Group #15674980 — subgroups

Subgroups — #15674980

Order 2944.

148 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 59 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	#13092465	C₂	1	1	yes	maximal, characteristic
2	#13092444	C₂	1	1	yes	maximal, characteristic
3	#13092187	C₂	1	1	yes	maximal, characteristic
4	#13092181	C₂	1	1	yes	maximal, characteristic
5	#13092181	C₂	1	1	yes	maximal, characteristic
6	C₂₃⋊#338	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#344	C₂	1	1	yes	maximal, characteristic
8	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×#117	C₂²	1	1	yes	characteristic
11	C₂×#67267	C₂²	1	1	yes	characteristic
12	C₂×#67267	C₂²	1	1	yes	characteristic
13	C₂×#67267	C₂²	1	1	yes	characteristic
14	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
15	#10625004	—	2	2	no	—
16	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
17	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	M₁₆×C₂₃	—	2	1	no	—
19	M₁₆×C₂₃	—	2	1	no	—
20	M₁₆×C₂₃	—	2	1	no	—
21	M₁₆×C₂₃	—	2	1	no	—
22	C₂×C₈×C₂₃	—	2	1	no	—
23	C₂×C₈×C₂₃	—	2	1	no	—
24	C₄⋊C₄×C₂₃	—	2	1	no	—
25	C₂²⋊C₄×C₂₃	Q₈	1	2	yes	—
26	C₄²×C₂₃	—	2	1	no	—
27	#67267	—	2	1	no	—
28	#67267	—	2	1	no	—
29	#67267	—	2	1	no	—
30	#67267	—	2	1	no	—
31	#67267	—	2	1	no	—
32	#67267	—	2	1	no	—
33	C₂×C₂₃⋊C₈	—	2	1	no	—
34	C₂×C₂₃⋊C₈	—	2	1	no	—
35	C₂×C₂₃⋊C₈	—	2	1	no	—
36	C₂×C₂₃⋊C₈	—	2	1	no	—
37	C₂×C₂₃⋊C₈	—	2	1	no	—
38	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
39	C₂³×C₂₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₂₃	—	2	2	no	—
41	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
43	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
44	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
45	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
46	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
47	C₈×C₂₃	—	4	1	no	—
48	C₈×C₂₃	—	4	1	no	—
49	C₂₃⋊C₈	—	2	2	no	—
50	C₂₃⋊C₈	—	4	1	no	—
51	C₂₃⋊C₈	—	4	1	no	—
52	C₂₃⋊C₈	—	4	1	no	—
53	#2103	—	23	1	no	maximal, Sylow
54	C₂²×C₂₃	—	4	1	no	—
55	C₂²×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
56	C₂²×C₂₃	#125	1	1	yes	characteristic
57	C₂²×C₂₃	#125	1	1	yes	characteristic
58	C₄×C₂₃	—	4	2	no	—
59	C₄×C₂₃	#123	1	1	yes	characteristic
60	C₄×C₂₃	#124	1	1	yes	characteristic
61	C₄×C₂₃	#126	1	1	yes	characteristic
62	C₄×C₂₃	#126	1	1	yes	characteristic
63	#430	—	23	1	no	—
64	#424	—	23	1	no	—
65	#344	—	23	1	no	—
66	#344	D₂₃	1	1	yes	characteristic
67	#341	—	23	1	no	—
68	#341	—	23	1	no	—
69	#338	—	23	1	no	—
70	C₂×C₂₃	—	2	1	no	—
71	C₂×C₂₃	—	2	1	no	—
72	C₂×C₂₃	—	2	1	no	—
73	C₂×C₂₃	#400	1	1	yes	characteristic
74	C₂×M₁₆	—	23	1	no	—
75	C₂×M₁₆	—	23	1	no	—
76	C₂×M₁₆	—	23	1	no	—
77	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
78	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
79	C₂²×C₈	—	23	1	no	—
80	#117	C₂×D₂₃	1	1	yes	characteristic
81	#108	—	46	2	no	—
82	C₄⋊C₈	—	46	2	no	—
83	C₂₃	#2103	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
85	M₁₆	—	2	1	no	—
86	M₁₆	—	2	1	no	—
87	M₁₆	—	2	1	no	—
88	M₁₆	—	2	1	no	—
89	M₁₆	—	46	1	no	—
90	M₁₆	—	46	1	no	—
91	M₁₆	—	46	1	no	—
92	M₁₆	—	46	1	no	—
93	M₁₆	—	46	1	no	—
94	M₁₆	—	46	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	23	2	no	—
98	C₂×C₈	—	46	1	no	—
99	C₂×C₈	—	46	1	no	—
100	C₂×C₈	—	46	1	no	—
101	C₂×C₈	—	46	1	no	—
102	C₂×C₈	—	46	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₂²⋊C₄	Dic₄₆	1	2	yes	—
105	C₄²	—	2	1	no	—
106	C₂³	C₂×Dic₄₆	1	1	yes	characteristic
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	#67288	1	1	yes	characteristic
109	C₂×C₄	#67288	1	1	yes	characteristic
110	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
111	C₂×C₄	#67290	1	1	yes	characteristic
112	C₂×C₄	#67290	1	1	yes	characteristic
113	C₂×C₄	#67292	1	1	yes	characteristic
114	C₈	—	4	1	no	—
115	C₈	—	4	1	no	—
116	C₈	—	46	2	no	—
117	C₈	—	92	1	no	—
118	C₈	—	92	1	no	—
119	C₈	—	92	1	no	—
120	C₂²	—	4	1	no	—
121	C₂²	#10625041	1	1	yes	characteristic
122	C₂²	#10625049	1	1	yes	characteristic
123	C₂²	#10625061	1	1	yes	characteristic
124	C₄	—	4	2	no	—
125	C₄	#10625050	1	1	yes	characteristic
126	C₄	#10625056	1	1	yes	characteristic
127	C₄	#10625065	1	1	yes	characteristic
128	C₄	#10625069	1	1	yes	characteristic
129	C₂	—	2	1	no	—
130	C₂	—	2	1	no	—
131	C₂	—	2	1	no	—
132	C₂	#13092359	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092465

