Group #15673247 — subgroups

Subgroups — C₄×Dic₂₃⋊C₈ #15673247

Order 2944.

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

normal subgroups only

This group has 316 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×Dic₂₃⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄²×Dic₂₃	C₂	1	1	yes	maximal, characteristic
4	#13092171	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₂₃	C₄	1	2	yes	—
9	C₂×C₄×Dic₂₃	C₄	1	2	yes	—
10	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
12	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
13	C₄×C₈×C₂₃	—	2	2	no	—
14	Dic₂₃⋊C₈	C₄	1	8	yes	direct product factor
15	C₄×C₂₃⋊C₈	—	2	2	no	—
16	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
20	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₂₃	—	2	2	no	—
22	C₂×C₈×C₂₃	—	2	2	no	—
23	C₂×C₈×C₂₃	C₂×C₄	1	4	yes	—
24	C₄²×C₂₃	D₄	1	2	yes	—
25	C₄²×C₂₃	Q₈	1	2	yes	—
26	C₄×Dic₂₃	—	2	4	no	—
27	C₄×Dic₂₃	—	2	2	no	—
28	C₄×Dic₂₃	C₈	1	8	yes	complemented
29	C₄×Dic₂₃	C₂×C₄	1	4	yes	—
30	C₂×C₂₃⋊C₈	—	2	2	no	—
31	C₂×C₂₃⋊C₈	—	2	2	no	—
32	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	—
33	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
35	C₂×C₄×C₂₃	C₂×C₈	1	4	yes	—
36	C₂×C₄×C₂₃	M₁₆	1	4	yes	—
37	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₂₃	#41	1	2	yes	—
41	C₂×Dic₂₃	—	2	2	no	—
42	C₂×Dic₂₃	—	2	2	no	—
43	C₂×Dic₂₃	—	2	2	no	—
44	C₂×Dic₂₃	C₄²	1	2	yes	—
45	C₂×Dic₂₃	C₄²	1	2	yes	—
46	C₂×Dic₂₃	C₂×C₈	1	4	yes	—
47	C₂×Dic₂₃	C₂×C₈	1	4	yes	—
48	C₈×C₂₃	—	2	4	no	—
49	C₂₃⋊C₈	—	2	4	no	—
50	C₄×C₄⋊C₈	—	23	1	no	maximal, Sylow
51	C₂²×C₂₃	C₂×C₄²	1	1	yes	characteristic
52	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₂₃	#117	1	1	yes	characteristic
54	C₂²×C₂₃	C₂²×C₈	1	2	yes	—
55	C₂²×C₂₃	C₂×M₁₆	1	2	yes	—
56	C₄×C₂₃	C₄⋊C₈	1	8	yes	complemented
57	C₄×C₂₃	C₄×D₄	1	2	yes	—
58	C₄×C₂₃	C₄×Q₈	1	2	yes	—
59	Dic₂₃	—	2	2	no	—
60	Dic₂₃	—	2	4	no	—
61	Dic₂₃	C₄×C₈	1	4	yes	complemented
62	C₂×C₄⋊C₈	—	23	2	no	—
63	C₂×C₄×C₈	—	23	1	no	—
64	C₂×C₄×C₈	D₂₃	1	1	yes	characteristic
