Group #15674346 — subgroups

Subgroups — #15674346

Order 2944.

292 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic
3	#13092179	C₂	1	4	yes	maximal
4	C₂×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₂₃	—	2	1	no	—
6	C₂₃×C₄⋊Q₈	—	2	2	no	—
7	C₂₃×C₄⋊Q₈	C₄	1	4	yes	complemented
8	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
9	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
10	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
12	C₂×C₄×Dic₂₃	—	2	1	no	—
13	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
14	C₂₃⋊#95	—	2	1	no	—
15	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
16	C₂×Q₈×C₂₃	—	2	4	no	—
17	C₂×Q₈×C₂₃	—	4	4	no	—
18	C₂²×C₄×C₂₃	—	2	2	no	—
19	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₂₃	—	2	2	no	—
23	C₄⋊C₄×C₂₃	—	2	4	no	—
24	C₄⋊C₄×C₂₃	—	2	2	no	—
25	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
26	C₄²×C₂₃	C₂×C₄	1	2	yes	—
27	C₄²×C₂₃	D₄	1	2	yes	—
28	C₄⋊Dic₂₃	—	2	2	no	—
29	C₄⋊Dic₂₃	Q₈	1	4	yes	complemented
30	C₄×Dic₂₃	—	4	2	no	—
31	C₂×C₂₃⋊C₈	—	2	2	no	—
32	C₂×C₂₃⋊C₈	—	2	4	no	—
33	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
34	Q₈×C₂₃	—	4	8	no	—
35	C₂×C₄×C₂₃	—	2	2	no	—
36	C₂×C₄×C₂₃	—	2	4	no	—
37	C₂×C₄×C₂₃	—	4	4	no	—
38	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₂₃	SD₁₆	1	2	yes	—
41	C₂×C₄×C₂₃	Q₁₆	1	2	yes	—
42	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₂₃	#41	1	2	yes	—
46	C₂×Dic₂₃	—	2	4	no	—
47	C₂×Dic₂₃	—	4	2	no	—
48	C₂×Dic₂₃	—	4	2	no	—
49	C₂₃⋊C₈	—	4	2	no	—
50	#1987	—	23	1	no	maximal, Sylow
51	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
53	C₂²×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
54	C₂²×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
55	C₂²×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
56	C₄×C₂₃	—	2	2	no	—
57	C₄×C₂₃	—	4	4	no	—
58	C₄×C₂₃	#103	1	4	yes	—
59	C₄×C₂₃	C₄×Q₈	1	2	yes	—
60	C₄×C₂₃	#124	1	1	yes	characteristic
61	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
62	Dic₂₃	—	4	4	no	—
63	C₂×C₄⋊Q₈	D₂₃	1	1	yes	characteristic
64	C₂×C₄⋊C₈	—	23	1	no	—
65	C₄×C₄⋊C₄	—	23	1	no	—
66	#340	—	23	4	no	—
67	C₂×C₂₃	#391	1	1	yes	characteristic
68	C₂×C₂₃	C₂×#103	1	1	yes	characteristic
69	C₂×C₂₃	#418	1	1	yes	characteristic
70	C₂×C₂₃	#476	1	2	yes	—
71	C₂×C₂₃	#477	1	2	yes	—
72	C₂²×Q₈	—	2	1	no	—
73	C₂²×C₈	—	23	2	no	—
74	C₄⋊Q₈	—	2	2	no	—
75	C₄⋊Q₈	Dic₂₃	1	4	yes	complemented
76	C₂×C₄⋊C₄	—	2	1	no	—
77	C₂×C₄⋊C₄	—	23	2	no	—
78	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
79	C₂×C₄²	—	46	1	no	—
80	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
81	C₄⋊C₈	—	46	2	no	—
82	#95	—	46	1	no	—
83	C₂₃	#1987	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₈	—	2	4	no	—
85	C₂×Q₈	—	4	4	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	46	2	no	—
88	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
89	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
91	C₂×C₈	—	46	4	no	—
92	C₂×C₈	—	46	2	no	—
93	C₄⋊C₄	—	2	4	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	23	4	no	—
96	C₄⋊C₄	—	46	2	no	—
97	C₄⋊C₄	C₂×Dic₂₃	1	4	yes	—
98	C₄²	—	92	2	no	—
99	C₄²	C₂×Dic₂₃	1	2	yes	—
100	C₄²	C₂²⋊D₂₃	1	2	yes	—
101	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
102	Q₈	—	4	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	4	4	no	—
106	C₂×C₄	—	46	4	no	—
107	C₂×C₄	—	92	2	no	—
108	C₂×C₄	—	92	2	no	—
109	C₂×C₄	#67274	1	2	yes	—
110	C₂×C₄	#67275	1	2	yes	—
111	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
112	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
113	C₂×C₄	Q₈×D₂₃	1	2	yes	—
114	C₂×C₄	#67292	1	2	yes	—
115	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
116	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
117	C₈	—	92	2	no	—
118	C₂²	#10625103	1	2	yes	—
119	C₂²	C₂×#67274	1	1	yes	characteristic
120	C₂²	C₂×#67275	1	1	yes	characteristic
121	C₂²	#10625117	1	2	yes	—
122	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	92	4	no	—
126	C₄	#10625017	1	4	yes	—
127	C₄	#10625115	1	1	yes	characteristic
128	C₄	Q₈×Dic₂₃	1	2	yes	—
129	C₄	#10625118	1	1	yes	characteristic
130	C₂	#13092324	1	1	yes	characteristic
131	C₂	C₂×#10625017	1	1	yes	characteristic
132	C₂	#13092422	1	1	yes	characteristic
133	C₂	#13092704	1	2	yes	—
134	C₂	#13092709	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₂₃⋊C₈)

