Group #15673616 — subgroups

Subgroups — #15673616

Order 2944.

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

normal subgroups only

This group has 252 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	#13093184	C₂	1	1	yes	maximal, characteristic
2	#13092465	C₂	1	2	yes	maximal, complemented
3	#13092401	C₂	1	1	yes	maximal, characteristic
4	#13092182	C₂	1	2	yes	maximal, complemented
5	C₂₃×#408	C₂	1	1	yes	maximal, characteristic
6	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	#10625119	—	2	4	no	—
10	C₂×#67267	—	2	2	no	—
11	C₂×#67267	C₂²	1	2	yes	—
12	C₂×#67267	C₂²	1	2	yes	—
13	C₂₃×C₂²⋊C₈	—	2	2	no	—
14	#10625018	C₄	1	2	yes	—
15	#10625015	C₄	1	2	yes	—
16	#10625004	—	2	4	no	—
17	#41×C₂₃	—	2	2	no	—
18	#41×C₂₃	—	2	2	no	—
19	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×D₄×C₂₃	Q₈	1	2	yes	—
22	C₂²×C₄×C₂₃	D₄	1	2	yes	—
23	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	M₁₆×C₂₃	—	2	2	no	—
25	C₂×C₈×C₂₃	—	2	1	no	—
26	C₂×C₈×C₂₃	—	2	1	no	—
27	C₂×C₈×C₂₃	—	2	2	no	—
28	#67267	—	2	4	no	—
29	#67267	—	2	2	no	—
30	#67267	—	2	2	no	—
31	#67267	C₂×C₄	1	4	yes	—
32	#67267	D₄	1	4	yes	—
33	C₂×C₂₃⋊C₈	—	2	4	no	—
34	C₂×C₂₃⋊C₈	—	2	2	no	—
35	C₂×C₂₃⋊C₈	—	2	2	no	—
36	C₂³×C₂₃	C₄⋊C₄	1	2	yes	—
37	C₂³×C₂₃	C₂×Q₈	1	1	yes	characteristic
38	Q₈×C₂₃	—	2	2	no	—
39	D₄×C₂₃	—	2	4	no	—
40	D₄×C₂₃	—	2	2	no	—
41	C₂×C₄×C₂₃	—	2	2	no	—
42	C₂×C₄×C₂₃	—	2	2	no	—
43	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₃	#41	1	2	yes	—
48	C₈×C₂₃	—	4	1	no	—
49	C₈×C₂₃	—	4	1	no	—
50	C₂₃⋊C₈	—	2	4	no	—
51	C₂₃⋊C₈	—	2	4	no	—
52	#1858	—	23	1	no	maximal, Sylow
53	C₂²×C₂₃	—	2	2	no	—
54	C₂²×C₂₃	—	4	1	no	—
55	C₂²×C₂₃	—	4	2	no	—
56	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₂₃	—	2	2	no	—
59	C₄×C₂₃	C₄×D₄	1	2	yes	—
60	C₄×C₂₃	#120	1	1	yes	characteristic
61	C₄×C₂₃	#123	1	1	yes	characteristic
62	#568	—	23	1	no	—
63	#430	—	23	2	no	—
64	#413	—	23	1	no	—
65	#408	D₂₃	1	1	yes	characteristic
66	#341	—	23	2	no	—
67	C₂×C₂₃	—	2	1	no	—
68	C₂×C₂₃	—	2	2	no	—
69	C₂×C₂₃	—	4	2	no	—
70	C₂×C₂₃	#385	1	1	yes	characteristic
71	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
72	#131	—	46	4	no	—
73	C₂×M₁₆	—	23	2	no	—
74	C₂×M₁₆	—	23	2	no	—
75	C₂×M₁₆	—	46	2	no	—
76	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
77	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
78	#108	—	46	4	no	—
79	#101	—	23	2	no	—
80	#100	—	23	2	no	—
81	C₂²⋊C₈	—	2	2	no	—
82	C₂₃	#1858	1	1	yes	characteristic, complemented, Sylow
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	C₄×D₂₃	1	1	yes	characteristic
86	C₂×D₄	Dic₄₆	1	2	yes	—
87	C₂×D₄	C₄×D₂₃	1	1	yes	characteristic
88	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
89	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	23	4	no	—
92	M₁₆	—	23	4	no	—
93	M₁₆	—	46	4	no	—
94	M₁₆	—	46	2	no	—
95	M₁₆	—	46	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	46	4	no	—
100	C₂×C₈	—	46	2	no	—
101	C₂×C₈	—	46	2	no	—
102	C₂³	Dic₂₃⋊C₄	1	2	yes	—
103	C₂³	C₂×Dic₄₆	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
111	C₂×C₄	D₄×D₂₃	1	2	yes	—
112	C₂×C₄	#67290	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
114	C₈	—	4	1	no	—
115	C₈	—	4	1	no	—
116	C₈	—	46	4	no	—
117	C₈	—	46	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	2	no	—
121	C₂²	#10625049	1	2	yes	—
122	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	(C₂²⋊D₂₃)⋊C₄	1	2	yes	—
125	C₄	#10625106	1	1	yes	characteristic
126	C₄	#10625108	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	2	no	—
129	C₂	—	4	2	no	—
130	C₂	#13092289	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13093184

