Group #27180520 — subgroups

Subgroups — C₂×#20265367 #27180520

Order 12928.

794 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 165 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×#15692236	C₂	1	3	yes	maximal, complemented
2	#20265367	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15692170	C₂	1	3	yes	maximal, complemented
4	C₂×H(5,𝔽₂)×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic, complemented
6	#15692236	—	2	6	no	—
7	#15692236	C₂²	1	12	yes	complemented
8	C₂×#13108053	—	2	3	no	—
9	C₂×C₂²⋊Dic₁₀₁	—	2	3	no	—
10	C₂×C₂²⋊Dic₁₀₁	C₂²	1	3	yes	complemented
11	#15692170	C₂²	1	12	yes	complemented
12	H(5,𝔽₂)×C₁₀₁	—	2	6	no	—
13	H(5,𝔽₂)×C₁₀₁	C₂²	1	4	yes	—
14	C₂×#41×C₁₀₁	—	2	3	no	—
15	C₂²×D₄×C₁₀₁	—	2	3	no	—
16	C₂²×D₄×C₁₀₁	C₂²	1	3	yes	—
17	C₂²×Dic₁₀₁	—	2	3	no	—
18	C₂²×Dic₁₀₁	—	2	3	no	—
19	C₂²×Dic₁₀₁	—	2	3	no	—
20	C₂²×Dic₁₀₁	C₂³	1	2	yes	complemented
21	#13108053	—	2	12	no	—
22	C₂²⋊Dic₁₀₁	—	2	12	no	—
23	C₂²⋊Dic₁₀₁	—	2	3	no	—
24	C₂²⋊Dic₁₀₁	C₂³	1	6	yes	complemented
25	C₂⁴×C₁₀₁	—	2	1	no	—
26	C₂⁴×C₁₀₁	D₄	1	3	yes	—
27	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
28	#41×C₁₀₁	—	2	12	no	—
29	#41×C₁₀₁	—	2	12	no	—
30	C₂×Q₈×C₁₀₁	—	2	1	no	—
31	C₂×D₄×C₁₀₁	—	2	3	no	—
32	C₂×D₄×C₁₀₁	—	2	12	no	—
33	C₂×D₄×C₁₀₁	—	2	6	no	—
34	C₂×D₄×C₁₀₁	—	2	12	no	—
35	C₂×D₄×C₁₀₁	—	2	6	no	—
36	C₂×D₄×C₁₀₁	D₄	1	6	yes	—
37	C₂×D₄×C₁₀₁	C₂³	1	6	yes	—
38	C₂²×C₄×C₁₀₁	—	2	3	no	—
39	C₂²×C₄×C₁₀₁	D₄	1	3	yes	—
40	C₂³×C₁₀₁	—	2	4	no	—
41	C₂³×C₁₀₁	—	2	3	no	—
42	C₂³×C₁₀₁	—	2	3	no	—
43	C₂³×C₁₀₁	—	4	2	no	—
44	C₂³×C₁₀₁	—	4	6	no	—
45	C₂³×C₁₀₁	—	8	2	no	—
46	C₂³×C₁₀₁	C₂×D₄	1	3	yes	—
47	C₂³×C₁₀₁	C₂×D₄	1	6	yes	—
48	C₂³×C₁₀₁	C₂×D₄	1	3	yes	—
49	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
50	Q₈×C₁₀₁	—	2	4	no	—
51	D₄×C₁₀₁	—	2	12	no	—
52	D₄×C₁₀₁	—	2	12	no	—
53	D₄×C₁₀₁	—	2	12	no	—
54	C₂×C₄×C₁₀₁	—	2	12	no	—
55	C₂×C₄×C₁₀₁	—	2	3	no	—
56	C₂×C₄×C₁₀₁	—	2	3	no	—
57	C₂×C₄×C₁₀₁	C₂×D₄	1	6	yes	—
58	C₂×Dic₁₀₁	—	2	6	no	—
59	C₂×Dic₁₀₁	—	2	6	no	—
60	C₂×Dic₁₀₁	—	4	6	no	—
61	C₂×Dic₁₀₁	—	4	3	no	—
62	C₂×Dic₁₀₁	—	4	1	no	—
63	C₂²×C₁₀₁	—	2	6	no	—
64	C₂²×C₁₀₁	—	2	6	no	—
65	C₂²×C₁₀₁	—	4	3	no	—
66	C₂²×C₁₀₁	—	4	1	no	—
67	C₂²×C₁₀₁	—	4	6	no	—
68	C₂²×C₁₀₁	—	4	6	no	—
69	C₂²×C₁₀₁	—	8	3	no	—
70	C₂²×C₁₀₁	—	8	4	no	—
71	C₂²×C₁₀₁	#120	1	3	yes	—
72	C₂²×C₁₀₁	#120	1	1	yes	characteristic
73	C₂²×C₁₀₁	C₂²×D₄	1	3	yes	—
74	C₄×C₁₀₁	—	2	6	no	—
75	Dic₁₀₁	—	4	6	no	—
