Group #23752 — subgroups

Subgroups — C₂×#2103 #23752

Order 256.

331 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 151 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×#430	C₂	1	1	yes	maximal, characteristic
2	C₂×#424	C₂	1	1	yes	maximal, characteristic
3	#2103	C₂	1	8	yes	maximal, direct product factor
4	C₂×#344	C₂	1	2	yes	maximal
5	C₂×#341	C₂	1	2	yes	maximal
6	C₂×#338	C₂	1	1	yes	maximal, characteristic
7	C₂²×M₁₆	C₂²	1	2	yes	—
8	C₂²×M₁₆	C₂²	1	2	yes	—
9	C₂²×M₁₆	C₂²	1	1	yes	characteristic
10	C₂³×C₈	C₂²	1	1	yes	characteristic
11	C₂×#117	C₂²	1	1	yes	characteristic
12	#430	—	2	2	no	—
13	#430	C₂²	1	4	yes	—
14	C₂×#108	—	2	2	no	—
15	#424	—	2	2	no	—
16	#424	C₂²	1	4	yes	—
17	C₂×C₄⋊C₈	—	2	2	no	—
18	#344	C₂²	1	8	yes	—
19	#341	C₂²	1	8	yes	—
20	#338	C₂²	1	4	yes	—
21	C₂³×C₄	C₂³	1	1	yes	characteristic
22	C₂×M₁₆	—	2	2	no	—
23	C₂×M₁₆	—	2	2	no	—
24	C₂×M₁₆	—	2	2	no	—
25	C₂×M₁₆	—	2	2	no	—
26	C₂×M₁₆	—	2	2	no	—
27	C₂×M₁₆	—	2	2	no	—
28	C₂×M₁₆	—	2	2	no	—
29	C₂×M₁₆	—	2	2	no	—
30	C₂×M₁₆	—	2	2	no	—
31	C₂×M₁₆	—	2	2	no	—
32	C₂×M₁₆	—	2	2	no	—
33	C₂×M₁₆	—	2	2	no	—
34	C₂×M₁₆	D₄	1	2	yes	—
35	C₂×M₁₆	C₂³	1	4	yes	—
36	C₂×M₁₆	C₂³	1	4	yes	—
37	C₂×M₁₆	C₂³	1	2	yes	—
38	C₂²×C₈	—	2	1	no	—
39	C₂²×C₈	—	2	1	no	—
40	C₂²×C₈	—	2	2	no	—
41	C₂²×C₈	—	2	2	no	—
42	C₂²×C₈	—	2	2	no	—
43	C₂²×C₈	—	2	2	no	—
44	C₂²×C₈	D₄	1	2	yes	—
45	C₂²×C₈	C₂³	1	2	yes	—
46	#117	—	2	1	no	—
47	#117	—	2	1	no	—
48	#117	Q₈	1	2	yes	—
49	#117	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄	—	2	1	no	—
51	C₂×C₂²⋊C₄	Q₈	1	2	yes	—
52	C₂×C₄²	—	2	1	no	—
53	#108	—	2	8	no	—
54	C₄⋊C₈	—	2	8	no	—
55	C₂⁴	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄	—	2	2	no	—
57	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄	#41	1	1	yes	characteristic
61	C₂²×C₄	#41	1	1	yes	characteristic
62	C₂²×C₄	#41	1	2	yes	—
63	C₂²×C₄	#41	1	2	yes	—
64	C₂²×C₄	#41	1	2	yes	—
65	C₂²×C₄	#41	1	2	yes	—
66	C₂²×C₄	C₂⁴	1	1	yes	characteristic
67	M₁₆	—	2	4	no	—
68	M₁₆	—	2	4	no	—
69	M₁₆	—	2	4	no	—
70	M₁₆	—	2	4	no	—
71	M₁₆	—	2	4	no	—
72	M₁₆	—	2	4	no	—
73	M₁₆	—	4	2	no	—
74	M₁₆	—	4	2	no	—
75	M₁₆	—	4	2	no	—
76	M₁₆	—	4	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	—	2	4	no	—
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	—	4	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	4	2	no	—
86	C₂×C₈	—	4	2	no	—
87	C₂×C₈	—	4	2	no	—
88	C₂×C₈	—	4	2	no	—
89	C₂×C₈	—	4	2	no	—
90	C₂×C₈	C₂×D₄	1	4	yes	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₂²⋊C₄	C₂×Q₈	1	4	yes	—
95	C₄²	—	2	2	no	—
96	C₄²	—	2	2	no	—
97	C₂³	—	4	2	no	—
98	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂³	#125	1	2	yes	—
101	C₂³	#125	1	2	yes	—
102	C₂³	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄	#123	1	1	yes	characteristic
108	C₂×C₄	#123	1	1	yes	characteristic
109	C₂×C₄	#123	1	2	yes	—
110	C₂×C₄	#124	1	2	yes	—
111	C₂×C₄	#124	1	1	yes	characteristic
112	C₂×C₄	#124	1	1	yes	characteristic
113	C₂×C₄	#125	1	2	yes	—
114	C₂×C₄	#125	1	2	yes	—
115	C₂×C₄	#126	1	2	yes	—
116	C₂×C₄	#126	1	2	yes	—
117	C₂×C₄	#126	1	2	yes	—
118	C₂×C₄	#126	1	2	yes	—
119	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄	C₂×#41	1	2	yes	—
121	C₂×C₄	C₂×#41	1	2	yes	—
122	C₂×C₄	C₂×#41	1	1	yes	characteristic
123	C₈	—	2	4	no	—
124	C₈	—	4	2	no	—
125	C₈	—	4	4	no	—
126	C₈	—	4	4	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	2	4	no	—
129	C₂²	—	4	1	no	—
130	C₂²	—	4	2	no	—
131	C₂²	—	4	1	no	—
132	C₂²	#400	1	1	yes	characteristic
133	C₂²	#400	1	2	yes	—
134	C₂²	#400	1	1	yes	characteristic
135	C₂²	C₂×C₂²⋊Q₈	1	1	yes	characteristic
136	C₂²	C₂×#125	1	2	yes	—
137	C₄	—	4	4	no	—
138	C₄	#400	1	1	yes	characteristic
139	C₄	#400	1	2	yes	—
140	C₄	#400	1	1	yes	characteristic
141	C₄	C₂×#123	1	1	yes	characteristic
142	C₄	C₂×#124	1	1	yes	characteristic
143	C₄	C₂×#126	1	2	yes	—
144	C₂	—	2	1	no	—
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	2	2	no	—
148	C₂	#2103	1	2	yes	direct product factor
149	C₂	C₂×#400	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#430

