Group #15673513 — subgroups

Subgroups — #15673513

Order 2944.

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

normal subgroups only

This group has 260 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	C₄²×Dic₂₃	C₂	1	1	yes	maximal, characteristic
3	#13092171	C₂	1	4	yes	maximal
4	C₂×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
6	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₂₃	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₂₃	C₄	1	2	yes	—
9	C₂×C₄×Dic₂₃	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
11	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
12	C₂₃×C₄⋊C₈	—	2	2	no	—
13	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
14	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
17	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
18	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₂₃	—	2	2	no	—
20	C₂×C₈×C₂₃	—	2	4	no	—
21	C₄²×C₂₃	D₄	1	2	yes	—
22	C₄²×C₂₃	Q₈	1	2	yes	—
23	C₄×Dic₂₃	—	2	2	no	—
24	C₄×Dic₂₃	—	2	2	no	—
25	C₄×Dic₂₃	—	2	2	no	—
26	C₄×Dic₂₃	—	2	4	no	—
27	C₄×Dic₂₃	C₈	1	4	yes	complemented
28	C₂×C₂₃⋊C₈	—	2	2	no	—
29	C₂×C₂₃⋊C₈	—	2	4	no	—
30	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₂₃	—	2	2	no	—
32	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
33	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₂₃	C₂×C₈	1	2	yes	—
35	C₂×C₄×C₂₃	M₁₆	1	2	yes	—
36	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₂₃	#41	1	2	yes	—
39	C₂×C₄×C₂₃	#41	1	2	yes	—
40	C₂×Dic₂₃	—	2	2	no	—
41	C₂×Dic₂₃	—	2	2	no	—
42	C₂×Dic₂₃	—	2	2	no	—
43	C₂×Dic₂₃	—	2	2	no	—
44	C₂×Dic₂₃	C₂×C₈	1	4	yes	—
45	C₂×Dic₂₃	M₁₆	1	4	yes	—
46	C₈×C₂₃	—	4	2	no	—
47	C₂₃⋊C₈	—	4	2	no	—
48	#1831	—	23	1	no	maximal, Sylow
49	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₂₃	#117	1	2	yes	—
51	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
52	C₂²×C₂₃	C₂×M₁₆	1	1	yes	characteristic
53	C₂²×C₂₃	C₂×M₁₆	1	2	yes	—
54	C₄×C₂₃	—	2	2	no	—
55	C₄×C₂₃	C₄⋊C₈	1	4	yes	—
56	C₄×C₂₃	#124	1	1	yes	characteristic
57	C₄×C₂₃	#124	1	1	yes	characteristic
58	C₄×C₂₃	#125	1	2	yes	—
59	Dic₂₃	—	2	4	no	—
60	Dic₂₃	—	2	4	no	—
61	C₂×C₄⋊C₈	—	23	1	no	—
62	C₂×C₄⋊C₈	D₂₃	1	1	yes	characteristic
63	C₄³	—	23	1	no	—
64	#336	—	23	4	no	—
65	C₂×C₂₃	#382	1	1	yes	characteristic
66	C₂×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
67	C₂×C₂₃	#423	1	1	yes	characteristic
68	C₂×C₂₃	#431	1	2	yes	—
69	C₂×C₂₃	#432	1	2	yes	—
70	C₂²×C₈	—	23	2	no	—
71	C₂²×C₈	C₂×D₂₃	1	2	yes	—
72	C₂×C₄²	—	23	1	no	—
73	C₂×C₄²	—	23	2	no	—
74	C₂×C₄²	—	23	1	no	—
75	C₂×C₄²	—	23	2	no	—
76	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
77	C₄⋊C₈	—	2	2	no	—
78	C₄⋊C₈	—	46	2	no	—
79	C₂₃	#1831	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₄	—	23	2	no	—
81	C₂²×C₄	—	23	2	no	—
82	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
83	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
84	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
85	C₂×C₈	—	2	4	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	46	4	no	—
88	C₂×C₈	—	46	2	no	—
89	C₄²	—	23	4	no	—
90	C₄²	—	46	4	no	—
91	C₄²	—	46	2	no	—
92	C₄²	—	46	2	no	—
93	C₄²	—	46	2	no	—
94	C₄²	Dic₄₆	1	2	yes	—
95	C₄²	C₂²⋊D₂₃	1	2	yes	—
96	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	23	4	no	—
99	C₂×C₄	—	23	4	no	—
100	C₂×C₄	—	46	2	no	—
101	C₂×C₄	—	46	2	no	—
102	C₂×C₄	—	46	2	no	—
103	C₂×C₄	—	46	2	no	—
104	C₂×C₄	C₈×D₂₃	1	2	yes	—
105	C₂×C₄	#67262	1	2	yes	—
106	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
107	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
108	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
109	C₂×C₄	#67290	1	2	yes	—
110	C₂×C₄	#67292	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
112	C₈	—	4	2	no	—
113	C₈	—	92	2	no	—
114	C₂²	#10625063	1	2	yes	—
115	C₂²	C₂×C₈×D₂₃	1	1	yes	characteristic
116	C₂²	C₂×#67262	1	1	yes	characteristic
117	C₂²	M₁₆×D₂₃	1	2	yes	—
118	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	—	46	4	no	—
121	C₄	—	46	4	no	—
122	C₄	Dic₂₃⋊C₈	1	4	yes	—
123	C₄	#10625061	1	2	yes	—
124	C₄	#10625107	1	1	yes	characteristic
125	C₄	#10625118	1	1	yes	characteristic
126	C₂	#13092282	1	1	yes	characteristic
127	C₂	C₂×Dic₂₃⋊C₈	1	1	yes	characteristic
128	C₂	#13092440	1	1	yes	characteristic
129	C₂	#13092469	1	2	yes	—
130	C₂	#13092473	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₂₃⋊C₈)

