Group #15674222 — subgroups

Subgroups — #15674222

Order 2944.

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

normal subgroups only

This group has 412 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×#10625014	C₂	1	4	yes	maximal, complemented
2	C₂×#10624985	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092283	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#127	C₂	1	1	yes	maximal, characteristic
5	C₂²×D₄×C₂₃	—	2	2	no	—
6	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
7	C₂₃×#127	—	2	2	no	—
8	C₂₃×#127	C₄	1	4	yes	complemented
9	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
11	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
12	#10625014	—	2	8	no	—
13	C₂₃⋊#95	—	2	2	no	—
14	#10624985	—	2	2	no	—
15	C₂⁴×C₂₃	—	2	2	no	—
16	C₂×D₄×C₂₃	—	2	8	no	—
17	C₂×D₄×C₂₃	—	2	4	no	—
18	C₂×D₄×C₂₃	—	2	2	no	—
19	C₂×D₄×C₂₃	—	2	4	no	—
20	C₂×D₄×C₂₃	—	4	8	no	—
21	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₂₃	D₄	1	2	yes	—
23	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₂₃	—	2	2	no	—
25	C₄²×C₂₃	C₂×C₄	1	2	yes	—
26	C₄²×C₂₃	D₄	1	2	yes	—
27	C₄⋊Dic₂₃	—	2	4	no	—
28	C₂×C₂₃⋊C₈	—	2	2	no	—
29	C₂×C₂₃⋊C₈	—	2	2	no	—
30	C₂×C₂₃⋊C₈	D₄	1	4	yes	complemented
31	C₂³×C₂₃	—	2	4	no	—
32	C₂³×C₂₃	—	4	2	no	—
33	C₂³×C₂₃	—	4	2	no	—
34	C₂³×C₂₃	—	4	8	no	—
35	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
36	D₄×C₂₃	—	2	8	no	—
37	D₄×C₂₃	—	4	16	no	—
38	D₄×C₂₃	—	4	4	no	—
39	C₂×C₄×C₂₃	—	2	4	no	—
40	C₂×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₃	#41	1	2	yes	—
45	C₂×Dic₂₃	—	4	2	no	—
46	C₂×Dic₂₃	—	4	4	no	—
47	C₂₃⋊C₈	—	2	4	no	—
48	#1961	—	23	1	no	maximal, Sylow
49	C₂²×C₂₃	—	4	4	no	—
50	C₂²×C₂₃	—	4	8	no	—
51	C₂²×C₂₃	—	8	4	no	—
52	C₂²×C₂₃	—	8	4	no	—
53	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₂₃	#124	1	1	yes	characteristic
55	C₂²×C₂₃	#127	1	1	yes	characteristic
56	C₂²×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
57	C₄×C₂₃	—	2	4	no	—
58	C₄×C₂₃	C₄×D₄	1	2	yes	—
59	C₄×C₂₃	#121	1	2	yes	—
60	Dic₂₃	—	8	2	no	—
61	C₂×#127	D₂₃	1	1	yes	characteristic
62	C₂×#102	—	23	4	no	—
63	C₂×#97	—	23	1	no	—
64	#382	—	23	1	no	—
65	C₂×C₂₃	—	8	4	no	—
66	C₂×C₂₃	#390	1	1	yes	characteristic
67	C₂×C₂₃	#418	1	2	yes	—
68	C₂×C₂₃	#490	1	2	yes	—
69	C₂×C₂₃	#496	1	2	yes	—
70	C₂²×D₄	—	2	2	no	—
71	C₂²×D₄	C₂×D₂₃	1	2	yes	—
72	C₂²×C₈	—	23	2	no	—
73	#127	—	2	2	no	—
74	#127	Dic₂₃	1	4	yes	complemented
75	C₂×C₄⋊C₄	—	23	2	no	—
76	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
77	#102	—	46	8	no	—
78	#97	—	46	2	no	—
79	#95	—	46	2	no	—
80	C₂₃	#1961	1	1	yes	characteristic, complemented, Sylow
81	C₂⁴	—	2	2	no	—
82	C₂×D₄	—	2	8	no	—
83	C₂×D₄	—	2	4	no	—
84	C₂×D₄	—	2	4	no	—
85	C₂×D₄	—	2	2	no	—
86	C₂×D₄	—	4	8	no	—
87	C₂×D₄	C₂×Dic₂₃	1	4	yes	—
88	C₂²×C₄	—	46	2	no	—
89	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
90	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
91	C₂×C₈	—	23	4	no	—
92	C₂×C₈	—	46	2	no	—
93	C₂×C₈	—	46	2	no	—
94	C₄⋊C₄	—	46	4	no	—
95	C₄²	C₂×Dic₂₃	1	2	yes	—
96	C₄²	C₂²⋊D₂₃	1	2	yes	—
97	C₂³	—	2	4	no	—
98	C₂³	—	4	8	no	—
99	C₂³	—	4	2	no	—
100	C₂³	—	4	2	no	—
101	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
102	D₄	—	2	8	no	—
103	D₄	—	4	16	no	—
104	D₄	—	4	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	92	4	no	—
107	C₂×C₄	—	92	2	no	—
108	C₂×C₄	C₂²⋊Dic₂₃	1	4	yes	—
109	C₂×C₄	D₄×D₂₃	1	2	yes	—
110	C₂×C₄	#67290	1	2	yes	—
111	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
113	C₈	—	46	4	no	—
114	C₂²	—	4	8	no	—
115	C₂²	—	4	4	no	—
116	C₂²	—	8	4	no	—
117	C₂²	—	8	4	no	—
118	C₂²	#10625103	1	4	yes	—
119	C₂²	#10625107	1	1	yes	characteristic
120	C₂²	#10625111	1	1	yes	characteristic
121	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	—	184	2	no	—
124	C₄	D₄×Dic₂₃	1	2	yes	—
125	C₄	#10625109	1	2	yes	—
126	C₂	—	8	4	no	—
127	C₂	#13092320	1	1	yes	characteristic
128	C₂	#13092422	1	2	yes	—
129	C₂	#13092781	1	2	yes	—
130	C₂	#13092804	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625014