C₂

yes

maximal, characteristic

C₂×#424

C₂

yes

maximal, characteristic

#2103

C₂

yes

maximal, direct product factor

C₂×#344

C₂

yes

maximal

C₂×#341

C₂

yes

maximal

C₂×#338

C₂

yes

maximal, characteristic

C₂²×M₁₆

C₂²

yes

—

C₂²×M₁₆

C₂²

yes

—

C₂²×M₁₆

C₂²

yes

characteristic

C₂³×C₈

C₂²

yes

characteristic

C₂×#117

C₂²

yes

characteristic

#430

—

#430

C₂²

yes

—

C₂×#108

—

#424

—

#424

C₂²

yes

—

C₂×C₄⋊C₈

—

#344

C₂²

yes

—

#341

C₂²

yes

—

#338

C₂²

yes

—

C₂³×C₄

C₂³

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₄

yes

—

C₂×M₁₆

C₂³

yes

—

C₂×M₁₆

C₂³

yes

—

C₂×M₁₆

C₂³

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

C₂³

yes

—

#117

—

#117

—

#117

Q₈

yes

—

#117

C₂³

yes

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

Q₈

yes

—

C₂×C₄²

—

#108

—

C₄⋊C₈

—

C₂⁴

C₂×Q₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

—

C₂²×C₄

#41

yes

—

C₂²×C₄

#41

yes

—

C₂²×C₄

#41

yes

—

C₂²×C₄

C₂⁴

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×Q₈

yes

—

C₄²

—

C₄²

—

C₂³

—

C₂³

C₂²⋊Q₈

yes

characteristic

C₂³

C₂²⋊Q₈

yes

characteristic

100

C₂³

#125

yes

—

101

C₂³

#125

yes

—

102

C₂³

C₂²×Q₈

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²⋊Q₈

yes

characteristic

106

C₂×C₄

C₂²⋊Q₈

yes

characteristic

107

C₂×C₄

#123

yes

characteristic

108

C₂×C₄

#123

yes

characteristic

109

C₂×C₄

#123

yes

—

110

C₂×C₄

#124

yes

—

111

C₂×C₄

#124

yes

characteristic

112

C₂×C₄

#124

yes

characteristic

113

C₂×C₄

#125

yes

—

114

C₂×C₄

#125

yes

—

115

C₂×C₄

#126

yes

—

116

C₂×C₄

#126

yes

—

117

C₂×C₄

#126

yes

—

118

C₂×C₄

#126

yes

—

119

C₂×C₄

C₂²×D₄

yes

characteristic

120

C₂×C₄

C₂×#41

yes

—

121

C₂×C₄

C₂×#41

yes

—

122

C₂×C₄

C₂×#41

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#400

yes

characteristic

133

C₂²

#400

yes

—

134

C₂²

#400

yes

characteristic

135

C₂²

C₂×C₂²⋊Q₈

yes

characteristic

136

C₂²

C₂×#125

yes

—

137

C₄

—

138

C₄

#400

yes

characteristic

139

C₄

#400

yes

—

140

C₄

#400

yes

characteristic

141

C₄

C₂×#123

yes

characteristic

142

C₄

C₂×#124

yes

characteristic

143

C₄

C₂×#126

yes

—

144

C₂

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#2103

yes

direct product factor

149

C₂

C₂×#400

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	#400
prime radical	C₂⁴	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#124

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴

socle

C₂²

#400

prime radical

C₂⁴

C₂×Q₈

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
3. #2103	C₂	direct (N × H)
148. C₂	#2103	direct (N × H)

H ≅ G/N

type

3. #2103

C₂

direct (N × H)

148. C₂

#2103

direct (N × H)

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

subgroup	index
C₂×#430	2
C₂×#424	2
#2103	2
C₂×#344	2
C₂×#341	2
C₂×#338	2

subgroup

index

Subgroups — C₂×#2103 #23752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#2103 #23752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2103 #23752