Group #15682184 — subgroups

Subgroups — C₂×#13092524 #15682184

Order 2944.

480 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 237 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×Q₈×Dic₂₃	C₂	1	2	yes	maximal
2	#13092524	C₂	1	8	yes	maximal, direct product factor
3	C₂×#10624999	C₂	1	1	yes	maximal, characteristic
4	C₂×#10625017	C₂	1	2	yes	maximal
5	C₂×#10624997	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
8	C₂×Q₁₆×C₂₃	C₄	1	8	yes	complemented
9	C₂×Q₁₆×C₂₃	C₂²	1	4	yes	—
10	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	Q₈×Dic₂₃	—	2	4	no	—
12	Q₈×Dic₂₃	C₂²	1	8	yes	—
13	C₂×C₄⋊Dic₂₃	—	2	2	no	—
14	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
15	C₂×C₄×Dic₂₃	—	2	2	no	—
16	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
17	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
18	#10625017	C₂²	1	8	yes	—
19	#10624999	C₂²	1	4	yes	—
20	#10624997	C₂²	1	4	yes	—
21	C₂×Q₈×C₂₃	—	2	2	no	—
22	C₂×Q₈×C₂₃	—	2	4	no	—
23	C₂×Q₈×C₂₃	C₂×C₄	1	8	yes	—
24	C₂×Q₈×C₂₃	C₂³	1	4	yes	—
25	C₂²×C₄×C₂₃	—	2	2	no	—
26	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
27	C₂²×Dic₂₃	—	2	2	no	—
28	C₂²×Dic₂₃	D₄	1	2	yes	—
29	Q₁₆×C₂₃	C₂×C₄	1	16	yes	complemented
30	C₂×C₈×C₂₃	C₂×C₄	1	4	yes	—
31	C₂×C₈×C₂₃	C₂³	1	2	yes	—
32	C₄⋊Dic₂₃	—	2	2	no	—
33	C₄⋊Dic₂₃	—	2	8	no	—
34	C₄⋊Dic₂₃	C₂³	1	4	yes	—
35	C₄×Dic₂₃	—	2	8	no	—
36	C₄×Dic₂₃	D₄	1	2	yes	—
37	C₄×Dic₂₃	C₂³	1	2	yes	—
38	C₂×C₂₃⋊C₈	—	2	2	no	—
39	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
40	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
41	Q₈×C₂₃	—	2	8	no	—
42	Q₈×C₂₃	—	2	4	no	—
43	Q₈×C₂₃	C₂²×C₄	1	8	yes	—
44	C₂×C₄×C₂₃	—	2	8	no	—
45	C₂×C₄×C₂₃	—	2	4	no	—
46	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₂₃	#41	1	2	yes	—
49	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₂₃	—	2	4	no	—
51	C₂×Dic₂₃	—	2	4	no	—
52	C₂×Dic₂₃	—	4	4	no	—
53	C₂×Dic₂₃	C₂×D₄	1	4	yes	—
54	C₈×C₂₃	C₂²×C₄	1	4	yes	—
55	C₂₃⋊C₈	—	2	2	no	—
56	C₂×#441	—	23	1	no	maximal, Sylow
57	C₂²×C₂₃	C₄×D₄	1	2	yes	—
58	C₂²×C₂₃	#137	1	4	yes	—
59	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
60	C₄×C₂₃	—	2	8	no	—
61	C₄×C₂₃	C₄×D₄	1	2	yes	—
62	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
63	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
