Group #15671817 — subgroups

Subgroups — #15671817

Order 2944.

218 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 77 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	#13092974	C₂	1	1	yes	maximal, characteristic
2	C₄×#67267	C₂	1	1	yes	maximal, characteristic
3	#13092147	C₂	1	4	yes	maximal, complemented
4	C₂₃×#431	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
6	#10625097	—	2	1	no	—
7	#10625095	—	2	1	no	—
8	#10625095	C₄	1	2	yes	—
9	C₂×#67267	—	2	1	no	—
10	#10625041	—	2	1	no	—
11	#10625040	C₄	1	1	yes	characteristic
12	#10625040	C₄	1	1	yes	characteristic
13	C₄×Dic₄₆	—	2	1	no	—
14	C₄×Dic₄₆	C₂²	1	2	yes	—
15	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂₃×C₂²⋊C₈	—	2	1	no	—
17	C₄×C₈×C₂₃	—	2	1	no	—
18	#10624985	—	2	1	no	—
19	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
20	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
23	C₂×Dic₄₆	—	2	1	no	—
24	C₂×Dic₄₆	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₂₃	—	2	2	no	—
26	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₂₃	D₄	1	1	yes	characteristic
28	C₄²×C₂₃	D₄	1	1	yes	characteristic
29	C₄²×C₂₃	C₂³	1	1	yes	characteristic
30	C₂²⋊Dic₂₃	—	2	2	no	—
31	C₄⋊Dic₂₃	—	2	1	no	—
32	C₄⋊Dic₂₃	—	2	1	no	—
33	C₄⋊Dic₂₃	—	2	1	no	—
34	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
35	Dic₂₃⋊C₄	—	2	2	no	—
36	Dic₂₃⋊C₄	—	2	2	no	—
37	C₄×Dic₂₃	—	2	2	no	—
38	#67267	—	2	4	no	—
39	C₂×C₂₃⋊C₈	—	2	2	no	—
40	C₂³×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
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	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
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×Dic₂₃	—	2	2	no	—
51	C₂×Dic₂₃	—	2	2	no	—
52	Dic₄₆	—	2	4	no	—
53	Dic₄₆	—	4	2	no	—
54	C₈×C₂₃	—	4	2	no	—
55	C₂₃⋊C₈	—	2	4	no	—
56	#1485	—	23	1	no	maximal, Sylow
57	C₂²×C₂₃	—	2	1	no	—
58	C₂²×C₂₃	—	2	1	no	—
59	C₂²×C₂₃	—	2	1	no	—
60	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₄×C₂₃	—	2	1	no	—
62	C₄×C₂₃	—	2	1	no	—
63	C₄×C₂₃	—	2	1	no	—
64	C₄×C₂₃	#104	1	2	yes	—
65	C₄×C₂₃	#131	1	2	yes	—
66	C₄×C₂₃	#137	1	2	yes	—
67	Dic₂₃	—	4	2	no	—
68	Dic₂₃	—	4	2	no	—
69	#533	—	23	1	no	—
70	#431	D₂₃	1	1	yes	characteristic
71	C₄×M₁₆	—	23	1	no	—
72	Q₈⋊C₈	—	23	4	no	—
73	C₂×C₂₃	—	4	1	no	—
74	C₂×C₂₃	#408	1	1	yes	characteristic
75	C₂×C₂₃	#419	1	1	yes	characteristic
76	C₂×C₂₃	C₂×#104	1	1	yes	characteristic
77	C₂×M₁₆	—	46	1	no	—
78	C₄⋊Q₈	—	23	1	no	—
79	C₄⋊Q₈	—	23	1	no	—
80	#125	—	46	1	no	—
81	C₂²⋊Q₈	—	23	2	no	—
82	C₂²⋊Q₈	—	46	1	no	—
83	C₄×Q₈	—	23	2	no	—
84	C₄×Q₈	—	46	1	no	—
85	#117	—	46	1	no	—
86	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
87	C₄⋊C₈	C₂×D₂₃	1	2	yes	—
88	C₂²⋊C₈	—	2	1	no	—
89	#97	—	46	1	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	—	23	2	no	—
92	C₂₃	#1485	1	1	yes	characteristic, complemented, Sylow
93	C₂×Q₈	—	23	2	no	—
94	C₂×Q₈	—	46	1	no	—
95	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
96	C₂²×C₄	D₉₂	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
98	M₁₆	—	46	4	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	46	2	no	—
101	C₄⋊C₄	—	23	2	no	—
102	C₄⋊C₄	—	46	1	no	—
103	C₄⋊C₄	—	46	1	no	—
104	C₄⋊C₄	—	46	2	no	—
105	C₄⋊C₄	—	46	1	no	—
106	C₄⋊C₄	—	46	2	no	—
107	C₂²⋊C₄	—	46	2	no	—
108	C₄²	—	46	2	no	—
109	C₄²	C₄×D₂₃	1	1	yes	characteristic
110	C₄²	D₉₂	1	1	yes	characteristic
111	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
112	C₄²	C₂²×D₂₃	1	1	yes	characteristic
113	C₂³	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
114	Q₈	—	46	4	no	—
115	Q₈	—	92	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	46	2	no	—
120	C₂×C₄	—	46	2	no	—
121	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
122	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
123	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
124	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
125	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₈	—	46	4	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	92	2	no	—
137	C₄	—	92	2	no	—
138	C₄	#10625007	1	2	yes	—
139	C₄	#10625072	1	2	yes	—
140	C₄	#10625122	1	2	yes	—
141	C₂	—	4	1	no	—
142	C₂	#13092383	1	1	yes	characteristic
143	C₂	#13092425	1	1	yes	characteristic
144	C₂	C₂×#10625007	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092974

