Group #15672516 — subgroups

Subgroups — #15672516

Order 2944.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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	#13093153	C₂	1	1	yes	maximal, characteristic
2	#13092709	C₂	1	2	yes	maximal, complemented
3	#13092161	C₂	1	1	yes	maximal, characteristic
4	#13092149	C₂	1	2	yes	maximal, complemented
5	C₂₃×#492	C₂	1	1	yes	maximal, characteristic
6	C₂×SD₁₆×C₂₃	—	2	2	no	—
7	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₂₃×#127	C₂²	1	1	yes	characteristic
9	#10625107	—	2	2	no	—
10	#10625106	—	2	2	no	—
11	D₄×Dic₂₃	—	2	2	no	—
12	C₄×Dic₄₆	C₂²	1	2	yes	—
13	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	#10624992	—	2	2	no	—
15	#10624989	—	2	2	no	—
16	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
17	C₂×Q₈×C₂₃	—	2	1	no	—
18	C₂×D₄×C₂₃	—	2	2	no	—
19	C₂×D₄×C₂₃	D₄	1	2	yes	—
20	C₂²×Dic₂₃	—	2	2	no	—
21	C₂×Dic₄₆	D₄	1	2	yes	—
22	SD₁₆×C₂₃	—	4	4	no	—
23	C₂×C₈×C₂₃	—	2	1	no	—
24	C₄⋊C₄×C₂₃	—	2	2	no	—
25	C₄²×C₂₃	C₂³	1	1	yes	characteristic
26	C₂²⋊Dic₂₃	—	2	4	no	—
27	C₂²⋊Dic₂₃	—	2	2	no	—
28	C₄⋊Dic₂₃	D₄	1	2	yes	—
29	Dic₂₃⋊C₄	—	2	2	no	—
30	C₄×Dic₂₃	—	2	2	no	—
31	C₂×C₂₃⋊C₈	—	2	2	no	—
32	C₂³×C₂₃	—	2	2	no	—
33	Q₈×C₂₃	—	4	2	no	—
34	D₄×C₂₃	—	4	2	no	—
35	D₄×C₂₃	—	4	2	no	—
36	D₄×C₂₃	—	4	2	no	—
37	C₂×C₄×C₂₃	—	4	1	no	—
38	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₂₃	—	2	2	no	—
41	C₂×Dic₂₃	—	2	2	no	—
42	C₂×Dic₂₃	—	8	2	no	—
43	Dic₄₆	—	2	2	no	—
44	Dic₄₆	—	2	2	no	—
45	C₈×C₂₃	—	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	—	8	1	no	—
53	C₄×C₂₃	#135	1	2	yes	—
54	C₄×C₂₃	#137	1	2	yes	—
55	Dic₂₃	—	4	2	no	—
56	Dic₂₃	—	4	2	no	—
57	#559	—	23	1	no	—
58	#492	D₂₃	1	1	yes	characteristic
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	—	2	2	no	—
66	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
67	#127	C₂×D₂₃	1	1	yes	characteristic
68	#124	—	46	2	no	—
69	#123	—	46	2	no	—
70	C₄×Q₈	—	23	2	no	—
71	C₄×D₄	—	46	2	no	—
72	#107	—	46	2	no	—
73	C₄⋊C₈	—	23	2	no	—
74	#103	—	46	2	no	—
75	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
76	C₂₃	#1630	1	1	yes	characteristic, complemented, Sylow
77	C₂×Q₈	—	2	1	no	—
78	C₂×Q₈	—	23	2	no	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	C₂²⋊D₂₃	1	2	yes	—
81	C₂²×C₄	—	46	2	no	—
82	SD₁₆	—	4	4	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	46	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	23	2	no	—
87	C₄⋊C₄	—	46	2	no	—
88	C₂²⋊C₄	—	46	4	no	—
89	C₂²⋊C₄	—	46	2	no	—
90	C₄²	—	46	2	no	—
91	C₄²	C₂²×D₂₃	1	1	yes	characteristic
92	C₂³	—	2	2	no	—
93	Q₈	—	4	2	no	—
94	Q₈	—	46	2	no	—
95	Q₈	—	46	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	C₂×C₄	—	4	1	no	—
100	C₂×C₄	—	46	2	no	—
101	C₂×C₄	—	46	2	no	—
102	C₂×C₄	—	184	2	no	—
103	C₂×C₄	D₄×D₂₃	1	2	yes	—
104	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	—	92	2	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	4	2	no	—
109	C₂²	#10625108	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	8	1	no	—
112	C₄	—	92	2	no	—
113	C₄	—	92	2	no	—
114	C₄	#10625086	1	2	yes	—
115	C₄	#10625087	1	2	yes	—
116	C₂	—	8	2	no	—
117	C₂	#13092576	1	2	yes	—
118	C₂	#13092584	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13093153

