Group #44401348 — subgroups

Subgroups — #44401348

Order 29056.

228 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 109 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	1	yes	maximal, characteristic, complemented
2	C₂²×C₂₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊(C₂₂₇⋊C₁₆)	C₂	1	4	yes	maximal, complemented
4	C₂×#131×C₂₂₇	C₂	1	1	yes	maximal, characteristic
5	#20300228	—	2	2	no	—
6	C₂×C₂₂₇⋊C₁₆	—	2	2	no	—
7	C₂×C₂₂₇⋊C₁₆	C₂²	1	2	yes	—
8	C₂×C₂₂₇⋊C₁₆	C₂²	1	2	yes	—
9	C₂×#41×C₂₂₇	C₄	1	1	yes	characteristic
10	#131×C₂₂₇	—	2	2	no	—
11	#131×C₂₂₇	—	2	2	no	—
12	C₂×M₁₆×C₂₂₇	C₄	1	1	yes	characteristic
13	C₂×M₁₆×C₂₂₇	C₄	1	2	yes	—
14	C₂²×C₈×C₂₂₇	C₂²	1	2	yes	—
15	C₂²×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
16	C₂₂₇⋊C₁₆	—	2	2	no	—
17	C₂₂₇⋊C₁₆	—	2	2	no	—
18	#41×C₂₂₇	—	2	2	no	—
19	#41×C₂₂₇	—	2	2	no	—
20	C₂×Q₈×C₂₂₇	C₈	1	1	yes	characteristic
21	C₂×D₄×C₂₂₇	C₈	1	2	yes	—
22	C₂×D₄×C₂₂₇	C₈	1	1	yes	characteristic
23	C₂²×C₄×C₂₂₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
25	M₁₆×C₂₂₇	—	2	2	no	—
26	M₁₆×C₂₂₇	—	2	4	no	—
27	C₂×C₈×C₂₂₇	—	2	2	no	—
28	C₂×C₈×C₂₂₇	—	2	2	no	—
29	C₂×C₈×C₂₂₇	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₂₂₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₂₂₇	D₄	1	2	yes	—
32	C₂×C₈×C₂₂₇	D₄	1	2	yes	—
33	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
34	C₂³×C₂₂₇	C₂×C₈	1	2	yes	—
35	C₂³×C₂₂₇	C₂×C₈	1	1	yes	characteristic
36	Q₈×C₂₂₇	—	2	2	no	—
37	D₄×C₂₂₇	—	2	4	no	—
38	D₄×C₂₂₇	—	2	2	no	—
39	C₂×C₄×C₂₂₇	—	2	2	no	—
40	C₂×C₄×C₂₂₇	—	2	2	no	—
41	C₂×C₄×C₂₂₇	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₂₂₇	C₂×C₈	1	2	yes	—
43	C₂×C₄×C₂₂₇	C₂×C₈	1	1	yes	characteristic
44	C₂×C₄×C₂₂₇	M₁₆	1	2	yes	—
45	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
46	C₈×C₂₂₇	—	2	2	no	—
47	C₈×C₂₂₇	C₂²⋊C₄	1	2	yes	—
48	C₈×C₂₂₇	C₂×D₄	1	1	yes	characteristic
49	C₈×C₂₂₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₂₂₇	—	2	2	no	—
51	C₂²×C₂₂₇	—	2	2	no	—
52	C₂²×C₂₂₇	—	2	4	no	—
53	C₂²×C₂₂₇	C₂²⋊C₈	1	2	yes	—
54	C₂²×C₂₂₇	C₂²×C₈	1	1	yes	characteristic
55	C₄×C₂₂₇	—	2	2	no	—
56	C₄×C₂₂₇	C₂²⋊C₈	1	2	yes	—
57	C₄×C₂₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₂₂₇	C₂×M₁₆	1	1	yes	characteristic
