Group #15674345 — subgroups

Subgroups — #15674345

Order 2944.

292 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 125 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×C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#13092179	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₄₆	—	2	1	no	—
6	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₃	—	2	1	no	—
8	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₂₃	—	2	1	no	—
10	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
11	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
12	#10625040	—	2	2	no	—
13	#10625040	C₄	1	4	yes	complemented
14	C₂₃×#95	—	2	1	no	—
15	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
16	C₂²×C₄×C₂₃	—	2	2	no	—
17	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₂₃	—	2	2	no	—
21	C₂×Dic₄₆	—	2	4	no	—
22	C₂×Dic₄₆	—	4	4	no	—
23	C₄⋊C₄×C₂₃	—	2	2	no	—
24	C₄⋊C₄×C₂₃	Q₈	1	4	yes	complemented
25	C₄²×C₂₃	—	4	2	no	—
26	C₄²×C₂₃	C₂×C₄	1	2	yes	—
27	C₄²×C₂₃	D₄	1	2	yes	—
28	C₄⋊Dic₂₃	—	2	4	no	—
29	C₄⋊Dic₂₃	—	2	2	no	—
30	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	—
31	C₂×C₂₃⋊C₈	—	2	2	no	—
32	C₂×C₂₃⋊C₈	—	2	4	no	—
33	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₃	—	2	2	no	—
35	C₂×C₄×C₂₃	—	2	4	no	—
36	C₂×C₄×C₂₃	—	4	2	no	—
37	C₂×C₄×C₂₃	—	4	2	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	4	no	—
48	Dic₄₆	—	4	8	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	—	23	1	no	—
64	C₂×C₄⋊C₈	—	23	1	no	—
65	C₄×C₄⋊C₄	D₂₃	1	1	yes	characteristic
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	—	46	1	no	—
73	C₂²×C₈	—	23	2	no	—
74	C₄⋊Q₈	—	23	4	no	—
75	C₄⋊Q₈	—	46	2	no	—
76	C₂×C₄⋊C₄	—	23	2	no	—
77	C₂×C₄⋊C₄	—	46	1	no	—
78	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
79	C₂×C₄²	—	2	1	no	—
80	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
81	C₄⋊C₈	—	46	2	no	—
82	#95	—	2	1	no	—
83	C₂₃	#1987	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₈	—	46	4	no	—
85	C₂×Q₈	—	92	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	2	no	—
92	C₂×C₈	—	46	4	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	23	4	no	—
95	C₄⋊C₄	—	46	4	no	—
96	C₄⋊C₄	—	46	2	no	—
97	C₄⋊C₄	Dic₄₆	1	4	yes	complemented
98	C₄²	—	4	2	no	—
99	C₄²	C₄×D₂₃	1	2	yes	—
100	C₄²	D₉₂	1	2	yes	—
101	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
102	Q₈	—	92	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	46	4	no	—
108	C₂×C₄	—	92	4	no	—
109	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
110	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
111	C₂×C₄	#67273	1	2	yes	—
112	C₂×C₄	#67275	1	2	yes	—
113	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
114	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
115	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
116	C₂×C₄	#67288	1	2	yes	—
117	C₈	—	92	2	no	—
118	C₂²	#10625095	1	2	yes	—
119	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
120	C₂²	C₂×#67273	1	1	yes	characteristic
121	C₂²	C₂×#67275	1	1	yes	characteristic
122	C₂²	#10625120	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	92	4	no	—
126	C₄	#10624992	1	4	yes	—
127	C₄	C₄×Dic₄₆	1	2	yes	—
128	C₄	#10625040	1	1	yes	characteristic
129	C₄	#10625046	1	1	yes	characteristic
130	C₂	#13092323	1	1	yes	characteristic
131	C₂	C₂×#10624992	1	1	yes	characteristic
132	C₂	#13092421	1	1	yes	characteristic
133	C₂	#13092707	1	2	yes	—
134	C₂	#13092712	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625040

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₂₃⋊C₈)

C₂

yes

maximal, characteristic

#13092179

C₂

yes

maximal

C₄×C₄⋊C₄×C₂₃

C₂

yes

maximal, characteristic

C₂²×Dic₄₆

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

#10625040

—

#10625040

C₄

yes

complemented

C₂₃×#95

—

C₄⋊(C₂₃⋊C₈)

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

Q₈

yes

complemented

C₄²×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

D₄

yes

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

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

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

—

Dic₄₆

—

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

—

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

D₂₃

yes

characteristic

#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

—

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

Dic₄₆

yes

complemented

C₄²

—

C₄²

C₄×D₂₃

yes

—

100

C₄²

D₉₂

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

(C₂×D₂₃)⋊C₄

yes

—

110

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

111

C₂×C₄

#67273

yes

—

112

C₂×C₄

#67275

yes

—

113

C₂×C₄

C₂×Dic₄₆

yes

—

114

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

115

C₂×C₄

C₂×D₉₂

yes

characteristic

116

C₂×C₄

#67288

yes

—

117

C₈

—

118

C₂²

#10625095

yes

—

119

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

120

C₂²

C₂×#67273

yes

characteristic

121

C₂²

C₂×#67275

yes

characteristic

122

C₂²

#10625120

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#10624992

yes

—

127

C₄

C₄×Dic₄₆

yes

—

128

C₄

#10625040

yes

characteristic

129

C₄

#10625046

yes

characteristic

130

C₂

#13092323

yes

characteristic

131

C₂

C₂×#10624992

yes

characteristic

132

C₂

#13092421

yes

characteristic

133

C₂

#13092707

yes

—

134

C₂

#13092712

yes

—

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

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₄×C₄⋊C₄×C₂₃

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₄×C₄⋊C₄

D₂₃

hypocenter

C₂₃

#1987

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
13. #10625040	C₄	semidirect
24. C₄⋊C₄×C₂₃	Q₈	semidirect
83. C₂₃	#1987	semidirect
97. C₄⋊C₄	Dic₄₆	semidirect

H ≅ G/N

type

13. #10625040

C₄

semidirect

24. C₄⋊C₄×C₂₃

Q₈

semidirect

83. C₂₃

#1987

semidirect

97. C₄⋊C₄

Dic₄₆

semidirect

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

subgroup	index
C₂×#10625040	2
C₂×C₄⋊(C₂₃⋊C₈)	2
#13092179	2
C₄×C₄⋊C₄×C₂₃	2
#1987	23

subgroup

index

C₂×#10625040

C₂×C₄⋊(C₂₃⋊C₈)

#13092179

C₄×C₄⋊C₄×C₂₃

#1987

Subgroups — #15674345

Conjugacy classes of subgroups

Complements

Maximal subgroups