Group #15675983 — subgroups

Subgroups — #15675983

Order 2944.

784 conjugacy classes of subgroups; only the 309 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 309 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×#10625097	C₂	1	2	yes	maximal
2	#13092324	C₂	1	8	yes	maximal, complemented
3	#13092280	C₂	1	4	yes	maximal
4	C₂³×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₂₃	—	2	8	no	—
6	C₂²×Q₈×C₂₃	C₄	1	8	yes	complemented
7	C₂²×Q₈×C₂₃	C₂²	1	4	yes	—
8	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
9	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊Dic₂₃	C₂²	1	4	yes	—
11	#10625097	—	2	8	no	—
12	C₂×C₄⋊Dic₂₃	C₂²	1	4	yes	—
13	C₂×C₄×Dic₂₃	C₂²	1	4	yes	—
14	C₂₃⋊#95	C₂²	1	16	yes	—
15	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
16	C₂×Q₈×C₂₃	—	2	32	no	—
17	C₂×Q₈×C₂₃	—	2	8	no	—
18	C₂×Q₈×C₂₃	—	2	8	no	—
19	C₂×Q₈×C₂₃	C₂×C₄	1	16	yes	—
20	C₂²×C₄×C₂₃	—	2	8	no	—
21	C₂²×C₄×C₂₃	C₂×C₄	1	8	yes	—
22	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
23	C₂²×C₄×C₂₃	D₄	1	8	yes	—
24	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
25	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
26	C₂²×Dic₂₃	C₂³	1	8	yes	—
27	C₂²⋊Dic₂₃	—	2	8	no	—
28	C₄⋊Dic₂₃	—	2	8	no	—
29	C₄×Dic₂₃	—	2	8	no	—
30	C₂³×C₂₃	—	2	4	no	—
31	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
32	C₂³×C₂₃	C₂×D₄	1	4	yes	—
33	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
34	Q₈×C₂₃	—	2	32	no	—
35	C₂×C₄×C₂₃	—	2	16	no	—
36	C₂×C₄×C₂₃	—	2	8	no	—
37	C₂×C₄×C₂₃	C₂²⋊C₄	1	16	yes	—
38	C₂×C₄×C₂₃	C₂²×C₄	1	8	yes	—
39	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
40	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
41	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
42	C₂×Dic₂₃	—	2	8	no	—
43	C₂×Dic₂₃	—	2	16	no	—
44	#2310	—	23	1	no	maximal, Sylow
45	C₂²×C₂₃	—	2	4	no	—
46	C₂²×C₂₃	—	2	8	no	—
47	C₂²×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
48	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
49	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
50	C₂²×C₂₃	2₋¹⁺⁴	1	4	yes	—
51	C₄×C₂₃	—	2	8	no	—
52	C₄×C₂₃	C₂×C₂²⋊C₄	1	8	yes	—
53	Dic₂₃	—	4	8	no	—
54	C₂³×Q₈	D₂₃	1	1	yes	characteristic
55	C₂×#117	—	23	2	no	—
56	#391	—	23	8	no	—
57	#381	—	23	4	no	—
58	C₂×C₂₃	—	2	4	no	—
59	C₂×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₂₃	#519	1	2	yes	—
61	C₂×C₂₃	#536	1	4	yes	—
62	C₂²×Q₈	—	2	8	no	—
63	C₂²×Q₈	Dic₂₃	1	8	yes	complemented
64	C₂²×Q₈	C₂×D₂₃	1	4	yes	—
65	C₂³×C₄	C₂×D₂₃	1	2	yes	—
66	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
67	#117	—	46	8	no	—
68	C₂×C₄⋊C₄	—	23	4	no	—
69	C₂×C₂²⋊C₄	—	23	4	no	—
70	C₂×C₄²	—	23	4	no	—
71	#95	—	23	16	no	—
72	C₂₃	#2310	1	1	yes	characteristic, complemented, Sylow
73	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
74	C₂×Q₈	—	2	32	no	—
75	C₂×Q₈	—	2	8	no	—
76	C₂×Q₈	—	2	8	no	—
77	C₂×Q₈	C₂×Dic₂₃	1	16	yes	—
78	C₂²×C₄	—	2	8	no	—
79	C₂²×C₄	—	23	8	no	—
80	C₂²×C₄	C₂×Dic₂₃	1	8	yes	—
81	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
82	C₂²×C₄	C₂²⋊D₂₃	1	8	yes	—
83	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
84	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
85	C₄⋊C₄	—	46	8	no	—
86	C₂²⋊C₄	—	46	8	no	—
87	C₄²	—	46	8	no	—
88	C₂³	—	2	4	no	—
89	C₂³	C₂²×Dic₂₃	1	2	yes	—
90	C₂³	C₂×C₂²⋊D₂₃	1	4	yes	—
91	C₂³	C₂³×D₂₃	1	1	yes	characteristic
92	Q₈	—	2	32	no	—
93	C₂×C₄	—	2	16	no	—
94	C₂×C₄	—	2	8	no	—
95	C₂×C₄	—	46	16	no	—
96	C₂×C₄	—	46	8	no	—
97	C₂×C₄	C₂²⋊Dic₂₃	1	16	yes	—
98	C₂×C₄	C₂²×Dic₂₃	1	8	yes	—
99	C₂×C₄	C₂²×Dic₂₃	1	4	yes	—
100	C₂×C₄	C₂×C₂²⋊D₂₃	1	4	yes	—
101	C₂×C₄	C₂×C₂²⋊D₂₃	1	4	yes	—
102	C₂²	—	2	4	no	—
103	C₂²	—	2	8	no	—
104	C₂²	C₂×C₂²⋊Dic₂₃	1	4	yes	—
105	C₂²	#10625158	1	4	yes	—
106	C₂²	C₂³×Dic₂₃	1	1	yes	characteristic
107	C₂²	C₂²×C₂²⋊D₂₃	1	2	yes	—
108	C₄	—	2	8	no	—
109	C₄	—	92	8	no	—
110	C₄	C₂×C₂²⋊Dic₂₃	1	8	yes	—
111	C₂	—	2	4	no	—
112	C₂	C₂²×C₂²⋊Dic₂₃	1	1	yes	characteristic
113	C₂	#13092895	1	2	yes	—
114	C₂	#13092991	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625097

