Group #44560710 — subgroups

Subgroups — #44560710

Order 32128.

392 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 133 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	#30632244	C₂	1	1	yes	maximal, characteristic, complemented
2	#30632055	C₂	1	2	yes	maximal, complemented
3	#30632051	C₂	1	2	yes	maximal, complemented
4	#30632046	C₂	1	4	yes	maximal, complemented
5	#30631683	C₂	1	4	yes	maximal, complemented
6	C₂×#22548626	C₂	1	1	yes	maximal, characteristic, complemented
7	#532×C₂₅₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₂₅₁⋊C₈	C₂²	1	2	yes	—
9	#22548715	C₂²	1	2	yes	—
10	#22548707	C₂²	1	2	yes	—
11	#22548694	—	2	2	no	—
12	C₄×Dic₅₀₂	—	2	2	no	—
13	#22548674	C₂²	1	2	yes	—
14	#22548639	C₂²	1	8	yes	—
15	#22548636	C₂²	1	8	yes	—
16	#22548626	C₂²	1	2	yes	—
17	#22548626	C₂²	1	2	yes	—
18	C₂×#41×C₂₅₁	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
20	#127×C₂₅₁	C₂²	1	1	yes	characteristic
21	#124×C₂₅₁	C₂²	1	2	yes	—
22	#121×C₂₅₁	—	2	2	no	—
23	C₄×D₄×C₂₅₁	—	2	2	no	—
24	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
25	C₂×Dic₅₀₂	—	2	2	no	—
26	C₂×C₂₅₁⋊C₈	—	2	4	no	—
27	C₂×C₂₅₁⋊C₈	C₂³	1	4	yes	—
28	C₄⋊Dic₂₅₁	—	2	2	no	—
29	C₄⋊Dic₂₅₁	C₂³	1	4	yes	—
30	Dic₂₅₁⋊C₄	—	2	4	no	—
31	C₂²⋊Dic₂₅₁	—	2	2	no	—
32	C₄×Dic₂₅₁	—	2	2	no	—
33	#41×C₂₅₁	—	2	8	no	—
34	C₂×Q₈×C₂₅₁	C₂³	1	2	yes	—
35	C₂×D₄×C₂₅₁	—	2	2	no	—
36	C₂×D₄×C₂₅₁	—	2	2	no	—
37	C₂×D₄×C₂₅₁	C₂³	1	2	yes	—
38	C₂²×C₄×C₂₅₁	—	2	2	no	—
39	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
40	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₂₅₁	—	2	2	no	—
42	C₂²⋊C₄×C₂₅₁	—	2	4	no	—
43	C₄²×C₂₅₁	D₄	1	2	yes	—
44	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
45	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
46	Dic₅₀₂	—	4	4	no	—
47	C₂×Dic₂₅₁	—	2	4	no	—
48	C₂₅₁⋊C₈	—	2	4	no	—
49	C₂³×C₂₅₁	—	2	2	no	—
50	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₂₅₁	—	2	4	no	—
52	D₄×C₂₅₁	—	2	4	no	—
53	D₄×C₂₅₁	—	4	4	no	—
54	D₄×C₂₅₁	—	4	4	no	—
55	C₂×C₄×C₂₅₁	—	2	2	no	—
56	C₂×C₄×C₂₅₁	—	4	2	no	—
57	C₂×C₄×C₂₅₁	—	4	4	no	—
58	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₂₅₁	#41	1	4	yes	—
62	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
63	C₂²×C₂₅₁	—	2	2	no	—
64	C₂²×C₂₅₁	—	4	2	no	—
65	C₂²×C₂₅₁	—	4	4	no	—
66	C₂²×C₂₅₁	#124	1	2	yes	—
67	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₂₅₁	—	2	2	no	—
69	C₄×C₂₅₁	—	2	2	no	—
70	C₄×C₂₅₁	—	4	2	no	—