C₂

yes

maximal, characteristic

#13092444

C₂

yes

maximal, characteristic

#13092187

C₂

yes

maximal, characteristic

#13092181

C₂

yes

maximal, characteristic

#13092181

C₂

yes

maximal, characteristic

C₂₃⋊#338

C₂

yes

maximal, characteristic

C₂₃×#344

C₂

yes

maximal, characteristic

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

C₂₃×#117

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

#10625004

—

C₄⋊(C₂₃⋊C₈)

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

Q₈

yes

—

C₄²×C₂₃

—

—

—

—

—

—

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂³×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#2103

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₂²⋊Q₈

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

—

—

D₂₃

yes

—

—

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#400

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

characteristic

C₂×M₁₆

C₂×D₂₃

yes

characteristic

C₂²×C₈

—

#117

C₂×D₂₃

yes

characteristic

#108

—

C₄⋊C₈

—

C₂₃

#2103

yes

characteristic, complemented, Sylow

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

Dic₄₆

yes

—

105

C₄²

—

106

C₂³

C₂×Dic₄₆

yes

characteristic

107

C₂×C₄

—

108

C₂×C₄

#67288

yes

characteristic

109

C₂×C₄

#67288

yes

characteristic

110

C₂×C₄

D₄×D₂₃

yes

characteristic

111

C₂×C₄

#67290

yes

characteristic

112

C₂×C₄

#67290

yes

characteristic

113

C₂×C₄

#67292

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₂²

—

121

C₂²

#10625041

yes

characteristic

122

C₂²

#10625049

yes

characteristic

123

C₂²

#10625061

yes

characteristic

124

C₄

—

125

C₄

#10625050

yes

characteristic

126

C₄

#10625056

yes

characteristic

127

C₄

#10625065

yes

characteristic

128

C₄

#10625069

yes

characteristic

129

C₂

—

130

C₂

—

131

C₂

—

132

C₂

#13092359

yes

characteristic

	subgroup	G/H
center	C₄	#10625056
commutator, cosocle	C₂²×C₄×C₂₃	C₂³
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#344	C₂
socle	C₂×C₂₃	#400
prime radical	C₂³×C₂₃	C₂×Q₈
hypercenter	#344	D₂₃
hypocenter	C₂₃	#2103
LCS	G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₄

#10625056

commutator, cosocle

C₂²×C₄×C₂₃

C₂³

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#344

C₂

socle

C₂×C₂₃

#400

prime radical

C₂³×C₂₃

C₂×Q₈

D₂₃

C₂₃

G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
83. C₂₃	#2103	semidirect

H ≅ G/N

type

83. C₂₃

#2103

semidirect

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

subgroup	index
#13092465	2
#13092444	2
#13092187	2
#13092181	2
C₂₃⋊#338	2
C₂₃×#344	2
#2103	23

subgroup

index

C₂₃⋊#338

C₂₃×#344

#2103

Subgroups — #15674980

Conjugacy classes of subgroups

Complements

Maximal subgroups