Group #44559082 — subgroups

Subgroups — #44559082

Order 32128.

484 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 181 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×#22548723	C₂	1	2	yes	maximal, complemented
2	#30632237	C₂	1	1	yes	maximal, characteristic
3	#30631757	C₂	1	8	yes	maximal, complemented
4	#30631655	C₂	1	2	yes	maximal, complemented
5	C₂×#22548626	C₂	1	1	yes	maximal, characteristic, complemented
6	#423×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16860810	C₂²	1	2	yes	—
8	#22548723	—	2	8	no	—
9	C₂×#16860793	C₂²	1	4	yes	—
10	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₂₅₁	C₂²	1	4	yes	—
13	#22548715	C₄	1	1	yes	characteristic
14	#22548712	C₄	1	1	yes	characteristic
15	#22548703	C₄	1	2	yes	—
16	#22548689	C₄	1	4	yes	—
17	C₄×D₁₀₀₄	C₂²	1	4	yes	—
18	(C₂×D₂₅₁)⋊C₈	C₂²	1	8	yes	—
19	#22548626	C₂²	1	4	yes	—
20	C₂×M₁₆×C₂₅₁	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₂₅₁	C₂²	1	4	yes	—
23	#16860810	—	2	4	no	—
24	#16860810	—	2	4	no	—
25	C₂×Dic₅₀₂	C₂×C₄	1	2	yes	—
26	C₂×D₁₀₀₄	C₂×C₄	1	2	yes	—
27	C₂×D₁₀₀₄	C₂×C₄	1	4	yes	—
28	C₂×C₂²⋊D₂₅₁	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₂₅₁	C₂³	1	4	yes	—
30	#16860793	—	2	8	no	—
31	C₂×C₂₅₁⋊C₈	—	2	2	no	—
32	C₂×C₂₅₁⋊C₈	D₄	1	4	yes	complemented
33	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
34	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
35	C₈×D₂₅₁	—	2	8	no	—
36	C₄⋊Dic₂₅₁	C₂×C₄	1	4	yes	—
37	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	8	yes	—
38	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
40	M₁₆×C₂₅₁	—	2	4	no	—
41	C₂×C₈×C₂₅₁	C₂³	1	4	yes	—
42	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₂₅₁	C₂³	1	2	yes	—
45	C₂²×D₂₅₁	C₂²×C₄	1	4	yes	—
46	Dic₅₀₂	—	2	4	no	—
47	D₁₀₀₄	—	2	4	no	—
48	D₁₀₀₄	—	4	4	no	—
49	C₂²⋊D₂₅₁	—	2	8	no	—
50	C₂×Dic₂₅₁	C₂²×C₄	1	4	yes	—
51	C₄×D₂₅₁	—	2	4	no	—
52	C₄×D₂₅₁	—	2	4	no	—
53	C₂₅₁⋊C₈	—	2	2	no	—
54	C₂₅₁⋊C₈	C₂×D₄	1	4	yes	—
55	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₂₅₁	—	4	2	no	—
57	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₂₅₁	#41	1	2	yes	—
62	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
63	C₈×C₂₅₁	—	2	4	no	—
64	C₂²×C₂₅₁	—	2	2	no	—
65	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
66	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
67	C₂×D₂₅₁	—	2	4	no	—
68	C₂×D₂₅₁	—	2	8	no	—
69	C₄×C₂₅₁	—	2	2	no	—
70	C₄×C₂₅₁	—	2	2	no	—
71	C₄×C₂₅₁	C₄×D₄	1	2	yes	—
72	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
74	Dic₂₅₁	—	2	4	no	—
75	C₂×C₂₅₁	—	2	2	no	—
76	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
77	C₂×C₂₅₁	#568	1	2	yes	—
78	D₂₅₁	—	4	4	no	—
79	C₂₅₁	#2932	1	1	yes	characteristic, complemented, Sylow
80	#2932	—	251	1	no	maximal, Sylow
81	C₂×#131	—	251	2	no	—
82	#532	—	251	1	no	—
83	#435	—	251	8	no	—
84	#423	D₂₅₁	1	1	yes	characteristic, complemented
85	#408	—	251	2	no	—
86	C₂×#97	—	251	1	no	—
87	C₂×#41	—	251	2	no	—
88	#131	—	502	8	no	—
89	C₂×M₁₆	—	251	4	no	—
90	C₂×M₁₆	C₂×D₂₅₁	1	2	yes	—
91	C₂²×C₈	—	251	1	no	—
92	C₂²×C₈	—	251	4	no	—
93	C₂²×C₈	—	251	1	no	—
94	C₄⋊Q₈	—	251	1	no	—
95	#127	—	251	1	no	—
96	#124	—	251	2	no	—
97	#121	—	251	4	no	—
98	C₄×D₄	—	251	4	no	—
99	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
100	C₄⋊C₈	C₂×D₂₅₁	1	4	yes	—
101	C₂²⋊C₈	—	251	8	no	—
102	#97	—	251	4	no	—
103	#41	—	502	4	no	—
104	#41	—	502	4	no	—
105	C₂×Q₈	—	251	2	no	—
106	C₂×D₄	—	251	4	no	—
107	C₂×D₄	—	251	4	no	—
108	C₂×D₄	—	251	2	no	—
109	C₂²×C₄	—	251	4	no	—
110	C₂²×C₄	C₄×D₂₅₁	1	2	yes	—
111	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
112	M₁₆	—	2	4	no	—
113	M₁₆	—	502	8	no	—
114	C₂×C₈	—	502	8	no	—
115	C₂×C₈	—	502	2	no	—
116	C₂×C₈	—	251	2	no	—
117	C₂×C₈	—	251	4	no	—
118	C₂×C₈	—	251	2	no	—
119	C₂×C₈	C₂²×D₂₅₁	1	4	yes	—
120	C₄⋊C₄	—	251	4	no	—
121	C₂²⋊C₄	—	251	8	no	—
122	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
123	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
124	C₄²	C₂²×D₂₅₁	1	2	yes	—
125	C₂³	—	251	4	no	—
126	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
127	Q₈	—	502	4	no	—
128	D₄	—	1004	4	no	—
129	D₄	—	502	8	no	—
130	D₄	—	502	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	502	4	no	—
133	C₂×C₄	—	502	4	no	—
134	C₂×C₄	—	251	4	no	—
135	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
136	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
138	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
139	C₂×C₄	#16860811	1	2	yes	—
140	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
141	C₈	—	2	4	no	—
142	C₈	—	502	2	no	—
143	C₈	—	251	4	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	502	8	no	—
146	C₂²	—	502	4	no	—
147	C₂²	#22548677	1	2	yes	—
148	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	502	4	no	—
152	C₄	#22548677	1	2	yes	—
153	C₄	C₂×D₄×D₂₅₁	1	1	yes	characteristic
154	C₄	C₂×#16860811	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	1004	4	no	—
157	C₂	C₂×#22548677	1	1	yes	characteristic
158	C₂	#30632453	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22548723

