Group #15674371 — subgroups

Subgroups — #15674371

Order 2944.

276 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 113 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×#10624985	C₂	1	1	yes	maximal, characteristic
2	#13092283	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	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
13	C₂₃⋊#95	—	2	1	no	—
14	C₂₃⋊#95	—	2	1	no	—
15	#10624985	—	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	4	no	—
29	C₂×C₂₃⋊C₈	—	2	2	no	—
30	C₂×C₂₃⋊C₈	—	2	2	no	—
31	C₂×C₂₃⋊C₈	Q₈	1	4	yes	complemented
32	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
33	Q₈×C₂₃	—	4	8	no	—
34	C₂×C₄×C₂₃	—	2	2	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	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₂₃	#41	1	2	yes	—
44	C₂×Dic₂₃	—	4	2	no	—
45	C₂×Dic₂₃	—	4	2	no	—
46	C₂×Dic₂₃	—	4	2	no	—
47	C₂₃⋊C₈	—	2	4	no	—
48	#1993	—	23	1	no	maximal, Sylow
49	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₂₃	#124	1	1	yes	characteristic
51	C₂²×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
52	C₂²×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
53	C₂²×C₂₃	#137	1	2	yes	—
54	C₄×C₂₃	—	2	2	no	—
55	C₄×C₂₃	—	2	2	no	—
56	C₄×C₂₃	—	4	4	no	—
57	C₄×C₂₃	C₄×Q₈	1	2	yes	—
58	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
59	Dic₂₃	—	8	2	no	—
60	C₂×C₄⋊Q₈	D₂₃	1	1	yes	characteristic
61	C₂×#97	—	23	1	no	—
62	#382	—	23	1	no	—
63	#340	—	23	4	no	—
64	C₂×C₂₃	#391	1	1	yes	characteristic
65	C₂×C₂₃	#418	1	1	yes	characteristic
66	C₂×C₂₃	#419	1	1	yes	characteristic
67	C₂×C₂₃	#489	1	2	yes	—
68	C₂×C₂₃	#501	1	2	yes	—
69	C₂²×Q₈	—	2	1	no	—
70	C₂²×C₈	—	23	2	no	—
71	C₄⋊Q₈	—	2	2	no	—
72	C₄⋊Q₈	Dic₂₃	1	4	yes	complemented
73	C₂×C₄⋊C₄	—	2	1	no	—
74	C₂×C₄⋊C₄	—	23	2	no	—
75	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
76	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
77	#97	—	46	2	no	—
78	#95	—	46	1	no	—
79	#95	—	46	1	no	—
80	C₂₃	#1993	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₈	—	2	4	no	—
82	C₂×Q₈	—	4	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	46	2	no	—
85	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
88	C₂×C₈	—	23	4	no	—
89	C₂×C₈	—	46	2	no	—
90	C₂×C₈	—	46	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	46	4	no	—
94	C₄⋊C₄	C₂×Dic₂₃	1	4	yes	—
95	C₄²	C₂×Dic₂₃	1	2	yes	—
96	C₄²	C₂²⋊D₂₃	1	2	yes	—
97	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
98	Q₈	—	4	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	4	4	no	—
103	C₂×C₄	—	92	2	no	—
104	C₂×C₄	—	92	2	no	—
105	C₂×C₄	—	92	2	no	—
106	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
107	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
108	C₂×C₄	Q₈×D₂₃	1	2	yes	—
109	C₂×C₄	#67292	1	2	yes	—
110	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
111	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
112	C₈	—	46	4	no	—
113	C₂²	#10625103	1	2	yes	—
114	C₂²	#10625113	1	2	yes	—
115	C₂²	#10625115	1	1	yes	characteristic
116	C₂²	#10625118	1	1	yes	characteristic
117	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	4	4	no	—
121	C₄	—	184	2	no	—
122	C₄	Q₈×Dic₂₃	1	2	yes	—
123	C₄	#10625117	1	2	yes	—
124	C₂	#13092324	1	1	yes	characteristic
125	C₂	#13092422	1	1	yes	characteristic
126	C₂	#13092426	1	1	yes	characteristic
127	C₂	#13092779	1	2	yes	—
128	C₂	#13092823	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10624985

C₂

yes

maximal, characteristic

#13092283

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

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂₃⋊#95

—

C₂₃⋊#95

—

#10624985

—

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

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

Q₈

yes

complemented

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

C₂×C₄×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

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

—

#1993

—

maximal, Sylow

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

#124

yes

characteristic

C₂²×C₂₃

C₄⋊Q₈

yes

characteristic

C₂²×C₂₃

AGL(1,ℤ/8)

yes

—

C₂²×C₂₃

#137

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×Q₈

yes

—

C₄×C₂₃

C₂²⋊Q₈

yes

—

Dic₂₃

—

C₂×C₄⋊Q₈

D₂₃

yes

characteristic

C₂×#97

—

#382

—

#340

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#501

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×D₂₃

yes

—

—

—

C₂₃

#1993

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

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₂₃

yes

—

C₄²

C₂×Dic₂₃

yes

—

C₄²

C₂²⋊D₂₃

yes

—

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂²⋊Dic₂₃

yes

—

107

C₂×C₄

C₂²⋊Dic₂₃

yes

—

108

C₂×C₄

Q₈×D₂₃

yes

—

109

C₂×C₄

#67292

yes

—

110

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

111

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

112

C₈

—

113

C₂²

#10625103

yes

—

114

C₂²

#10625113

yes

—

115

C₂²

#10625115

yes

characteristic

116

C₂²

#10625118

yes

characteristic

117

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

184

—

122

C₄

Q₈×Dic₂₃

yes

—

123

C₄

#10625117

yes

—

124

C₂

#13092324

yes

characteristic

125

C₂

#13092422

yes

characteristic

126

C₂

#13092426

yes

characteristic

127

C₂

#13092779

yes

—

128

C₂

#13092823

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	#1993
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

#1993

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
31. C₂×C₂₃⋊C₈	Q₈	semidirect
72. C₄⋊Q₈	Dic₂₃	semidirect
80. C₂₃	#1993	semidirect

H ≅ G/N

type

7. C₂₃×C₄⋊Q₈

C₄

semidirect

31. C₂×C₂₃⋊C₈

Q₈

semidirect

72. C₄⋊Q₈

Dic₂₃

semidirect

80. C₂₃

#1993

semidirect

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

subgroup	index
C₂×#10624985	2
#13092283	2
#13092179	2
C₂×C₂₃×C₄⋊Q₈	2
#1993	23

subgroup

index

C₂×C₂₃×C₄⋊Q₈

Subgroups — #15674371

Conjugacy classes of subgroups

Complements

Maximal subgroups