Group #15683708 — subgroups

Subgroups — C₂×#13092768 #15683708

Order 2944.

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

normal subgroups only

This group has 424 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×#10625040	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625040	C₂	1	1	yes	maximal, characteristic
3	#13092768	C₂	1	8	yes	maximal, direct product factor
4	C₂×#10624998	C₂	1	4	yes	maximal
5	C₂×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₆	—	2	1	no	—
7	C₂²×Dic₄₆	C₂²	1	2	yes	—
8	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
9	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₂₃	—	2	1	no	—
11	C₂×C₄⋊Dic₂₃	—	2	1	no	—
12	C₂×C₄⋊Dic₂₃	—	2	1	no	—
13	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
14	#10625040	—	2	2	no	—
15	#10625040	—	2	2	no	—
16	#10625040	C₂²	1	4	yes	—
17	#10625040	C₂²	1	4	yes	—
18	C₄×C₈×C₂₃	C₂²	1	4	yes	—
19	#10624998	C₂²	1	16	yes	—
20	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₂₃	—	2	2	no	—
24	C₂²×Dic₂₃	—	2	2	no	—
25	C₂×Dic₄₆	—	2	8	no	—
26	C₂×Dic₄₆	—	2	2	no	—
27	C₂×Dic₄₆	—	2	4	no	—
28	C₂×Dic₄₆	—	4	4	no	—
29	C₂×Dic₄₆	C₂³	1	4	yes	—
30	C₂×C₈×C₂₃	—	2	4	no	—
31	C₂×C₈×C₂₃	C₂³	1	4	yes	—
32	C₄²×C₂₃	D₄	1	2	yes	—
33	C₄²×C₂₃	C₂³	1	2	yes	—
34	C₄⋊Dic₂₃	—	2	2	no	—
35	C₄⋊Dic₂₃	—	2	4	no	—
36	C₄⋊Dic₂₃	—	2	4	no	—
37	C₄⋊Dic₂₃	—	2	4	no	—
38	C₄⋊Dic₂₃	C₂³	1	4	yes	—
39	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₃	SD₁₆	1	4	yes	—
41	C₂×C₄×C₂₃	Q₁₆	1	4	yes	—
42	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₂₃	#41	1	4	yes	—
46	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₂₃	—	2	4	no	—
48	C₂×Dic₂₃	—	2	4	no	—
49	C₂×Dic₂₃	—	4	4	no	—
50	C₂×Dic₂₃	—	4	4	no	—
51	Dic₄₆	—	2	8	no	—
52	Dic₄₆	—	4	4	no	—
53	Dic₄₆	—	4	8	no	—
54	C₈×C₂₃	—	2	4	no	—
55	C₂×#487	—	23	1	no	maximal, Sylow
56	C₂²×C₂₃	#124	1	2	yes	—
57	C₂²×C₂₃	C₂×SD₁₆	1	2	yes	—
58	C₂²×C₂₃	C₂×Q₁₆	1	2	yes	—
59	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
60	C₄×C₂₃	#124	1	2	yes	—
61	C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
62	C₄×C₂₃	C₂×Q₁₆	1	4	yes	—
63	C₄×C₂₃	C₂×#41	1	2	yes	—
64	Dic₂₃	—	4	4	no	—
65	Dic₂₃	—	4	4	no	—
66	C₂×C₄⋊Q₈	—	23	1	no	—
67	C₂×C₄⋊Q₈	—	23	1	no	—
68	#487	—	23	8	no	—
69	C₂×#103	—	23	4	no	—
70	C₂×C₄×C₈	D₂₃	1	1	yes	characteristic
71	C₂×C₂₃	#487	1	4	yes	complemented
72	C₂×C₂₃	C₂×#124	1	1	yes	characteristic
73	C₂×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
74	C₂×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
75	C₂²×Q₈	—	23	2	no	—
76	C₂²×Q₈	—	46	1	no	—
77	C₂²×C₈	C₂×D₂₃	1	2	yes	—
78	C₄⋊Q₈	—	23	4	no	—
79	C₄⋊Q₈	—	23	4	no	—
80	C₄⋊Q₈	—	46	2	no	—
81	C₄⋊Q₈	—	46	2	no	—
82	C₂×C₄⋊C₄	—	23	2	no	—
83	C₂×C₄⋊C₄	—	46	1	no	—
84	C₂×C₄⋊C₄	—	46	1	no	—
85	C₂×C₄⋊C₄	—	46	1	no	—
86	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
87	#103	—	23	16	no	—
88	C₄×C₈	C₂×D₂₃	1	4	yes	—
89	C₂₃	C₂×#487	1	1	yes	characteristic, complemented, Sylow
90	C₂×Q₈	—	23	4	no	—
91	C₂×Q₈	—	46	4	no	—
92	C₂×Q₈	—	46	2	no	—
93	C₂×Q₈	—	46	8	no	—
94	C₂×Q₈	—	92	4	no	—
95	C₂²×C₄	—	46	2	no	—
96	C₂²×C₄	—	46	2	no	—
97	C₂²×C₄	D₉₂	1	1	yes	characteristic
98	C₂²×C₄	D₉₂	1	1	yes	characteristic
99	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	C₂²×D₂₃	1	4	yes	—
102	C₄⋊C₄	—	23	4	no	—
103	C₄⋊C₄	—	46	2	no	—
104	C₄⋊C₄	—	46	4	no	—
105	C₄⋊C₄	—	46	4	no	—
106	C₄⋊C₄	—	46	4	no	—
107	C₄²	D₉₂	1	2	yes	—
108	C₄²	C₂²×D₂₃	1	2	yes	—
109	C₂³	C₂×D₉₂	1	1	yes	characteristic
110	Q₈	—	46	8	no	—
111	Q₈	—	92	4	no	—
112	Q₈	—	92	8	no	—
113	C₂×C₄	—	46	4	no	—
114	C₂×C₄	—	46	4	no	—
115	C₂×C₄	—	92	4	no	—
116	C₂×C₄	—	92	4	no	—
117	C₂×C₄	#67263	1	4	yes	—
118	C₂×C₄	Dic₉₂	1	4	yes	—
119	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
120	C₂×C₄	C₂×D₉₂	1	2	yes	—
121	C₂×C₄	C₂×D₉₂	1	2	yes	—
122	C₂×C₄	#67288	1	4	yes	—
123	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
124	C₈	—	2	4	no	—
125	C₂²	#10625046	1	2	yes	—
126	C₂²	C₂×#67263	1	2	yes	—
127	C₂²	C₂×Dic₉₂	1	2	yes	—
128	C₂²	C₂²×D₉₂	1	1	yes	characteristic
129	C₄	—	92	4	no	—
130	C₄	—	92	4	no	—
131	C₄	#10625046	1	2	yes	—
132	C₄	C₂×#67263	1	4	yes	—
133	C₄	C₂×Dic₉₂	1	4	yes	—
134	C₄	C₂×#67288	1	2	yes	—
135	C₂	#13092768	1	4	yes	direct product factor
136	C₂	C₂×#10625046	1	1	yes	characteristic
137	C₂	C₂²×#67263	1	1	yes	characteristic
138	C₂	C₂²×Dic₉₂	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625040

