Group #15674120 — subgroups

Subgroups — #15674120

Order 2944.

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

normal subgroups only

This group has 248 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×C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₂₃⋊C₈	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic
4	#13092171	C₂	1	2	yes	maximal
5	C₂×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₂₃	C₄	1	2	yes	—
10	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
11	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
12	C₂₃×#97	—	2	2	no	—
13	C₂₃⋊#95	C₄	1	1	yes	characteristic
14	C₂₃⋊#95	C₄	1	1	yes	characteristic
15	Dic₂₃⋊C₈	—	2	4	no	—
16	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
17	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
21	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₂₃	—	2	1	no	—
23	C₂×C₈×C₂₃	—	2	1	no	—
24	C₂×C₈×C₂₃	—	2	1	no	—
25	C₂×C₈×C₂₃	—	2	1	no	—
26	C₂×C₈×C₂₃	D₄	1	2	yes	—
27	C₂×C₈×C₂₃	Q₈	1	2	yes	—
28	C₄²×C₂₃	D₄	1	2	yes	—
29	C₄²×C₂₃	Q₈	1	2	yes	—
30	C₄⋊Dic₂₃	—	2	4	no	—
31	C₄×Dic₂₃	—	2	2	no	—
32	C₄×Dic₂₃	—	2	2	no	—
33	C₂×C₂₃⋊C₈	—	2	2	no	—
34	C₂×C₂₃⋊C₈	—	2	2	no	—
35	C₂×C₂₃⋊C₈	—	2	2	no	—
36	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₃	—	2	2	no	—
38	C₂×C₄×C₂₃	—	2	2	no	—
39	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
46	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
47	C₂×Dic₂₃	—	2	2	no	—
48	C₂×Dic₂₃	—	2	2	no	—
49	C₂×Dic₂₃	—	2	2	no	—
50	C₂×Dic₂₃	—	2	2	no	—
51	C₂×Dic₂₃	—	2	2	no	—
52	C₂×Dic₂₃	M₁₆	1	4	yes	—
53	C₈×C₂₃	—	2	2	no	—
54	C₈×C₂₃	—	2	2	no	—
55	C₂₃⋊C₈	—	4	2	no	—
56	#1940	—	23	1	no	maximal, Sylow
57	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
59	C₂²×C₂₃	C₄×Q₈	1	1	yes	characteristic
60	C₂²×C₂₃	C₂×M₁₆	1	2	yes	—
61	C₂²×C₂₃	#131	1	2	yes	—
62	C₄×C₂₃	—	2	2	no	—
63	C₄×C₂₃	—	2	2	no	—
64	C₄×C₂₃	#121	1	1	yes	characteristic
65	C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
66	C₄×C₂₃	#125	1	1	yes	characteristic
67	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
68	Dic₂₃	—	2	4	no	—
69	Dic₂₃	—	4	2	no	—
70	C₂×C₄⋊C₈	—	23	2	no	—
71	C₂×C₄⋊C₈	—	23	1	no	—
72	C₂×#97	D₂₃	1	1	yes	characteristic
73	C₄×C₄⋊C₄	—	23	1	no	—
74	#336	—	23	2	no	—
75	C₂×C₂₃	#389	1	1	yes	characteristic
76	C₂×C₂₃	#423	1	1	yes	characteristic
77	C₂×C₂₃	#424	1	1	yes	characteristic
78	C₂×C₂₃	#435	1	2	yes	—
79	C₂×C₂₃	#446	1	2	yes	—
80	C₂²×C₈	—	23	2	no	—
81	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
82	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
83	C₂×C₄⋊C₄	—	23	2	no	—
84	C₂×C₄²	—	23	2	no	—
85	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
86	C₄⋊C₈	—	46	4	no	—
87	C₄⋊C₈	—	46	2	no	—
88	#97	—	2	2	no	—
89	#95	—	23	1	no	—
90	#95	—	23	1	no	—
91	C₂₃	#1940	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₄	—	23	2	no	—
93	C₂²×C₄	—	23	2	no	—
94	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
95	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	46	2	no	—
102	C₂×C₈	—	46	2	no	—
103	C₂×C₈	—	46	2	no	—
104	C₂×C₈	Dic₄₆	1	2	yes	—
105	C₂×C₈	D₉₂	1	2	yes	—
106	C₄⋊C₄	—	46	4	no	—
107	C₄²	—	46	2	no	—
108	C₄²	—	46	2	no	—
109	C₄²	Dic₄₆	1	2	yes	—
110	C₄²	C₂²⋊D₂₃	1	2	yes	—
111	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	23	4	no	—
115	C₂×C₄	—	46	2	no	—
116	C₂×C₄	—	46	2	no	—
117	C₂×C₄	—	46	2	no	—
118	C₂×C₄	—	46	2	no	—
119	C₂×C₄	—	46	2	no	—
120	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
121	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
122	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
123	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
124	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
125	C₂×C₄	#67288	1	1	yes	characteristic
126	C₂×C₄	#67288	1	1	yes	characteristic
127	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	—	2	2	no	—
130	C₈	—	92	2	no	—
131	C₂²	C₄×Dic₄₆	1	1	yes	characteristic
132	C₂²	C₄×D₉₂	1	1	yes	characteristic
133	C₂²	M₁₆×D₂₃	1	2	yes	—
134	C₂²	#10625078	1	2	yes	—
135	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	46	4	no	—
139	C₄	—	92	2	no	—
140	C₄	#10625040	1	1	yes	characteristic
141	C₄	#10625041	1	1	yes	characteristic
142	C₄	#10625095	1	1	yes	characteristic
143	C₄	#10625101	1	1	yes	characteristic
144	C₂	#13092315	1	1	yes	characteristic
145	C₂	#13092440	1	1	yes	characteristic
146	C₂	#13092445	1	1	yes	characteristic
147	C₂	#13092490	1	2	yes	—
148	C₂	#13092545	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₂₃⋊C₈)