71	C₄×C₂₅₁	#124	1	2	yes	—
72	C₄×C₂₅₁	C₂×#41	1	2	yes	—
73	Dic₂₅₁	—	4	4	no	—
74	C₂×C₂₅₁	—	2	2	no	—
75	C₂×C₂₅₁	—	8	2	no	—
76	C₂×C₂₅₁	C₂×#124	1	1	yes	characteristic
77	C₂×C₂₅₁	#575	1	2	yes	—
78	C₂₅₁	#3135	1	1	yes	characteristic, complemented, Sylow
79	#3135	—	251	1	no	maximal, Sylow
80	#533	—	251	1	no	—
81	#532	D₂₅₁	1	1	yes	characteristic
82	#491	—	251	2	no	—
83	#490	—	251	2	no	—
84	#489	—	251	4	no	—
85	#416	—	251	4	no	—
86	C₂×#97	—	251	1	no	—
87	C₂×#41	C₂×D₂₅₁	1	2	yes	—
88	C₂²×C₈	—	251	2	no	—
89	C₄⋊Q₈	—	251	2	no	—
90	C₄⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
91	#127	C₂×D₂₅₁	1	1	yes	characteristic
92	#125	—	251	2	no	—
93	#124	C₂×D₂₅₁	1	2	yes	—
94	C₂²⋊Q₈	—	502	2	no	—
95	#121	—	2	2	no	—
96	C₄×Q₈	—	502	2	no	—
97	C₄×D₄	—	2	2	no	—
98	#117	—	251	2	no	—
99	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
100	#103	—	251	8	no	—
101	#102	—	251	8	no	—
102	#97	—	251	2	no	—
103	#97	—	251	2	no	—
104	#41	—	2	8	no	—
105	C₂×Q₈	—	502	2	no	—
106	C₂×Q₈	C₂²×D₂₅₁	1	2	yes	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	C₂²×D₂₅₁	1	2	yes	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	C₂²⋊D₂₅₁	1	2	yes	—
112	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
113	C₂×C₈	—	502	4	no	—
114	C₂×C₈	—	251	4	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	502	4	no	—
117	C₄⋊C₄	—	502	2	no	—
118	C₄⋊C₄	—	251	4	no	—
119	C₂²⋊C₄	—	2	4	no	—
120	C₂²⋊C₄	—	502	2	no	—
121	C₄²	—	502	2	no	—
122	C₄²	C₂²⋊D₂₅₁	1	2	yes	—
123	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
124	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
125	C₂³	—	2	2	no	—
126	C₂³	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
127	Q₈	—	2	4	no	—
128	Q₈	—	1004	4	no	—
129	D₄	—	2	4	no	—
130	D₄	—	4	4	no	—
131	D₄	—	4	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	502	4	no	—
136	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
139	C₂×C₄	#16860809	1	4	yes	—
140	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
141	C₈	—	502	4	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	4	4	no	—
145	C₂²	#22548702	1	2	yes	—
146	C₂²	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	2	no	—
150	C₄	—	1004	4	no	—
151	C₄	#22548702	1	2	yes	—
152	C₄	C₂×#16860809	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	—	8	2	no	—
155	C₂	C₂×#22548702	1	1	yes	characteristic
156	C₂	#30632485	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#30632244

