Group #15674186 — subgroups

Subgroups — #15674186

Order 2944.

184 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 79 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	#13092466	C₂	1	1	yes	maximal, characteristic
2	#13092465	C₂	1	1	yes	maximal, characteristic
3	C₂×#10625004	C₂	1	1	yes	maximal, characteristic
4	C₂×#10625004	C₂	1	1	yes	maximal, characteristic
5	#13092373	C₂	1	1	yes	maximal, characteristic
6	#13092187	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	#10625004	—	2	1	no	—
17	#10625004	—	2	1	no	—
18	#10625004	C₄	1	4	yes	complemented
19	#10625001	—	2	2	no	—
20	#10624985	—	2	1	no	—
21	C₄×C₂₃⋊C₈	—	2	1	no	—
22	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	M₁₆×C₂₃	—	2	1	no	—
24	M₁₆×C₂₃	—	2	1	no	—
25	M₁₆×C₂₃	—	2	1	no	—
26	M₁₆×C₂₃	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₂₃	—	2	1	no	—
28	C₂×C₈×C₂₃	—	2	1	no	—
29	C₄⋊C₄×C₂₃	—	2	1	no	—
30	C₂²⋊C₄×C₂₃	Q₈	1	2	yes	—
31	C₄²×C₂₃	—	2	1	no	—
32	#67267	—	2	1	no	—
33	#67267	—	2	1	no	—
34	#67267	—	2	1	no	—
35	#67267	—	2	1	no	—
36	#67267	C₂×C₄	1	2	yes	—
37	#67267	D₄	1	2	yes	—
38	C₂×C₂₃⋊C₈	—	2	1	no	—
39	C₂×C₂₃⋊C₈	—	2	1	no	—
40	C₂×C₂₃⋊C₈	—	2	1	no	—
41	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
42	C₂×C₂₃⋊C₈	D₄	1	1	yes	characteristic
43	C₂×C₂₃⋊C₈	D₄	1	1	yes	characteristic
44	C₂×C₂₃⋊C₈	Q₈	1	2	yes	—
45	C₂³×C₂₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₂₃	—	2	2	no	—
47	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
52	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
53	C₈×C₂₃	—	2	2	no	—
54	C₈×C₂₃	—	4	1	no	—
55	C₂₃⋊C₈	—	2	2	no	—
56	C₂₃⋊C₈	—	2	2	no	—
57	C₂₃⋊C₈	—	4	1	no	—
58	C₂₃⋊C₈	C₄⋊C₄	1	2	yes	—
59	C₂₃⋊C₈	C₄⋊C₄	1	2	yes	—
60	#1953	—	23	1	no	maximal, Sylow
61	C₂²×C₂₃	—	4	1	no	—
62	C₂²×C₂₃	C₄×Q₈	1	1	yes	characteristic
63	C₂²×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂²×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₄×C₂₃	—	4	2	no	—
66	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
68	C₄×C₂₃	#121	1	1	yes	characteristic
69	C₄×C₂₃	#125	1	1	yes	characteristic
70	#430	—	23	1	no	—
71	#430	—	23	1	no	—
72	C₂×#108	—	23	1	no	—
73	C₂×#108	—	23	1	no	—
74	#405	—	23	1	no	—
75	#344	—	23	1	no	—
76	#344	D₂₃	1	1	yes	characteristic
77	C₂×C₂₃	—	2	1	no	—
78	C₂×C₂₃	—	2	1	no	—
79	C₂×C₂₃	—	2	1	no	—
80	C₂×C₂₃	#389	1	1	yes	characteristic
81	C₂×M₁₆	—	23	1	no	—
82	C₂×M₁₆	—	23	1	no	—
83	C₂×M₁₆	—	23	1	no	—
84	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
85	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
86	C₂²×C₈	—	23	1	no	—
87	#117	C₂×D₂₃	1	1	yes	characteristic
88	#108	—	23	4	no	—
89	#108	—	46	2	no	—
90	#108	—	46	1	no	—
91	#108	—	46	1	no	—
92	#108	—	46	2	no	—
93	#97	—	46	1	no	—
94	C₄×C₈	—	46	1	no	—
95	C₂₃	#1953	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
97	M₁₆	—	2	1	no	—
98	M₁₆	—	2	1	no	—
99	M₁₆	—	2	1	no	—
100	M₁₆	—	23	2	no	—
101	M₁₆	—	23	2	no	—
102	M₁₆	—	46	1	no	—
103	M₁₆	—	46	1	no	—
104	M₁₆	—	46	1	no	—
105	M₁₆	—	46	1	no	—
106	M₁₆	C₄×D₂₃	1	2	yes	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	23	2	no	—
110	C₂×C₈	—	23	1	no	—
111	C₂×C₈	—	23	1	no	—
112	C₂×C₈	—	23	2	no	—
113	C₂×C₈	—	46	1	no	—
114	C₂×C₈	—	46	1	no	—
115	C₂×C₈	—	46	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₂²⋊C₄	Dic₄₆	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₂³	C₂×Dic₄₆	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
122	C₂×C₄	#67288	1	1	yes	characteristic
123	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
124	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
125	C₂×C₄	#67290	1	1	yes	characteristic
126	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	—	4	1	no	—
129	C₈	—	23	2	no	—
130	C₈	—	23	2	no	—
131	C₈	—	46	2	no	—
132	C₈	—	46	2	no	—
133	C₈	—	92	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	C₄×Dic₄₆	1	1	yes	characteristic
136	C₂²	#10625049	1	1	yes	characteristic
137	C₂²	#10625059	1	1	yes	characteristic
138	C₄	—	4	2	no	—
139	C₄	(C₂²⋊D₂₃)⋊C₄	1	1	yes	characteristic
140	C₄	#10625055	1	1	yes	characteristic
141	C₄	#10625060	1	1	yes	characteristic
142	C₄	C₄⋊C₄×D₂₃	1	1	yes	characteristic
143	C₂	—	2	1	no	—
144	C₂	—	2	1	no	—
145	C₂	—	2	1	no	—
146	C₂	#13092314	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092466

