Group #15673438 — subgroups

Subgroups — #15673438

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	C₂²×#67262	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625004	C₂	1	2	yes	maximal, complemented
3	#13092380	C₂	1	2	yes	maximal
4	#13092182	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#341	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×#67267	C₂²	1	2	yes	—
10	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×#67262	—	2	4	no	—
12	C₂×#67262	C₄	1	4	yes	—
13	#10625004	—	2	4	no	—
14	(C₂×D₂₃)⋊C₈	—	2	4	no	—
15	C₂³×D₂₃	Q₈	1	1	yes	characteristic
16	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
17	C₂²×Dic₂₃	Q₈	1	1	yes	characteristic
18	C₂×C₄×D₂₃	—	2	2	no	—
19	C₂×C₄×D₂₃	—	2	2	no	—
20	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
21	C₂×C₄×D₂₃	D₄	1	2	yes	—
22	M₁₆×C₂₃	—	2	2	no	—
23	M₁₆×C₂₃	—	2	2	no	—
24	C₂×C₈×C₂₃	—	2	2	no	—
25	C₂×C₈×C₂₃	—	2	2	no	—
26	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
27	#67267	—	2	2	no	—
28	#67267	—	2	2	no	—
29	C₂×C₂₃⋊C₈	—	2	2	no	—
30	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
31	C₂×C₂₃⋊C₈	D₄	1	4	yes	—
32	#67262	—	2	8	no	—
33	C₂³×C₂₃	C₂×Q₈	1	1	yes	characteristic
34	C₂²×D₂₃	—	2	2	no	—
35	C₂²×D₂₃	—	4	2	no	—
36	C₂²×D₂₃	C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₃	#41	1	2	yes	—
41	C₂×Dic₂₃	—	2	2	no	—
42	C₂×Dic₂₃	C₄⋊C₄	1	2	yes	—
43	C₄×D₂₃	—	2	2	no	—
44	C₄×D₂₃	—	2	2	no	—
45	C₄×D₂₃	—	2	4	no	—
46	C₈×C₂₃	—	2	2	no	—
47	C₈×C₂₃	—	4	2	no	—
48	C₂₃⋊C₈	—	4	2	no	—
49	C₂₃⋊C₈	C₂²⋊C₄	1	4	yes	—
50	#1811	—	23	1	no	maximal, Sylow
51	C₂²×C₂₃	—	2	2	no	—
52	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
54	C₂×D₂₃	—	2	2	no	—
55	C₂×D₂₃	—	2	4	no	—
56	C₂×D₂₃	—	4	4	no	—
57	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₂₃	#117	1	1	yes	characteristic
59	C₄×C₂₃	#121	1	2	yes	—
60	Dic₂₃	—	2	2	no	—
61	C₂²×M₁₆	—	23	1	no	—
62	C₂×#108	—	23	2	no	—
63	#407	—	23	2	no	—
64	#341	—	23	1	no	—
65	#341	D₂₃	1	1	yes	characteristic, complemented
66	C₂×C₂₃	—	2	2	no	—
67	C₂×C₂₃	#380	1	1	yes	characteristic
68	C₂×C₂₃	#430	1	2	yes	—
69	D₂₃	—	4	2	no	—
70	C₂³×C₄	—	23	1	no	—
71	C₂×M₁₆	—	23	4	no	—
72	C₂×M₁₆	—	23	2	no	—
73	C₂×M₁₆	—	46	4	no	—
74	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
75	C₂²×C₈	—	23	1	no	—
76	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
77	#108	—	46	4	no	—
78	C₂²⋊C₈	—	46	4	no	—
79	C₂₃	#1811	1	1	yes	characteristic, complemented, Sylow
80	C₂⁴	—	23	1	no	—
81	C₂²×C₄	—	23	1	no	—
82	C₂²×C₄	—	23	2	no	—
83	C₂²×C₄	—	23	2	no	—
84	C₂²×C₄	—	46	2	no	—
85	C₂²×C₄	—	46	2	no	—
86	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
87	M₁₆	—	2	2	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	46	8	no	—
90	M₁₆	—	46	2	no	—
91	M₁₆	—	46	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	23	2	no	—
95	C₂×C₈	—	23	4	no	—
96	C₂×C₈	—	46	2	no	—
97	C₂×C₈	C₄×D₂₃	1	2	yes	—
98	C₂³	—	23	2	no	—
99	C₂³	—	46	2	no	—
100	C₂³	—	92	2	no	—
101	C₂³	Q₈×D₂₃	1	1	yes	characteristic
102	C₂×C₄	—	23	2	no	—
103	C₂×C₄	—	46	2	no	—
104	C₂×C₄	—	46	2	no	—
105	C₂×C₄	—	46	2	no	—
106	C₂×C₄	—	46	4	no	—
107	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
108	C₂×C₄	#67288	1	2	yes	—
109	C₂×C₄	D₄×D₂₃	1	2	yes	—
110	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	4	2	no	—
113	C₈	—	23	4	no	—
114	C₈	—	92	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	—	46	4	no	—
117	C₂²	—	46	2	no	—
118	C₂²	—	92	4	no	—
119	C₂²	C₄⋊C₄×D₂₃	1	1	yes	characteristic
120	C₂²	#10625067	1	2	yes	—
121	C₄	—	46	2	no	—
122	C₄	#10625043	1	1	yes	characteristic
123	C₄	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
124	C₄	#10625055	1	2	yes	—
125	C₂	—	2	2	no	—
126	C₂	—	92	2	no	—
127	C₂	#13092276	1	1	yes	characteristic
128	C₂	#13092464	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#67262

