Group #15673506 — subgroups

Subgroups — #15673506

Order 2944.

272 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 81 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	#13092892	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625015	C₂	1	1	yes	maximal, characteristic
3	#13092189	C₂	1	2	yes	maximal, complemented
4	#13092188	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#100	C₂	1	1	yes	maximal, characteristic
6	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂×C₂²⋊Dic₂₃	C₄	1	4	yes	—
9	D₄×Dic₂₃	—	2	2	no	—
10	D₄×Dic₂₃	—	2	2	no	—
11	#10625097	C₂²	1	1	yes	characteristic
12	#10625097	C₂²	1	1	yes	characteristic
13	C₂×#67267	C₂²	1	2	yes	—
14	C₂₃×#100	—	2	2	no	—
15	#10625015	—	2	2	no	—
16	C₂⁴×C₂₃	C₂×C₄	1	2	yes	—
17	C₂×D₄×C₂₃	—	2	2	no	—
18	C₂×D₄×C₂₃	—	2	2	no	—
19	C₂×D₄×C₂₃	D₄	1	2	yes	—
20	C₂×D₄×C₂₃	D₄	1	2	yes	—
21	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₂₃	—	2	4	no	—
23	M₁₆×C₂₃	—	2	2	no	—
24	M₁₆×C₂₃	—	2	2	no	—
25	C₂×C₈×C₂₃	—	2	2	no	—
26	C₂²⋊Dic₂₃	—	2	4	no	—
27	C₂²⋊Dic₂₃	C₂×C₄	1	2	yes	—
28	C₂²⋊Dic₂₃	C₂×C₄	1	2	yes	—
29	C₄⋊Dic₂₃	—	2	1	no	—
30	C₄⋊Dic₂₃	—	2	1	no	—
31	C₄×Dic₂₃	—	2	1	no	—
32	C₄×Dic₂₃	—	2	1	no	—
33	#67267	—	2	2	no	—
34	#67267	—	2	2	no	—
35	C₂×C₂₃⋊C₈	—	2	2	no	—
36	C₂³×C₂₃	—	2	2	no	—
37	C₂³×C₂₃	—	2	2	no	—
38	C₂³×C₂₃	—	8	2	no	—
39	C₂³×C₂₃	C₂²⋊C₄	1	4	yes	—
40	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
41	D₄×C₂₃	—	2	4	no	—
42	D₄×C₂₃	—	2	4	no	—
43	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	#41	1	2	yes	—
46	C₂×C₄×C₂₃	#41	1	2	yes	—
47	C₂×Dic₂₃	—	2	2	no	—
48	C₂×Dic₂₃	—	2	2	no	—
49	C₂×Dic₂₃	—	8	4	no	—
50	C₈×C₂₃	—	4	2	no	—
51	C₂₃⋊C₈	—	4	2	no	—
52	#1829	—	23	1	no	maximal, Sylow
53	C₂²×C₂₃	—	2	4	no	—
54	C₂²×C₂₃	—	4	1	no	—
55	C₂²×C₂₃	—	4	4	no	—
56	C₂²×C₂₃	—	8	2	no	—
57	C₂²×C₂₃	—	8	2	no	—
58	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₂₃	#117	1	1	yes	characteristic
60	C₂²×C₂₃	#117	1	1	yes	characteristic
61	C₄×C₂₃	#123	1	2	yes	—
62	C₄×C₂₃	#123	1	2	yes	—
63	Dic₂₃	—	4	2	no	—
64	Dic₂₃	—	4	2	no	—
65	#518	—	23	1	no	—
66	C₂×#100	—	23	1	no	—
67	C₂×#100	D₂₃	1	1	yes	characteristic
68	#344	—	23	2	no	—
69	#344	—	23	2	no	—
70	C₂×C₂₃	—	2	1	no	—
71	C₂×C₂₃	—	2	1	no	—
72	C₂×C₂₃	—	2	1	no	—
73	C₂×C₂₃	—	4	4	no	—
74	C₂×C₂₃	#381	1	1	yes	characteristic
75	C₂²×D₄	C₂×D₂₃	1	1	yes	characteristic
76	C₂×M₁₆	—	23	2	no	—
77	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
78	C₄×D₄	—	46	2	no	—
79	C₄×D₄	—	46	2	no	—
80	#117	—	23	1	no	—
81	#117	—	23	1	no	—
82	C₂×C₂²⋊C₄	—	23	4	no	—
83	#100	—	2	2	no	—
84	#100	—	46	2	no	—
85	C₂₃	#1829	1	1	yes	characteristic, complemented, Sylow
86	C₂⁴	C₄×D₂₃	1	2	yes	—
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	—	2	2	no	—
89	C₂×D₄	D₉₂	1	2	yes	—
90	C₂×D₄	C₂²⋊D₂₃	1	2	yes	—
91	C₂²×C₄	—	46	4	no	—
92	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	46	2	no	—
96	M₁₆	—	46	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	46	2	no	—
99	C₄⋊C₄	—	46	1	no	—
100	C₄⋊C₄	—	46	1	no	—
101	C₂²⋊C₄	—	23	2	no	—
102	C₂²⋊C₄	—	23	2	no	—
103	C₂²⋊C₄	—	46	4	no	—
104	C₄²	—	46	1	no	—
105	C₄²	—	46	1	no	—
106	C₂³	—	2	2	no	—
107	C₂³	—	2	2	no	—
108	C₂³	—	8	2	no	—
109	C₂³	(C₂×D₂₃)⋊C₄	1	4	yes	—
110	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
111	D₄	—	2	4	no	—
112	D₄	—	2	4	no	—
113	C₂×C₄	—	46	2	no	—
114	C₂×C₄	—	46	2	no	—
115	C₂×C₄	—	184	4	no	—
116	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
117	C₂×C₄	#67290	1	2	yes	—
118	C₂×C₄	#67290	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
120	C₈	—	4	2	no	—
121	C₈	—	92	2	no	—
122	C₂²	—	2	4	no	—
123	C₂²	—	4	1	no	—
124	C₂²	—	4	4	no	—
125	C₂²	—	8	2	no	—
126	C₂²	—	8	2	no	—
127	C₂²	#10625048	1	1	yes	characteristic
128	C₂²	#10625048	1	1	yes	characteristic
129	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
130	C₄	—	92	2	no	—
131	C₄	—	92	2	no	—
132	C₄	#10625057	1	2	yes	—
133	C₄	#10625106	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	—	2	1	no	—
137	C₂	—	4	4	no	—
138	C₂	#13092278	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092892