C₂

yes

maximal, characteristic

#13092465

C₂

yes

maximal, complemented

#13092401

C₂

yes

maximal, characteristic

#13092182

C₂

yes

maximal, complemented

C₂₃×#408

C₂

yes

maximal, characteristic

C₂×#41×C₂₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

—

—

C₂²

yes

—

C₂×#67267

C₂²

yes

—

C₂₃×C₂²⋊C₈

—

#10625018

C₄

yes

—

#10625015

C₄

yes

—

#10625004

—

#41×C₂₃

—

#41×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₄

yes

characteristic

C₂×D₄×C₂₃

C₂×C₄

yes

characteristic

C₂×D₄×C₂₃

Q₈

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

—

—

—

C₂×C₄

yes

—

#67267

D₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₄⋊C₄

yes

—

C₂³×C₂₃

C₂×Q₈

yes

characteristic

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄⋊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

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1858

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

C₂²⋊Q₈

yes

—

C₄×C₂₃

—

C₄×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

yes

C₄×C₂₃

yes

—

—

—

D₂₃

yes

characteristic

#341

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#385

yes

characteristic

C₂×#41

C₂×D₂₃

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

characteristic

C₂²×C₈

C₂×D₂₃

yes

—

—

—

C₂²⋊C₈

—

C₂₃

#1858

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

C₄×D₂₃

yes

characteristic

C₂×D₄

Dic₄₆

yes

—

C₂×D₄

C₄×D₂₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂³

Dic₂₃⋊C₄

yes

—

103

C₂³

C₂×Dic₄₆

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

Dic₂₃⋊C₄

yes

—

110

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

111

C₂×C₄

D₄×D₂₃

yes

—

112

C₂×C₄

#67290

yes

—

113

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

#10625049

yes

—

122

C₂²

C₂×Dic₂₃⋊C₄

yes

characteristic

123

C₄

—

124

C₄

(C₂²⋊D₂₃)⋊C₄

yes

—

125

C₄

#10625106

yes

characteristic

126

C₄

#10625108

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#13092289

yes

characteristic

	subgroup	G/H
center	C₄	#10625108
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#408	C₂
socle	C₂×C₂₃	#385
prime radical	C₂×#41×C₂₃	C₂²
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#408	D₂₃
hypocenter	C₂₃	#1858
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₄

#10625108

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#408

C₂

socle

C₂×C₂₃

#385

prime radical

C₂×#41×C₂₃

C₂²

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
2. #13092465	C₂	semidirect
4. #13092182	C₂	semidirect
82. C₂₃	#1858	semidirect

H ≅ G/N

type

2. #13092465

C₂

semidirect

4. #13092182

C₂

semidirect

82. C₂₃

#1858

semidirect

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

subgroup	index
#13093184	2
#13092465	2
#13092401	2
#13092182	2
C₂₃×#408	2
#1858	23

subgroup

index

C₂₃×#408

Subgroups — #15673616

Conjugacy classes of subgroups

Complements

Maximal subgroups