Group #15675010 — subgroups

Subgroups — #15675010

Order 2944.

228 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 93 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×#10625072	C₂	1	1	yes	maximal, characteristic
2	C₂×#10624994	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₂₃)⋊C₁₆	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#67288	C₄	1	1	yes	characteristic
6	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×#67267	C₄	1	1	yes	characteristic
8	#10625072	—	2	2	no	—
9	#10625072	—	2	2	no	—
10	C₂×#67262	C₄	1	2	yes	—
11	C₂×C₈×D₂₃	C₂²	1	2	yes	—
12	C₂×C₁₆×C₂₃	—	2	2	no	—
13	C₂×C₁₆×C₂₃	C₂²	1	2	yes	—
14	#10624994	—	2	2	no	—
15	C₂×C₂₃⋊C₁₆	C₂²	1	2	yes	—
16	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₂²⋊D₂₃	C₈	1	2	yes	—
18	#67288	—	2	2	no	—
19	#67288	—	2	2	no	—
20	C₂×D₉₂	C₈	1	1	yes	characteristic
21	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
22	C₂×Dic₄₆	C₈	1	1	yes	characteristic
23	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₂₃	D₄	1	2	yes	—
25	C₂×C₈×C₂₃	D₄	1	2	yes	—
26	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
27	#67267	—	2	2	no	—
28	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
29	#67262	—	2	4	no	—
30	C₈×D₂₃	—	2	2	no	—
31	C₈×D₂₃	—	2	2	no	—
32	C₁₆×C₂₃	—	2	2	no	—
33	C₂₃⋊C₁₆	—	2	2	no	—
34	C₂³×C₂₃	C₂×C₈	1	1	yes	characteristic
35	C₂²×D₂₃	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×C₂₃	M₁₆	1	2	yes	—
39	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²⋊D₂₃	—	2	4	no	—
41	C₂×Dic₂₃	C₂×C₈	1	2	yes	—
42	D₉₂	—	2	2	no	—
43	C₄×D₂₃	—	2	2	no	—
44	C₄×D₂₃	—	2	2	no	—
45	Dic₄₆	—	2	2	no	—
46	C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
47	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
48	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂₃⋊C₈	—	2	2	no	—
50	#2113	—	23	1	no	maximal, Sylow
51	C₂²×C₂₃	—	2	2	no	—
52	C₂²×C₂₃	C₂²⋊C₈	1	2	yes	—
53	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
54	C₂×D₂₃	—	2	2	no	—
55	C₂×D₂₃	—	2	4	no	—
56	C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
57	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
59	Dic₂₃	—	2	2	no	—
60	C₂×#131	—	23	1	no	—
61	C₂×M₃₂	—	23	1	no	—
62	C₂²×C₁₆	D₂₃	1	1	yes	characteristic, complemented
63	C₂²⋊C₁₆	—	23	4	no	—
64	C₂×C₂₃	—	2	2	no	—
65	C₂×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
66	C₂×C₂₃	#504	1	2	yes	—
67	D₂₃	—	4	2	no	—
68	C₂×#41	—	23	1	no	—
69	#131	—	46	2	no	—
70	#131	—	46	2	no	—
71	C₂×M₁₆	—	23	1	no	—
72	C₂×M₁₆	—	23	2	no	—
73	C₂²×C₈	—	23	2	no	—
74	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
75	M₃₂	—	46	2	no	—
76	C₂×C₁₆	—	2	2	no	—
77	C₂×C₁₆	—	23	2	no	—
78	C₂×C₁₆	C₂×D₂₃	1	2	yes	—
79	C₂₃	#2113	1	1	yes	characteristic, complemented, Sylow
80	#41	—	46	2	no	—
81	#41	—	46	2	no	—
82	C₂×Q₈	—	23	1	no	—
83	C₂×D₄	—	23	2	no	—
84	C₂×D₄	—	23	1	no	—
85	C₂²×C₄	—	23	2	no	—
86	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
87	M₁₆	—	46	2	no	—
88	M₁₆	—	46	4	no	—
89	C₂×C₈	—	23	2	no	—
90	C₂×C₈	—	46	2	no	—
91	C₂×C₈	—	46	2	no	—
92	C₂×C₈	C₄×D₂₃	1	1	yes	characteristic
93	C₂×C₈	D₉₂	1	2	yes	—
94	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
95	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
96	C₁₆	—	2	2	no	—
97	C₁₆	—	46	2	no	—
98	C₂³	—	23	2	no	—
99	C₂³	C₈×D₂₃	1	1	yes	characteristic
100	Q₈	—	46	2	no	—
101	D₄	—	46	4	no	—
102	D₄	—	46	2	no	—
103	C₂×C₄	—	23	2	no	—
104	C₂×C₄	—	46	2	no	—
105	C₂×C₄	—	46	2	no	—
106	C₂×C₄	C₈×D₂₃	1	1	yes	characteristic
107	C₂×C₄	#67262	1	2	yes	—
108	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
109	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
110	C₈	—	46	2	no	—
111	C₈	(C₂×D₂₃)⋊C₄	1	2	yes	—
112	C₈	C₂×D₉₂	1	1	yes	characteristic
113	C₈	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
114	C₂²	—	2	2	no	—
115	C₂²	—	46	2	no	—
116	C₂²	—	46	4	no	—
117	C₂²	(C₂×D₂₃)⋊C₈	1	2	yes	—
118	C₂²	C₂×C₈×D₂₃	1	1	yes	characteristic
119	C₄	—	46	2	no	—
120	C₄	(C₂×D₂₃)⋊C₈	1	2	yes	—
121	C₄	C₂×#67262	1	1	yes	characteristic
122	C₄	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	92	2	no	—
125	C₂	C₂×(C₂×D₂₃)⋊C₈	1	1	yes	characteristic
126	C₂	#13092833	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625072

