Group #15672518 — subgroups

Subgroups — #15672518

Order 2944.

218 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 57 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	#13092789	C₂	1	1	yes	maximal, characteristic
2	#13092712	C₂	1	2	yes	maximal, complemented
3	#13092161	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊(C₂₃⋊C₈)	C₂	1	2	yes	maximal, complemented
5	C₂₃×#559	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#127	C₂²	1	1	yes	characteristic
7	C₂₃×#124	—	2	2	no	—
8	C₂₃×#123	—	2	2	no	—
9	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
10	C₄×D₄×C₂₃	—	2	2	no	—
11	C₂×#67273	—	2	2	no	—
12	#10625040	C₂²	1	1	yes	characteristic
13	#10625017	—	2	2	no	—
14	#10624989	—	2	2	no	—
15	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
16	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₂₃	D₄	1	2	yes	—
18	C₂×D₄×C₂₃	—	2	2	no	—
19	C₂×D₄×C₂₃	D₄	1	2	yes	—
20	C₂²×C₄×C₂₃	—	2	2	no	—
21	C₂×Dic₄₆	—	2	1	no	—
22	C₄⋊C₄×C₂₃	—	2	2	no	—
23	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
24	C₂²⋊C₄×C₂₃	—	2	4	no	—
25	C₂²⋊C₄×C₂₃	—	2	2	no	—
26	C₄²×C₂₃	—	2	2	no	—
27	C₄²×C₂₃	C₂³	1	1	yes	characteristic
28	#67273	—	4	4	no	—
29	C₄⋊Dic₂₃	—	2	2	no	—
30	C₂×C₂₃⋊C₈	—	2	1	no	—
31	C₂×C₂₃⋊C₈	—	2	2	no	—
32	C₂³×C₂₃	—	2	2	no	—
33	Q₈×C₂₃	—	2	2	no	—
34	Q₈×C₂₃	—	2	2	no	—
35	D₄×C₂₃	—	4	2	no	—
36	D₄×C₂₃	—	4	2	no	—
37	D₄×C₂₃	—	4	2	no	—
38	C₂×C₄×C₂₃	—	2	2	no	—
39	C₂×C₄×C₂₃	—	2	2	no	—
40	C₂×C₄×C₂₃	—	8	2	no	—
41	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
43	C₂×Dic₂₃	—	4	1	no	—
44	Dic₄₆	—	4	2	no	—
45	C₂₃⋊C₈	—	2	2	no	—
46	C₂₃⋊C₈	—	4	2	no	—
47	#1630	—	23	1	no	maximal, Sylow
48	C₂²×C₂₃	—	4	4	no	—
49	C₂²×C₂₃	—	4	2	no	—
50	C₂²×C₂₃	#120	1	1	yes	characteristic
51	C₄×C₂₃	—	2	1	no	—
52	C₄×C₂₃	—	4	2	no	—
53	C₄×C₂₃	—	4	2	no	—
54	C₄×C₂₃	#135	1	2	yes	—
55	C₄×C₂₃	#137	1	2	yes	—
56	Dic₂₃	—	8	1	no	—
57	#559	D₂₃	1	1	yes	characteristic
58	#492	—	23	1	no	—
59	#477	—	23	2	no	—
60	#331	—	23	1	no	—
61	Q₈⋊C₈	—	23	2	no	—
62	C₂×C₂₃	—	8	2	no	—
63	C₂×C₂₃	#452	1	2	yes	—
64	C₂×C₂₃	#453	1	1	yes	characteristic
65	C₂×SD₁₆	—	46	2	no	—
66	C₄⋊Q₈	—	23	1	no	—
67	#127	C₂×D₂₃	1	1	yes	characteristic
68	#124	—	2	2	no	—
69	#123	—	2	2	no	—
70	C₄×Q₈	C₂×D₂₃	1	2	yes	—
71	C₄×D₄	—	2	2	no	—
72	#107	—	46	2	no	—
73	C₄⋊C₈	—	23	2	no	—
74	#103	—	46	2	no	—
75	C₄×C₈	—	23	1	no	—
76	C₂₃	#1630	1	1	yes	characteristic, complemented, Sylow
77	C₂×Q₈	—	46	1	no	—
78	C₂×Q₈	C₂²⋊D₂₃	1	2	yes	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	C₂²⋊D₂₃	1	2	yes	—
81	C₂²×C₄	—	2	2	no	—
82	SD₁₆	—	92	4	no	—
83	C₂×C₈	—	46	1	no	—
84	C₂×C₈	—	46	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	46	2	no	—
87	C₄⋊C₄	C₂²⋊D₂₃	1	2	yes	—
88	C₂²⋊C₄	—	2	4	no	—
89	C₂²⋊C₄	—	2	2	no	—
90	C₄²	—	2	2	no	—
91	C₄²	C₂²×D₂₃	1	1	yes	characteristic
92	C₂³	—	2	2	no	—
93	Q₈	—	2	2	no	—
94	Q₈	—	2	2	no	—
95	Q₈	—	92	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	8	2	no	—
102	C₂×C₄	—	92	1	no	—
103	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
104	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
105	C₈	—	46	2	no	—
106	C₈	—	92	2	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	4	2	no	—
109	C₂²	#10625124	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	4	2	no	—
112	C₄	—	4	2	no	—
113	C₄	—	184	1	no	—
114	C₄	#10625121	1	2	yes	—
115	C₄	#10625122	1	2	yes	—
116	C₂	—	8	2	no	—
117	C₂	#13092579	1	2	yes	—
118	C₂	#13092585	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092789