C₂

yes

maximal, complemented

#30632237

C₂

yes

maximal, characteristic

#30631757

C₂

yes

maximal, complemented

#30631655

C₂

yes

maximal, complemented

C₂×#22548626

C₂

yes

maximal, characteristic, complemented

#423×C₂₅₁

C₂

yes

maximal, characteristic, complemented

C₂×#16860810

C₂²

yes

—

#22548723

—

C₂×#16860793

C₂²

yes

—

C₂²×C₂₅₁⋊C₈

C₂²

yes

characteristic

C₂²×C₂₅₁⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₂₅₁

C₂²

yes

—

#22548715

C₄

yes

characteristic

#22548712

C₄

yes

characteristic

#22548703

C₄

yes

—

#22548689

C₄

yes

—

C₄×D₁₀₀₄

C₂²

yes

—

(C₂×D₂₅₁)⋊C₈

C₂²

yes

—

#22548626

C₂²

yes

—

C₂×M₁₆×C₂₅₁

C₂²

yes

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₅₁

C₂²

yes

—

#16860810

—

#16860810

—

C₂×Dic₅₀₂

C₂×C₄

yes

—

C₂×D₁₀₀₄

C₂×C₄

yes

—

C₂×D₁₀₀₄

C₂×C₄

yes

—

C₂×C₂²⋊D₂₅₁

C₂×C₄

yes

—

C₂×C₄×D₂₅₁

C₂³

yes

—

#16860793

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

D₄

yes

complemented

C₂×C₂₅₁⋊C₈

C₂³

yes

—

C₂×C₂₅₁⋊C₈

C₂³

yes

—

C₈×D₂₅₁

—

C₄⋊Dic₂₅₁

C₂×C₄

yes

—

(C₂×D₂₅₁)⋊C₄

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

M₁₆×C₂₅₁

—

C₂×C₈×C₂₅₁

C₂³

yes

—

C₄²×C₂₅₁

C₂×C₄

yes

characteristic

C₄²×C₂₅₁

C₂×C₄

yes

characteristic

C₄²×C₂₅₁

C₂³

yes

—

C₂²×D₂₅₁

C₂²×C₄

yes

—

Dic₅₀₂

—

D₁₀₀₄

—

D₁₀₀₄

—

C₂²⋊D₂₅₁

—

C₂×Dic₂₅₁

C₂²×C₄

yes

—

C₄×D₂₅₁

—

C₄×D₂₅₁

—

C₂₅₁⋊C₈

—

C₂₅₁⋊C₈

C₂×D₄

yes

—

C₂³×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

#41

yes

—

C₂×C₄×C₂₅₁

C₂⁴

yes

characteristic

C₈×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₄×D₄

yes

—

C₂²×C₂₅₁

C₂³×C₄

yes

characteristic

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

C₄×D₄

yes

—

C₄×C₂₅₁

C₂²×D₄

yes

characteristic

C₄×C₂₅₁

C₂×#41

yes

characteristic

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

C₂×C₄×D₄

yes

characteristic

C₂×C₂₅₁

#568

yes

—

D₂₅₁

—

C₂₅₁

#2932

yes

characteristic, complemented, Sylow

#2932

—

251

maximal, Sylow

C₂×#131

—

251

—

#532

—

251

—

#435

—

251

—

#423

D₂₅₁

yes

characteristic, complemented

#408

—

251

—

C₂×#97

—

251

—

C₂×#41

—

251

—

#131

—

502

—

C₂×M₁₆

—

251

—

C₂×M₁₆

C₂×D₂₅₁

yes

—

C₂²×C₈

—

251

—

C₂²×C₈

—

251

—

C₂²×C₈

—

251

—

C₄⋊Q₈

—

251

—

#127

—

251

—

#124

—

251

—

#121

—

251

—

C₄×D₄

—

251

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

100

C₄⋊C₈

C₂×D₂₅₁

yes

—

101

C₂²⋊C₈

—

251

—

102

#97

—

251

—

103

#41

—

502

—

104

#41

—

502

—

105

C₂×Q₈

—

251

—

106

C₂×D₄

—

251

—

107

