Group #15673522 — subgroups

Subgroups — #15673522

Order 2944.

172 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 81 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	#13092455	C₂	1	1	yes	maximal, characteristic
2	#13092372	C₂	1	1	yes	maximal, characteristic
3	#13092189	C₂	1	2	yes	maximal
4	#13092174	C₂	1	2	yes	maximal
5	C₂×C₂₃×#108	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	#10625097	C₂²	1	2	yes	—
9	#10625097	C₂²	1	1	yes	characteristic
10	C₂×#67267	C₂²	1	1	yes	characteristic
11	C₂₃×#108	—	2	1	no	—
12	C₂₃×#108	—	2	1	no	—
13	#10625000	—	2	1	no	—
14	#10624999	—	2	1	no	—
15	#10624997	C₄	1	2	yes	complemented
16	C₈×Dic₂₃	C₄	1	2	yes	complemented
17	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	M₁₆×C₂₃	—	2	2	no	—
19	M₁₆×C₂₃	—	2	2	no	—
20	C₂×C₈×C₂₃	—	2	2	no	—
21	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
24	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
25	C₂×C₈×C₂₃	Q₈	1	1	yes	characteristic
26	C₂×C₈×C₂₃	Q₈	1	1	yes	characteristic
27	C₂²⋊Dic₂₃	—	2	1	no	—
28	C₂²⋊Dic₂₃	—	2	2	no	—
29	C₄⋊Dic₂₃	—	2	1	no	—
30	C₄⋊Dic₂₃	—	2	2	no	—
31	C₄×Dic₂₃	—	2	2	no	—
32	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
33	#67267	—	2	1	no	—
34	#67267	—	2	1	no	—
35	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
36	C₂³×C₂₃	#41	1	1	yes	characteristic
37	C₂×C₄×C₂₃	C₂²×C₄	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	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
43	C₂×Dic₂₃	—	2	2	no	—
44	C₂×Dic₂₃	—	4	2	no	—
45	C₈×C₂₃	—	4	2	no	—
46	C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
47	C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
48	C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
49	C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂₃⋊C₈	—	2	2	no	—
51	#1833	—	23	1	no	maximal, Sylow
52	C₂²×C₂₃	—	4	1	no	—
53	C₂²×C₂₃	#117	1	1	yes	characteristic
54	C₂²×C₂₃	#124	1	1	yes	characteristic
55	C₂²×C₂₃	#125	1	1	yes	characteristic
56	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₂₃	#117	1	1	yes	characteristic
58	C₄×C₂₃	#124	1	1	yes	characteristic
59	C₄×C₂₃	#125	1	1	yes	characteristic
60	Dic₂₃	—	4	2	no	—
61	Dic₂₃	—	8	2	no	—
62	C₂×#108	D₂₃	1	1	yes	characteristic
63	#427	—	23	1	no	—
64	#405	—	23	1	no	—
65	#344	—	23	2	no	—
66	H(3,ℤ/4)	—	23	2	no	—
67	C₂×C₂₃	—	2	1	no	—
68	C₂×C₂₃	—	2	1	no	—
69	C₂×C₂₃	—	2	1	no	—
70	C₂×C₂₃	#382	1	1	yes	characteristic
71	C₂×M₁₆	—	23	1	no	—
72	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
73	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
74	#117	—	23	1	no	—
75	#117	—	23	2	no	—
76	#108	—	2	1	no	—
77	#108	—	2	1	no	—
78	#107	—	46	1	no	—
79	#106	—	46	1	no	—
80	#97	—	23	2	no	—
81	C₄×C₈	—	23	2	no	—
82	C₂₃	#1833	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	2	2	no	—
86	M₁₆	—	46	1	no	—
87	M₁₆	—	46	1	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	23	2	no	—
90	C₂×C₈	Dic₄₆	1	1	yes	characteristic
91	C₂×C₈	Dic₄₆	1	1	yes	characteristic
92	C₂×C₈	C₄×D₂₃	1	1	yes	characteristic
93	C₂×C₈	C₄×D₂₃	1	1	yes	characteristic
94	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
95	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
96	C₄⋊C₄	—	46	1	no	—
97	C₄⋊C₄	—	46	2	no	—
98	C₂²⋊C₄	—	46	1	no	—
99	C₂²⋊C₄	—	46	2	no	—
100	C₄²	—	23	2	no	—
101	C₄²	—	46	2	no	—
102	C₂³	#67292	1	1	yes	characteristic
103	C₂×C₄	—	46	2	no	—
104	C₂×C₄	—	92	2	no	—
105	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
106	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
107	C₂×C₄	#67290	1	1	yes	characteristic
108	C₂×C₄	#67290	1	1	yes	characteristic
109	C₂×C₄	#67292	1	1	yes	characteristic
110	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
111	C₈	—	4	2	no	—
112	C₈	—	46	2	no	—
113	C₈	Dic₂₃⋊C₄	1	1	yes	characteristic
114	C₈	Dic₂₃⋊C₄	1	1	yes	characteristic
115	C₈	Dic₂₃⋊C₄	1	1	yes	characteristic
116	C₈	Dic₂₃⋊C₄	1	1	yes	characteristic
117	C₂²	—	4	1	no	—
118	C₂²	#10625061	1	1	yes	characteristic
119	C₂²	#10625063	1	1	yes	characteristic
120	C₂²	#10625118	1	1	yes	characteristic
121	C₄	—	92	2	no	—
122	C₄	—	184	2	no	—
123	C₄	#10625061	1	1	yes	characteristic
124	C₄	#10625063	1	1	yes	characteristic
125	C₄	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
126	C₄	#10625107	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#13092282	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092455