C₂

yes

maximal, characteristic

C₂×#10625015

C₂

yes

maximal, characteristic

#13092189

C₂

yes

maximal, complemented

#13092188

C₂

yes

maximal, complemented

C₂×C₂₃×#100

C₂

yes

maximal, characteristic

C₂²×D₄×C₂₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂×C₂²⋊Dic₂₃

C₄

yes

—

D₄×Dic₂₃

—

D₄×Dic₂₃

—

#10625097

C₂²

yes

characteristic

#10625097

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂₃×#100

—

#10625015

—

C₂⁴×C₂₃

C₂×C₄

yes

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

D₄

yes

—

C₂×D₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂²⋊Dic₂₃

—

C₂²⋊Dic₂₃

C₂×C₄

yes

—

C₂²⋊Dic₂₃

C₂×C₄

yes

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²⋊C₄

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1829

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

yes

—

C₄×C₂₃

#123

yes

—

Dic₂₃

—

Dic₂₃

—

#518

—

C₂×#100

—

C₂×#100

D₂₃

yes

characteristic

#344

—

#344

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#381

yes

characteristic

C₂²×D₄

C₂×D₂₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

—

C₄×D₄

—

C₄×D₄

—

#117

—

#117

—

C₂×C₂²⋊C₄

—

#100

—

#100

—

C₂₃

#1829

yes

characteristic, complemented, Sylow

C₂⁴

C₄×D₂₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₉₂

yes

—

C₂×D₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₄²

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

(C₂×D₂₃)⋊C₄

yes

—

110

C₂³

C₂×C₄×D₂₃

yes

characteristic

111

D₄

—

112

D₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

184

—

116

C₂×C₄

C₂×D₉₂

yes

characteristic

117

C₂×C₄

#67290

yes

—

118

C₂×C₄

#67290

yes

—

119

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

120

C₈

—

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

#10625048

yes

characteristic

128

C₂²

#10625048

yes

characteristic

129

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

#10625057

yes

—

133

C₄

#10625106

yes

—

134

C₂

—

135

C₂

—

136

C₂

—

137

C₂

—

138

C₂

#13092278

yes

characteristic

	subgroup	G/H
center	C₂	#13092278
commutator	C₂³×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×#100	C₂
socle	C₂×C₂₃	#381
prime radical	C₂²×D₄×C₂₃	C₂²
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×#100	D₂₃
hypocenter	C₂₃	#1829
LCS	G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#100

subgroup

G/H

center

C₂

#13092278

commutator

C₂³×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₂₃×#100

C₂

socle

C₂×C₂₃

#381

prime radical

C₂²×D₄×C₂₃

C₂²

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#100

N	H ≅ G/N	type
3. #13092189	C₂	semidirect
4. #13092188	C₂	semidirect
85. C₂₃	#1829	semidirect

H ≅ G/N

type

3. #13092189

C₂

semidirect

4. #13092188

C₂

semidirect

85. C₂₃

#1829

semidirect

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

subgroup	index
#13092892	2
C₂×#10625015	2
#13092189	2
#13092188	2
C₂×C₂₃×#100	2
#1829	23

subgroup

index

C₂×C₂₃×#100

Subgroups — #15673506

Conjugacy classes of subgroups

Complements

Maximal subgroups