Group #44560888 — subgroups

Subgroups — #44560888

Order 32128.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#30632240	C₂	1	2	yes	maximal
2	#30632095	C₂	1	4	yes	maximal, complemented
3	#30632094	C₂	1	4	yes	maximal, complemented
4	#30631725	C₂	1	1	yes	maximal, characteristic
5	#30631723	C₂	1	2	yes	maximal
6	C₂×#22548625	C₂	1	1	yes	maximal, characteristic
7	#427×C₂₅₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
9	#22548714	C₂²	1	4	yes	—
10	#22548706	C₂²	1	4	yes	—
11	#22548692	—	2	4	no	—
12	C₄×Dic₅₀₂	—	2	4	no	—
13	#22548674	C₂²	1	2	yes	—
14	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
15	#22548649	C₂²	1	2	yes	—
16	#22548648	C₂²	1	4	yes	—
17	#22548647	C₂²	1	2	yes	—
18	#22548646	C₂²	1	4	yes	—
19	#22548625	C₂²	1	2	yes	—
20	#22548625	C₂²	1	2	yes	—
21	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
22	#117×C₂₅₁	C₂²	1	2	yes	—
23	#107×C₂₅₁	C₂²	1	2	yes	—
24	#106×C₂₅₁	C₂²	1	2	yes	—
25	C₂×Dic₅₀₂	—	2	4	no	—
26	C₂²×Dic₂₅₁	D₄	1	2	yes	—
27	C₂×C₂₅₁⋊C₈	Q₈	1	4	yes	complemented
28	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
29	C₄⋊Dic₂₅₁	—	2	4	no	—
30	C₄⋊Dic₂₅₁	C₂³	1	4	yes	—
31	Dic₂₅₁⋊C₄	—	2	4	no	—
32	Dic₂₅₁⋊C₄	—	2	4	no	—
33	C₂²⋊Dic₂₅₁	—	2	2	no	—
34	C₄×Dic₂₅₁	—	2	2	no	—
35	C₄×Dic₂₅₁	D₄	1	2	yes	—
36	C₄×Dic₂₅₁	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₂₅₁	Q₈	1	4	yes	complemented
39	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₂₅₁	C₂³	1	4	yes	—
42	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
43	C₄²×C₂₅₁	—	2	2	no	—
44	Dic₅₀₂	—	4	8	no	—
45	C₂×Dic₂₅₁	—	2	4	no	—
46	C₂×Dic₂₅₁	—	4	2	no	—
47	C₂×Dic₂₅₁	C₂×D₄	1	4	yes	—
48	C₂₅₁⋊C₈	C₂×Q₈	1	4	yes	—
49	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₅₁	—	2	4	no	—
51	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
55	C₈×C₂₅₁	C₂×Q₈	1	2	yes	—
56	C₈×C₂₅₁	C₂×Q₈	1	2	yes	—
57	C₂²×C₂₅₁	—	2	2	no	—
58	C₂²×C₂₅₁	C₄⋊Q₈	1	2	yes	—
59	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
60	C₄×C₂₅₁	—	4	4	no	—
61	C₄×C₂₅₁	C₄⋊Q₈	1	2	yes	—
62	C₄×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
63	C₄×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
64	Dic₂₅₁	—	2	4	no	—
65	Dic₂₅₁	—	4	4	no	—
66	C₂×C₂₅₁	—	2	2	no	—
67	C₂×C₂₅₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₂₅₁	#575	1	2	yes	—
69	C₂₅₁	#3162	1	1	yes	characteristic, complemented, Sylow
70	#3162	—	251	1	no	maximal, Sylow
71	#533	—	251	2	no	—
72	#501	—	251	4	no	—
73	#501	—	251	4	no	—
74	#427	—	251	1	no	—
75	#427	—	251	2	no	—
76	#427	D₂₅₁	1	1	yes	characteristic
77	C₂×#97	—	251	1	no	—
78	C₂²×C₈	—	251	1	no	—
79	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
80	C₄⋊Q₈	—	251	4	no	—
81	#125	—	251	4	no	—
82	C₂²⋊Q₈	—	502	4	no	—
83	C₄×Q₈	—	502	4	no	—
84	#117	—	251	2	no	—
85	#117	C₂×D₂₅₁	1	2	yes	—
86	C₂×C₄²	—	251	1	no	—
87	#107	—	251	2	no	—
88	#107	—	251	4	no	—
89	#107	C₂×D₂₅₁	1	2	yes	—
90	#106	—	251	2	no	—
91	#106	—	251	4	no	—
92	#106	C₂×D₂₅₁	1	2	yes	—
93	#97	—	251	2	no	—
94	#97	—	251	2	no	—
95	C₂×Q₈	—	502	4	no	—
96	C₂²×C₄	—	251	2	no	—
97	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
98	C₂×C₈	—	251	4	no	—
99	C₂×C₈	—	251	2	no	—
100	C₂×C₈	Dic₅₀₂	1	4	yes	complemented
101	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
103	C₄⋊C₄	—	502	4	no	—
104	C₄⋊C₄	—	502	4	no	—
105	C₄⋊C₄	—	502	4	no	—
106	C₄⋊C₄	—	251	4	no	—
107	C₄⋊C₄	C₂²×D₂₅₁	1	4	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₂²⋊C₄	—	502	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	502	2	no	—
112	C₄²	—	251	2	no	—
113	C₄²	—	251	2	no	—
114	C₂³	D₄×D₂₅₁	1	1	yes	characteristic
115	Q₈	—	1004	8	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	1004	2	no	—
118	C₂×C₄	—	502	4	no	—
119	C₂×C₄	—	251	4	no	—
120	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
121	C₂×C₄	Q₈×D₂₅₁	1	2	yes	—
122	C₂×C₄	C₂×Dic₅₀₂	1	2	yes	—
123	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
124	C₈	—	251	4	no	—
125	C₈	C₂×Dic₅₀₂	1	2	yes	—
126	C₈	C₂×Dic₅₀₂	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	#22548714	1	2	yes	—
129	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
130	C₄	—	4	4	no	—
131	C₄	—	1004	4	no	—
132	C₄	—	502	4	no	—
133	C₄	#22548714	1	2	yes	—
134	C₄	C₂×Q₈×D₂₅₁	1	1	yes	characteristic
135	C₄	C₂²×Dic₅₀₂	1	1	yes	characteristic
136	C₂	—	2	2	no	—
137	C₂	C₂×#22548714	1	1	yes	characteristic
138	C₂	#30632484	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#30632240

