Group #15673122 — subgroups

Subgroups — #15673122

Order 2944.

372 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 137 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	C₂²×#67262	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×D₂₃	C₂	1	2	yes	maximal
3	#13092182	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092181	C₂	1	2	yes	maximal, complemented
5	C₂₃×#341	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×#67267	C₂²	1	2	yes	—
10	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
11	M₁₆×D₂₃	—	2	4	no	—
12	M₁₆×D₂₃	C₄	1	8	yes	—
13	C₂×#67262	—	2	2	no	—
14	C₂×#67262	—	2	4	no	—
15	C₂×#67262	C₄	1	4	yes	—
16	C₂×C₈×D₂₃	—	2	2	no	—
17	C₂³×D₂₃	Q₈	1	1	yes	characteristic
18	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₂₃	Q₈	1	1	yes	characteristic
20	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
21	C₂×C₄×D₂₃	C₂×C₄	1	4	yes	—
22	C₂×C₄×D₂₃	D₄	1	2	yes	—
23	C₂×C₄×D₂₃	D₄	1	4	yes	—
24	M₁₆×C₂₃	—	2	2	no	—
25	M₁₆×C₂₃	C₂×C₄	1	4	yes	—
26	C₂×C₈×C₂₃	—	2	2	no	—
27	C₂×C₈×C₂₃	—	2	2	no	—
28	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
29	#67267	—	2	2	no	—
30	#67267	C₂×C₄	1	4	yes	—
31	C₂×C₂₃⋊C₈	—	2	2	no	—
32	C₂×C₂₃⋊C₈	—	2	2	no	—
33	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
34	#67262	—	2	8	no	—
35	#67262	—	2	8	no	—
36	C₈×D₂₃	—	2	8	no	—
37	C₂³×C₂₃	C₂×Q₈	1	1	yes	characteristic
38	C₂²×D₂₃	—	4	2	no	—
39	C₂²×D₂₃	C₄⋊C₄	1	2	yes	—
40	C₂²×D₂₃	C₄⋊C₄	1	4	yes	—
41	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₂₃	C₄⋊C₄	1	4	yes	—
46	C₂×Dic₂₃	C₄⋊C₄	1	2	yes	—
47	C₄×D₂₃	C₄²	1	4	yes	—
48	C₄×D₂₃	C₂²⋊C₄	1	4	yes	—
49	C₄×D₂₃	C₂²⋊C₄	1	4	yes	—
50	C₄×D₂₃	C₂²⋊C₄	1	4	yes	—
51	C₈×C₂₃	—	2	4	no	—
52	C₈×C₂₃	—	2	2	no	—
53	C₂₃⋊C₈	—	2	4	no	—
54	C₂₃⋊C₈	—	2	2	no	—
55	#1738	—	23	1	no	maximal, Sylow
56	C₂²×C₂₃	—	2	2	no	—
57	C₂²×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
58	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×D₂₃	—	2	4	no	—
60	C₂×D₂₃	—	4	4	no	—
61	C₂×D₂₃	#95	1	4	yes	—
62	C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
63	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
65	Dic₂₃	#95	1	4	yes	—
66	C₂²×M₁₆	—	23	1	no	—
67	C₂²×M₁₆	—	23	2	no	—
68	#341	—	23	2	no	—
69	#341	—	23	1	no	—
70	#341	D₂₃	1	1	yes	characteristic, complemented
71	C₂×C₂₃	—	2	2	no	—
72	C₂×C₂₃	C₂×#95	1	1	yes	characteristic
73	C₂×C₂₃	#430	1	2	yes	—
74	D₂₃	—	2	4	no	—
75	C₂³×C₄	—	23	1	no	—
76	C₂×M₁₆	—	23	4	no	—
77	C₂×M₁₆	—	23	2	no	—
78	C₂×M₁₆	—	23	8	no	—
79	C₂×M₁₆	—	46	4	no	—
80	C₂×M₁₆	—	46	2	no	—
81	C₂×M₁₆	—	46	4	no	—
82	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
83	C₂²×C₈	—	23	1	no	—
84	C₂²×C₈	—	46	2	no	—
85	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
86	C₂₃	#1738	1	1	yes	characteristic, complemented, Sylow
87	C₂⁴	—	23	1	no	—
88	C₂²×C₄	—	23	1	no	—
89	C₂²×C₄	—	23	4	no	—
90	C₂²×C₄	—	23	2	no	—
91	C₂²×C₄	—	23	4	no	—
92	C₂²×C₄	—	23	2	no	—
93	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
94	M₁₆	—	2	2	no	—
95	M₁₆	—	23	4	no	—
96	M₁₆	—	46	8	no	—
97	M₁₆	—	46	8	no	—
98	M₁₆	—	46	2	no	—
99	M₁₆	C₄×D₂₃	1	4	yes	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	23	2	no	—
103	C₂×C₈	—	46	8	no	—
104	C₂×C₈	—	46	2	no	—
105	C₂×C₈	—	46	2	no	—
106	C₂×C₈	C₄×D₂₃	1	2	yes	—
107	C₂³	—	23	2	no	—
108	C₂³	—	23	4	no	—
109	C₂³	—	92	2	no	—
110	C₂³	Q₈×D₂₃	1	1	yes	characteristic
111	C₂×C₄	—	23	4	no	—
112	C₂×C₄	—	23	2	no	—
113	C₂×C₄	—	23	4	no	—
114	C₂×C₄	—	23	4	no	—
115	C₂×C₄	—	23	4	no	—
116	C₂×C₄	—	23	4	no	—
117	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
118	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
119	C₂×C₄	D₄×D₂₃	1	2	yes	—
120	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	—	2	4	no	—
123	C₈	—	46	2	no	—
124	C₈	—	46	4	no	—
125	C₂²	—	2	2	no	—
126	C₂²	—	23	4	no	—
127	C₂²	—	46	4	no	—
128	C₂²	—	92	4	no	—
129	C₂²	C₄⋊C₄×D₂₃	1	2	yes	—
130	C₂²	C₄⋊C₄×D₂₃	1	1	yes	characteristic
131	C₄	—	23	4	no	—
132	C₄	C₄²×D₂₃	1	1	yes	characteristic
133	C₄	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
134	C₄	C₂²⋊C₄×D₂₃	1	2	yes	—
135	C₂	—	2	2	no	—
136	C₂	—	46	4	no	—
137	C₂	#95×D₂₃	1	1	yes	characteristic
138	C₂	#13092464	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#67262

