Group #15675092 — subgroups

Subgroups — #15675092

Order 2944.

230 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 73 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	#13092401	C₂	1	2	yes	maximal
2	#13092216	C₂	1	4	yes	maximal, complemented
3	C₂₃×#536	C₂	1	1	yes	maximal, characteristic
4	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
5	C₂²×Q₈×C₂₃	C₄	1	1	yes	characteristic
6	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂₃×#124	C₄	1	2	yes	—
8	C₂₃×#123	—	2	2	no	—
9	C₂₃×C₂²⋊Q₈	—	2	2	no	—
10	C₂₃×#117	C₄	1	1	yes	characteristic
11	C₂×#67267	—	2	2	no	—
12	#10625018	C₂²	1	4	yes	—
13	#10625015	—	2	2	no	—
14	#41×C₂₃	—	4	2	no	—
15	C₂×Q₈×C₂₃	—	2	1	no	—
16	C₂×Q₈×C₂₃	—	4	2	no	—
17	C₂×Q₈×C₂₃	C₂×C₄	1	2	yes	—
18	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
19	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×D₄×C₂₃	D₄	1	2	yes	—
21	C₂²×C₄×C₂₃	—	2	1	no	—
22	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₂₃	D₄	1	2	yes	—
24	C₄⋊C₄×C₂₃	—	2	1	no	—
25	C₄⋊C₄×C₂₃	—	2	4	no	—
26	C₂²⋊C₄×C₂₃	—	2	1	no	—
27	C₂²⋊C₄×C₂₃	—	2	4	no	—
28	C₄²×C₂₃	C₂×C₄	1	2	yes	—
29	#67267	—	2	4	no	—
30	#67267	—	4	2	no	—
31	C₂×C₂₃⋊C₈	—	4	2	no	—
32	C₂³×C₂₃	C₂²⋊C₄	1	2	yes	—
33	C₂³×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
34	Q₈×C₂₃	—	4	1	no	—
35	Q₈×C₂₃	—	4	2	no	—
36	Q₈×C₂₃	—	4	2	no	—
37	D₄×C₂₃	—	4	1	no	—
38	D₄×C₂₃	—	4	2	no	—
39	C₂×C₄×C₂₃	—	2	2	no	—
40	C₂×C₄×C₂₃	—	2	2	no	—
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	—	4	2	no	—
45	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
48	C₂₃⋊C₈	—	4	4	no	—
49	#2135	—	23	1	no	maximal, Sylow
50	C₂²×C₂₃	—	2	1	no	—
51	C₂²×C₂₃	—	2	2	no	—
52	C₂²×C₂₃	—	4	2	no	—
53	C₂²×C₂₃	—	4	1	no	—
54	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₄×C₂₃	—	2	2	no	—
56	C₄×C₂₃	—	2	1	no	—
57	C₄×C₂₃	—	4	2	no	—
58	C₄×C₂₃	—	4	2	no	—
59	C₄×C₂₃	#99	1	2	yes	—
60	#536	D₂₃	1	1	yes	characteristic
61	#413	—	23	2	no	—
62	#356	—	23	4	no	—
63	C₂×C₂₃	—	2	1	no	—
64	C₂×C₂₃	—	4	1	no	—
65	C₂×C₂₃	—	4	2	no	—
66	C₂×C₂₃	C₂×#99	1	1	yes	characteristic
67	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
68	C₂²×Q₈	Dic₂₃	1	1	yes	characteristic
69	C₂×M₁₆	—	46	2	no	—
70	C₄⋊Q₈	C₂×D₂₃	1	2	yes	—
71	#124	Dic₂₃	1	2	yes	—
72	#123	—	2	2	no	—
73	C₂²⋊Q₈	—	2	2	no	—
74	#117	Dic₂₃	1	1	yes	characteristic
75	#101	—	23	4	no	—
76	#100	—	46	2	no	—
77	C₂₃	#2135	1	1	yes	characteristic, complemented, Sylow
78	#41	—	4	2	no	—
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	4	2	no	—
81	C₂×Q₈	C₂×Dic₂₃	1	2	yes	—
82	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
83	C₂×D₄	C₂×Dic₂₃	1	1	yes	characteristic
84	C₂×D₄	C₂²⋊D₂₃	1	2	yes	—
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
87	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
88	M₁₆	—	46	4	no	—
89	M₁₆	—	92	2	no	—
90	C₂×C₈	—	92	2	no	—
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	—	2	4	no	—
93	C₂²⋊C₄	—	2	1	no	—
94	C₂²⋊C₄	—	2	4	no	—
95	C₄²	C₂×Dic₂₃	1	2	yes	—
96	C₂³	C₂²⋊Dic₂₃	1	2	yes	—
97	C₂³	C₂²⋊Dic₂₃	1	1	yes	characteristic
98	Q₈	—	4	1	no	—
99	Q₈	—	4	2	no	—
100	Q₈	—	4	2	no	—
101	D₄	—	4	1	no	—
102	D₄	—	4	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
110	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
111	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
112	C₈	—	92	4	no	—
113	C₂²	—	2	1	no	—
114	C₂²	—	2	2	no	—
115	C₂²	—	4	2	no	—
116	C₂²	—	4	1	no	—
117	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	1	no	—
120	C₄	—	4	2	no	—
121	C₄	—	4	2	no	—
122	C₄	#10625016	1	2	yes	—
123	C₂	—	2	1	no	—
124	C₂	—	4	1	no	—
125	C₂	—	4	2	no	—
126	C₂	C₂×#10625016	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092401