C₂

yes

maximal

#30632095

C₂

yes

maximal, complemented

#30632094

C₂

yes

maximal, complemented

#30631725

C₂

yes

maximal, characteristic

#30631723

C₂

yes

maximal

C₂×#22548625

C₂

yes

maximal, characteristic

#427×C₂₅₁

C₂

yes

maximal, characteristic

C₂²×C₂₅₁⋊C₈

C₂²

yes

characteristic

#22548714

C₂²

yes

—

#22548706

C₂²

yes

—

#22548692

—

C₄×Dic₅₀₂

—

#22548674

C₂²

yes

—

C₂×C₄×Dic₂₅₁

C₂²

yes

characteristic

#22548649

C₂²

yes

—

#22548648

C₂²

yes

—

#22548647

C₂²

yes

—

#22548646

C₂²

yes

—

#22548625

C₂²

yes

—

#22548625

C₂²

yes

—

C₂²×C₈×C₂₅₁

C₂²

yes

characteristic

#117×C₂₅₁

C₂²

yes

—

#107×C₂₅₁

C₂²

yes

—

#106×C₂₅₁

C₂²

yes

—

C₂×Dic₅₀₂

—

C₂²×Dic₂₅₁

D₄

yes

—

C₂×C₂₅₁⋊C₈

Q₈

yes

complemented

C₂×C₂₅₁⋊C₈

C₂³

yes

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

C₂³

yes

—

Dic₂₅₁⋊C₄

—

Dic₂₅₁⋊C₄

—

C₂²⋊Dic₂₅₁

—

C₄×Dic₂₅₁

—

C₄×Dic₂₅₁

D₄

yes

—

C₄×Dic₂₅₁

C₂³

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₂×C₈×C₂₅₁

Q₈

yes

complemented

C₂×C₈×C₂₅₁

C₂³

yes

characteristic

C₂×C₈×C₂₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₅₁

C₂³

yes

—

C₂²⋊C₄×C₂₅₁

—

C₄²×C₂₅₁

—

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

C₂×D₄

yes

—

C₂₅₁⋊C₈

C₂×Q₈

yes

—

C₂³×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×Q₈

yes

—

C₂×C₄×C₂₅₁

C₂×Q₈

yes

—

C₂×C₄×C₂₅₁

C₂⁴

yes

characteristic

C₈×C₂₅₁

C₂×Q₈

yes

—

C₈×C₂₅₁

C₂×Q₈

yes

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₄⋊Q₈

yes

—

C₂²×C₂₅₁

C₂²×D₄

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

C₄⋊Q₈

yes

—

C₄×C₂₅₁

C₂²×Q₈

yes

characteristic

C₄×C₂₅₁

C₂²×Q₈

yes

characteristic

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₅₁

#575

yes

—

C₂₅₁

#3162

yes

characteristic, complemented, Sylow

#3162

—

251

maximal, Sylow

#533

—

251

—

#501

—

251

—

#501

—

251

—

#427

—

251

—

#427

—

251

—

#427

D₂₅₁

yes

characteristic

C₂×#97

—

251

—

C₂²×C₈

—

251

—

C₂²×C₈

C₂×D₂₅₁

yes

characteristic

C₄⋊Q₈

—

251

—

#125

—

251

—

C₂²⋊Q₈

—

502

—

C₄×Q₈

—

502

—

#117

—

251

—

#117

C₂×D₂₅₁

yes

—

C₂×C₄²

—

251

—

#107

—

251

—

#107

—

251

—

#107

C₂×D₂₅₁

yes

—

#106

—

251

—

#106

—

251

—

#106

C₂×D₂₅₁

yes

—

#97

—

251

—

#97

—

251

—

C₂×Q₈

—

502

—

C₂²×C₄

—

251

—