C₂

yes

maximal, characteristic

#13092465

C₂

yes

maximal, characteristic

C₂×#10625004

C₂

yes

maximal, characteristic

C₂×#10625004

C₂

yes

maximal, characteristic

#13092373

C₂

yes

maximal, characteristic

#13092187

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

—

—

—

C₄

yes

complemented

#10625001

—

#10624985

—

C₄×C₂₃⋊C₈

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

Q₈

yes

—

C₄²×C₂₃

—

—

—

—

—

C₂×C₄

yes

—

#67267

D₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

D₄

yes

characteristic

C₂×C₂₃⋊C₈

D₄

yes

characteristic

C₂×C₂₃⋊C₈

Q₈

yes

—

C₂³×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

C₂₃⋊C₈

C₄⋊C₄

yes

—

C₂₃⋊C₈

C₄⋊C₄

yes

—

#1953

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₄×Q₈

yes

characteristic

C₂²×C₂₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₃

C₄⋊Q₈

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃

C₄×D₄

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

—

—

—

—

—

D₂₃

yes

characteristic

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#389

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

—

—

—

—

—

—

C₄×C₈

—

C₂₃

#1953

yes

characteristic, complemented, Sylow

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

106

M₁₆

C₄×D₂₃

yes

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

Dic₄₆

yes

—

118

C₄²

—

119

C₂³

C₂×Dic₄₆

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

122

C₂×C₄

#67288

yes

characteristic

123

C₂×C₄

D₄×D₂₃

yes

characteristic

124

C₂×C₄

D₄×D₂₃

yes

characteristic

125

C₂×C₄

#67290

yes

characteristic

126

C₂×C₄

Q₈×D₂₃

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₂²

—

135

C₂²

C₄×Dic₄₆

yes

characteristic

136

C₂²

#10625049

yes

characteristic

137

C₂²

#10625059

yes

characteristic

138

C₄

—

139

C₄

(C₂²⋊D₂₃)⋊C₄

yes

characteristic

140

C₄

#10625055

yes

characteristic

141

C₄

#10625060

yes

characteristic

142

C₄

C₄⋊C₄×D₂₃

yes

characteristic

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

#13092314

yes

characteristic

	subgroup	G/H
center	C₄	#10625055
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#344	C₂
socle	C₂×C₂₃	#389
prime radical	C₂³×C₂₃	C₂×Q₈
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#344	D₂₃
hypocenter	C₂₃	#1953
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₄

#10625055

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#344

C₂

socle

C₂×C₂₃

#389

prime radical

C₂³×C₂₃

C₂×Q₈

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
18. #10625004	C₄	semidirect
95. C₂₃	#1953	semidirect

H ≅ G/N

type

18. #10625004

C₄

semidirect

95. C₂₃

#1953

semidirect

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

subgroup	index
#13092466	2
#13092465	2
C₂×#10625004	2
#13092373	2
#13092187	2
C₂₃×#344	2
#1953	23

subgroup

index

C₂₃×#344

Subgroups — #15674186

Conjugacy classes of subgroups

Complements

Maximal subgroups