Group #15674338 — subgroups

Subgroups — #15674338

Order 2944.

280 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 121 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	C₂×#10624985	C₂	1	1	yes	maximal, characteristic
3	#13092172	C₂	1	4	yes	maximal
4	C₂×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₂₃	C₄	1	1	yes	characteristic
6	C₄×Q₈×C₂₃	—	2	2	no	—
7	C₄×Q₈×C₂₃	—	2	2	no	—
8	C₂×C₄⋊C₄×C₂₃	C₄	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
10	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
11	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
13	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
14	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
15	#10624985	—	2	2	no	—
16	C₂×Q₈×C₂₃	—	2	2	no	—
17	C₂×Q₈×C₂₃	—	2	4	no	—
18	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₂₃	—	2	4	no	—
24	C₄⋊C₄×C₂₃	—	2	2	no	—
25	C₄²×C₂₃	—	2	2	no	—
26	C₄²×C₂₃	—	2	2	no	—
27	C₄²×C₂₃	D₄	1	2	yes	—
28	C₄²×C₂₃	D₄	1	2	yes	—
29	C₂×C₂₃⋊C₈	—	2	2	no	—
30	C₂×C₂₃⋊C₈	—	2	2	no	—
31	C₂×C₂₃⋊C₈	—	2	2	no	—
32	C₂×C₂₃⋊C₈	—	2	4	no	—
33	C₂×C₂₃⋊C₈	Q₈	1	4	yes	complemented
34	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₂₃	—	2	8	no	—
36	C₂×C₄×C₂₃	—	2	2	no	—
37	C₂×C₄×C₂₃	—	2	2	no	—
38	C₂×C₄×C₂₃	—	2	2	no	—
39	C₂×C₄×C₂₃	—	2	2	no	—
40	C₂×C₄×C₂₃	—	2	2	no	—
41	C₂×C₄×C₂₃	—	2	4	no	—
42	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₂₃	M₁₆	1	4	yes	—
45	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₂₃	#41	1	2	yes	—
49	C₂₃⋊C₈	—	2	4	no	—
50	C₂₃⋊C₈	—	4	2	no	—
51	#1985	—	23	1	no	maximal, Sylow
52	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₂₃	C₄×Q₈	1	2	yes	—
54	C₂²×C₂₃	C₂×M₁₆	1	2	yes	—
55	C₂²×C₂₃	#131	1	2	yes	—
56	C₄×C₂₃	—	2	2	no	—
57	C₄×C₂₃	—	2	4	no	—
58	C₄×C₂₃	—	2	2	no	—
59	C₄×C₂₃	—	4	2	no	—
60	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₂₃	#124	1	1	yes	characteristic
62	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×Q₈	D₂₃	1	1	yes	characteristic
64	C₂×C₄⋊C₈	—	23	1	no	—
65	C₂×#97	—	23	1	no	—
66	#336	—	23	4	no	—
67	C₂×C₂₃	#391	1	1	yes	characteristic
68	C₂×C₂₃	#407	1	1	yes	characteristic
69	C₂×C₂₃	#408	1	1	yes	characteristic
70	C₂×C₂₃	#446	1	4	yes	—
71	C₂²×Q₈	Dic₂₃	1	1	yes	characteristic
72	C₂²×C₈	—	23	2	no	—
73	C₂²×C₈	—	23	2	no	—
74	C₄×Q₈	—	2	2	no	—
75	C₄×Q₈	—	2	2	no	—
76	C₂×C₄⋊C₄	Dic₂₃	1	1	yes	characteristic
77	C₂×C₄⋊C₄	Dic₂₃	1	2	yes	—
78	C₂×C₄²	C₂×D₂₃	1	2	yes	—
79	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
80	C₄⋊C₈	—	46	2	no	—
81	#97	—	46	2	no	—
82	C₂₃	#1985	1	1	yes	characteristic, complemented, Sylow
83	C₂×Q₈	—	2	4	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
86	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
87	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
88	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
89	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
90	C₂×C₈	—	23	4	no	—
91	C₂×C₈	—	46	2	no	—
92	C₂×C₈	—	46	2	no	—
93	C₂×C₈	—	46	4	no	—
94	C₂×C₈	—	46	2	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄²	—	2	2	no	—
98	C₄²	—	2	2	no	—
99	C₄²	C₂²⋊D₂₃	1	2	yes	—
100	C₄²	C₂²⋊D₂₃	1	2	yes	—
101	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
102	Q₈	—	2	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	#67267	1	4	yes	—
110	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
111	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
112	C₂×C₄	Q₈×D₂₃	1	2	yes	—
113	C₂×C₄	#67292	1	2	yes	—
114	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
115	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
116	C₈	—	46	4	no	—
117	C₈	—	92	2	no	—
118	C₂²	C₂×#67267	1	2	yes	—
119	C₂²	Q₈×Dic₂₃	1	2	yes	—
120	C₂²	#10625119	1	2	yes	—
121	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	4	2	no	—
126	C₄	#10625115	1	1	yes	characteristic
127	C₄	#10625117	1	2	yes	—
128	C₄	#10625118	1	1	yes	characteristic
129	C₂	#13092324	1	1	yes	characteristic
130	C₂	#13092381	1	1	yes	characteristic
131	C₂	#13092384	1	1	yes	characteristic
132	C₂	#13092548	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₂₃⋊C₈)