C₂

yes

maximal, characteristic

#13092372

C₂

yes

maximal, characteristic

#13092189

C₂

yes

maximal

#13092174

C₂

yes

maximal

C₂×C₂₃×#108

C₂

yes

maximal, characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂²×C₈×C₂₃

C₂²

yes

characteristic

#10625097

C₂²

yes

—

#10625097

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

characteristic

C₂₃×#108

—

C₂₃×#108

—

#10625000

—

#10624999

—

#10624997

C₄

yes

complemented

C₈×Dic₂₃

C₄

yes

complemented

C₂²×C₄×C₂₃

C₂³

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

D₄

yes

characteristic

C₂×C₈×C₂₃

D₄

yes

characteristic

C₂×C₈×C₂₃

Q₈

yes

characteristic

C₂×C₈×C₂₃

Q₈

yes

characteristic

C₂²⋊Dic₂₃

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

C₂×C₄

yes

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂³×C₂₃

#41

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₈×C₂₃

—

C₈×C₂₃

C₄⋊C₄

yes

characteristic

C₈×C₂₃

C₄⋊C₄

yes

characteristic

C₈×C₂₃

C₄⋊C₄

yes

characteristic

C₈×C₂₃

C₄⋊C₄

yes

characteristic

C₂₃⋊C₈

—

#1833

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

C₂×C₄⋊C₄

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

Dic₂₃

—

C₂×#108

D₂₃

yes

—

—

—

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#382

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

—

C₂²×C₈

C₂×D₂₃

yes

—

—

—

—

—

—

—

C₄×C₈

—

C₂₃

#1833

yes

characteristic, complemented, Sylow

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₄₆

yes

characteristic

C₂×C₈

Dic₄₆

yes

characteristic

C₂×C₈

C₄×D₂₃

yes

characteristic

C₂×C₈

C₄×D₂₃

yes

characteristic

C₂×C₈

C₂²⋊D₂₃

yes

characteristic

C₂×C₈

C₂²⋊D₂₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₂³

#67292

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂×Dic₄₆

yes

characteristic

106

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

107

C₂×C₄

#67290

yes

characteristic

108

C₂×C₄

#67290

yes

characteristic

109

C₂×C₄

#67292

yes

characteristic

110

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

Dic₂₃⋊C₄

yes

characteristic

114

C₈

Dic₂₃⋊C₄

yes

characteristic

115

C₈

Dic₂₃⋊C₄

yes

characteristic

116

C₈

Dic₂₃⋊C₄

yes

characteristic

117

C₂²

—

118

C₂²

#10625061

yes

characteristic

119

C₂²

#10625063

yes

characteristic

120

C₂²

#10625118

yes

characteristic

121

C₄

—

122

C₄

—

184

—

123

C₄

#10625061

yes

characteristic

124

C₄

#10625063

yes

characteristic

125

C₄

C₂×Dic₂₃⋊C₄

yes

characteristic

126

C₄

#10625107

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#13092282

yes

characteristic

	subgroup	G/H
center	C₄	#10625063
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×#108	C₂
socle	C₂×C₂₃	#382
prime radical	C₂³×C₂₃	#41
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×#108	D₂₃
hypocenter	C₂₃	#1833
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#10625063

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₂₃×#108

C₂

C₂×C₂₃

C₂³×C₂₃

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
15. #10624997	C₄	semidirect
16. C₈×Dic₂₃	C₄	semidirect
82. C₂₃	#1833	semidirect

H ≅ G/N

type

15. #10624997

C₄

semidirect

16. C₈×Dic₂₃

C₄

semidirect

82. C₂₃

#1833

semidirect

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

subgroup	index
#13092455	2
#13092372	2
#13092189	2
#13092174	2
C₂×C₂₃×#108	2
#1833	23

subgroup

index

C₂×C₂₃×#108

Subgroups — #15673522

Conjugacy classes of subgroups

Complements

Maximal subgroups