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

C₂×C₈

—

251

—

C₂×C₈

—

251

—

100

C₂×C₈

Dic₅₀₂

yes

complemented

101

C₂×C₈

C₂²×D₂₅₁

yes

characteristic

102

C₂×C₈

C₂²×D₂₅₁

yes

characteristic

103

C₄⋊C₄

—

502

—

104

C₄⋊C₄

—

502

—

105

C₄⋊C₄

—

502

—

106

C₄⋊C₄

—

251

—

107

C₄⋊C₄

C₂²×D₂₅₁

yes

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

502

—

110

C₄²

—

111

C₄²

—

502

—

112

C₄²

—

251

—

113

C₄²

—

251

—

114

C₂³

D₄×D₂₅₁

yes

characteristic

115

Q₈

—

1004

—

116

C₂×C₄

—

117

C₂×C₄

—

1004

—

118

C₂×C₄

—

502

—

119

C₂×C₄

—

251

—

120

C₂×C₄

D₄×D₂₅₁

yes

characteristic

121

C₂×C₄

Q₈×D₂₅₁

yes

—

122

C₂×C₄

C₂×Dic₅₀₂

yes

—

123

C₂×C₄

C₂³×D₂₅₁

yes

characteristic

124

C₈

—

251

—

125

C₈

C₂×Dic₅₀₂

yes

—

126

C₈

C₂×Dic₅₀₂

yes

—

127

C₂²

—

128

C₂²

#22548714

yes

—

129

C₂²

C₂×D₄×D₂₅₁

yes

characteristic

130

C₄

—

131

C₄

—

1004

—

132

C₄

—

502

—

133

C₄

#22548714

yes

—

134

C₄

C₂×Q₈×D₂₅₁

yes

characteristic

135

C₄

C₂²×Dic₅₀₂

yes

characteristic

136

C₂

—

137

C₂

C₂×#22548714

yes

characteristic

138

C₂

#30632484

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₅₁
commutator, cosocle	C₂×C₄×C₂₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#427×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂³×C₂₅₁	C₂×D₄
hypercenter	#427	D₂₅₁
hypocenter	C₂₅₁	#3162
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₂₅₁

commutator, cosocle

C₂×C₄×C₂₅₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₅₁

Fitting

#427×C₂₅₁

C₂

socle

C₂²×C₂₅₁

C₂²×D₄

prime radical

C₂³×C₂₅₁

C₂×D₄

D₂₅₁

C₂₅₁

G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #30632095	C₂	semidirect
3. #30632094	C₂	semidirect
27. C₂×C₂₅₁⋊C₈	Q₈	semidirect
38. C₂×C₈×C₂₅₁	Q₈	semidirect
69. C₂₅₁	#3162	semidirect
100. C₂×C₈	Dic₅₀₂	semidirect

H ≅ G/N

type

2. #30632095

C₂

semidirect

3. #30632094

C₂

semidirect

27. C₂×C₂₅₁⋊C₈

Q₈

semidirect

38. C₂×C₈×C₂₅₁

Q₈

semidirect

69. C₂₅₁

#3162

semidirect

100. C₂×C₈

Dic₅₀₂

semidirect

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

subgroup	index
#30632240	2
#30632095	2
#30632094	2
#30631725	2
#30631723	2
C₂×#22548625	2
#427×C₂₅₁	2
#3162	251

subgroup

index

#427×C₂₅₁

251

Subgroups — #44560888

Conjugacy classes of subgroups

Complements

Maximal subgroups