Group #15673321 — subgroups

Subgroups — #15673321

Order 2944.

564 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 133 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×#10625018	C₂	1	4	yes	maximal, complemented
2	#13092381	C₂	1	2	yes	maximal
3	C₂³×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₈×C₂₃	—	2	8	no	—
5	C₂²×Q₈×C₂₃	—	2	4	no	—
6	C₂²×Q₈×C₂₃	C₄	1	2	yes	—
7	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
8	C₂³×C₄×C₂₃	C₄	1	2	yes	—
9	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×#67267	C₂²	1	4	yes	—
11	#10625018	—	2	8	no	—
12	C₂²⋊(C₂₃⋊C₈)	—	2	4	no	—
13	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂×Q₈×C₂₃	—	2	32	no	—
15	C₂×Q₈×C₂₃	—	2	8	no	—
16	C₂×Q₈×C₂₃	—	2	4	no	—
17	C₂×Q₈×C₂₃	—	2	4	no	—
18	C₂×Q₈×C₂₃	—	2	4	no	—
19	C₂×Q₈×C₂₃	D₄	1	8	yes	—
20	C₂²×C₄×C₂₃	—	2	4	no	—
21	C₂²×C₄×C₂₃	—	2	8	no	—
22	C₂²×C₄×C₂₃	—	2	4	no	—
23	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₃	D₄	1	4	yes	—
26	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
27	#67267	—	2	8	no	—
28	C₂×C₂₃⋊C₈	—	2	4	no	—
29	C₂³×C₂₃	—	2	4	no	—
30	C₂³×C₂₃	—	2	2	no	—
31	C₂³×C₂₃	C₂²⋊C₄	1	2	yes	—
32	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
33	Q₈×C₂₃	—	2	16	no	—
34	Q₈×C₂₃	—	2	16	no	—
35	C₂×C₄×C₂₃	—	2	16	no	—
36	C₂×C₄×C₂₃	—	2	4	no	—
37	C₂×C₄×C₂₃	—	2	2	no	—
38	C₂×C₄×C₂₃	—	2	4	no	—
39	C₂×C₄×C₂₃	—	2	8	no	—
40	C₂×C₄×C₂₃	C₂²⋊C₄	1	8	yes	—
41	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
44	C₂₃⋊C₈	—	4	4	no	—
45	#1784	—	23	1	no	maximal, Sylow
46	C₂²×C₂₃	—	2	8	no	—
47	C₂²×C₂₃	—	2	2	no	—
48	C₂²×C₂₃	—	2	2	no	—
49	C₂²×C₂₃	—	2	2	no	—
50	C₂²×C₂₃	#101	1	4	yes	complemented
51	C₂²×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
52	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₂₃	—	2	8	no	—
54	C₄×C₂₃	—	2	2	no	—
55	C₄×C₂₃	#120	1	4	yes	—
56	C₂³×Q₈	D₂₃	1	1	yes	characteristic
57	C₂×#101	—	23	4	no	—
58	#407	—	23	2	no	—
59	C₂×C₂₃	—	2	4	no	—
60	C₂×C₂₃	—	2	2	no	—
61	C₂×C₂₃	#379	1	1	yes	characteristic
62	C₂×C₂₃	C₂×#101	1	2	yes	—
63	C₂²×Q₈	—	2	8	no	—
64	C₂²×Q₈	—	2	4	no	—
65	C₂²×Q₈	Dic₂₃	1	2	yes	—
66	C₂²×Q₈	C₂×D₂₃	1	2	yes	—
67	C₂³×C₄	Dic₂₃	1	2	yes	—
68	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
69	C₂×M₁₆	—	23	4	no	—
70	#101	—	46	8	no	—
71	C₂²⋊C₈	—	46	4	no	—
72	C₂₃	#1784	1	1	yes	characteristic, complemented, Sylow
73	C₂⁴	C₂×Dic₂₃	1	1	yes	characteristic
74	C₂×Q₈	—	2	32	no	—
75	C₂×Q₈	—	2	8	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	—	2	4	no	—
78	C₂×Q₈	—	2	4	no	—
79	C₂×Q₈	C₂²⋊D₂₃	1	8	yes	—
80	C₂²×C₄	—	2	4	no	—
81	C₂²×C₄	—	2	8	no	—
82	C₂²×C₄	—	2	4	no	—
83	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
84	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
85	C₂²×C₄	C₂²⋊D₂₃	1	4	yes	—
86	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
87	M₁₆	—	46	8	no	—
88	C₂×C₈	—	46	4	no	—
89	C₂³	—	2	2	no	—
90	C₂³	—	2	4	no	—
91	C₂³	C₂²⋊Dic₂₃	1	2	yes	—
92	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
93	Q₈	—	2	16	no	—
94	Q₈	—	2	16	no	—
95	C₂×C₄	—	2	16	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	8	no	—
100	C₂×C₄	C₂²⋊Dic₂₃	1	8	yes	—
101	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
102	C₂×C₄	C₂×C₂²⋊D₂₃	1	4	yes	—
103	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
104	C₈	—	92	4	no	—
105	C₂²	—	2	8	no	—
106	C₂²	—	2	2	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	2	2	no	—
109	C₂²	#10625018	1	4	yes	complemented
110	C₂²	C₂×C₂²⋊Dic₂₃	1	2	yes	—
111	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
112	C₄	—	2	8	no	—
113	C₄	—	2	2	no	—
114	C₄	#10625124	1	4	yes	—
115	C₂	—	2	2	no	—
116	C₂	—	2	4	no	—
117	C₂	#13092273	1	1	yes	characteristic
118	C₂	C₂×#10625018	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625018

