Group #15673320 — subgroups

Subgroups — #15673320

Order 2944.

564 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂³×Dic₄₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625006	C₂	1	4	yes	maximal, complemented
3	#13092381	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#407	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₂³×Dic₂₃	C₄	1	2	yes	—
7	C₂²×Dic₄₆	—	2	2	no	—
8	C₂²×Dic₄₆	—	2	2	no	—
9	C₂²×Dic₄₆	—	2	8	no	—
10	C₂²×Dic₄₆	C₄	1	2	yes	—
11	C₂²×Dic₄₆	C₂²	1	2	yes	—
12	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂×#67267	C₂²	1	2	yes	—
14	C₂₃×C₂²⋊C₈	—	2	2	no	—
15	C₂²⋊(C₂₃⋊C₈)	—	2	2	no	—
16	#10625006	—	2	8	no	—
17	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₃	—	2	4	no	—
19	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₃	D₄	1	2	yes	—
21	C₂²×C₄×C₂₃	D₄	1	2	yes	—
22	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₂₃	—	2	4	no	—
24	C₂²×Dic₂₃	—	2	8	no	—
25	C₂²×Dic₂₃	C₂×C₄	1	4	yes	—
26	C₂×Dic₄₆	—	2	16	no	—
27	C₂×Dic₄₆	—	2	16	no	—
28	C₂×Dic₄₆	—	2	4	no	—
29	C₂×Dic₄₆	—	2	2	no	—
30	C₂×Dic₄₆	—	2	4	no	—
31	C₂×Dic₄₆	—	2	2	no	—
32	C₂×Dic₄₆	—	2	8	no	—
33	C₂×Dic₄₆	D₄	1	8	yes	—
34	M₁₆×C₂₃	—	2	4	no	—
35	C₂×C₈×C₂₃	—	2	2	no	—
36	#67267	—	2	4	no	—
37	C₂×C₂₃⋊C₈	—	2	2	no	—
38	C₂³×C₂₃	—	2	4	no	—
39	C₂³×C₂₃	—	2	2	no	—
40	C₂³×C₂₃	C₂²⋊C₄	1	2	yes	—
41	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃	—	2	2	no	—
43	C₂×C₄×C₂₃	—	2	4	no	—
44	C₂×C₄×C₂₃	—	2	4	no	—
45	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
48	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₂₃	—	2	16	no	—
50	C₂×Dic₂₃	—	2	4	no	—
51	C₂×Dic₂₃	—	2	4	no	—
52	C₂×Dic₂₃	C₂²⋊C₄	1	8	yes	—
53	Dic₄₆	—	2	16	no	—
54	Dic₄₆	—	2	8	no	—
55	Dic₄₆	—	2	8	no	—
56	C₈×C₂₃	—	4	2	no	—
57	C₂₃⋊C₈	—	4	2	no	—
58	#1784	—	23	1	no	maximal, Sylow
59	C₂²×C₂₃	—	2	8	no	—
60	C₂²×C₂₃	—	2	2	no	—
61	C₂²×C₂₃	—	2	2	no	—
62	C₂²×C₂₃	—	2	2	no	—
63	C₂²×C₂₃	#101	1	4	yes	complemented
64	C₂²×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
65	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₄×C₂₃	—	2	2	no	—
67	C₄×C₂₃	#120	1	2	yes	—
68	C₄×C₂₃	#120	1	2	yes	—
69	Dic₂₃	—	2	8	no	—
70	C₂³×Q₈	—	23	1	no	—
71	C₂×#101	—	23	4	no	—
72	#407	—	23	1	no	—
73	#407	D₂₃	1	1	yes	characteristic
74	C₂×C₂₃	—	2	4	no	—
75	C₂×C₂₃	—	2	2	no	—
76	C₂×C₂₃	#379	1	1	yes	characteristic
77	C₂×C₂₃	C₂×#101	1	2	yes	—
78	C₂²×Q₈	—	23	2	no	—
79	C₂²×Q₈	—	23	2	no	—
80	C₂²×Q₈	—	46	8	no	—
81	C₂²×Q₈	—	46	2	no	—
82	C₂²×Q₈	—	46	2	no	—
83	C₂³×C₄	—	23	2	no	—
84	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
85	C₂×M₁₆	—	23	2	no	—
86	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
87	#101	—	46	8	no	—
88	C₂²⋊C₈	—	2	2	no	—
89	C₂²⋊C₈	—	46	2	no	—
90	C₂₃	#1784	1	1	yes	characteristic, complemented, Sylow
91	C₂⁴	C₄×D₂₃	1	1	yes	characteristic
92	C₂×Q₈	—	23	8	no	—
93	C₂×Q₈	—	46	8	no	—
94	C₂×Q₈	—	46	16	no	—
95	C₂×Q₈	—	46	16	no	—
96	C₂×Q₈	—	46	4	no	—
97	C₂×Q₈	—	46	2	no	—
98	C₂×Q₈	—	46	4	no	—
99	C₂×Q₈	—	46	2	no	—
100	C₂²×C₄	—	2	4	no	—
101	C₂²×C₄	—	23	4	no	—
102	C₂²×C₄	—	46	8	no	—
103	C₂²×C₄	—	46	4	no	—
104	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
105	C₂²×C₄	D₉₂	1	2	yes	—
106	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
107	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
108	M₁₆	—	2	4	no	—
109	M₁₆	—	46	4	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	46	2	no	—
112	C₂³	—	2	2	no	—
113	C₂³	—	2	4	no	—
114	C₂³	(C₂×D₂₃)⋊C₄	1	2	yes	—
115	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
116	Q₈	—	46	16	no	—
117	Q₈	—	46	8	no	—
118	Q₈	—	46	8	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	23	8	no	—
123	C₂×C₄	—	46	16	no	—
124	C₂×C₄	—	46	4	no	—
125	C₂×C₄	—	46	4	no	—
126	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
127	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
128	C₂×C₄	D₄×D₂₃	1	4	yes	—
129	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
130	C₈	—	4	2	no	—
131	C₈	—	92	2	no	—
132	C₂²	—	2	8	no	—
133	C₂²	—	2	2	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	#10625006	1	4	yes	complemented
137	C₂²	C₂²⋊C₄×D₂₃	1	2	yes	—
138	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	46	8	no	—
141	C₄	#10625053	1	2	yes	—
142	C₄	#10625108	1	2	yes	—
143	C₂	—	2	2	no	—
144	C₂	—	2	4	no	—
145	C₂	#13092272	1	1	yes	characteristic
146	C₂	C₂×#10625006	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Dic₄₆