59	C₂×C₂₂₇	—	2	2	no	—
60	C₂×C₂₂₇	—	4	2	no	—
61	C₂×C₂₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
62	C₂×C₂₂₇	#504	1	2	yes	—
63	C₂₂₇	#2113	1	1	yes	characteristic, complemented, Sylow
64	#2113	—	227	1	no	maximal, Sylow
65	C₂×#131	D₂₂₇	1	1	yes	characteristic
66	C₂×M₃₂	—	227	1	no	—
67	C₂²×C₁₆	—	227	1	no	—
68	C₂²⋊C₁₆	—	227	4	no	—
69	C₂×#41	Dic₂₂₇	1	1	yes	characteristic
70	#131	—	2	2	no	—
71	#131	—	2	2	no	—
72	C₂×M₁₆	Dic₂₂₇	1	1	yes	characteristic
73	C₂×M₁₆	Dic₂₂₇	1	2	yes	—
74	C₂²×C₈	C₂×D₂₂₇	1	2	yes	—
75	C₂²×C₈	C₂×D₂₂₇	1	1	yes	characteristic
76	M₃₂	—	454	2	no	—
77	C₂×C₁₆	—	454	2	no	—
78	C₂×C₁₆	—	227	2	no	—
79	C₂×C₁₆	—	227	2	no	—
80	#41	—	2	2	no	—
81	#41	—	2	2	no	—
82	C₂×Q₈	C₂₂₇⋊C₈	1	1	yes	characteristic
83	C₂×D₄	C₂₂₇⋊C₈	1	2	yes	—
84	C₂×D₄	C₂₂₇⋊C₈	1	1	yes	characteristic
85	C₂²×C₄	C₂×Dic₂₂₇	1	2	yes	—
86	C₂²×C₄	C₂×Dic₂₂₇	1	1	yes	characteristic
87	M₁₆	—	2	2	no	—
88	M₁₆	—	2	4	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	C₂×Dic₂₂₇	1	2	yes	—
92	C₂×C₈	C₂×Dic₂₂₇	1	1	yes	characteristic
93	C₂×C₈	C₂²⋊D₂₂₇	1	2	yes	—
94	C₂×C₈	C₂²⋊D₂₂₇	1	2	yes	—
95	C₂×C₈	C₂²×D₂₂₇	1	1	yes	characteristic
96	C₁₆	—	454	2	no	—
97	C₁₆	—	454	2	no	—
98	C₂³	C₂×C₂₂₇⋊C₈	1	2	yes	—
99	C₂³	C₂×C₂₂₇⋊C₈	1	1	yes	characteristic
100	Q₈	—	2	2	no	—
101	D₄	—	2	4	no	—
102	D₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₂²⋊Dic₂₂₇	1	2	yes	—
106	C₂×C₄	C₂×C₂₂₇⋊C₈	1	2	yes	—
107	C₂×C₄	C₂×C₂₂₇⋊C₈	1	1	yes	characteristic
108	C₂×C₄	#16813868	1	2	yes	—
109	C₂×C₄	C₂²×Dic₂₂₇	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	C₂²⋊Dic₂₂₇	1	2	yes	—
112	C₈	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
113	C₈	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
114	C₂²	—	2	2	no	—
115	C₂²	—	2	4	no	—
116	C₂²	—	2	2	no	—
117	C₂²	C₂²⋊(C₂₂₇⋊C₈)	1	2	yes	—
118	C₂²	C₂²×C₂₂₇⋊C₈	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	C₂²⋊(C₂₂₇⋊C₈)	1	2	yes	—
121	C₄	C₂×C₂²⋊Dic₂₂₇	1	1	yes	characteristic
122	C₄	C₂×#16813868	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	4	2	no	—
125	C₂	C₂×C₂²⋊(C₂₂₇⋊C₈)	1	1	yes	characteristic
126	C₂	#28409449	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20300228