65	C₄³	—	23	1	no	—
66	#336	—	23	2	no	—
67	C₂×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
69	C₂×C₂₃	C₄×M₁₆	1	1	yes	characteristic
70	C₂×C₂₃	C₂×C₄⋊C₈	1	2	yes	—
71	C₂×C₂₃	#431	1	2	yes	—
72	C₂²×C₈	—	23	2	no	—
73	C₂²×C₈	C₂×D₂₃	1	2	yes	—
74	C₂×C₄²	—	23	2	no	—
75	C₂×C₄²	—	23	1	no	—
76	C₂×C₄²	—	23	2	no	—
77	C₂×C₄²	—	23	1	no	—
78	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
79	C₄⋊C₈	—	23	8	no	—
80	C₄×C₈	—	2	2	no	—
81	C₄×C₈	—	46	2	no	—
82	C₂₃	C₄×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₄	—	23	1	no	—
84	C₂²×C₄	—	23	1	no	—
85	C₂²×C₄	—	23	2	no	—
86	C₂²×C₄	C₄×D₂₃	1	2	yes	—
87	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	23	4	no	—
91	C₂×C₈	—	46	2	no	—
92	C₂×C₈	—	46	2	no	—
93	C₂×C₈	C₄×D₂₃	1	4	yes	—
94	C₄²	—	23	8	no	—
95	C₄²	—	23	4	no	—
96	C₄²	—	46	4	no	—
97	C₄²	—	46	2	no	—
98	C₄²	Dic₄₆	1	2	yes	—
99	C₄²	C₂²⋊D₂₃	1	2	yes	—
100	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
101	C₂×C₄	—	23	2	no	—
102	C₂×C₄	—	23	4	no	—
103	C₂×C₄	—	23	2	no	—
104	C₂×C₄	—	23	4	no	—
105	C₂×C₄	—	46	2	no	—
106	C₂×C₄	—	46	2	no	—
107	C₂×C₄	—	46	2	no	—
108	C₂×C₄	C₈×D₂₃	1	4	yes	—
109	C₂×C₄	#67262	1	4	yes	—
110	C₂×C₄	Dic₂₃⋊C₄	1	4	yes	—
111	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
112	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
113	C₂×C₄	#67288	1	2	yes	—
114	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
115	C₈	—	2	4	no	—
116	C₈	—	46	4	no	—
117	C₂²	C₄²×D₂₃	1	1	yes	characteristic
118	C₂²	#10625043	1	1	yes	characteristic
119	C₂²	C₂×C₈×D₂₃	1	2	yes	—
120	C₂²	C₂×#67262	1	2	yes	—
121	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
122	C₄	—	23	4	no	—
123	C₄	—	46	2	no	—
124	C₄	—	46	4	no	—
125	C₄	Dic₂₃⋊C₈	1	8	yes	direct product factor
126	C₄	C₄×Dic₄₆	1	2	yes	—
127	C₄	C₄×C₂²⋊D₂₃	1	2	yes	—
128	C₂	C₄×Dic₂₃⋊C₄	1	1	yes	characteristic
129	C₂	C₄×C₈×D₂₃	1	1	yes	characteristic
130	C₂	C₄×#67262	1	1	yes	characteristic
131	C₂	C₂×Dic₂₃⋊C₈	1	2	yes	—
132	C₂	#13092468	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₂₃⋊C₈