C₂

yes

maximal, complemented

#13092381

C₂

yes

maximal

C₂³×Q₈×C₂₃

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

—

C₂²×Q₈×C₂₃

—

C₂²×Q₈×C₂₃

C₄

yes

—

C₂²×Q₈×C₂₃

C₂²

yes

—

C₂³×C₄×C₂₃

C₄

yes

—

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

#10625018

—

C₂²⋊(C₂₃⋊C₈)

—

C₂⁴×C₂₃

C₂×C₄

yes

characteristic

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

#67267

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²⋊C₄

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂₃⋊C₈

—

#1784

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#101

yes

complemented

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#120

yes

—

C₂³×Q₈

D₂₃

yes

characteristic

C₂×#101

—

#407

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#379

yes

characteristic

C₂×C₂₃

C₂×#101

yes

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

Dic₂₃

yes

—

C₂²×Q₈

C₂×D₂₃

yes

—

C₂³×C₄

Dic₂₃

yes

—

C₂³×C₄

C₂×D₂₃

yes

characteristic

C₂×M₁₆

—

#101

—

C₂²⋊C₈

—

C₂₃

#1784

yes

characteristic, complemented, Sylow

C₂⁴

C₂×Dic₂₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂³

—

C₂³

—

C₂³

C₂²⋊Dic₂₃

yes

—

C₂³

C₂²×Dic₂₃

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₂²⋊Dic₂₃

yes

—

101

C₂×C₄

C₂²⋊Dic₂₃

yes

—

102

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

103

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

104

C₈

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

#10625018

yes

complemented

110

C₂²

C₂×C₂²⋊Dic₂₃

yes

—

111

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

#10625124

yes

—

115

C₂

—

116

C₂

—

117

C₂

#13092273

yes

characteristic

118

C₂

C₂×#10625018

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₃
commutator	C₂³×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂³×Q₈×C₂₃	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₂₃	C₂×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂³×Q₈	D₂₃
hypocenter	C₂₃	#1784
LCS	G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₂₃

commutator

C₂³×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂³×Q₈×C₂₃

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₂₃

C₂×C₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂³×Q₈

D₂₃

hypocenter

C₂₃

#1784

LCS

G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×Q₈

N	H ≅ G/N	type
1. C₂×#10625018	C₂	semidirect
50. C₂²×C₂₃	#101	semidirect
72. C₂₃	#1784	semidirect
109. C₂²	#10625018	semidirect

H ≅ G/N

type

1. C₂×#10625018

C₂

semidirect

50. C₂²×C₂₃

#101

semidirect

72. C₂₃

#1784

semidirect

109. C₂²

#10625018

semidirect

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

subgroup	index
C₂×#10625018	2
#13092381	2
C₂³×Q₈×C₂₃	2
#1784	23

subgroup

index

C₂×#10625018

#13092381

C₂³×Q₈×C₂₃

#1784

Subgroups — #15673321

Conjugacy classes of subgroups

Complements

Maximal subgroups