76	Dic₁₀₁	—	4	2	no	—
77	C₂×C₁₀₁	—	2	3	no	—
78	C₂×C₁₀₁	—	2	3	no	—
79	C₂×C₁₀₁	—	4	6	no	—
80	C₂×C₁₀₁	#458	1	2	yes	complemented
81	C₂×C₁₀₁	C₂×#120	1	1	yes	characteristic
82	C₂×#458	—	101	1	no	maximal, Sylow
83	C₁₀₁	C₂×#458	1	1	yes	characteristic, complemented, Sylow
84	C₂×H(5,𝔽₂)	D₁₀₁	1	1	yes	characteristic
85	C₂×#123	—	101	3	no	—
86	#458	—	101	8	no	—
87	C₂×#99	—	101	3	no	—
88	H(5,𝔽₂)	—	2	6	no	—
89	H(5,𝔽₂)	C₂×D₁₀₁	1	4	yes	—
90	C₂×#41	—	2	3	no	—
91	C₂²×D₄	—	2	3	no	—
92	C₂²×D₄	C₂×D₁₀₁	1	3	yes	—
93	C₂³×C₄	—	101	1	no	—
94	#123	—	101	12	no	—
95	#123	—	202	6	no	—
96	C₂×C₄⋊C₄	—	202	3	no	—
97	C₂×C₂²⋊C₄	—	101	3	no	—
98	C₂×C₂²⋊C₄	—	202	3	no	—
99	#99	—	101	12	no	—
100	C₂⁴	—	2	1	no	—
101	C₂⁴	C₂²⋊D₁₀₁	1	3	yes	—
102	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
103	#41	—	2	12	no	—
104	#41	—	2	12	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×D₄	—	2	3	no	—
107	C₂×D₄	—	2	12	no	—
108	C₂×D₄	—	2	6	no	—
109	C₂×D₄	—	2	12	no	—
110	C₂×D₄	—	2	6	no	—
111	C₂×D₄	C₂²⋊D₁₀₁	1	6	yes	—
112	C₂×D₄	C₂²×D₁₀₁	1	6	yes	—
113	C₂²×C₄	—	2	3	no	—
114	C₂²×C₄	—	101	2	no	—
115	C₂²×C₄	—	202	3	no	—
116	C₂²×C₄	—	202	3	no	—
117	C₂²×C₄	—	202	3	no	—
118	C₂²×C₄	C₂²⋊D₁₀₁	1	3	yes	—
119	C₄⋊C₄	—	202	12	no	—
120	C₂²⋊C₄	—	101	6	no	—
121	C₂²⋊C₄	—	202	3	no	—
122	C₂²⋊C₄	—	202	12	no	—
123	C₂³	—	2	4	no	—
124	C₂³	—	2	3	no	—
125	C₂³	—	2	3	no	—
126	C₂³	—	4	2	no	—
127	C₂³	—	4	6	no	—
128	C₂³	—	8	2	no	—
129	C₂³	C₂×C₂²⋊D₁₀₁	1	3	yes	—
130	C₂³	C₂×C₂²⋊D₁₀₁	1	6	yes	—
131	C₂³	C₂×C₂²⋊D₁₀₁	1	3	yes	—
132	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
133	Q₈	—	2	4	no	—
134	D₄	—	2	12	no	—
135	D₄	—	2	12	no	—
136	D₄	—	2	12	no	—
137	C₂×C₄	—	2	12	no	—
138	C₂×C₄	—	2	3	no	—
139	C₂×C₄	—	2	3	no	—
140	C₂×C₄	—	404	6	no	—
141	C₂×C₄	—	404	3	no	—
142	C₂×C₄	—	404	1	no	—
143	C₂×C₄	—	202	6	no	—
144	C₂×C₄	—	202	6	no	—
145	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	6	yes	—
146	C₂²	—	2	6	no	—
147	C₂²	—	2	6	no	—
148	C₂²	—	4	3	no	—
149	C₂²	—	4	1	no	—
150	C₂²	—	4	6	no	—
151	C₂²	—	4	6	no	—
152	C₂²	—	8	3	no	—
153	C₂²	—	8	4	no	—
154	C₂²	#15692224	1	3	yes	—
155	C₂²	#15692224	1	1	yes	characteristic
156	C₂²	C₂²×C₂²⋊D₁₀₁	1	3	yes	—
157	C₄	—	2	6	no	—
158	C₄	—	404	6	no	—
159	C₄	—	404	2	no	—
160	C₂	—	2	3	no	—
161	C₂	—	2	3	no	—
162	C₂	—	4	6	no	—
163	C₂	#20265367	1	2	yes	direct product factor
164	C₂	C₂×#15692224	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692236

