Group #15674187 — subgroups

Subgroups — #15674187

Order 2944.

184 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 89 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	#13092465	C₂	1	2	yes	maximal
2	C₂×#10625001	C₂	1	1	yes	maximal, characteristic
3	C₂×#10625001	C₂	1	1	yes	maximal, characteristic
4	#13092187	C₂	1	2	yes	maximal
5	C₂₃×#405	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#117	C₂²	1	1	yes	characteristic
9	C₂×#67267	C₂²	1	2	yes	—
10	C₂×#67267	C₂²	1	2	yes	—
11	C₂₃×#97	—	2	1	no	—
12	C₄×C₈×C₂₃	—	2	1	no	—
13	#10625004	—	2	4	no	—
14	#10625001	—	2	1	no	—
15	#10625001	—	2	1	no	—
16	#10625001	C₄	1	4	yes	complemented
17	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	M₁₆×C₂₃	—	2	1	no	—
19	M₁₆×C₂₃	D₄	1	2	yes	—
20	C₂×C₈×C₂₃	—	2	1	no	—
21	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
23	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
24	C₂×C₈×C₂₃	Q₈	1	2	yes	—
25	C₄⋊C₄×C₂₃	—	2	1	no	—
26	C₂²⋊C₄×C₂₃	Q₈	1	2	yes	—
27	C₄²×C₂₃	—	2	1	no	—
28	#67267	—	2	2	no	—
29	#67267	—	2	2	no	—
30	#67267	—	2	2	no	—
31	#67267	C₂×C₄	1	4	yes	—
32	C₂×C₂₃⋊C₈	—	2	2	no	—
33	C₂×C₂₃⋊C₈	—	2	2	no	—
34	C₂³×C₂₃	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₂₃	—	2	2	no	—
36	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
41	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
42	C₈×C₂₃	—	2	2	no	—
43	C₈×C₂₃	C₄⋊C₄	1	2	yes	—
44	C₈×C₂₃	C₄⋊C₄	1	2	yes	—
45	C₂₃⋊C₈	—	2	4	no	—
46	C₂₃⋊C₈	—	4	2	no	—
47	#1953	—	23	1	no	maximal, Sylow
48	C₂²×C₂₃	—	4	1	no	—
49	C₂²×C₂₃	C₄×Q₈	1	1	yes	characteristic
50	C₂²×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
51	C₂²×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
52	C₄×C₂₃	—	4	2	no	—
53	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
55	C₄×C₂₃	#121	1	1	yes	characteristic
56	C₄×C₂₃	#125	1	1	yes	characteristic
57	#430	—	23	2	no	—
58	C₂×#108	—	23	1	no	—
59	C₂×#108	—	23	1	no	—
60	#405	D₂₃	1	1	yes	characteristic
61	#344	—	23	2	no	—
62	C₂×C₂₃	—	2	1	no	—
63	C₂×C₂₃	—	2	1	no	—
64	C₂×C₂₃	—	2	1	no	—
65	C₂×C₂₃	#389	1	1	yes	characteristic
66	C₂×M₁₆	—	23	2	no	—
67	C₂×M₁₆	—	23	2	no	—
68	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
69	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
70	#117	C₂×D₂₃	1	1	yes	characteristic
71	#108	—	23	4	no	—
72	#108	—	46	4	no	—
73	#108	—	46	1	no	—
74	#108	—	46	1	no	—
75	#97	—	2	1	no	—
76	C₄×C₈	—	2	1	no	—
77	C₂₃	#1953	1	1	yes	characteristic, complemented, Sylow
78	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
79	M₁₆	—	2	1	no	—
80	M₁₆	—	23	4	no	—
81	M₁₆	—	46	2	no	—
82	M₁₆	—	46	2	no	—
83	M₁₆	—	46	2	no	—
84	M₁₆	D₉₂	1	2	yes	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	46	2	no	—
87	C₂×C₈	—	46	2	no	—
88	C₂×C₈	Dic₄₆	1	2	yes	—
89	C₂×C₈	C₄×D₂₃	1	2	yes	—
90	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
91	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
92	C₄⋊C₄	—	2	1	no	—
93	C₂²⋊C₄	Dic₄₆	1	2	yes	—
94	C₄²	—	2	1	no	—
95	C₂³	C₂×Dic₄₆	1	1	yes	characteristic
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
98	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
99	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
100	C₂×C₄	#67288	1	1	yes	characteristic
101	C₂×C₄	#67288	1	1	yes	characteristic
102	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
103	C₈	—	2	2	no	—
104	C₈	—	46	4	no	—
105	C₈	—	92	2	no	—
106	C₈	Dic₂₃⋊C₄	1	2	yes	—
107	C₈	Dic₂₃⋊C₄	1	2	yes	—
108	C₂²	—	4	1	no	—
109	C₂²	C₄×Dic₄₆	1	1	yes	characteristic
110	C₂²	#10625040	1	1	yes	characteristic
111	C₂²	#10625095	1	1	yes	characteristic
112	C₄	—	4	2	no	—
113	C₄	#10625041	1	1	yes	characteristic
114	C₄	C₄×D₉₂	1	1	yes	characteristic
115	C₄	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
116	C₄	#10625101	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	#13092315	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092465