C₂

yes

maximal, characteristic

C₂×Dic₂₃⋊C₈

C₂

yes

maximal

C₄×C₄⋊Dic₂₃

C₂

yes

maximal, characteristic

#13092171

C₂

yes

maximal

C₂×C₂₃×#97

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₄

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂₃×#97

—

C₂₃⋊#95

C₄

yes

characteristic

C₂₃⋊#95

C₄

yes

characteristic

Dic₂₃⋊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

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

Q₈

yes

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

Q₈

yes

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

M₁₆

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1940

—

maximal, Sylow

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

C₄×D₄

yes

characteristic

C₂²×C₂₃

C₄×Q₈

yes

characteristic

C₂²×C₂₃

C₂×M₁₆

yes

—

C₂²×C₂₃

#131

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#121

yes

characteristic

C₄×C₂₃

C₂²⋊Q₈

yes

characteristic

C₄×C₂₃

#125

yes

characteristic

C₄×C₂₃

C₄⋊Q₈

yes

characteristic

Dic₂₃

—

Dic₂₃

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

D₂₃

yes

characteristic

C₄×C₄⋊C₄

—

#336

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#446

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#95

—

#95

—

C₂₃

#1940

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

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

Dic₄₆

yes

—

105

C₂×C₈

D₉₂

yes

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

—

109

C₄²

Dic₄₆

yes

—

110

C₄²

C₂²⋊D₂₃

yes

—

111

C₂³

C₂×C₄×D₂₃

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

Dic₂₃⋊C₄

yes

—

121

C₂×C₄

Dic₂₃⋊C₄

yes

—

122

C₂×C₄

C₂×Dic₄₆

yes

characteristic

123

C₂×C₄

C₂×Dic₄₆

yes

characteristic

124

C₂×C₄

C₂×D₉₂

yes

characteristic

125

C₂×C₄

#67288

yes

characteristic

126

C₂×C₄

#67288

yes

characteristic

127

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₂²

C₄×Dic₄₆

yes

characteristic

132

C₂²

C₄×D₉₂

yes

characteristic

133

C₂²

M₁₆×D₂₃

yes

—

134

C₂²

#10625078

yes

—

135

C₂²

C₂×Dic₂₃⋊C₄

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#10625040

yes

characteristic

141

C₄

#10625041

yes

characteristic

142

C₄

#10625095

yes

characteristic

143

C₄

#10625101

yes

characteristic

144

C₂

#13092315

yes

characteristic

145

C₂

#13092440

yes

characteristic

146

C₂

#13092445

yes

characteristic

147

C₂

#13092490

yes

—

148

C₂

#13092545

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator, socle, prime radical	C₂³×C₂₃	C₂²×C₄
Fitting	C₂×C₂₃×#97	C₂
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×#97	D₂₃
hypocenter	C₂₃	#1940
LCS	G ▷ C₂³×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃

commutator, socle, prime radical

C₂³×C₂₃

C₂²×C₄

Fitting

C₂×C₂₃×#97

C₂

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×#97

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂₃

C₁ ◁ C₂²×C₄ ◁ C₂×#97

N	H ≅ G/N	type
91. C₂₃	#1940	semidirect

H ≅ G/N

type

91. C₂₃

#1940

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
C₂×Dic₂₃⋊C₈	2
C₄×C₄⋊Dic₂₃	2
#13092171	2
C₂×C₂₃×#97	2
#1940	23

subgroup

index

C₂×C₄⋊(C₂₃⋊C₈)

C₂×Dic₂₃⋊C₈

C₄×C₄⋊Dic₂₃

#13092171

C₂×C₂₃×#97

#1940

Subgroups — #15674120

Conjugacy classes of subgroups

Complements

Maximal subgroups