C₂

yes

maximal, characteristic

C₂×#10625004

C₂

yes

maximal, complemented

#13092380

C₂

yes

maximal

#13092182

C₂

yes

maximal, characteristic, complemented

C₂₃×#341

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₂₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂×#67262

—

C₂×#67262

C₄

yes

—

#10625004

—

(C₂×D₂₃)⋊C₈

—

C₂³×D₂₃

Q₈

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

Q₈

yes

characteristic

C₂×C₄×D₂₃

—

C₂×C₄×D₂₃

—

C₂×C₄×D₂₃

C₂×C₄

yes

—

C₂×C₄×D₂₃

D₄

yes

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

D₄

yes

—

#67262

—

C₂³×C₂₃

C₂×Q₈

yes

characteristic

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

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

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₄⋊C₄

yes

—

C₄×D₂₃

—

C₄×D₂₃

—

C₄×D₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

C₂²⋊C₄

yes

—

#1811

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

C₂²⋊Q₈

yes

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

C₂×C₂²⋊C₄

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

Dic₂₃

—

C₂²×M₁₆

—

—

—

—

D₂₃

yes

characteristic, complemented

C₂×C₂₃

—

C₂×C₂₃

#380

yes

characteristic

C₂×C₂₃

#430

yes

—

D₂₃

—

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

#108

—

C₂²⋊C₈

—

C₂₃

#1811

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₂₃

yes

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

Q₈×D₂₃

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

108

C₂×C₄

#67288

yes

—

109

C₂×C₄

D₄×D₂₃

yes

—

110

C₂×C₄

D₄×D₂₃

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

C₄⋊C₄×D₂₃

yes

characteristic

120

C₂²

#10625067

yes

—

121

C₄

—

122

C₄

#10625043

yes

characteristic

123

C₄

C₂²⋊C₄×D₂₃

yes

characteristic

124

C₄

#10625055

yes

—

125

C₂

—

126

C₂

—

127

C₂

#13092276

yes

characteristic

128

C₂

#13092464

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#341	C₂
socle	C₂²×C₂₃	C₂×C₄⋊C₄
prime radical	C₂³×D₂₃	Q₈
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#341	D₂₃
hypocenter	C₂₃	#1811
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#341

C₂

socle

C₂²×C₂₃

C₂×C₄⋊C₄

prime radical

C₂³×D₂₃

Q₈

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
2. C₂×#10625004	C₂	semidirect
4. #13092182	C₂	semidirect
5. C₂₃×#341	C₂	semidirect
65. #341	D₂₃	semidirect
79. C₂₃	#1811	semidirect

H ≅ G/N

type

2. C₂×#10625004

C₂

semidirect

4. #13092182

C₂

semidirect

5. C₂₃×#341

C₂

semidirect

65. #341

D₂₃

semidirect

79. C₂₃

#1811

semidirect

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

subgroup	index
C₂²×#67262	2
C₂×#10625004	2
#13092380	2
#13092182	2
C₂₃×#341	2
#1811	23

subgroup

index

C₂²×#67262

C₂₃×#341

Subgroups — #15673438

Conjugacy classes of subgroups

Complements

Maximal subgroups