Group #15674184 — subgroups

Subgroups — #15674184

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	#13092465	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625004	C₂	1	1	yes	maximal, characteristic
3	C₂×#10625004	C₂	1	1	yes	maximal, characteristic
4	#13092375	C₂	1	1	yes	maximal, characteristic
5	#13092189	C₂	1	1	yes	maximal, characteristic
6	#13092188	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#430	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	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	#10625097	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	C₂₃×#108	—	2	2	no	—
16	#10625004	—	2	2	no	—
17	#10625004	—	2	1	no	—
18	#10625004	—	2	1	no	—
19	#10625004	C₄	1	4	yes	complemented
20	#10624997	—	2	1	no	—
21	C₈×Dic₂₃	—	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	—	2	1	no	—
27	M₁₆×C₂₃	C₂×C₄	1	2	yes	—
28	M₁₆×C₂₃	D₄	1	2	yes	—
29	C₂×C₈×C₂₃	—	2	1	no	—
30	C₂×C₈×C₂₃	—	2	1	no	—
31	C₂×C₈×C₂₃	—	2	1	no	—
32	C₂²⋊Dic₂₃	Q₈	1	2	yes	—
33	C₄⋊Dic₂₃	—	2	1	no	—
34	C₄×Dic₂₃	—	2	1	no	—
35	#67267	—	2	1	no	—
36	#67267	—	2	1	no	—
37	#67267	—	2	1	no	—
38	#67267	C₂×C₄	1	2	yes	—
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	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
51	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
52	C₂×Dic₂₃	—	2	2	no	—
53	C₈×C₂₃	—	2	2	no	—
54	C₈×C₂₃	—	2	2	no	—
55	C₈×C₂₃	—	4	1	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	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
67	C₄×C₂₃	#121	1	1	yes	characteristic
68	C₄×C₂₃	#125	1	1	yes	characteristic
69	Dic₂₃	—	4	2	no	—
70	#430	—	23	1	no	—
71	#430	D₂₃	1	1	yes	characteristic
72	C₂×#108	—	23	1	no	—
73	C₂×#108	—	23	1	no	—
74	#405	—	23	1	no	—
75	#344	—	23	1	no	—
76	#344	—	23	1	no	—
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	C₂×D₂₃	1	1	yes	characteristic
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	—	23	1	no	—
88	#108	—	2	2	no	—
89	#108	—	23	4	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	—	2	1	no	—
101	M₁₆	—	23	2	no	—
102	M₁₆	—	46	1	no	—
103	M₁₆	—	46	1	no	—
104	M₁₆	—	46	1	no	—
105	M₁₆	C₄×D₂₃	1	2	yes	—
106	M₁₆	C₂²⋊D₂₃	1	2	yes	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	23	2	no	—
111	C₂×C₈	—	23	1	no	—
112	C₂×C₈	—	23	1	no	—
113	C₂×C₈	—	23	2	no	—
114	C₂×C₈	—	46	1	no	—
115	C₂×C₈	—	46	1	no	—
116	C₄⋊C₄	—	46	1	no	—
117	C₂²⋊C₄	—	23	2	no	—
118	C₄²	—	46	1	no	—
119	C₂³	Q₈×D₂₃	1	1	yes	characteristic
120	C₂×C₄	—	46	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	#67290	1	1	yes	characteristic
125	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	—	2	2	no	—
129	C₈	—	4	1	no	—
130	C₈	—	23	2	no	—
131	C₈	—	23	2	no	—
132	C₈	—	46	2	no	—
133	C₈	—	92	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	#10625058	1	1	yes	characteristic
136	C₂²	#10625067	1	1	yes	characteristic
137	C₂²	#10625115	1	1	yes	characteristic
138	C₄	—	92	2	no	—
139	C₄	#10625060	1	1	yes	characteristic
140	C₄	C₄⋊C₄×D₂₃	1	1	yes	characteristic
141	C₄	C₄×C₂²⋊D₂₃	1	1	yes	characteristic
142	C₄	#10625110	1	1	yes	characteristic
143	C₂	—	2	1	no	—
144	C₂	—	2	1	no	—
145	C₂	—	2	1	no	—
146	C₂	#13092312	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092465

C₂

yes

maximal, characteristic

C₂×#10625004

C₂

yes

maximal, characteristic

C₂×#10625004

C₂

yes

maximal, characteristic

#13092375

C₂

yes

maximal, characteristic

#13092189

C₂

yes

maximal, characteristic

#13092188

C₂

yes

maximal, characteristic

C₂₃×#430

C₂

yes

maximal, characteristic

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

#10625097

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂₃×#108

—

—

—

—

C₄

yes

complemented

#10624997

—

C₈×Dic₂₃

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

C₂×C₄

yes

—

M₁₆×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂²⋊Dic₂₃

Q₈

yes

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

—

—

—

—

C₂×C₄

yes

—

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

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

C₂×Dic₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

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

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃

C₄×D₄

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

#430

—

#430

D₂₃

yes

—

—

—

—

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#389

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

characteristic

C₂×M₁₆

C₂×D₂₃

yes

characteristic

C₂×M₁₆

C₂×D₂₃

yes

characteristic

C₂²×C₈

—

—

—

—

—

—

—

—

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

C₄×D₂₃

yes

—

106

M₁₆

C₂²⋊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

—

118

C₄²

—

119

C₂³

Q₈×D₂₃

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

#67290

yes

characteristic

125

C₂×C₄

Q₈×D₂₃

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₂²

—

135

C₂²

#10625058

yes

characteristic

136

C₂²

#10625067

yes

characteristic

137

C₂²

#10625115

yes

characteristic

138

C₄

—

139

C₄

#10625060

yes

characteristic

140

C₄

C₄⋊C₄×D₂₃

yes

characteristic

141

C₄

C₄×C₂²⋊D₂₃

yes

characteristic

142

C₄

#10625110

yes

characteristic

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

#13092312

yes

characteristic

	subgroup	G/H
center	C₄	#10625110
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#430	C₂
socle	C₂×C₂₃	#389
prime radical	C₂³×C₂₃	C₂×Q₈
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#430	D₂₃
hypocenter	C₂₃	#1953
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

subgroup

G/H

center

C₄

#10625110

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#430

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 ◁ #430

N	H ≅ G/N	type
19. #10625004	C₄	semidirect
95. C₂₃	#1953	semidirect

H ≅ G/N

type

19. #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
#13092465	2
C₂×#10625004	2
#13092375	2
#13092189	2
#13092188	2
C₂₃×#430	2
#1953	23

subgroup

index

C₂₃×#430

Subgroups — #15674184

Conjugacy classes of subgroups

Complements

Maximal subgroups