C₂

yes

maximal, characteristic

C₂×#10625006

C₂

yes

maximal, complemented

#13092381

C₂

yes

maximal, characteristic

C₂₃×#407

C₂

yes

maximal, characteristic

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂³×Dic₂₃

C₄

yes

—

C₂²×Dic₄₆

—

C₂²×Dic₄₆

—

C₂²×Dic₄₆

—

C₂²×Dic₄₆

C₄

yes

—

C₂²×Dic₄₆

C₂²

yes

—

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂×#67267

C₂²

yes

—

C₂₃×C₂²⋊C₈

—

C₂²⋊(C₂₃⋊C₈)

—

#10625006

—

C₂⁴×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

C₂²×Dic₂₃

C₂×C₄

yes

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

D₄

yes

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²⋊C₄

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂²⋊C₄

yes

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1784

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#101

yes

complemented

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

#120

yes

—

C₄×C₂₃

#120

yes

—

Dic₂₃

—

C₂³×Q₈

—

C₂×#101

—

#407

—

#407

D₂₃

yes

characteristic

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#379

yes

characteristic

C₂×C₂₃

C₂×#101

yes

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂³×C₄

—

C₂³×C₄

C₂×D₂₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

—

#101

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂₃

#1784

yes

characteristic, complemented, Sylow

C₂⁴

C₄×D₂₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₄×D₂₃

yes

characteristic

105

C₂²×C₄

D₉₂

yes

—

106

C₂²×C₄

C₂²⋊D₂₃

yes

—

107

C₂²×C₄

C₂²×D₂₃

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂³

—

113

C₂³

—

114

C₂³

(C₂×D₂₃)⋊C₄

yes

—

115

C₂³

C₂×C₄×D₂₃

yes

characteristic

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

127

C₂×C₄

C₂×D₉₂

yes

characteristic

128

C₂×C₄

D₄×D₂₃

yes

—

129

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

130

C₈

—

131

C₈

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

#10625006

yes

complemented

137

C₂²

C₂²⋊C₄×D₂₃

yes

—

138

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

#10625053

yes

—

142

C₄

#10625108

yes

—

143

C₂

—

144

C₂

—

145

C₂

#13092272

yes

characteristic

146

C₂

C₂×#10625006

yes

—

	subgroup	G/H
center	C₂²	C₂×(C₂×D₂₃)⋊C₄
commutator	C₂³×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#407	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₂₃	C₂×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#407	D₂₃
hypocenter	C₂₃	#1784
LCS	G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₂×(C₂×D₂₃)⋊C₄

commutator

C₂³×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#407

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₂₃

C₂×C₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
2. C₂×#10625006	C₂	semidirect
63. C₂²×C₂₃	#101	semidirect
90. C₂₃	#1784	semidirect
136. C₂²	#10625006	semidirect

H ≅ G/N

type

2. C₂×#10625006

C₂

semidirect

63. C₂²×C₂₃

#101

semidirect

90. C₂₃

#1784

semidirect

136. C₂²

#10625006

semidirect

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

subgroup	index
C₂³×Dic₄₆	2
C₂×#10625006	2
#13092381	2
C₂₃×#407	2
#1784	23

subgroup

index

C₂³×Dic₄₆

C₂×#10625006

#13092381

C₂₃×#407

#1784

Subgroups — #15673320

Conjugacy classes of subgroups

Complements

Maximal subgroups