C₂

yes

maximal, characteristic, complemented

C₂²×C₂₂₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂²⋊(C₂₂₇⋊C₁₆)

C₂

yes

maximal, complemented

C₂×#131×C₂₂₇

C₂

yes

maximal, characteristic

#20300228

—

C₂×C₂₂₇⋊C₁₆

—

C₂×C₂₂₇⋊C₁₆

C₂²

yes

—

C₂×C₂₂₇⋊C₁₆

C₂²

yes

—

C₂×#41×C₂₂₇

C₄

yes

characteristic

#131×C₂₂₇

—

#131×C₂₂₇

—

C₂×M₁₆×C₂₂₇

C₄

yes

characteristic

C₂×M₁₆×C₂₂₇

C₄

yes

—

C₂²×C₈×C₂₂₇

C₂²

yes

—

C₂²×C₈×C₂₂₇

C₂²

yes

characteristic

C₂₂₇⋊C₁₆

—

C₂₂₇⋊C₁₆

—

#41×C₂₂₇

—

#41×C₂₂₇

—

C₂×Q₈×C₂₂₇

C₈

yes

characteristic

C₂×D₄×C₂₂₇

C₈

yes

—

C₂×D₄×C₂₂₇

C₈

yes

characteristic

C₂²×C₄×C₂₂₇

C₂×C₄

yes

—

C₂²×C₄×C₂₂₇

C₂×C₄

yes

characteristic

M₁₆×C₂₂₇

—

M₁₆×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

D₄

yes

—

C₂×C₈×C₂₂₇

C₂³

yes

characteristic

C₂³×C₂₂₇

C₂×C₈

yes

—

C₂³×C₂₂₇

C₂×C₈

yes

characteristic

Q₈×C₂₂₇

—

D₄×C₂₂₇

—

D₄×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

characteristic

C₈×C₂₂₇

C₂×D₄

yes

characteristic

C₂²×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

C₂²⋊C₈

yes

—

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

#504

yes

—

C₂₂₇

#2113

yes

characteristic, complemented, Sylow

#2113

—

227

maximal, Sylow

C₂×#131

D₂₂₇

yes

characteristic

C₂×M₃₂

—

227

—

C₂²×C₁₆

—

227

—

C₂²⋊C₁₆

—

227

—

C₂×#41

Dic₂₂₇

yes

characteristic

#131

—

#131

—

C₂×M₁₆

Dic₂₂₇

yes

characteristic

C₂×M₁₆

Dic₂₂₇

yes

—

C₂²×C₈

C₂×D₂₂₇

yes

—

C₂²×C₈

C₂×D₂₂₇

yes

characteristic

M₃₂

—

454

—

C₂×C₁₆

—

454

—

C₂×C₁₆

—

227

—

C₂×C₁₆

—

227

—

#41

—

#41

—

C₂×Q₈

C₂₂₇⋊C₈

yes

characteristic

C₂×D₄

C₂₂₇⋊C₈

yes

—

C₂×D₄

C₂₂₇⋊C₈

yes

characteristic

C₂²×C₄

C₂×Dic₂₂₇

yes

—

C₂²×C₄

C₂×Dic₂₂₇

yes

characteristic

M₁₆

—

M₁₆

—

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

—

C₂×C₈

C₂²×D₂₂₇

yes

characteristic

C₁₆

—

454

—

C₁₆

—

454

—

C₂³

C₂×C₂₂₇⋊C₈

yes

—

C₂³

C₂×C₂₂₇⋊C₈

yes

characteristic

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²⋊Dic₂₂₇

yes

—

106

C₂×C₄

C₂×C₂₂₇⋊C₈

yes

—

107

C₂×C₄

C₂×C₂₂₇⋊C₈

yes

characteristic

108

C₂×C₄

#16813868

yes

—

109

C₂×C₄

C₂²×Dic₂₂₇

yes

characteristic

110

C₈

—

111

C₈

C₂²⋊Dic₂₂₇

yes

—

112

C₈

C₂×C₂²⋊D₂₂₇

yes

characteristic

113

C₈

C₂×C₂²⋊D₂₂₇

yes

characteristic

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

C₂²⋊(C₂₂₇⋊C₈)

yes

—

118

C₂²

C₂²×C₂₂₇⋊C₈

yes

characteristic

119

C₄

—

120

C₄

C₂²⋊(C₂₂₇⋊C₈)

yes

—

121

C₄

C₂×C₂²⋊Dic₂₂₇

yes

characteristic

122

C₄

C₂×#16813868

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

C₂×C₂²⋊(C₂₂₇⋊C₈)

yes

characteristic

126

C₂

#28409449

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₂₂₇
commutator, socle	C₂²×C₂₂₇	C₂²×C₈
Fitting	C₂×#131×C₂₂₇	C₂
prime radical	C₂×#41×C₂₂₇	C₄
cosocle	C₂×C₈×C₂₂₇	C₂³
hypercenter	C₂×#131	D₂₂₇
hypocenter	C₂₂₇	#2113
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₂₂₇

commutator, socle

C₂²×C₂₂₇

C₂²×C₈

Fitting

C₂×#131×C₂₂₇

C₂

prime radical

C₂×#41×C₂₂₇

C₄

cosocle

C₂×C₈×C₂₂₇

C₂³

D₂₂₇

C₂₂₇

G ▷ C₂²×C₂₂₇ ▷ C₂₂₇

C₁ ◁ C₂×C₈ ◁ C₂×#131

N	H ≅ G/N	type
1. C₂×#20300228	C₂	semidirect
2. C₂²×C₂₂₇⋊C₁₆	C₂	semidirect
3. C₂²⋊(C₂₂₇⋊C₁₆)	C₂	semidirect
63. C₂₂₇	#2113	semidirect

H ≅ G/N

type

1. C₂×#20300228

C₂

semidirect

2. C₂²×C₂₂₇⋊C₁₆

C₂

semidirect

3. C₂²⋊(C₂₂₇⋊C₁₆)

C₂

semidirect

63. C₂₂₇

#2113

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
C₂²⋊(C₂₂₇⋊C₁₆)	2
C₂×#131×C₂₂₇	2
#2113	227

subgroup

index

C₂×#20300228

C₂²×C₂₂₇⋊C₁₆

C₂²⋊(C₂₂₇⋊C₁₆)

C₂×#131×C₂₂₇

#2113

227

Subgroups — #44401348

Conjugacy classes of subgroups

Complements

Maximal subgroups