C₂

yes

maximal, characteristic

#13092712

C₂

yes

maximal, complemented

#13092161

C₂

yes

maximal, characteristic

Q₈⋊(C₂₃⋊C₈)

C₂

yes

maximal, complemented

C₂₃×#559

C₂

yes

maximal, characteristic

C₂₃×#127

C₂²

yes

characteristic

C₂₃×#124

—

C₂₃×#123

—

C₄×Q₈×C₂₃

C₂²

yes

—

C₄×D₄×C₂₃

—

C₂×#67273

—

#10625040

C₂²

yes

characteristic

#10625017

—

#10624989

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

—

C₄×C₂₃⋊C₈

C₂²

yes

characteristic

C₂×Q₈×C₂₃

D₄

yes

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

—

C₂×Dic₄₆

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

D₄

yes

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

#67273

—

C₄⋊Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×Dic₂₃

—

Dic₄₆

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1630

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#120

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#135

yes

—

C₄×C₂₃

#137

yes

—

Dic₂₃

—

#559

D₂₃

yes

—

—

—

Q₈⋊C₈

—

C₂×C₂₃

—

C₂×C₂₃

#452

yes

—

C₂×C₂₃

#453

yes

characteristic

C₂×SD₁₆

—

C₄⋊Q₈

—

#127

C₂×D₂₃

yes

characteristic

#124

—

#123

—

C₄×Q₈

C₂×D₂₃

yes

—

C₄×D₄

—

#107

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₂₃

#1630

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₂₃

yes

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₂₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₂₃

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

104

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

105

C₈

—

106

C₈

—

107

C₂²

—

108

C₂²

—

109

C₂²

#10625124

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

184

—

114

C₄

#10625121

yes

—

115

C₄

#10625122

yes

—

116

C₂

—

117

C₂

#13092579

yes

—

118

C₂

#13092585

yes

characteristic

	subgroup	G/H
center	C₂²	#10625124
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#559	C₂
socle	C₂²×C₂₃	#120
prime radical	C₂₃×#127	C₂²
hypercenter	#559	D₂₃
hypocenter	C₂₃	#1630
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #559

subgroup

G/H

center

C₂²

#10625124

commutator, cosocle

C₄²×C₂₃

C₂³

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#559

C₂

socle

C₂²×C₂₃

#120

prime radical

C₂₃×#127

C₂²

D₂₃

C₂₃

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #559

N	H ≅ G/N	type
2. #13092712	C₂	semidirect
4. Q₈⋊(C₂₃⋊C₈)	C₂	semidirect
76. C₂₃	#1630	semidirect

H ≅ G/N

type

2. #13092712

C₂

semidirect

4. Q₈⋊(C₂₃⋊C₈)

C₂

semidirect

76. C₂₃

#1630

semidirect

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

subgroup	index
#13092789	2
#13092712	2
#13092161	2
Q₈⋊(C₂₃⋊C₈)	2
C₂₃×#559	2
#1630	23

subgroup

index

#13092789

#13092712

#13092161

Q₈⋊(C₂₃⋊C₈)

C₂₃×#559

#1630

Subgroups — #15672518

Conjugacy classes of subgroups

Complements

Maximal subgroups