C₂×D₄

—

251

—

108

C₂×D₄

—

251

—

109

C₂²×C₄

—

251

—

110

C₂²×C₄

C₄×D₂₅₁

yes

—

111

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

112

M₁₆

—

113

M₁₆

—

502

—

114

C₂×C₈

—

502

—

115

C₂×C₈

—

502

—

116

C₂×C₈

—

251

—

117

C₂×C₈

—

251

—

118

C₂×C₈

—

251

—

119

C₂×C₈

C₂²×D₂₅₁

yes

—

120

C₄⋊C₄

—

251

—

121

C₂²⋊C₄

—

251

—

122

C₄²

C₄×D₂₅₁

yes

characteristic

123

C₄²

C₄×D₂₅₁

yes

characteristic

124

C₄²

C₂²×D₂₅₁

yes

—

125

C₂³

—

251

—

126

C₂³

C₂×C₄×D₂₅₁

yes

characteristic

127

Q₈

—

502

—

128

D₄

—

1004

—

129

D₄

—

502

—

130

D₄

—

502

—

131

C₂×C₄

—

132

C₂×C₄

—

502

—

133

C₂×C₄

—

502

—

134

C₂×C₄

—

251

—

135

C₂×C₄

C₂×C₄×D₂₅₁

yes

—

136

C₂×C₄

C₂×C₄×D₂₅₁

yes

—

137

C₂×C₄

C₂×C₄×D₂₅₁

yes

characteristic

138

C₂×C₄

D₄×D₂₅₁

yes

—

139

C₂×C₄

#16860811

yes

—

140

C₂×C₄

C₂³×D₂₅₁

yes

characteristic

141

C₈

—

142

C₈

—

502

—

143

C₈

—

251

—

144

C₂²

—

145

C₂²

—

502

—

146

C₂²

—

502

—

147

C₂²

#22548677

yes

—

148

C₂²

C₂²×C₄×D₂₅₁

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

502

—

152

C₄

#22548677

yes

—

153

C₄

C₂×D₄×D₂₅₁

yes

characteristic

154

C₄

C₂×#16860811

yes

characteristic

155

C₂

—

156

C₂

—

1004

—

157

C₂

C₂×#22548677

yes

characteristic

158

C₂

#30632453

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₅₁
commutator, socle	C₂²×C₂₅₁	C₂³×C₄
Fitting	#423×C₂₅₁	C₂
prime radical	#30632237	C₂
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	#423	D₂₅₁
hypocenter	C₂₅₁	#2932
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₂₅₁

commutator, socle

C₂²×C₂₅₁

C₂³×C₄

Fitting

#423×C₂₅₁

C₂

prime radical

#30632237

C₂

cosocle

C₂×C₄×C₂₅₁

C₂⁴

D₂₅₁

C₂₅₁

G ▷ C₂²×C₂₅₁ ▷ C₂₅₁

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
1. C₂×#22548723	C₂	semidirect
3. #30631757	C₂	semidirect
4. #30631655	C₂	semidirect
5. C₂×#22548626	C₂	semidirect
6. #423×C₂₅₁	C₂	semidirect
32. C₂×C₂₅₁⋊C₈	D₄	semidirect
79. C₂₅₁	#2932	semidirect
84. #423	D₂₅₁	semidirect

H ≅ G/N

type

1. C₂×#22548723

C₂

semidirect

3. #30631757

C₂

semidirect

4. #30631655

C₂

semidirect

5. C₂×#22548626

C₂

semidirect

6. #423×C₂₅₁

C₂

semidirect

32. C₂×C₂₅₁⋊C₈

D₄

semidirect

79. C₂₅₁

#2932

semidirect

84. #423

D₂₅₁

semidirect

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

subgroup	index
C₂×#22548723	2
#30632237	2
#30631757	2
#30631655	2
C₂×#22548626	2
#423×C₂₅₁	2
#2932	251

subgroup

index

#423×C₂₅₁

251

Subgroups — #44559082

Conjugacy classes of subgroups

Complements

Maximal subgroups