C₂

yes

maximal, characteristic

C₂×#10624994

C₂

yes

maximal, characteristic, complemented

(C₂×D₂₃)⋊C₁₆

C₂

yes

maximal, complemented

C₂²×C₁₆×C₂₃

C₂

yes

maximal, characteristic, complemented

C₂×#67288

C₄

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂×#67267

C₄

yes

—

—

C₄

yes

—

C₂×C₈×D₂₃

C₂²

yes

—

C₂×C₁₆×C₂₃

—

C₂×C₁₆×C₂₃

C₂²

yes

—

#10624994

—

C₂×C₂₃⋊C₁₆

C₂²

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₂₃

C₈

yes

—

#67288

—

#67288

—

C₂×D₉₂

C₈

yes

characteristic

C₂×C₄×D₂₃

C₂×C₄

yes

—

C₂×Dic₄₆

C₈

yes

characteristic

C₂×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

C₂³

yes

characteristic

#67267

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

#67262

—

C₈×D₂₃

—

C₈×D₂₃

—

C₁₆×C₂₃

—

C₂₃⋊C₁₆

—

C₂³×C₂₃

C₂×C₈

yes

characteristic

C₂²×D₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×C₈

yes

characteristic

C₂×C₄×C₂₃

M₁₆

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂²⋊D₂₃

—

C₂×Dic₂₃

C₂×C₈

yes

—

D₉₂

—

C₄×D₂₃

—

C₄×D₂₃

—

Dic₄₆

—

C₈×C₂₃

C₂²⋊C₄

yes

—

C₈×C₂₃

C₂×D₄

yes

characteristic

C₈×C₂₃

C₂×D₄

yes

characteristic

C₂₃⋊C₈

—

#2113

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₂²⋊C₈

yes

—

C₂²×C₂₃

C₂²×C₈

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

C₂²⋊C₈

yes

—

C₄×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

C₂×M₁₆

yes

characteristic

Dic₂₃

—

C₂×#131

—

C₂×M₃₂

—

C₂²×C₁₆

D₂₃

yes

characteristic, complemented

C₂²⋊C₁₆

—

C₂×C₂₃

—

C₂×C₂₃

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₂₃

#504

yes

—

D₂₃

—

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₂₃

yes

—

C₂₃

#2113

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₂₃

yes

characteristic

C₂×C₈

D₉₂

yes

—

C₂×C₈

C₂²⋊D₂₃

yes

—

C₂×C₈

C₂²×D₂₃

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

—

C₂³

C₈×D₂₃

yes

characteristic

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₈×D₂₃

yes

characteristic

107

C₂×C₄

#67262

yes

—

108

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

109

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

110

C₈

—

111

C₈

(C₂×D₂₃)⋊C₄

yes

—

112

C₈

C₂×D₉₂

yes

characteristic

113

C₈

C₂×C₂²⋊D₂₃

yes

characteristic

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

(C₂×D₂₃)⋊C₈

yes

—

118

C₂²

C₂×C₈×D₂₃

yes

characteristic

119

C₄

—

120

C₄

(C₂×D₂₃)⋊C₈

yes

—

121

C₄

C₂×#67262

yes

characteristic

122

C₄

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

C₂×(C₂×D₂₃)⋊C₈

yes

characteristic

126

C₂

#13092833

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₂₃
commutator, socle	C₂²×C₂₃	C₂²×C₈
Fitting	C₂²×C₁₆×C₂₃	C₂
prime radical	C₂×#67288	C₄
cosocle	C₂×C₈×C₂₃	C₂³
hypercenter	C₂²×C₁₆	D₂₃
hypocenter	C₂₃	#2113
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₂₃

commutator, socle

C₂²×C₂₃

C₂²×C₈

Fitting

C₂²×C₁₆×C₂₃

C₂

prime radical

C₂×#67288

C₄

cosocle

C₂×C₈×C₂₃

C₂³

hypercenter

C₂²×C₁₆

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
2. C₂×#10624994	C₂	semidirect
3. (C₂×D₂₃)⋊C₁₆	C₂	semidirect
4. C₂²×C₁₆×C₂₃	C₂	semidirect
62. C₂²×C₁₆	D₂₃	semidirect
79. C₂₃	#2113	semidirect

H ≅ G/N

type

2. C₂×#10624994

C₂

semidirect

3. (C₂×D₂₃)⋊C₁₆

C₂

semidirect

4. C₂²×C₁₆×C₂₃

C₂

semidirect

62. C₂²×C₁₆

D₂₃

semidirect

79. C₂₃

#2113

semidirect

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

subgroup	index
C₂×#10625072	2
C₂×#10624994	2
(C₂×D₂₃)⋊C₁₆	2
C₂²×C₁₆×C₂₃	2
#2113	23

subgroup

index

C₂×#10625072

C₂×#10624994

(C₂×D₂₃)⋊C₁₆

C₂²×C₁₆×C₂₃

#2113

Subgroups — #15675010

Conjugacy classes of subgroups

Complements

Maximal subgroups