C₂

yes

maximal, characteristic

C₄×#67267

C₂

yes

maximal, characteristic

#13092147

C₂

yes

maximal, complemented

C₂₃×#431

C₂

yes

maximal, characteristic

C₂×C₄²×C₂₃

C₂²

yes

—

—

C₄

yes

—

C₂×#67267

—

#10625041

—

#10625040

C₄

yes

characteristic

#10625040

C₄

yes

characteristic

C₄×Dic₄₆

—

C₄×Dic₄₆

C₂²

yes

—

C₂₃×C₄⋊C₈

C₂²

yes

—

C₂₃×C₂²⋊C₈

—

C₄×C₈×C₂₃

—

#10624985

—

C₄×C₂₃⋊C₈

C₂²

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂×Dic₄₆

—

C₂×Dic₄₆

C₂×C₄

yes

—

C₂×C₈×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

C₂³

yes

characteristic

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂×C₄

yes

—

Dic₂₃⋊C₄

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1485

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#104

yes

—

C₄×C₂₃

#131

yes

—

C₄×C₂₃

#137

yes

—

Dic₂₃

—

Dic₂₃

—

#533

—

#431

D₂₃

yes

characteristic

C₄×M₁₆

—

Q₈⋊C₈

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

C₂×D₂₃

yes

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂₃

#1485

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₄×D₂₃

yes

characteristic

C₂²×C₄

D₉₂

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

M₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₄²

C₄×D₂₃

yes

characteristic

110

C₄²

D₉₂

yes

characteristic

111

C₄²

C₂²⋊D₂₃

yes

characteristic

112

C₄²

C₂²×D₂₃

yes

characteristic

113

C₂³

(C₂×D₂₃)⋊C₄

yes

characteristic

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

characteristic

122

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

characteristic

123

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

characteristic

124

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

125

C₂×C₄

C₂×D₉₂

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

127

C₈

—

128

C₈

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#10625007

yes

—

139

C₄

#10625072

yes

—

140

C₄

#10625122

yes

—

141

C₂

—

142

C₂

#13092383

yes

characteristic

143

C₂

#13092425

yes

characteristic

144

C₂

C₂×#10625007

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#431	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₃	C₂²⋊C₄
cosocle	C₄²×C₂₃	C₂³
hypercenter	#431	D₂₃
hypocenter	C₂₃	#1485
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#431

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₃

C₂²⋊C₄

cosocle

C₄²×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #13092147	C₂	semidirect
92. C₂₃	#1485	semidirect

H ≅ G/N

type

3. #13092147

C₂

semidirect

92. C₂₃

#1485

semidirect

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

subgroup	index
#13092974	2
C₄×#67267	2
#13092147	2
C₂₃×#431	2
#1485	23

subgroup

index

C₂₃×#431

Subgroups — #15671817

Conjugacy classes of subgroups

Complements

Maximal subgroups