Group #15673200 — subgroups

Subgroups — #15673200

Order 2944.

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

normal subgroups only

This group has 284 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×#10625119	C₂	1	1	yes	maximal, characteristic
2	#13092391	C₂	1	1	yes	maximal, characteristic
3	#13092375	C₂	1	2	yes	maximal, complemented
4	#13092189	C₂	1	2	yes	maximal, complemented
5	C₂₃×#413	C₂	1	1	yes	maximal, characteristic
6	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
8	#10625119	—	2	2	no	—
9	#10625119	—	2	2	no	—
10	#10625097	C₂²	1	2	yes	—
11	C₂×#67267	C₄	1	2	yes	complemented
12	C₂×#67267	C₂²	1	1	yes	characteristic
13	C₂²×C₂₃⋊C₈	C₄	1	2	yes	complemented
14	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂₃×#101	C₄	1	2	yes	complemented
16	C₂₃×#100	C₄	1	2	yes	complemented
17	#10625016	C₄	1	4	yes	—
18	#10624997	—	2	2	no	—
19	C₈×Dic₂₃	—	2	2	no	—
20	#41×C₂₃	—	2	2	no	—
21	#41×C₂₃	—	2	2	no	—
22	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
24	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
27	M₁₆×C₂₃	—	2	2	no	—
28	M₁₆×C₂₃	C₂×C₄	1	4	yes	—
29	C₂×C₈×C₂₃	—	2	2	no	—
30	C₂²⋊Dic₂₃	C₂×C₄	1	4	yes	—
31	C₄⋊Dic₂₃	—	2	2	no	—
32	C₄×Dic₂₃	—	2	2	no	—
33	#67267	—	2	2	no	—
34	#67267	—	2	4	no	—
35	C₂×C₂₃⋊C₈	—	2	2	no	—
36	C₂×C₂₃⋊C₈	—	2	2	no	—
37	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
38	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
41	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
42	C₂³×C₂₃	C₄²	1	2	yes	—
43	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
44	Q₈×C₂₃	—	2	2	no	—
45	D₄×C₂₃	—	2	4	no	—
46	D₄×C₂₃	—	2	2	no	—
47	C₂×C₄×C₂₃	—	2	2	no	—
48	C₂×C₄×C₂₃	—	2	2	no	—
49	C₂×C₄×C₂₃	C₄²	1	2	yes	—
50	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₂₃	#41	1	2	yes	—
54	C₂×Dic₂₃	—	2	4	no	—
55	C₈×C₂₃	—	2	4	no	—
56	C₂₃⋊C₈	—	2	2	no	—
57	C₂₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
58	C₂₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
59	C₂₃⋊C₈	C₂²⋊C₄	1	2	yes	—
60	#1757	—	23	1	no	maximal, Sylow
61	C₂²×C₂₃	—	2	2	no	—
62	C₂²×C₂₃	—	4	1	no	—
63	C₂²×C₂₃	—	4	2	no	—
64	C₂²×C₂₃	C₂×C₄²	1	1	yes	characteristic
65	C₂²×C₂₃	C₄×D₄	1	2	yes	—
66	C₄×C₂₃	—	2	2	no	—
67	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₄×C₂₃	#117	1	1	yes	characteristic
69	C₄×C₂₃	C₄×D₄	1	2	yes	—
70	Dic₂₃	—	4	4	no	—
71	C₂×#131	—	23	1	no	—
72	#413	D₂₃	1	1	yes	characteristic
73	#410	—	23	1	no	—
74	#405	—	23	2	no	—
75	#344	—	23	2	no	—
76	C₂×C₂₃	—	2	2	no	—
77	C₂×C₂₃	—	2	1	no	—
78	C₂×C₂₃	—	4	2	no	—
79	C₂×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
81	#131	—	46	2	no	—
82	#131	—	46	2	no	—
83	C₂×M₁₆	—	23	2	no	—
84	C₂×M₁₆	—	23	1	no	—
85	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
86	C₂²×C₈	—	23	1	no	—
87	C₂²×C₈	—	23	2	no	—
88	#117	—	23	2	no	—
89	#101	Dic₂₃	1	2	yes	complemented
90	#100	Dic₂₃	1	2	yes	complemented
91	#99	—	23	4	no	—
92	#97	—	46	2	no	—
93	C₄×C₈	—	46	2	no	—
94	C₂₃	#1757	1	1	yes	characteristic, complemented, Sylow
95	#41	—	2	2	no	—
96	#41	—	2	2	no	—
97	C₂×Q₈	C₂×Dic₂₃	1	1	yes	characteristic
98	C₂×D₄	C₄×D₂₃	1	2	yes	—
99	C₂×D₄	C₂×Dic₂₃	1	1	yes	characteristic
100	C₂²×C₄	C₄×D₂₃	1	2	yes	—
101	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
102	M₁₆	—	2	2	no	—
103	M₁₆	—	46	4	no	—
104	M₁₆	—	46	2	no	—
105	M₁₆	C₂×Dic₂₃	1	4	yes	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	23	1	no	—
108	C₂×C₈	—	23	1	no	—
109	C₂×C₈	—	23	2	no	—
110	C₂×C₈	—	23	2	no	—
111	C₂×C₈	—	23	2	no	—
112	C₂×C₈	—	46	2	no	—
113	C₂×C₈	—	46	2	no	—
114	C₄⋊C₄	—	46	2	no	—
115	C₂²⋊C₄	—	23	4	no	—
116	C₄²	—	46	2	no	—
117	C₂³	C₄×Dic₂₃	1	2	yes	—
118	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
119	Q₈	—	2	2	no	—
120	D₄	—	2	4	no	—
121	D₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	46	4	no	—
125	C₂×C₄	C₄×Dic₂₃	1	2	yes	—
126	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
127	C₂×C₄	D₄×D₂₃	1	2	yes	—
128	C₂×C₄	#67290	1	2	yes	—
129	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
130	C₈	—	2	4	no	—
131	C₈	—	23	1	no	—
132	C₈	—	23	1	no	—
133	C₈	—	23	2	no	—
134	C₈	—	46	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	4	1	no	—
137	C₂²	—	4	2	no	—
138	C₂²	(C₂²⋊D₂₃)⋊C₄	1	2	yes	—
139	C₂²	C₂×C₄×Dic₂₃	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	92	4	no	—
142	C₄	#10625048	1	1	yes	characteristic
143	C₄	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
144	C₄	D₄×Dic₂₃	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	4	2	no	—
148	C₂	C₂²⋊C₄×Dic₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625119

