Group #15673380 — subgroups

Subgroups — #15673380

Order 2944.

312 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 101 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×#10625043	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₂₃⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₂×(C₂×D₂₃)⋊C₈	C₂	1	2	yes	maximal
4	#13092172	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#336	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
8	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×(C₂×D₂₃)⋊C₄	C₄	1	2	yes	—
10	C₂×Dic₂₃⋊C₄	C₄	1	2	yes	—
11	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
12	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
13	#10625043	—	2	4	no	—
14	(C₂×D₂₃)⋊C₈	—	2	4	no	—
15	Dic₂₃⋊C₈	—	2	4	no	—
16	C₂³×D₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
20	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×D₂₃	—	2	2	no	—
22	C₂×C₄×D₂₃	—	2	2	no	—
23	C₂×C₄×D₂₃	D₄	1	2	yes	—
24	C₂×C₄×D₂₃	Q₈	1	2	yes	—
25	C₂×C₈×C₂₃	—	2	2	no	—
26	C₂×C₈×C₂₃	—	2	2	no	—
27	C₂×C₈×C₂₃	—	2	2	no	—
28	C₄²×C₂₃	—	2	2	no	—
29	(C₂×D₂₃)⋊C₄	—	2	4	no	—
30	Dic₂₃⋊C₄	—	2	4	no	—
31	C₄×Dic₂₃	D₄	1	4	yes	—
32	C₂×C₂₃⋊C₈	—	2	2	no	—
33	C₂×C₂₃⋊C₈	—	2	2	no	—
34	C₂×C₂₃⋊C₈	—	2	2	no	—
35	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×D₂₃	—	2	1	no	—
37	C₂²×D₂₃	—	2	1	no	—
38	C₂²×D₂₃	—	2	2	no	—
39	C₂²×D₂₃	—	2	2	no	—
40	C₂²×D₂₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₂₃	—	2	2	no	—
42	C₂×C₄×C₂₃	—	2	2	no	—
43	C₂×C₄×C₂₃	—	2	2	no	—
44	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₂₃	#41	1	2	yes	—
48	C₂×Dic₂₃	—	2	1	no	—
49	C₂×Dic₂₃	—	2	1	no	—
50	C₂×Dic₂₃	—	2	2	no	—
51	C₂×Dic₂₃	—	2	2	no	—
52	C₂×Dic₂₃	C₂²⋊C₄	1	4	yes	—
53	C₂×Dic₂₃	C₄⋊C₄	1	2	yes	—
54	C₄×D₂₃	—	2	4	no	—
55	C₄×D₂₃	—	2	4	no	—
56	C₈×C₂₃	—	4	2	no	—
57	C₂₃⋊C₈	—	4	2	no	—
58	#1798	—	23	1	no	maximal, Sylow
59	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₂₃	#117	1	1	yes	characteristic
62	C₂²×C₂₃	#131	1	2	yes	—
63	C₂²×C₂₃	#131	1	2	yes	—
64	C₂×D₂₃	—	2	2	no	—
65	C₂×D₂₃	—	2	4	no	—
66	C₂×D₂₃	—	4	2	no	—
67	C₂×D₂₃	—	4	1	no	—
68	C₂×D₂₃	—	4	1	no	—
69	C₄×C₂₃	—	4	2	no	—
70	C₄×C₂₃	#121	1	2	yes	—
71	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
72	Dic₂₃	—	2	2	no	—
73	Dic₂₃	—	2	4	no	—
74	C₂×#117	—	23	1	no	—
75	C₂×C₄⋊C₈	—	23	2	no	—
76	C₂×C₂²⋊C₈	—	23	2	no	—
77	#336	—	23	1	no	—
78	#336	D₂₃	1	1	yes	characteristic, complemented
79	C₂×C₂₃	#380	1	1	yes	characteristic
80	C₂×C₂₃	#408	1	2	yes	—
81	C₂×C₂₃	#424	1	2	yes	—
82	C₂×C₂₃	#433	1	1	yes	characteristic
83	C₂×C₂₃	#433	1	1	yes	characteristic
84	D₂₃	—	4	2	no	—
85	C₂³×C₄	—	23	1	no	—
86	C₂²×C₈	—	23	2	no	—
87	C₂²×C₈	C₂×D₂₃	1	2	yes	—
88	#117	—	46	4	no	—
89	C₂×C₄⋊C₄	—	23	2	no	—
90	C₂×C₂²⋊C₄	—	23	2	no	—
91	C₂×C₄²	—	23	1	no	—
92	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
93	C₄⋊C₈	—	46	4	no	—
94	C₂²⋊C₈	—	46	4	no	—
95	C₂₃	#1798	1	1	yes	characteristic, complemented, Sylow
96	C₂⁴	—	23	1	no	—
97	C₂²×C₄	—	23	2	no	—
98	C₂²×C₄	—	23	1	no	—
99	C₂²×C₄	—	23	2	no	—
100	C₂²×C₄	—	23	2	no	—
101	C₂²×C₄	—	46	2	no	—
102	C₂²×C₄	—	46	2	no	—
103	C₂²×C₄	C₄×D₂₃	1	2	yes	—
104	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	46	2	no	—
109	C₂×C₈	—	46	2	no	—
110	C₂×C₈	—	46	2	no	—
111	C₄⋊C₄	—	46	4	no	—
112	C₂²⋊C₄	—	46	4	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	23	4	no	—
115	C₂³	—	23	2	no	—
116	C₂³	—	46	1	no	—
117	C₂³	—	46	1	no	—
118	C₂³	—	46	2	no	—
119	C₂³	—	46	2	no	—
120	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	23	4	no	—
125	C₂×C₄	—	23	2	no	—
126	C₂×C₄	—	46	1	no	—
127	C₂×C₄	—	46	1	no	—
128	C₂×C₄	—	46	2	no	—
129	C₂×C₄	—	46	2	no	—
130	C₂×C₄	—	46	4	no	—
131	C₂×C₄	—	46	4	no	—
132	C₂×C₄	#67288	1	2	yes	—
133	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
134	C₂×C₄	D₄×D₂₃	1	2	yes	—
135	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
136	C₈	—	4	2	no	—
137	C₈	—	92	2	no	—
138	C₂²	—	46	4	no	—
139	C₂²	—	46	2	no	—
140	C₂²	—	92	2	no	—
141	C₂²	—	92	1	no	—
142	C₂²	—	92	1	no	—
143	C₂²	#10625043	1	1	yes	characteristic
144	C₂²	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
145	C₂²	C₄⋊C₄×D₂₃	1	1	yes	characteristic
146	C₂²	#10625072	1	2	yes	—
147	C₂²	#10625078	1	2	yes	—
148	C₄	—	4	2	no	—
149	C₄	—	46	4	no	—
150	C₄	—	46	2	no	—
151	C₄	#10625055	1	2	yes	—
152	C₄	#10625067	1	2	yes	—
153	C₂	—	92	2	no	—
154	C₂	#13092276	1	1	yes	characteristic
155	C₂	#13092382	1	2	yes	—
156	C₂	#13092442	1	2	yes	—
157	C₂	#13092476	1	1	yes	characteristic
158	C₂	#13092479	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625043