C₂

yes

maximal, characteristic

#13092709

C₂

yes

maximal, complemented

#13092161

C₂

yes

maximal, characteristic

#13092149

C₂

yes

maximal, complemented

C₂₃×#492

C₂

yes

maximal, characteristic

C₂×SD₁₆×C₂₃

—

C₂₃×C₄⋊Q₈

C₂²

yes

characteristic

C₂₃×#127

C₂²

yes

characteristic

#10625107

—

#10625106

—

D₄×Dic₂₃

—

C₄×Dic₄₆

C₂²

yes

—

C₄×C₈×C₂₃

C₂²

yes

characteristic

#10624992

—

#10624989

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

—

C₂×Q₈×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

D₄

yes

—

C₂²×Dic₂₃

—

C₂×Dic₄₆

D₄

yes

—

SD₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

C₂²⋊Dic₂₃

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

D₄

yes

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1630

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#120

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#135

yes

—

C₄×C₂₃

#137

yes

—

Dic₂₃

—

Dic₂₃

—

#559

—

#492

D₂₃

yes

characteristic

#477

—

#331

—

Q₈⋊C₈

—

C₂×C₂₃

—

C₂×C₂₃

#452

yes

—

C₂×C₂₃

#453

yes

characteristic

C₂×SD₁₆

—

C₄⋊Q₈

C₂×D₂₃

yes

characteristic

#127

C₂×D₂₃

yes

characteristic

#124

—

#123

—

C₄×Q₈

—

C₄×D₄

—

#107

—

C₄⋊C₈

—

#103

—

C₄×C₈

C₂×D₂₃

yes

characteristic

C₂₃

#1630

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

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

—

184

—

103

C₂×C₄

D₄×D₂₃

yes

—

104

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

105

C₈

—

106

C₈

—

107

C₂²

—

108

C₂²

—

109

C₂²

#10625108

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

#10625086

yes

—

115

C₄

#10625087

yes

—

116

C₂

—

117

C₂

#13092576

yes

—

118

C₂

#13092584

yes

characteristic

	subgroup	G/H
center	C₂²	#10625108
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#492	C₂
socle	C₂²×C₂₃	#120
prime radical	C₂₃×#127	C₂²
hypercenter	#492	D₂₃
hypocenter	C₂₃	#1630
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#10625108

commutator, cosocle

C₄²×C₂₃

C₂³

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#492

C₂

socle

C₂²×C₂₃

#120

prime radical

C₂₃×#127

C₂²

D₂₃

C₂₃

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
2. #13092709	C₂	semidirect
4. #13092149	C₂	semidirect
76. C₂₃	#1630	semidirect

H ≅ G/N

type

2. #13092709

C₂

semidirect

4. #13092149

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
#13093153	2
#13092709	2
#13092161	2
#13092149	2
C₂₃×#492	2
#1630	23

subgroup

index

C₂₃×#492

Subgroups — #15672516

Conjugacy classes of subgroups

Complements

Maximal subgroups