C₂

yes

maximal, characteristic

C₄²×Dic₂₃

C₂

yes

maximal, characteristic

#13092171

C₂

yes

maximal

C₂×C₂₃×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₄

yes

characteristic

C₂×C₄×Dic₂₃

C₄

yes

—

C₂×C₄×Dic₂₃

C₄

yes

characteristic

C₂×C₄×Dic₂₃

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂₃×C₄⋊C₈

—

C₄⋊(C₂₃⋊C₈)

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

—

C₂²×Dic₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

Q₈

yes

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

C₈

yes

complemented

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₃

M₁₆

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂×C₈

yes

—

C₂×Dic₂₃

M₁₆

yes

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1831

—

maximal, Sylow

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

#117

yes

—

C₂²×C₂₃

C₂²×C₈

yes

characteristic

C₂²×C₂₃

C₂×M₁₆

yes

characteristic

C₂²×C₂₃

C₂×M₁₆

yes

—

C₄×C₂₃

—

C₄×C₂₃

C₄⋊C₈

yes

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

Dic₂₃

—

Dic₂₃

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₂₃

yes

characteristic

C₄³

—

#336

—

C₂×C₂₃

#382

yes

characteristic

C₂×C₂₃

C₂×C₄⋊C₈

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#432

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

—

C₂₃

#1831

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₂₃

yes

characteristic

C₂²×C₄

C₄×D₂₃

yes

characteristic

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

Dic₄₆

yes

—

C₄²

C₂²⋊D₂₃

yes

—

C₂³

C₂×C₄×D₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₈×D₂₃

yes

—

105

C₂×C₄

#67262

yes

—

106

C₂×C₄

Dic₂₃⋊C₄

yes

—

107

C₂×C₄

Dic₂₃⋊C₄

yes

—

108

C₂×C₄

C₂×Dic₄₆

yes

characteristic

109

C₂×C₄

#67290

yes

—

110

C₂×C₄

#67292

yes

—

111

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

112

C₈

—

113

C₈

—

114

C₂²

#10625063

yes

—

115

C₂²

C₂×C₈×D₂₃

yes

characteristic

116

C₂²

C₂×#67262

yes

characteristic

117

C₂²

M₁₆×D₂₃

yes

—

118

C₂²

C₂×Dic₂₃⋊C₄

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

Dic₂₃⋊C₈

yes

—

123

C₄

#10625061

yes

—

124

C₄

#10625107

yes

characteristic

125

C₄

#10625118

yes

characteristic

126

C₂

#13092282

yes

characteristic

127

C₂

C₂×Dic₂₃⋊C₈

yes

characteristic

128

C₂

#13092440

yes

characteristic

129

C₂

#13092469

yes

—

130

C₂

#13092473

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂²×C₈
Fitting	C₂×C₂₃×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₂₃	C₂²×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×C₄⋊C₈	D₂₃
hypocenter	C₂₃	#1831
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃

commutator

C₂²×C₂₃

C₂²×C₈

Fitting

C₂×C₂₃×C₄⋊C₈

C₂

socle, prime radical

C₂³×C₂₃

C₂²×C₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×C₄⋊C₈

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
27. C₄×Dic₂₃	C₈	semidirect
79. C₂₃	#1831	semidirect

H ≅ G/N

type

27. C₄×Dic₂₃

C₈

semidirect

79. C₂₃

#1831

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
C₄²×Dic₂₃	2
#13092171	2
C₂×C₂₃×C₄⋊C₈	2
#1831	23

subgroup

index

C₂×C₄⋊(C₂₃⋊C₈)

C₄²×Dic₂₃

#13092171

C₂×C₂₃×C₄⋊C₈

#1831

Subgroups — #15673513

Conjugacy classes of subgroups

Complements

Maximal subgroups