆)	C₂	semidirect
5. C₂×C₂₂₇⋊C₁₆	C₂²	semidirect
48. C₂²×C₂₂₇	M₃₂	semidirect
58. C₂₂₇	#2112	semidirect
107. C₂²	#20300228	semidirect

H ≅ G/N

type

1. C₂×#20300228

C₂

semidirect

2. C₂²⋊(C₂₂₇⋊C₁₆)

C₂

semidirect

5. C₂×C₂₂₇⋊C₁₆

C₂²

semidirect

48. C₂²×C₂₂₇

M₃₂

semidirect

58. C₂₂₇

#2112

semidirect

107. C₂²

#20300228

semidirect

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

subgroup	index
C₂×#20300228	2
C₂²⋊(C₂₂₇⋊C₁₆)	2
C₂³×C₈×C₂₂₇	2
#2112	227

subgroup

index

C₂×#20300228

C₂²⋊(C₂₂₇⋊C₁₆)

C₂³×C₈×C₂₂₇

#2112

227

Subgroups — #44401345

Conjugacy classes of subgroups

Complements

Maximal subgroups