C₂

yes

maximal, complemented

#20265367

C₂

yes

maximal, direct product factor

C₂×#15692170

C₂

yes

maximal, complemented

C₂×H(5,𝔽₂)×C₁₀₁

C₂

yes

maximal, characteristic

C₂³×Dic₁₀₁

C₂²

yes

characteristic, complemented

#15692236

—

#15692236

C₂²

yes

complemented

C₂×#13108053

—

C₂×C₂²⋊Dic₁₀₁

—

C₂×C₂²⋊Dic₁₀₁

C₂²

yes

complemented

#15692170

C₂²

yes

complemented

H(5,𝔽₂)×C₁₀₁

—

H(5,𝔽₂)×C₁₀₁

C₂²

yes

—

C₂×#41×C₁₀₁

—

C₂²×D₄×C₁₀₁

—

C₂²×D₄×C₁₀₁

C₂²

yes

—

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

C₂³

yes

complemented

#13108053

—

C₂²⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

C₂³

yes

complemented

C₂⁴×C₁₀₁

—

C₂⁴×C₁₀₁

D₄

yes

—

C₂⁴×C₁₀₁

C₂³

yes

characteristic

#41×C₁₀₁

—

#41×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

D₄

yes

—

C₂×D₄×C₁₀₁

C₂³

yes

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂×D₄

yes

—

C₂³×C₁₀₁

C₂×D₄

yes

—

C₂³×C₁₀₁

C₂×D₄

yes

—

C₂³×C₁₀₁

C₂⁴

yes

characteristic

Q₈×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#120

yes

—

C₂²×C₁₀₁

#120

yes

characteristic

C₂²×C₁₀₁

C₂²×D₄

yes

—

C₄×C₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

101

maximal, Sylow

C₁₀₁

C₂×#458

yes

characteristic, complemented, Sylow

C₂×H(5,𝔽₂)

D₁₀₁

yes

characteristic

C₂×#123

—

101

—

#458

—

101

—

C₂×#99

—

101

—

H(5,𝔽₂)

—

H(5,𝔽₂)