C₂

yes

maximal, characteristic, complemented

#30632055

C₂

yes

maximal, complemented

#30632051

C₂

yes

maximal, complemented

#30632046

C₂

yes

maximal, complemented

#30631683

C₂

yes

maximal, complemented

C₂×#22548626

C₂

yes

maximal, characteristic, complemented

#532×C₂₅₁

C₂

yes

maximal, characteristic

C₂²×C₂₅₁⋊C₈

C₂²

yes

—

#22548715

C₂²

yes

—

#22548707

C₂²

yes

—

#22548694

—

C₄×Dic₅₀₂

—

#22548674

C₂²

yes

—

#22548639

C₂²

yes

—

#22548636

C₂²

yes

—

#22548626

C₂²

yes

—

#22548626

C₂²

yes

—

C₂×#41×C₂₅₁

C₂²

yes

—

C₄⋊Q₈×C₂₅₁

C₂²

yes

characteristic

#127×C₂₅₁

C₂²

yes

characteristic

#124×C₂₅₁

C₂²

yes

—

#121×C₂₅₁

—

C₄×D₄×C₂₅₁

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

C₂×Dic₅₀₂

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

C₂³

yes

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

C₂³

yes

—

Dic₂₅₁⋊C₄

—

C₂²⋊Dic₂₅₁

—

C₄×Dic₂₅₁

—

#41×C₂₅₁

—

C₂×Q₈×C₂₅₁

C₂³

yes

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

C₂³

yes

—

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

D₄

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₅₁

—

C₂²⋊C₄×C₂₅₁

—

C₄²×C₂₅₁

D₄

yes

—

C₄²×C₂₅₁

C₂³

yes

characteristic

C₄²×C₂₅₁

C₂³

yes

characteristic

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

C₂×D₄

yes

characteristic

Q₈×C₂₅₁

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

#41

yes

—

C₂×C₄×C₂₅₁

C₂⁴

yes

characteristic

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

#124

yes

—

C₂²×C₂₅₁

C₂²×D₄

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

#124

yes

—

C₄×C₂₅₁

C₂×#41

yes

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

C₂×#124

yes

characteristic

C₂×C₂₅₁

#575

yes

—

C₂₅₁

#3135

yes

characteristic, complemented, Sylow

#3135

—

251

maximal, Sylow

#533

—

251

—

#532

D₂₅₁

yes

characteristic

#491

—

251

—

#490

—

251

—

#489

—

251

—

#416

—

251

—

C₂×#97

—

251

—

C₂×#41

C₂×D₂₅₁

yes

—

C₂²×C₈

—

251

—

C₄⋊Q₈

—

251

—

C₄⋊Q₈

C₂×D₂₅₁

yes

characteristic

#127

C₂×D₂₅₁

yes

characteristic

#125

—

251

—

#124

C₂×D₂₅₁

yes

—

C₂²⋊Q₈

—

502

—

#121

—

C₄×Q₈

—

502

—

C₄×D₄

—

#117

—

251

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

100

#103

—

251

—

101

#102

—

251

—

102

#97

—

251

—

103

#97

—

251

—

104

#41

—

105

C₂×Q₈

—

502

—

106

C₂×Q₈

C₂²×D₂₅₁

yes

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

C₂²×D₂₅₁

yes

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊D₂₅₁

yes

—

112

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

113

C₂×C₈

—

502

—

114

C₂×C₈

—

251

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

502

—

117

C₄⋊C₄

—

502

—

118

C₄⋊C₄

—

251

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

502

—

121

C₄²

—

502

—

122

C₄²

C₂²⋊D₂₅₁

yes

—

123

C₄²

C₂²×D₂₅₁

yes

characteristic

124

C₄²

C₂²×D₂₅₁

yes

characteristic

125

C₂³

—

126

C₂³

C₂×C₂²⋊D₂₅₁

yes

characteristic

127

Q₈

—

128

Q₈

—

1004

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

502

—

136

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

—

137

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

—

138

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

139

C₂×C₄

#16860809

yes

—

140

C₂×C₄

C₂³×D₂₅₁

yes

characteristic

141

C₈

—

502

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

#22548702

yes

—

146

C₂²

C₂²×C₂²⋊D₂₅₁

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

1004

—

151

C₄

#22548702

yes

—

152

C₄

C₂×#16860809

yes

—

153

C₂

—

154

C₂

—

155

C₂

C₂×#22548702

yes

characteristic

156

C₂

#30632485

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₅₁
commutator, cosocle	C₂×C₄×C₂₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting, prime radical	#532×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
hypercenter	#532	D₂₅₁
hypocenter	C₂₅₁	#3135
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₅₁

commutator, cosocle

C₂×C₄×C₂₅₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₅₁

Fitting, prime radical

#532×C₂₅₁

C₂

socle

C₂²×C₂₅₁

C₂²×D₄

D₂₅₁

C₂₅₁

G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #30632244	C₂	semidirect
2. #30632055	C₂	semidirect
3. #30632051	C₂	semidirect
4. #30632046	C₂	semidirect
5. #30631683	C₂	semidirect
6. C₂×#22548626	C₂	semidirect
78. C₂₅₁	#3135	semidirect

H ≅ G/N

type

1. #30632244

C₂

semidirect

2. #30632055

C₂

semidirect

3. #30632051

C₂

semidirect

4. #30632046

C₂

semidirect

5. #30631683

C₂

semidirect

6. C₂×#22548626

C₂

semidirect

78. C₂₅₁

#3135

semidirect

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

subgroup	index
#30632244	2
#30632055	2
#30632051	2
#30632046	2
#30631683	2
C₂×#22548626	2
#532×C₂₅₁	2
#3135	251

subgroup

index

#532×C₂₅₁

251

Subgroups — #44560710

Conjugacy classes of subgroups

Complements

Maximal subgroups