64	Dic₂₃	—	2	4	no	—
65	Dic₂₃	—	4	4	no	—
66	C₂²×Q₁₆	D₂₃	1	1	yes	characteristic
67	C₂×C₄×Q₈	—	23	2	no	—
68	#441	—	23	8	no	—
69	C₂×#106	—	23	1	no	—
70	C₂×#103	—	23	2	no	—
71	C₂×#97	—	23	1	no	—
72	C₂×C₂₃	#441	1	4	yes	complemented
73	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
74	C₂×C₂₃	C₂×#137	1	2	yes	—
75	C₂²×Q₈	C₂×D₂₃	1	2	yes	—
76	C₂×Q₁₆	Dic₂₃	1	8	yes	complemented
77	C₂×Q₁₆	C₂×D₂₃	1	4	yes	—
78	C₂²×C₈	—	23	1	no	—
79	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
80	C₄×Q₈	—	23	8	no	—
81	C₄×Q₈	—	46	4	no	—
82	C₂×C₄⋊C₄	—	23	2	no	—
83	C₂×C₄⋊C₄	—	46	2	no	—
84	C₂×C₄²	—	23	1	no	—
85	C₂×C₄²	—	46	2	no	—
86	#106	—	23	4	no	—
87	#103	—	23	8	no	—
88	#97	—	23	4	no	—
89	C₂₃	C₂×#441	1	1	yes	characteristic, complemented, Sylow
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	2	4	no	—
92	C₂×Q₈	C₂×Dic₂₃	1	8	yes	—
93	C₂×Q₈	C₂²×D₂₃	1	4	yes	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	23	2	no	—
96	C₂²×C₄	—	46	2	no	—
97	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
98	Q₁₆	C₂×Dic₂₃	1	16	yes	complemented
99	C₂×C₈	—	23	2	no	—
100	C₂×C₈	—	46	2	no	—
101	C₂×C₈	C₂×Dic₂₃	1	4	yes	—
102	C₂×C₈	C₂²×D₂₃	1	2	yes	—
103	C₄⋊C₄	—	23	4	no	—
104	C₄⋊C₄	—	46	8	no	—
105	C₄⋊C₄	—	46	2	no	—
106	C₄²	—	23	2	no	—
107	C₄²	—	23	2	no	—
108	C₄²	—	46	8	no	—
109	C₂³	D₄×D₂₃	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	Q₈	—	2	4	no	—
112	Q₈	C₂²×Dic₂₃	1	8	yes	—
113	C₂×C₄	—	2	8	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	23	4	no	—
116	C₂×C₄	—	46	4	no	—
117	C₂×C₄	—	46	4	no	—
118	C₂×C₄	—	92	4	no	—
119	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
120	C₂×C₄	#67290	1	2	yes	—
121	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
122	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
123	C₈	—	46	2	no	—
124	C₈	C₂²×Dic₂₃	1	4	yes	—
125	C₂²	#10625089	1	4	yes	—
126	C₂²	D₄×Dic₂₃	1	2	yes	—
127	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
128	C₄	—	2	8	no	—
129	C₄	—	46	4	no	—
130	C₄	—	92	4	no	—
131	C₄	D₄×Dic₂₃	1	2	yes	—
132	C₄	C₂×#67290	1	1	yes	characteristic
133	C₄	C₂³×Dic₂₃	1	1	yes	characteristic
134	C₂	#13092524	1	4	yes	direct product factor
135	C₂	C₂×D₄×Dic₂₃	1	1	yes	characteristic
136	C₂	C₂×#10625089	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₂₃