C₂×D₁₀₁

yes

—

C₂×#41

—

C₂²×D₄

—

C₂²×D₄

C₂×D₁₀₁

yes

—

C₂³×C₄

—

101

—

#123

—

101

—

#123

—

202

—

C₂×C₄⋊C₄

—

202

—

C₂×C₂²⋊C₄

—

101

—

C₂×C₂²⋊C₄

—

202

—

#99

—

101

—

100

C₂⁴

—

101

C₂⁴

C₂²⋊D₁₀₁

yes

—

102

C₂⁴

C₂²×D₁₀₁

yes

characteristic

103

#41

—

104

#41

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

C₂²⋊D₁₀₁

yes

—

112

C₂×D₄

C₂²×D₁₀₁

yes

—

113

C₂²×C₄

—

114

C₂²×C₄

—

101

—

115

C₂²×C₄

—

202

—

116

C₂²×C₄

—

202

—

117

C₂²×C₄

—

202

—

118

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

119

C₄⋊C₄

—

202

—

120

C₂²⋊C₄

—

101

—

121

C₂²⋊C₄

—

202

—

122

C₂²⋊C₄

—

202

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

C₂×C₂²⋊D₁₀₁

yes

—

130

C₂³

C₂×C₂²⋊D₁₀₁

yes

—

131

C₂³

C₂×C₂²⋊D₁₀₁

yes

—

132

C₂³

C₂³×D₁₀₁

yes

characteristic

133

Q₈

—

134

D₄

—

135

D₄

—

136

D₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

404

—

141

C₂×C₄

—

404

—

142

C₂×C₄

—

404

—

143

C₂×C₄

—

202

—

144

C₂×C₄

—

202

—

145

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

#15692224

yes

—

155

C₂²

#15692224

yes

characteristic

156

C₂²

C₂²×C₂²⋊D₁₀₁

yes

—

157

C₄

—

158

C₄

—

404

—

159

C₄

—

404

—

160

C₂

—

161

C₂

—

162

C₂

—

163

C₂

#20265367

yes

direct product factor

164

C₂

C₂×#15692224

yes

characteristic

	subgroup	G/H
center	C₂²	#15692224
commutator, cosocle	C₂³×C₁₀₁	C₂⁴
Frattini	C₂³	C₂³×D₁₀₁
Fitting, prime radical	C₂×H(5,𝔽₂)×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#120
hypercenter	C₂×H(5,𝔽₂)	D₁₀₁
hypocenter	C₁₀₁	C₂×#458
LCS	G ▷ C₂³×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×H(5,𝔽₂)

subgroup

G/H

center

C₂²

#15692224

commutator, cosocle

C₂³×C₁₀₁

C₂⁴

Frattini

C₂³

C₂³×D₁₀₁

Fitting, prime radical

C₂×H(5,𝔽₂)×C₁₀₁

C₂

socle

C₂²×C₁₀₁

#120

hypercenter

C₂×H(5,𝔽₂)

D₁₀₁

hypocenter

C₁₀₁

C₂×#458

LCS

G ▷ C₂³×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×H(5,𝔽₂)

N	H ≅ G/N	type
1. C₂×#15692236	C₂	semidirect
2. #20265367	C₂	direct (N × H)
3. C₂×#15692170	C₂	semidirect
5. C₂³×Dic₁₀₁	C₂²	semidirect
7. #15692236	C₂²	semidirect
10. C₂×C₂²⋊Dic₁₀₁	C₂²	semidirect
11. #15692170	C₂²	semidirect
20. C₂²×Dic₁₀₁	C₂³	semidirect
24. C₂²⋊Dic₁₀₁	C₂³	semidirect
80. C₂×C₁₀₁	#458	semidirect
83. C₁₀₁	C₂×#458	semidirect
163. C₂	#20265367	direct (N × H)

H ≅ G/N

type

1. C₂×#15692236

C₂

semidirect

2. #20265367

C₂

direct (N × H)

3. C₂×#15692170

C₂

semidirect

5. C₂³×Dic₁₀₁

C₂²

semidirect

7. #15692236

C₂²

semidirect

10. C₂×C₂²⋊Dic₁₀₁

C₂²

semidirect

11. #15692170

C₂²

semidirect

20. C₂²×Dic₁₀₁

C₂³

semidirect

24. C₂²⋊Dic₁₀₁

C₂³

semidirect

80. C₂×C₁₀₁

#458

semidirect

83. C₁₀₁

C₂×#458

semidirect

163. C₂

#20265367

direct (N × H)

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

subgroup	index
C₂×#15692236	2
#20265367	2
C₂×#15692170	2
C₂×H(5,𝔽₂)×C₁₀₁	2
C₂×#458	101

subgroup

index

C₂×H(5,𝔽₂)×C₁₀₁

101

Subgroups — C₂×#20265367 #27180520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20265367 #27180520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265367 #27180520