Group #15673478 — subgroups

Subgroups — #15673478

Order 2944.

272 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 85 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	1	yes	maximal, characteristic
2	#13092381	C₂	1	1	yes	maximal, characteristic
3	#13092185	C₂	1	4	yes	maximal, complemented
4	C₂₃×#407	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂×C₂²⋊Dic₂₃	C₄	1	2	yes	—
8	#10625097	—	2	2	no	—
9	#10625097	—	2	2	no	—
10	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
11	C₂×C₄×Dic₂₃	C₄	1	2	yes	—
12	C₂×#67267	C₂²	1	2	yes	—
13	C₂₃×C₂²⋊C₈	—	2	2	no	—
14	C₂²⋊(C₂₃⋊C₈)	—	2	2	no	—
15	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₂₃	—	2	4	no	—
17	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₃	D₄	1	2	yes	—
19	C₂²×C₄×C₂₃	D₄	1	2	yes	—
20	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
21	C₂²×Dic₂₃	C₂×C₄	1	4	yes	—
22	M₁₆×C₂₃	—	2	4	no	—
23	C₂×C₈×C₂₃	—	2	2	no	—
24	C₂²⋊Dic₂₃	—	2	4	no	—
25	C₄⋊Dic₂₃	—	2	2	no	—
26	C₄⋊Dic₂₃	—	2	2	no	—
27	C₄×Dic₂₃	—	2	4	no	—
28	#67267	—	2	4	no	—
29	C₂×C₂₃⋊C₈	—	2	2	no	—
30	C₂³×C₂₃	—	2	2	no	—
31	C₂³×C₂₃	—	4	2	no	—
32	C₂³×C₂₃	C₂²⋊C₄	1	2	yes	—
33	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₃	—	2	2	no	—
35	C₂×C₄×C₂₃	—	2	4	no	—
36	C₂×C₄×C₂₃	—	2	4	no	—
37	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₃	#41	1	4	yes	—
41	C₂×Dic₂₃	—	2	4	no	—
42	C₂×Dic₂₃	—	4	4	no	—
43	C₈×C₂₃	—	4	2	no	—
44	C₂₃⋊C₈	—	4	2	no	—
45	#1822	—	23	1	no	maximal, Sylow
46	C₂²×C₂₃	—	2	2	no	—
47	C₂²×C₂₃	—	2	2	no	—
48	C₂²×C₂₃	—	2	4	no	—
49	C₂²×C₂₃	—	4	4	no	—
50	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₂₃	#117	1	2	yes	—
52	C₄×C₂₃	—	2	2	no	—
53	C₄×C₂₃	#123	1	2	yes	—
54	C₄×C₂₃	#123	1	2	yes	—
55	Dic₂₃	—	4	4	no	—
56	C₂×#117	—	23	1	no	—
57	#407	—	23	1	no	—
58	#407	D₂₃	1	1	yes	characteristic
59	#343	—	23	4	no	—
60	C₂×C₂₃	—	2	2	no	—
61	C₂×C₂₃	—	4	2	no	—
62	C₂×C₂₃	#381	1	1	yes	characteristic
63	C₂×C₂₃	#413	1	2	yes	—
64	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
65	C₂×M₁₆	—	23	2	no	—
66	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
67	#117	—	46	2	no	—
68	#117	—	46	2	no	—
69	C₂×C₄⋊C₄	—	23	2	no	—
70	C₂×C₂²⋊C₄	—	23	2	no	—
71	C₂×C₄²	—	23	2	no	—
72	C₂²⋊C₈	—	2	2	no	—
73	C₂²⋊C₈	—	46	2	no	—
74	C₂₃	#1822	1	1	yes	characteristic, complemented, Sylow
75	C₂⁴	C₄×D₂₃	1	1	yes	characteristic
76	C₂²×C₄	—	2	4	no	—
77	C₂²×C₄	—	23	4	no	—
78	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
79	C₂²×C₄	D₉₂	1	2	yes	—
80	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
81	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
82	M₁₆	—	2	4	no	—
83	M₁₆	—	46	4	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	46	2	no	—
86	C₄⋊C₄	—	46	2	no	—
87	C₄⋊C₄	—	46	2	no	—
88	C₂²⋊C₄	—	46	4	no	—
89	C₄²	—	46	4	no	—
90	C₂³	—	2	2	no	—
91	C₂³	—	4	2	no	—
92	C₂³	(C₂×D₂₃)⋊C₄	1	2	yes	—
93	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	46	4	no	—
98	C₂×C₄	—	92	4	no	—
99	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
100	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
101	C₂×C₄	#67290	1	4	yes	—
102	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
103	C₈	—	4	2	no	—
104	C₈	—	92	2	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	2	4	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	4	4	no	—
109	C₂²	#10625048	1	2	yes	—
110	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
111	C₄	—	2	2	no	—
112	C₄	—	92	4	no	—
113	C₄	#10625057	1	2	yes	—
114	C₄	#10625106	1	2	yes	—
115	C₂	—	2	2	no	—
116	C₂	—	4	2	no	—
117	C₂	#13092278	1	1	yes	characteristic
118	C₂	#13092399	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625097

C₂

yes

maximal, characteristic

#13092381

C₂

yes

maximal, characteristic

#13092185

C₂

yes

maximal, complemented

C₂₃×#407

C₂

yes

maximal, characteristic

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂×C₂²⋊Dic₂₃

C₄

yes

—

#10625097

—

#10625097

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×C₄×Dic₂₃

C₄

yes

—

C₂×#67267

C₂²

yes

—

C₂₃×C₂²⋊C₈

—

C₂²⋊(C₂₃⋊C₈)

—

C₂⁴×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²⋊C₄

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1822

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

#117

yes

—

C₄×C₂₃

—

C₄×C₂₃

#123

yes

—

C₄×C₂₃

#123

yes

—

Dic₂₃

—

C₂×#117

—

#407

—

#407

D₂₃

yes

characteristic

#343

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#381

yes

characteristic

C₂×C₂₃

#413

yes

—

C₂³×C₄

C₂×D₂₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂₃

#1822

yes

characteristic, complemented, Sylow

C₂⁴

C₄×D₂₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₂₃

yes

characteristic

C₂²×C₄

D₉₂

yes

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

(C₂×D₂₃)⋊C₄

yes

—

C₂³

C₂×C₄×D₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

100

C₂×C₄

C₂×D₉₂

yes

characteristic

101

C₂×C₄

#67290

yes

—

102

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

103

C₈

—

104

C₈

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

#10625048

yes

—

110

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

#10625057

yes

—

114

C₄

#10625106

yes

—

115

C₂

—

116

C₂

—

117

C₂

#13092278

yes

characteristic

118

C₂

#13092399

yes

—

	subgroup	G/H
center	C₂²	C₂×(C₂×D₂₃)⋊C₄
commutator	C₂³×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#407	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₂₃	C₂×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#407	D₂₃
hypocenter	C₂₃	#1822
LCS	G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₂×(C₂×D₂₃)⋊C₄

commutator

C₂³×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#407

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₂₃

C₂×C₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
3. #13092185	C₂	semidirect
74. C₂₃	#1822	semidirect

H ≅ G/N

type

3. #13092185

C₂

semidirect

74. C₂₃

#1822

semidirect

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

subgroup	index
C₂×#10625097	2
#13092381	2
#13092185	2
C₂₃×#407	2
#1822	23

subgroup

index

C₂₃×#407

Subgroups — #15673478

Conjugacy classes of subgroups

Complements

Maximal subgroups