C₂

yes

maximal, characteristic

#13092391

C₂

yes

maximal, characteristic

#13092375

C₂

yes

maximal, complemented

#13092189

C₂

yes

maximal, complemented

C₂₃×#413

C₂

yes

maximal, characteristic

C₂×#41×C₂₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

#10625119

—

#10625119

—

#10625097

C₂²

yes

—

C₂×#67267

C₄

yes

complemented

C₂×#67267

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₄

yes

complemented

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂₃×#101

C₄

yes

complemented

C₂₃×#100

C₄

yes

complemented

#10625016

C₄

yes

—

#10624997

—

C₈×Dic₂₃

—

#41×C₂₃

—

#41×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₄

yes

characteristic

C₂×D₄×C₂₃

C₂×C₄

yes

—

C₂×D₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

—

C₂²⋊Dic₂₃

C₂×C₄

yes

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₃⋊C₈

D₄

yes

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂³×C₂₃

C₄²

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄²

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

C₂²⋊C₄

yes

characteristic

C₂₃⋊C₈

C₂²⋊C₄

yes

characteristic

C₂₃⋊C₈

C₂²⋊C₄

yes

—

#1757

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₄²

yes

characteristic

C₂²×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

—

C₄×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

#117

yes

characteristic

C₄×C₂₃

C₄×D₄

yes

—

Dic₂₃

—

C₂×#131

—

#413

D₂₃

yes

—

—

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₄×C₂²⋊C₄

yes

characteristic

C₂×#41

C₂×D₂₃

yes

characteristic

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

#117

—

#101

Dic₂₃

yes

complemented

#100

Dic₂₃

yes

complemented

#99

—

#97

—

C₄×C₈

—

C₂₃

#1757

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

C₂×Dic₂₃

yes

characteristic

C₂×D₄

C₄×D₂₃

yes

—

C₂×D₄

C₂×Dic₂₃

yes

characteristic

100

C₂²×C₄

C₄×D₂₃

yes

—

101

C₂²×C₄

C₂²×D₂₃

yes

characteristic

102

M₁₆

—

103

M₁₆

—

104

M₁₆

—

105

M₁₆

C₂×Dic₂₃

yes

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

—

117

C₂³

C₄×Dic₂₃

yes

—

118

C₂³

C₂×C₄×D₂₃

yes

characteristic

119

Q₈

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₄×Dic₂₃

yes

—

126

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

127

C₂×C₄

D₄×D₂₃

yes

—

128

C₂×C₄

#67290

yes

—

129

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

(C₂²⋊D₂₃)⋊C₄

yes

—

139

C₂²

C₂×C₄×Dic₂₃

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

#10625048

yes

characteristic

143

C₄

C₂²⋊C₄×D₂₃

yes

characteristic

144

C₄

D₄×Dic₂₃

yes

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

C₂²⋊C₄×Dic₂₃

yes

characteristic

	subgroup	G/H
center	C₄	C₂²⋊C₄×D₂₃
commutator	C₂²×C₂₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#413	C₂
socle	C₂×C₂₃	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₂₃	C₂²
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#413	D₂₃
hypocenter	C₂₃	#1757
LCS	G ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

C₂²⋊C₄×D₂₃

commutator

C₂²×C₂₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#413

C₂

socle

C₂×C₂₃

C₄×C₂²⋊C₄

prime radical

C₂×#41×C₂₃

C₂²

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
3. #13092375	C₂	semidirect
4. #13092189	C₂	semidirect
11. C₂×#67267	C₄	semidirect
13. C₂²×C₂₃⋊C₈	C₄	semidirect
15. C₂₃×#101	C₄	semidirect
16. C₂₃×#100	C₄	semidirect
89. #101	Dic₂₃	semidirect
90. #100	Dic₂₃	semidirect
94. C₂₃	#1757	semidirect

H ≅ G/N

type

3. #13092375

C₂

semidirect

4. #13092189

C₂

semidirect

11. C₂×#67267

C₄

semidirect

13. C₂²×C₂₃⋊C₈

C₄

semidirect

15. C₂₃×#101

C₄

semidirect

16. C₂₃×#100

C₄

semidirect

89. #101

Dic₂₃

semidirect

90. #100

Dic₂₃

semidirect

94. C₂₃

#1757

semidirect

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

subgroup	index
C₂×#10625119	2
#13092391	2
#13092375	2
#13092189	2
C₂₃×#413	2
#1757	23

subgroup

index

C₂₃×#413

Subgroups — #15673200

Conjugacy classes of subgroups

Complements

Maximal subgroups