C₂

yes

maximal, complemented

C₂×#10624985

C₂

yes

maximal, characteristic, complemented

#13092283

C₂

yes

maximal, characteristic, complemented

C₂×C₂₃×#127

C₂

yes

maximal, characteristic

C₂²×D₄×C₂₃

—

C₂²×D₄×C₂₃

C₂²

yes

—

C₂₃×#127

—

C₂₃×#127

C₄

yes

complemented

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

#10625014

—

C₂₃⋊#95

—

#10624985

—

C₂⁴×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

D₄

yes

—

C₄⋊Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

D₄

yes

complemented

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂₃⋊C₈

—

#1961

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₄×C₂₃

—

C₄×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

#121

yes

—

Dic₂₃

—

C₂×#127

D₂₃

yes

characteristic

C₂×#102

—

C₂×#97

—

#382

—

C₂×C₂₃

—

C₂×C₂₃

#390

yes

characteristic

C₂×C₂₃

#418

yes

—

C₂×C₂₃

#490

yes

—

C₂×C₂₃

#496

yes

—

C₂²×D₄

—

C₂²×D₄

C₂×D₂₃

yes

—

C₂²×C₈

—

#127

—

#127

Dic₂₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₂₃

yes

—

—

—

C₂₃

#1961

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄²

C₂×Dic₂₃

yes

—

C₄²

C₂²⋊D₂₃

yes

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

102

D₄

—

103

D₄

—

104

D₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂²⋊Dic₂₃

yes

—

109

C₂×C₄

D₄×D₂₃

yes

—

110

C₂×C₄

#67290

yes

—

111

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

113

C₈

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

#10625103

yes

—

119

C₂²

#10625107

yes

characteristic

120

C₂²

#10625111

yes

characteristic

121

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

122

C₄

—

123

C₄

—

184

—

124

C₄

D₄×Dic₂₃

yes

—

125

C₄

#10625109

yes

—

126

C₂

—

127

C₂

#13092320

yes

characteristic

128

C₂

#13092422

yes

—

129

C₂

#13092781

yes

—

130

C₂

#13092804

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting, prime radical	C₂×C₂₃×#127	C₂
socle	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×#127	D₂₃
hypocenter	C₂₃	#1961
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting, prime radical

C₂×C₂₃×#127

C₂

socle

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#10625014	C₂	semidirect
2. C₂×#10624985	C₂	semidirect
3. #13092283	C₂	semidirect
8. C₂₃×#127	C₄	semidirect
30. C₂×C₂₃⋊C₈	D₄	semidirect
74. #127	Dic₂₃	semidirect
80. C₂₃	#1961	semidirect

H ≅ G/N

type

1. C₂×#10625014

C₂

semidirect

2. C₂×#10624985

C₂

semidirect

3. #13092283

C₂

semidirect

8. C₂₃×#127

C₄

semidirect

30. C₂×C₂₃⋊C₈

D₄

semidirect

74. #127

Dic₂₃

semidirect

80. C₂₃

#1961

semidirect

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

subgroup	index
C₂×#10625014	2
C₂×#10624985	2
#13092283	2
C₂×C₂₃×#127	2
#1961	23

subgroup

index

C₂×C₂₃×#127

Subgroups — #15674222

Conjugacy classes of subgroups

Complements

Maximal subgroups