C₂

yes

maximal

C₂×#10625001

C₂

yes

maximal, characteristic

C₂×#10625001

C₂

yes

maximal, characteristic

#13092187

C₂

yes

maximal

C₂₃×#405

C₂

yes

maximal, characteristic

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#117

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂×#67267

C₂²

yes

—

C₂₃×#97

—

C₄×C₈×C₂₃

—

—

—

—

C₄

yes

complemented

C₂²×C₄×C₂₃

C₂³

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

D₄

yes

characteristic

C₂×C₈×C₂₃

D₄

yes

characteristic

C₂×C₈×C₂₃

Q₈

yes

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

Q₈

yes

—

C₄²×C₂₃

—

—

—

—

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₈×C₂₃

—

C₈×C₂₃

C₄⋊C₄

yes

—

C₈×C₂₃

C₄⋊C₄

yes

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1953

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₄×Q₈

yes

characteristic

C₂²×C₂₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₃

C₄⋊Q₈

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃

C₄×D₄

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

—

—

D₂₃

yes

characteristic

#344

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#389

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

characteristic

C₂²×C₈

C₂×D₂₃

yes

characteristic

#117

C₂×D₂₃

yes

—

—

—

—

—

C₄×C₈

—

C₂₃

#1953

yes

characteristic, complemented, Sylow

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

D₉₂

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₄₆

yes

—

C₂×C₈

C₄×D₂₃

yes

—

C₂×C₈

C₂²⋊D₂₃

yes

characteristic

C₂×C₈

C₂²⋊D₂₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

Dic₄₆

yes

—

C₄²

—

C₂³

C₂×Dic₄₆

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×Dic₄₆

yes

characteristic

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

C₂×C₄

C₂×D₉₂

yes

characteristic

100

C₂×C₄

#67288

yes

characteristic

101

C₂×C₄

#67288

yes

characteristic

102

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

Dic₂₃⋊C₄

yes

—

107

C₈

Dic₂₃⋊C₄

yes

—

108

C₂²

—

109

C₂²

C₄×Dic₄₆

yes

characteristic

110

C₂²

#10625040

yes

characteristic

111

C₂²

#10625095

yes

characteristic

112

C₄

—

113

C₄

#10625041

yes

characteristic

114

C₄

C₄×D₉₂

yes

characteristic

115

C₄

C₂×Dic₂₃⋊C₄

yes

characteristic

116

C₄

#10625101

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

#13092315

yes

characteristic

	subgroup	G/H
center	C₄	#10625101
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#405	C₂
socle	C₂×C₂₃	#389
prime radical	C₂³×C₂₃	C₂×Q₈
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#405	D₂₃
hypocenter	C₂₃	#1953
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #405

subgroup

G/H

center

C₄

#10625101

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#405

C₂

socle

C₂×C₂₃

#389

prime radical

C₂³×C₂₃

C₂×Q₈

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #405

N	H ≅ G/N	type
16. #10625001	C₄	semidirect
77. C₂₃	#1953	semidirect

H ≅ G/N

type

16. #10625001

C₄

semidirect

77. C₂₃

#1953

semidirect

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

subgroup	index
#13092465	2
C₂×#10625001	2
#13092187	2
C₂₃×#405	2
#1953	23

subgroup

index

C₂₃×#405

Subgroups — #15674187

Conjugacy classes of subgroups

Complements

Maximal subgroups