Group #15674218 — subgroups

Subgroups — #15674218

Order 2944.

308 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 113 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×#10625017	C₂	1	4	yes	maximal
2	C₂×#10624985	C₂	1	1	yes	maximal, characteristic
3	#13092283	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
6	C₂₃×C₄⋊Q₈	—	2	2	no	—
7	C₂₃×C₄⋊Q₈	C₄	1	4	yes	complemented
8	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
9	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
11	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
12	#10625017	—	2	8	no	—
13	C₂₃⋊#95	—	2	2	no	—
14	#10624985	—	2	2	no	—
15	C₂×Q₈×C₂₃	—	2	2	no	—
16	C₂×Q₈×C₂₃	—	2	4	no	—
17	C₂×Q₈×C₂₃	—	2	4	no	—
18	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₂₃	—	2	2	no	—
20	C₂²×C₄×C₂₃	D₄	1	2	yes	—
21	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₂₃	—	2	2	no	—
23	C₄⋊C₄×C₂₃	—	2	8	no	—
24	C₄²×C₂₃	C₂×C₄	1	2	yes	—
25	C₄²×C₂₃	D₄	1	2	yes	—
26	C₄⋊Dic₂₃	—	2	4	no	—
27	C₂×C₂₃⋊C₈	—	2	2	no	—
28	C₂×C₂₃⋊C₈	—	2	2	no	—
29	C₂×C₂₃⋊C₈	D₄	1	4	yes	—
30	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
31	Q₈×C₂₃	—	2	8	no	—
32	Q₈×C₂₃	—	4	4	no	—
33	C₂×C₄×C₂₃	—	2	4	no	—
34	C₂×C₄×C₂₃	—	2	4	no	—
35	C₂×C₄×C₂₃	—	4	2	no	—
36	C₂×C₄×C₂₃	—	4	2	no	—
37	C₂×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₃	#41	1	2	yes	—
42	C₂×Dic₂₃	—	4	2	no	—
43	C₂×Dic₂₃	—	4	4	no	—
44	C₂₃⋊C₈	—	2	4	no	—
45	#1960	—	23	1	no	maximal, Sylow
46	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₂₃	#124	1	1	yes	characteristic
48	C₂²×C₂₃	#127	1	1	yes	characteristic
49	C₂²×C₂₃	#137	1	4	yes	—
50	C₄×C₂₃	—	2	4	no	—
51	C₄×C₂₃	—	4	4	no	—
52	C₄×C₂₃	C₄×D₄	1	2	yes	—
53	C₄×C₂₃	#121	1	2	yes	—
54	Dic₂₃	—	8	2	no	—
55	C₂×C₄⋊Q₈	D₂₃	1	1	yes	characteristic
56	C₂×#103	—	23	4	no	—
57	C₂×#97	—	23	1	no	—
58	#382	—	23	1	no	—
59	C₂×C₂₃	#390	1	1	yes	characteristic
60	C₂×C₂₃	#419	1	2	yes	—
61	C₂×C₂₃	#491	1	2	yes	—
62	C₂×C₂₃	#497	1	2	yes	—
63	C₂²×Q₈	C₂×D₂₃	1	2	yes	—
64	C₂²×C₈	—	23	2	no	—
65	C₄⋊Q₈	—	2	2	no	—
66	C₄⋊Q₈	Dic₂₃	1	4	yes	complemented
67	C₂×C₄⋊C₄	—	2	2	no	—
68	C₂×C₄⋊C₄	—	23	2	no	—
69	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
70	#103	—	46	8	no	—
71	#97	—	46	2	no	—
72	#95	—	46	2	no	—
73	C₂₃	#1960	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	C₂×Dic₂₃	1	4	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	46	2	no	—
80	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
81	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
82	C₂×C₈	—	23	4	no	—
83	C₂×C₈	—	46	2	no	—
84	C₂×C₈	—	46	2	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	46	4	no	—
87	C₄²	C₂×Dic₂₃	1	2	yes	—
88	C₄²	C₂²⋊D₂₃	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
90	Q₈	—	2	8	no	—
91	Q₈	—	4	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	92	2	no	—
97	C₂×C₄	—	92	4	no	—
98	C₂×C₄	C₂²⋊Dic₂₃	1	4	yes	—
99	C₂×C₄	D₄×D₂₃	1	2	yes	—
100	C₂×C₄	#67290	1	2	yes	—
101	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
102	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
103	C₈	—	46	4	no	—
104	C₂²	#10625107	1	1	yes	characteristic
105	C₂²	#10625111	1	1	yes	characteristic
106	C₂²	#10625113	1	4	yes	—
107	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
108	C₄	—	2	4	no	—
109	C₄	—	4	4	no	—
110	C₄	—	184	2	no	—
111	C₄	D₄×Dic₂₃	1	2	yes	—
112	C₄	#10625109	1	2	yes	—
113	C₂	#13092320	1	1	yes	characteristic
114	C₂	#13092426	1	2	yes	—
115	C₂	#13092785	1	2	yes	—
116	C₂	#13092809	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625017

C₂

yes

maximal

C₂×#10624985

C₂

yes

maximal, characteristic

#13092283

C₂

yes

maximal, characteristic

C₂×C₂₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

C₂²

yes

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

#10625017

—

C₂₃⋊#95

—

#10624985

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

D₄

yes

—

C₄⋊Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

Q₈×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

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂₃⋊C₈

—

#1960

—

maximal, Sylow

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

#121

yes

—

Dic₂₃

—

C₂×C₄⋊Q₈

D₂₃

yes

characteristic

C₂×#103

—

C₂×#97

—

#382

—

C₂×C₂₃

#390

yes

characteristic

C₂×C₂₃

#419

yes

—

C₂×C₂₃

#491

yes

—

C₂×C₂₃

#497

yes

—

C₂²×Q₈

C₂×D₂₃

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₂₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₂₃

yes

—

—

—

C₂₃

#1960

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×Dic₂₃

yes

—

C₄²

C₂²⋊D₂₃

yes

—

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₂₃

yes

—

C₂×C₄

D₄×D₂₃

yes

—

100

C₂×C₄

#67290

yes

—

101

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

102

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

103

C₈

—

104

C₂²

#10625107

yes

characteristic

105

C₂²

#10625111

yes

characteristic

106

C₂²

#10625113

yes

—

107

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

184

—

111

C₄

D₄×Dic₂₃

yes

—

112

C₄

#10625109

yes

—

113

C₂

#13092320

yes

characteristic

114

C₂

#13092426

yes

—

115

C₂

#13092785

yes

—

116

C₂

#13092809

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×C₄⋊Q₈	D₂₃
hypocenter	C₂₃	#1960
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₂₃×C₄⋊Q₈

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×C₄⋊Q₈

D₂₃

hypocenter

C₂₃

#1960

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
7. C₂₃×C₄⋊Q₈	C₄	semidirect
66. C₄⋊Q₈	Dic₂₃	semidirect
73. C₂₃	#1960	semidirect

H ≅ G/N

type

7. C₂₃×C₄⋊Q₈

C₄

semidirect

66. C₄⋊Q₈

Dic₂₃

semidirect

73. C₂₃

#1960

semidirect

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

subgroup	index
C₂×#10625017	2
C₂×#10624985	2
#13092283	2
C₂×C₂₃×C₄⋊Q₈	2
#1960	23

subgroup

index

C₂×C₂₃×C₄⋊Q₈

Subgroups — #15674218

Conjugacy classes of subgroups

Complements

Maximal subgroups