C₂

yes

maximal

#13092324

C₂

yes

maximal, complemented

#13092280

C₂

yes

maximal

C₂³×Q₈×C₂₃

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

—

C₂²×Q₈×C₂₃

C₄

yes

complemented

C₂²×Q₈×C₂₃

C₂²

yes

—

C₂³×C₄×C₂₃

C₂²

yes

—

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂×C₂²⋊Dic₂₃

C₂²

yes

—

#10625097

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

—

C₂₃⋊#95

C₂²

yes

—

C₂⁴×C₂₃

C₂³

yes

characteristic

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×Dic₂₃

C₂³

yes

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²×C₄

yes

—

C₂³×C₂₃

C₂×D₄

yes

—

C₂³×C₂₃

C₂⁴

yes

characteristic

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

#2310

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

—

C₂²×C₂₃

C₂³×C₄

yes

characteristic

C₂²×C₂₃

C₂²×D₄

yes

—

C₂²×C₂₃

2₋¹⁺⁴

yes

—

C₄×C₂₃

—

C₄×C₂₃

C₂×C₂²⋊C₄

yes

—

Dic₂₃

—

C₂³×Q₈

D₂₃

yes

—

—

—

C₂×C₂₃

—

C₂×C₂₃

C₂²×C₂²⋊C₄

yes

characteristic

C₂×C₂₃

#519

yes

—

C₂×C₂₃

#536

yes

—

C₂²×Q₈

—

C₂²×Q₈

Dic₂₃

yes

complemented

C₂²×Q₈

C₂×D₂₃

yes

—

C₂³×C₄

C₂×D₂₃

yes

—

C₂³×C₄

C₂×D₂₃

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

C₂₃

#2310

yes

characteristic, complemented, Sylow

C₂⁴

C₂²×D₂₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

C₂²×Dic₂₃

yes

—

C₂³

C₂×C₂²⋊D₂₃

yes

—

C₂³

C₂³×D₂₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₂₃

yes

—

C₂×C₄

C₂²×Dic₂₃

yes

—

C₂×C₄

C₂²×Dic₂₃

yes

—

100

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

101

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂×C₂²⋊Dic₂₃

yes

—

105

C₂²

#10625158

yes

—

106

C₂²

C₂³×Dic₂₃

yes

characteristic

107

C₂²

C₂²×C₂²⋊D₂₃

yes

—

108

C₄

—

109

C₄

—

110

C₄

C₂×C₂²⋊Dic₂₃

yes

—

111

C₂

—

112

C₂

C₂²×C₂²⋊Dic₂₃

yes

characteristic

113

C₂

#13092895

yes

—

114

C₂

#13092991

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Fitting	C₂³×Q₈×C₂₃	C₂
socle, cosocle	C₂³×C₂₃	C₂⁴
prime radical	C₂⁴×C₂₃	C₂³
hypercenter	C₂³×Q₈	D₂₃
hypocenter	C₂₃	#2310
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₃

commutator

C₂²×C₂₃

C₂³×C₄

Fitting

C₂³×Q₈×C₂₃

C₂

socle, cosocle

C₂³×C₂₃

C₂⁴

prime radical

C₂⁴×C₂₃

C₂³

hypercenter

C₂³×Q₈

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂³ ◁ C₂³×Q₈

N	H ≅ G/N	type
2. #13092324	C₂	semidirect
6. C₂²×Q₈×C₂₃	C₄	semidirect
63. C₂²×Q₈	Dic₂₃	semidirect
72. C₂₃	#2310	semidirect

H ≅ G/N

type

2. #13092324

C₂

semidirect

6. C₂²×Q₈×C₂₃

C₄

semidirect

63. C₂²×Q₈

Dic₂₃

semidirect

72. C₂₃

#2310

semidirect

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

subgroup	index
C₂×#10625097	2
#13092324	2
#13092280	2
C₂³×Q₈×C₂₃	2
#2310	23

subgroup

index

C₂³×Q₈×C₂₃

Subgroups — #15675983

Conjugacy classes of subgroups

Complements

Maximal subgroups