C₂

yes

maximal, characteristic

C₂×#10625040

C₂

yes

maximal, characteristic

#13092768

C₂

yes

maximal, direct product factor

C₂×#10624998

C₂

yes

maximal

C₂×C₄×C₈×C₂₃

C₂

yes

maximal, characteristic

C₂²×Dic₄₆

—

C₂²×Dic₄₆

C₂²

yes

—

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

—

C₂×C₄⋊Dic₂₃

—

C₂×C₄⋊Dic₂₃

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

#10625040

—

#10625040

—

#10625040

C₂²

yes

—

#10625040

C₂²

yes

—

C₄×C₈×C₂₃

C₂²

yes

—

#10624998

C₂²

yes

—

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

C₂³

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂³

yes

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

C₂³

yes

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂³

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

SD₁₆

yes

—

C₂×C₄×C₂₃

Q₁₆

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

—

C₂×#487

—

maximal, Sylow

C₂²×C₂₃

#124

yes

—

C₂²×C₂₃

C₂×SD₁₆

yes

—

C₂²×C₂₃

C₂×Q₁₆

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

#124

yes

—

C₄×C₂₃

C₂×SD₁₆

yes

—

C₄×C₂₃

C₂×Q₁₆

yes

—

C₄×C₂₃

C₂×#41

yes

—

Dic₂₃

—

Dic₂₃

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

#487

—

C₂×#103

—

C₂×C₄×C₈

D₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

C₂²×SD₁₆

yes

characteristic

C₂×C₂₃

C₂²×Q₁₆

yes

characteristic

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₂₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#103

—

C₄×C₈

C₂×D₂₃

yes

—

C₂₃

C₂×#487

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₉₂

yes

characteristic

C₂²×C₄

D₉₂

yes

characteristic

C₂²×C₄

C₂²×D₂₃

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

C₂²×D₂₃

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄²

D₉₂

yes

—

108

C₄²

C₂²×D₂₃

yes

—

109

C₂³

C₂×D₉₂

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#67263

yes

—

118

C₂×C₄

Dic₉₂

yes

—

119

C₂×C₄

C₂×D₉₂

yes

characteristic

120

C₂×C₄

C₂×D₉₂

yes

—

121

C₂×C₄

C₂×D₉₂

yes

—

122

C₂×C₄

#67288

yes

—

123

C₂×C₄

C₂³×D₂₃

yes

characteristic

124

C₈

—

125

C₂²

#10625046

yes

—

126

C₂²

C₂×#67263

yes

—

127

C₂²

C₂×Dic₉₂

yes

—

128

C₂²

C₂²×D₉₂

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

#10625046

yes

—

132

C₄

C₂×#67263

yes

—

133

C₄

C₂×Dic₉₂

yes

—

134

C₄

C₂×#67288

yes

—

135

C₂

#13092768

yes

direct product factor

136

C₂

C₂×#10625046

yes

characteristic

137

C₂

C₂²×#67263

yes

characteristic

138

C₂

C₂²×Dic₉₂

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₉₂
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂×C₄×C₈×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
hypercenter	C₂×C₄×C₈	D₂₃
hypocenter	C₂₃	C₂×#487
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₉₂

commutator, cosocle

C₂×C₄×C₂₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₃

Fitting

C₂×C₄×C₈×C₂₃

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

hypercenter

C₂×C₄×C₈

D₂₃

hypocenter

C₂₃

C₂×#487

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
3. #13092768	C₂	direct (N × H)
71. C₂×C₂₃	#487	semidirect
89. C₂₃	C₂×#487	semidirect
135. C₂	#13092768	direct (N × H)

H ≅ G/N

type

3. #13092768

C₂

direct (N × H)

71. C₂×C₂₃

#487

semidirect

89. C₂₃

C₂×#487

semidirect

135. C₂

#13092768

direct (N × H)

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

subgroup	index
C₂×#10625040	2
#13092768	2
C₂×#10624998	2
C₂×C₄×C₈×C₂₃	2
C₂×#487	23

subgroup

index

C₂×C₄×C₈×C₂₃

Subgroups — C₂×#13092768 #15683708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#13092768 #15683708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13092768 #15683708