C₂

yes

maximal

C₂×C₄×C₂₃⋊C₈

C₂

yes

maximal, characteristic

C₄²×Dic₂₃

C₂

yes

maximal, characteristic

#13092171

C₂

yes

maximal

C₂×C₄×C₈×C₂₃

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₄

yes

—

C₂×C₄×Dic₂₃

C₄

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₄×C₈×C₂₃

—

Dic₂₃⋊C₈

C₄

yes

direct product factor

C₄×C₂₃⋊C₈

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

—

C₂²×Dic₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

Q₈

yes

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

C₈

yes

complemented

C₄×Dic₂₃

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₃

M₁₆

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₄²

yes

—

C₂×Dic₂₃

C₄²

yes

—

C₂×Dic₂₃

C₂×C₈

yes

—

C₂×Dic₂₃

C₂×C₈

yes

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₄×C₄⋊C₈

—

maximal, Sylow

C₂²×C₂₃

C₂×C₄²

yes

characteristic

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

#117

yes

characteristic

C₂²×C₂₃

C₂²×C₈

yes

—

C₂²×C₂₃

C₂×M₁₆

yes

—

C₄×C₂₃

C₄⋊C₈

yes

complemented

C₄×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

C₄×Q₈

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

C₄×C₈

yes

complemented

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

D₂₃

yes

characteristic

C₄³

—

#336

—

C₂×C₂₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₃

C₂×C₄×C₈

yes

characteristic

C₂×C₂₃

C₄×M₁₆

yes

characteristic

C₂×C₂₃

C₂×C₄⋊C₈

yes

—

C₂×C₂₃

#431

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂₃

C₄×C₄⋊C₈

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₂₃

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

Dic₄₆

yes

—

C₄²

C₂²⋊D₂₃

yes

—

100

C₂³

C₂×C₄×D₂₃

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₈×D₂₃

yes

—

109

C₂×C₄

#67262

yes

—

110

C₂×C₄

Dic₂₃⋊C₄

yes

—

111

C₂×C₄

C₂×Dic₄₆

yes

characteristic

112

C₂×C₄

C₂×C₄×D₂₃

yes

—

113

C₂×C₄

#67288

yes

—

114

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

115

C₈

—

116

C₈

—

117

C₂²

C₄²×D₂₃

yes

characteristic

118

C₂²

#10625043

yes

characteristic

119

C₂²

C₂×C₈×D₂₃

yes

—

120

C₂²

C₂×#67262

yes

—

121

C₂²

C₂×Dic₂₃⋊C₄

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

Dic₂₃⋊C₈

yes

direct product factor

126

C₄

C₄×Dic₄₆

yes

—

127

C₄

C₄×C₂²⋊D₂₃

yes

—

128

C₂

C₄×Dic₂₃⋊C₄

yes

characteristic

129

C₂

C₄×C₈×D₂₃

yes

characteristic

130

C₂

C₄×#67262

yes

characteristic

131

C₂

C₂×Dic₂₃⋊C₈

yes

—

132

C₂

#13092468

yes

—

	subgroup	G/H
center	C₂×C₄²	C₂×D₂₃
commutator	C₂×C₂₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₄×C₈×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	C₂²×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×C₄×C₈	D₂₃
hypocenter	C₂₃	C₄×C₄⋊C₈
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄²

C₂×D₂₃

commutator

C₂×C₂₃

C₂×C₄×C₈

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₄×C₈×C₂₃

C₂

socle, prime radical

C₂³×C₂₃

C₂²×C₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×C₄×C₈

D₂₃

hypocenter

C₂₃

C₄×C₄⋊C₈

LCS

G ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
14. Dic₂₃⋊C₈	C₄	direct (N × H)
28. C₄×Dic₂₃	C₈	semidirect
56. C₄×C₂₃	C₄⋊C₈	semidirect
61. Dic₂₃	C₄×C₈	semidirect
82. C₂₃	C₄×C₄⋊C₈	semidirect
125. C₄	Dic₂₃⋊C₈	direct (N × H)

H ≅ G/N

type

14. Dic₂₃⋊C₈

C₄

direct (N × H)

28. C₄×Dic₂₃

C₈

semidirect

56. C₄×C₂₃

C₄⋊C₈

semidirect

61. Dic₂₃

C₄×C₈

semidirect

82. C₂₃

C₄×C₄⋊C₈

semidirect

125. C₄

Dic₂₃⋊C₈

direct (N × H)

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

subgroup	index
C₂×Dic₂₃⋊C₈	2
C₂×C₄×C₂₃⋊C₈	2
C₄²×Dic₂₃	2
#13092171	2
C₂×C₄×C₈×C₂₃	2
C₄×C₄⋊C₈	23

subgroup

index

C₂×Dic₂₃⋊C₈

C₂×C₄×C₂₃⋊C₈

C₄²×Dic₂₃

#13092171

C₂×C₄×C₈×C₂₃

C₄×C₄⋊C₈

Subgroups — C₄×Dic₂₃⋊C₈ #15673247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×Dic23⋊C8 #15673247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Dic₂₃⋊C₈ #15673247