C₂

yes

maximal

#13092216

C₂

yes

maximal, complemented

C₂₃×#536

C₂

yes

maximal, characteristic

C₂×#41×C₂₃

C₂²

yes

characteristic

C₂²×Q₈×C₂₃

C₄

yes

characteristic

C₂₃×C₄⋊Q₈

C₂²

yes

—

C₂₃×#124

C₄

yes

—

C₂₃×#123

—

C₂₃×C₂²⋊Q₈

—

C₂₃×#117

C₄

yes

characteristic

C₂×#67267

—

#10625018

C₂²

yes

—

#10625015

—

#41×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₄

yes

—

C₂×Q₈×C₂₃

C₂³

yes

characteristic

C₂×D₄×C₂₃

C₂×C₄

yes

characteristic

C₂×D₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂²⋊C₄

yes

—

C₂³×C₂₃

C₂²⋊C₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

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

C₂×D₄

yes

—

C₂₃⋊C₈

—

#2135

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#99

yes

—

#536

D₂₃

yes

characteristic

#413

—

#356

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#99

yes

characteristic

C₂×#41

C₂×D₂₃

yes

characteristic

C₂²×Q₈

Dic₂₃

yes

characteristic

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₂₃

yes

—

#124

Dic₂₃

yes

—

#123

—

C₂²⋊Q₈

—

#117

Dic₂₃

yes

characteristic

#101

—

#100

—

C₂₃

#2135

yes

characteristic, complemented, Sylow

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₂₃

yes

—

C₂×Q₈

C₂²×D₂₃

yes

characteristic

C₂×D₄

C₂×Dic₂₃

yes

characteristic

C₂×D₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃

yes

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×Dic₂₃

yes

—

C₂³

C₂²⋊Dic₂₃

yes

—

C₂³

C₂²⋊Dic₂₃

yes

characteristic

Q₈

—

Q₈

—

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂²⋊Dic₂₃

yes

characteristic

110

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

111

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

112

C₈

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

#10625016

yes

—

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

C₂×#10625016

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#10625016
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂×Q₈	C₂²×D₂₃
Fitting	C₂₃×#536	C₂
socle	C₂×C₂₃	C₂×#99
prime radical	C₂×#41×C₂₃	C₂²
cosocle	C₂×Q₈×C₂₃	C₂³
hypercenter	#536	D₂₃
hypocenter	C₂₃	#2135
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #536

subgroup

G/H

center

C₂

C₂×#10625016

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂×Q₈

C₂²×D₂₃

Fitting

C₂₃×#536

C₂

socle

C₂×C₂₃

C₂×#99

prime radical

C₂×#41×C₂₃

C₂²

cosocle

C₂×Q₈×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #536

N	H ≅ G/N	type
2. #13092216	C₂	semidirect
77. C₂₃	#2135	semidirect

H ≅ G/N

type

2. #13092216

C₂

semidirect

77. C₂₃

#2135

semidirect

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

subgroup	index
#13092401	2
#13092216	2
C₂₃×#536	2
#2135	23

subgroup

index

#13092401

#13092216

C₂₃×#536

#2135

Subgroups — #15675092

Conjugacy classes of subgroups

Complements

Maximal subgroups