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₂₃

C₂

yes

maximal, characteristic

#13092179

C₂

yes

maximal

C₂×C₂₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×C₄×Dic₂₃

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂₃⋊#95

—

C₄⋊(C₂₃⋊C₈)

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

D₄

yes

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

Q₈

yes

complemented

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

SD₁₆

yes

—

C₂×C₄×C₂₃

Q₁₆

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂₃⋊C₈

—

#1987

—

maximal, Sylow

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

C₂²⋊Q₈

yes

—

C₂²×C₂₃

C₂×SD₁₆

yes

characteristic

C₂²×C₂₃

C₂×Q₁₆

yes

characteristic

C₂²×C₂₃

AGL(1,ℤ/8)

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#103

yes

—

C₄×C₂₃

C₄×Q₈

yes

—

C₄×C₂₃

#124

yes

characteristic

C₄×C₂₃

C₄⋊Q₈

yes

characteristic

Dic₂₃

—

C₂×C₄⋊Q₈

D₂₃

yes

characteristic

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

—

#340

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#477

yes

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₂₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

#95

—

C₂₃

#1987

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₂₃

yes

—

C₄²

—

C₄²

C₂×Dic₂₃

yes

—

100

C₄²

C₂²⋊D₂₃

yes

—

101

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#67274

yes

—

110

C₂×C₄

#67275

yes

—

111

C₂×C₄

C₂²⋊Dic₂₃

yes

—

112

C₂×C₄

C₂²⋊Dic₂₃

yes

—

113

C₂×C₄

Q₈×D₂₃

yes

—

114

C₂×C₄

#67292

yes

—

115

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

116

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

117

C₈

—

118

C₂²

#10625103

yes

—

119

C₂²

C₂×#67274

yes

characteristic

120

C₂²

C₂×#67275

yes

characteristic

121

C₂²

#10625117

yes

—

122

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#10625017

yes

—

127

C₄

#10625115

yes

characteristic

128

C₄

Q₈×Dic₂₃

yes

—

129

C₄

#10625118

yes

characteristic

130

C₂

#13092324

yes

characteristic

131

C₂

C₂×#10625017

yes

characteristic

132

C₂

#13092422

yes

characteristic

133

C₂

#13092704

yes

—

134

C₂

#13092709

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×C₄⋊Q₈	D₂₃
hypocenter	C₂₃	#1987
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₂₃×C₄⋊Q₈

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×C₄⋊Q₈

D₂₃

hypocenter

C₂₃

#1987

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
7. C₂₃×C₄⋊Q₈	C₄	semidirect
29. C₄⋊Dic₂₃	Q₈	semidirect
75. C₄⋊Q₈	Dic₂₃	semidirect
83. C₂₃	#1987	semidirect

H ≅ G/N

type

7. C₂₃×C₄⋊Q₈

C₄

semidirect

29. C₄⋊Dic₂₃

Q₈

semidirect

75. C₄⋊Q₈

Dic₂₃

semidirect

83. C₂₃

#1987

semidirect

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

subgroup	index
C₂×C₄⋊(C₂₃⋊C₈)	2
C₄×C₄⋊Dic₂₃	2
#13092179	2
C₂×C₂₃×C₄⋊Q₈	2
#1987	23

subgroup

index

C₂×C₄⋊(C₂₃⋊C₈)

C₄×C₄⋊Dic₂₃

#13092179

C₂×C₂₃×C₄⋊Q₈

#1987

Subgroups — #15674346

Conjugacy classes of subgroups

Complements

Maximal subgroups