C₂

yes

maximal

#13092524

C₂

yes

maximal, direct product factor

C₂×#10624999

C₂

yes

maximal, characteristic

C₂×#10625017

C₂

yes

maximal

C₂×#10624997

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₂₃

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

C₂²

yes

—

C₂×Q₁₆×C₂₃

C₄

yes

complemented

C₂×Q₁₆×C₂₃

C₂²

yes

—

C₂²×C₈×C₂₃

C₂²

yes

characteristic

Q₈×Dic₂₃

—

Q₈×Dic₂₃

C₂²

yes

—

C₂×C₄⋊Dic₂₃

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×C₄×Dic₂₃

—

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

#10625017

C₂²

yes

—

#10624999

C₂²

yes

—

#10624997

C₂²

yes

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₄

yes

—

C₂×Q₈×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

D₄

yes

—

Q₁₆×C₂₃

C₂×C₄

yes

complemented

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

C₂³

yes

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂³

yes

—

C₄×Dic₂₃

—

C₄×Dic₂₃

D₄

yes

—

C₄×Dic₂₃

C₂³

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂×D₄

yes

—

C₈×C₂₃

C₂²×C₄

yes

—

C₂₃⋊C₈

—

C₂×#441

—

maximal, Sylow

C₂²×C₂₃

C₄×D₄

yes

—

C₂²×C₂₃

#137

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

C₂³×C₄

yes

characteristic

C₄×C₂₃

C₂×#41

yes

characteristic

Dic₂₃

—

Dic₂₃

—

C₂²×Q₁₆

D₂₃

yes

characteristic

C₂×C₄×Q₈

—

#441

—

C₂×#106

—

C₂×#103

—

C₂×#97

—

C₂×C₂₃

#441

yes

complemented

C₂×C₂₃

C₂×C₄×D₄

yes

characteristic

C₂×C₂₃

C₂×#137

yes

—

C₂²×Q₈

C₂×D₂₃

yes

—

C₂×Q₁₆

Dic₂₃

yes

complemented

C₂×Q₁₆

C₂×D₂₃

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

#106

—

#103

—

#97

—

C₂₃

C₂×#441

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₂₃

yes

—

C₂×Q₈

C₂²×D₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

Q₁₆

C₂×Dic₂₃

yes

complemented

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂×Dic₂₃

yes

—

102

C₂×C₈

C₂²×D₂₃

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₂³

D₄×D₂₃

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

C₂²×Dic₂₃

yes

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

D₄×D₂₃

yes

characteristic

120

C₂×C₄

#67290

yes

—

121

C₂×C₄

C₂²×Dic₂₃

yes

—

122

C₂×C₄

C₂³×D₂₃

yes

characteristic

123

C₈

—

124

C₈

C₂²×Dic₂₃

yes

—

125

C₂²

#10625089

yes

—

126

C₂²

D₄×Dic₂₃

yes

—

127

C₂²

C₂×D₄×D₂₃

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

D₄×Dic₂₃

yes

—

132

C₄

C₂×#67290

yes

characteristic

133

C₄

C₂³×Dic₂₃

yes

characteristic

134

C₂

#13092524

yes

direct product factor

135

C₂

C₂×D₄×Dic₂₃

yes

characteristic

136

C₂

C₂×#10625089

yes

—

	subgroup	G/H
center	C₂³	D₄×D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂²×Q₁₆×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂²×Q₁₆	D₂₃
hypocenter	C₂₃	C₂×#441
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂³

D₄×D₂₃

commutator

C₄×C₂₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₂₃

Fitting

C₂²×Q₁₆×C₂₃

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂×C₄×C₂₃

C₂⁴

hypercenter

C₂²×Q₁₆

D₂₃

hypocenter

C₂₃

C₂×#441

LCS

G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
2. #13092524	C₂	direct (N × H)
8. C₂×Q₁₆×C₂₃	C₄	semidirect
29. Q₁₆×C₂₃	C₂×C₄	semidirect
72. C₂×C₂₃	#441	semidirect
76. C₂×Q₁₆	Dic₂₃	semidirect
89. C₂₃	C₂×#441	semidirect
98. Q₁₆	C₂×Dic₂₃	semidirect
134. C₂	#13092524	direct (N × H)

H ≅ G/N

type

2. #13092524

C₂

direct (N × H)

8. C₂×Q₁₆×C₂₃

C₄

semidirect

29. Q₁₆×C₂₃

C₂×C₄

semidirect

72. C₂×C₂₃

#441

semidirect

76. C₂×Q₁₆

Dic₂₃

semidirect

89. C₂₃

C₂×#441

semidirect

98. Q₁₆

C₂×Dic₂₃

semidirect

134. C₂

#13092524

direct (N × H)

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

subgroup	index
C₂×Q₈×Dic₂₃	2
#13092524	2
C₂×#10624999	2
C₂×#10625017	2
C₂×#10624997	2
C₂²×Q₁₆×C₂₃	2
C₂×#441	23

subgroup

index

C₂×Q₈×Dic₂₃

C₂²×Q₁₆×C₂₃

Subgroups — C₂×#13092524 #15682184

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#13092524 #15682184

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13092524 #15682184