C₂

yes

maximal, characteristic

C₂×Dic₂₃⋊C₈

C₂

yes

maximal, complemented

C₂×(C₂×D₂₃)⋊C₈

C₂

yes

maximal

#13092172

C₂

yes

maximal, characteristic, complemented

C₂₃×#336

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×(C₂×D₂₃)⋊C₄

C₄

yes

—

C₂×Dic₂₃⋊C₄

C₄

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

—

#10625043

—

(C₂×D₂₃)⋊C₈

—

Dic₂₃⋊C₈

—

C₂³×D₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

—

C₂²×Dic₂₃

C₂×C₄

yes

characteristic

C₂×C₄×D₂₃

—

C₂×C₄×D₂₃

—

C₂×C₄×D₂₃

D₄

yes

—

C₂×C₄×D₂₃

Q₈

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄²×C₂₃

—

(C₂×D₂₃)⋊C₄

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

D₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂²⋊C₄

yes

—

C₂×Dic₂₃

C₄⋊C₄

yes

—

C₄×D₂₃

—

C₄×D₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1798

—

maximal, Sylow

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

C₂×C₄⋊C₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₂²×C₂₃

#131

yes

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#121

yes

—

C₄×C₂₃

C₂²⋊Q₈

yes

—

Dic₂₃

—

Dic₂₃

—

C₂×#117

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

#336

—

#336

D₂₃

yes

characteristic, complemented

C₂×C₂₃

#380

yes

characteristic

C₂×C₂₃

#408

yes

—

C₂×C₂₃

#424

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

D₂₃

—

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₂₃

#1798

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₄×D₂₃

yes

—

104

C₂²×C₄

C₂²×D₂₃

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

—

115

C₂³

—

116

C₂³

—

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

C₂×C₄×D₂₃

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

#67288

yes

—

133

C₂×C₄

D₄×D₂₃

yes

characteristic

134

C₂×C₄

D₄×D₂₃

yes

—

135

C₂×C₄

Q₈×D₂₃

yes

characteristic

136

C₈

—

137

C₈

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#10625043

yes

characteristic

144

C₂²

C₂²⋊C₄×D₂₃

yes

characteristic

145

C₂²

C₄⋊C₄×D₂₃

yes

characteristic

146

C₂²

#10625072

yes

—

147

C₂²

#10625078

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#10625055

yes

—

152

C₄

#10625067

yes

—

153

C₂

—

154

C₂

#13092276

yes

characteristic

155

C₂

#13092382

yes

—

156

C₂

#13092442

yes

—

157

C₂

#13092476

yes

characteristic

158

C₂

#13092479

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator, socle	C₂³×C₂₃	C₂²×C₄
Fitting	C₂₃×#336	C₂
prime radical	C₂³×D₂₃	C₂×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#336	D₂₃
hypocenter	C₂₃	#1798
LCS	G ▷ C₂³×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃

commutator, socle

C₂³×C₂₃

C₂²×C₄

Fitting

C₂₃×#336

C₂

prime radical

C₂³×D₂₃

C₂×C₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂₃

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
2. C₂×Dic₂₃⋊C₈	C₂	semidirect
4. #13092172	C₂	semidirect
5. C₂₃×#336	C₂	semidirect
78. #336	D₂₃	semidirect
95. C₂₃	#1798	semidirect

H ≅ G/N

type

2. C₂×Dic₂₃⋊C₈

C₂

semidirect

4. #13092172

C₂

semidirect

5. C₂₃×#336

C₂

semidirect

78. #336

D₂₃

semidirect

95. C₂₃

#1798

semidirect

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

subgroup	index
C₂×#10625043	2
C₂×Dic₂₃⋊C₈	2
C₂×(C₂×D₂₃)⋊C₈	2
#13092172	2
C₂₃×#336	2
#1798	23

subgroup

index

C₂×#10625043

C₂×Dic₂₃⋊C₈

C₂×(C₂×D₂₃)⋊C₈

#13092172

C₂₃×#336

#1798

Subgroups — #15673380

Conjugacy classes of subgroups

Complements

Maximal subgroups