C₂

yes

maximal, characteristic

C₂×#10624985

C₂

yes

maximal, characteristic

#13092172

C₂

yes

maximal

C₂×C₄×Q₈×C₂₃

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

C₄

yes

characteristic

C₄×Q₈×C₂₃

—

C₄×Q₈×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₄

yes

characteristic

C₂×C₄⋊C₄×C₂₃

C₄

yes

—

C₂×C₄²×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₄⋊(C₂₃⋊C₈)

—

#10624985

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

D₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

Q₈

yes

complemented

C₂³×C₂₃

C₂²×C₄

yes

characteristic

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×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

M₁₆

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1985

—

maximal, Sylow

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

C₄×Q₈

yes

—

C₂²×C₂₃

C₂×M₁₆

yes

—

C₂²×C₂₃

#131

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂²⋊Q₈

yes

—

C₄×C₂₃

#124

yes

characteristic

C₄×C₂₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×Q₈

D₂₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#97

—

#336

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂²×Q₈

Dic₂₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₂₃

yes

characteristic

C₂×C₄⋊C₄

Dic₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

#97

—

C₂₃

#1985

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₂₃

yes

—

100

C₄²

C₂²⋊D₂₃

yes

—

101

C₂³

C₂²×Dic₂₃

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

#67267

yes

—

110

C₂×C₄

C₂²⋊Dic₂₃

yes

—

111

C₂×C₄

C₂²⋊Dic₂₃

yes

—

112

C₂×C₄

Q₈×D₂₃

yes

—

113

C₂×C₄

#67292

yes

—

114

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

115

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

116

C₈

—

117

C₈

—

118

C₂²

C₂×#67267

yes

—

119

C₂²

Q₈×Dic₂₃

yes

—

120

C₂²

#10625119

yes

—

121

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#10625115

yes

characteristic

127

C₄

#10625117

yes

—

128

C₄

#10625118

yes

characteristic

129

C₂

#13092324

yes

characteristic

130

C₂

#13092381

yes

characteristic

131

C₂

#13092384

yes

characteristic

132

C₂

#13092548

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator, socle, prime radical	C₂³×C₂₃	C₂²×C₄
Fitting	C₂×C₄×Q₈×C₂₃	C₂
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×C₄×Q₈	D₂₃
hypocenter	C₂₃	#1985
LCS	G ▷ C₂³×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃

commutator, socle, prime radical

C₂³×C₂₃

C₂²×C₄

Fitting

C₂×C₄×Q₈×C₂₃

C₂

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×C₄×Q₈

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
33. C₂×C₂₃⋊C₈	Q₈	semidirect
82. C₂₃	#1985	semidirect

H ≅ G/N

type

33. C₂×C₂₃⋊C₈

Q₈

semidirect

82. C₂₃

#1985

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
C₂×#10624985	2
#13092172	2
C₂×C₄×Q₈×C₂₃	2
#1985	23

subgroup

index

C₂×C₄⋊(C₂₃⋊C₈)

C₂×#10624985

#13092172

C₂×C₄×Q₈×C₂₃

#1985

Subgroups — #15674338

Conjugacy classes of subgroups

Complements

Maximal subgroups