C₂

yes

maximal, characteristic

C₂×M₁₆×D₂₃

C₂

yes

maximal

#13092182

C₂

yes

maximal, characteristic, complemented

#13092181

C₂

yes

maximal, complemented

C₂₃×#341

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₂₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

M₁₆×D₂₃

—

M₁₆×D₂₃

C₄

yes

—

C₂×#67262

—

C₂×#67262

—

C₂×#67262

C₄

yes

—

C₂×C₈×D₂₃

—

C₂³×D₂₃

Q₈

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

Q₈

yes

characteristic

C₂×C₄×D₂₃

C₂×C₄

yes

—

C₂×C₄×D₂₃

C₂×C₄

yes

—

C₂×C₄×D₂₃

D₄

yes

—

C₂×C₄×D₂₃

D₄

yes

—

M₁₆×C₂₃

—

M₁₆×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

#67267

—

#67267

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

#67262

—

#67262

—

C₈×D₂₃

—

C₂³×C₂₃

C₂×Q₈

yes

characteristic

C₂²×D₂₃

—

C₂²×D₂₃

C₄⋊C₄

yes

—

C₂²×D₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×Dic₂₃

C₄⋊C₄

yes

—

C₂×Dic₂₃

C₄⋊C₄

yes

—

C₄×D₂₃

C₄²

yes

—

C₄×D₂₃

C₂²⋊C₄

yes

—

C₄×D₂₃

C₂²⋊C₄

yes

—

C₄×D₂₃

C₂²⋊C₄

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1738

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₄⋊C₄

yes

—

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

#95

yes

—

C₄×C₂₃

C₂×C₄²

yes

characteristic

C₄×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

C₂×C₂²⋊C₄

yes

—

Dic₂₃

#95

yes

—

C₂²×M₁₆

—

C₂²×M₁₆

—

#341

—

#341

—

#341

D₂₃

yes

characteristic, complemented

C₂×C₂₃

—

C₂×C₂₃

C₂×#95

yes

characteristic

C₂×C₂₃

#430

yes

—

D₂₃

—

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₂₃

#1738

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

C₄×D₂₃

yes

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₄×D₂₃

yes

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

Q₈×D₂₃

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

118

C₂×C₄

C₂×C₄×D₂₃

yes

—

119

C₂×C₄

D₄×D₂₃

yes

—

120

C₂×C₄

D₄×D₂₃

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₄⋊C₄×D₂₃

yes

—

130

C₂²

C₄⋊C₄×D₂₃

yes

characteristic

131

C₄

—

132

C₄

C₄²×D₂₃

yes

characteristic

133

C₄

C₂²⋊C₄×D₂₃

yes

characteristic

134

C₄

C₂²⋊C₄×D₂₃

yes

—

135

C₂

—

136

C₂

—

137

C₂

#95×D₂₃

yes

characteristic

138

C₂

#13092464

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₃
commutator	C₄×C₂₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#341	C₂
socle	C₂²×C₂₃	C₂×C₄⋊C₄
prime radical	C₂³×D₂₃	Q₈
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#341	D₂₃
hypocenter	C₂₃	#1738
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₂₃

commutator

C₄×C₂₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#341

C₂

socle

C₂²×C₂₃

C₂×C₄⋊C₄

prime radical

C₂³×D₂₃

Q₈

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
3. #13092182	C₂	semidirect
4. #13092181	C₂	semidirect
5. C₂₃×#341	C₂	semidirect
70. #341	D₂₃	semidirect
86. C₂₃	#1738	semidirect

H ≅ G/N

type

3. #13092182

C₂

semidirect

4. #13092181

C₂

semidirect

5. C₂₃×#341

C₂

semidirect

70. #341

D₂₃

semidirect

86. C₂₃

#1738

semidirect

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

subgroup	index
C₂²×#67262	2
C₂×M₁₆×D₂₃	2
#13092182	2
#13092181	2
C₂₃×#341	2
#1738	23

subgroup

index

C₂²×#67262

C₂×M₁₆×D₂₃

#13092182

#13092181

C₂₃×#341

#1738

Subgroups — #15673122

Conjugacy classes of subgroups

Complements

Maximal subgroups