Group #15683733 — subgroups

Subgroups — C₂×#13092783 #15683733

Order 2944.

488 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 149 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×#10625045	C₂	1	1	yes	maximal, characteristic
2	#13092783	C₂	1	8	yes	maximal, direct product factor
3	C₂×#10624991	C₂	1	4	yes	maximal, complemented
4	C₂×#10624985	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₉₂	—	2	2	no	—
7	C₂²×D₉₂	C₂²	1	2	yes	—
8	C₂₃×#125	—	2	2	no	—
9	C₂₃×#125	C₂²	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
11	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
14	#10625045	—	2	2	no	—
15	#10625045	C₂²	1	4	yes	—
16	#10624991	C₂²	1	16	yes	complemented
17	#10624985	C₂²	1	4	yes	complemented
18	C₂³×D₂₃	—	2	2	no	—
19	C₂²×C₄×C₂₃	—	2	2	no	—
20	C₂²×C₄×C₂₃	D₄	1	2	yes	—
21	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₂×D₉₂	—	2	8	no	—
23	C₂×D₉₂	—	2	2	no	—
24	C₂×D₉₂	—	2	8	no	—
25	C₂×D₉₂	—	4	8	no	—
26	C₂×D₉₂	C₂³	1	4	yes	—
27	C₄⋊C₄×C₂₃	—	2	8	no	—
28	C₄⋊C₄×C₂₃	—	2	2	no	—
29	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
30	C₄²×C₂₃	D₄	1	2	yes	—
31	C₄²×C₂₃	C₂³	1	2	yes	—
32	C₂×C₂₃⋊C₈	—	2	4	no	—
33	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
34	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×D₂₃	—	2	4	no	—
36	C₂²×D₂₃	—	4	4	no	—
37	C₂²×D₂₃	—	4	8	no	—
38	C₂×C₄×C₂₃	—	2	4	no	—
39	C₂×C₄×C₂₃	—	2	4	no	—
40	C₂×C₄×C₂₃	—	4	4	no	—
41	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
42	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₃	#41	1	4	yes	—
44	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
45	D₉₂	—	2	8	no	—
46	D₉₂	—	4	16	no	—
47	D₉₂	—	4	4	no	—
48	C₂₃⋊C₈	—	2	4	no	—
49	C₂×#490	—	23	1	no	maximal, Sylow
50	C₂²×C₂₃	#124	1	2	yes	—
51	C₂²×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
52	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
53	C₂×D₂₃	—	4	4	no	—
54	C₂×D₂₃	—	4	8	no	—
55	C₂×D₂₃	—	8	8	no	—
56	C₄×C₂₃	—	2	4	no	—
57	C₄×C₂₃	—	4	4	no	—
58	C₄×C₂₃	#124	1	2	yes	—
59	C₄×C₂₃	C₂×#41	1	2	yes	—
60	C₂×#127	—	23	1	no	—
61	C₂×#125	D₂₃	1	1	yes	characteristic, complemented
62	#490	—	23	8	no	—
63	C₂×#102	—	23	4	no	—
64	C₂×#97	—	23	1	no	—
65	C₂×C₂₃	#490	1	4	yes	complemented
66	C₂×C₂₃	C₂×#124	1	1	yes	characteristic
67	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
68	D₂₃	—	8	4	no	—
69	C₂²×D₄	—	23	2	no	—
70	C₂²×D₄	—	46	2	no	—
71	C₂²×C₈	—	23	2	no	—
72	#127	—	23	4	no	—
73	#127	—	46	2	no	—
74	#125	—	2	2	no	—
75	#125	C₂×D₂₃	1	4	yes	complemented
76	C₂×C₄⋊C₄	—	2	2	no	—
77	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
78	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
79	#102	—	23	16	no	—
80	#97	—	23	4	no	—
81	C₂₃	C₂×#490	1	1	yes	characteristic, complemented, Sylow
82	C₂⁴	—	46	2	no	—
83	C₂×D₄	—	23	4	no	—
84	C₂×D₄	—	46	8	no	—
85	C₂×D₄	—	46	8	no	—
86	C₂×D₄	—	46	2	no	—
87	C₂×D₄	—	92	8	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
90	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
91	C₂×C₈	—	23	4	no	—
92	C₂×C₈	—	46	4	no	—
93	C₄⋊C₄	—	2	8	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
96	C₄²	C₂²⋊D₂₃	1	2	yes	—
97	C₄²	C₂²×D₂₃	1	2	yes	—
98	C₂³	—	46	4	no	—
99	C₂³	—	92	4	no	—
100	C₂³	—	92	8	no	—
101	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
102	D₄	—	46	8	no	—
103	D₄	—	92	16	no	—
104	D₄	—	92	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	#67292	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₂₃	1	4	yes	—
110	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
111	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
112	C₈	—	46	4	no	—
113	C₂²	—	92	4	no	—
114	C₂²	—	92	8	no	—
115	C₂²	—	184	8	no	—
116	C₂²	#10625118	1	2	yes	—
117	C₂²	#10625120	1	4	yes	—
118	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	—	4	4	no	—
121	C₄	#10625118	1	2	yes	—
122	C₄	C₂×#67292	1	2	yes	—
123	C₂	—	184	4	no	—
124	C₂	#13092783	1	4	yes	direct product factor
125	C₂	C₂×#10625118	1	1	yes	characteristic
126	C₂	C₂×#10625120	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625045

C₂

yes

maximal, characteristic

#13092783

C₂

yes

maximal, direct product factor

C₂×#10624991

C₂

yes

maximal, complemented

C₂×#10624985

C₂

yes

maximal, characteristic, complemented

C₂×C₂₃×#125

C₂

yes

maximal, characteristic, complemented

C₂²×D₉₂

—

C₂²×D₉₂

C₂²

yes

—

C₂₃×#125

—

C₂₃×#125

C₂²

yes

complemented

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

—

#10625045

—

#10625045

C₂²

yes

—

#10624991

C₂²

yes

complemented

#10624985

C₂²

yes

complemented

C₂³×D₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

C₂³

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

C₂³

yes

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

C₂³

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

D₉₂

—

D₉₂

—

D₉₂

—

C₂₃⋊C₈

—

C₂×#490

—

maximal, Sylow

C₂²×C₂₃

#124

yes

—

C₂²×C₂₃

AGL(1,ℤ/8)

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#124

yes

—

C₄×C₂₃

C₂×#41

yes

—

C₂×#127

—

C₂×#125

D₂₃

yes

characteristic, complemented

#490

—

C₂×#102

—

C₂×#97

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

D₂₃

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×C₈

—

—

—

—

C₂×D₂₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#102

—

#97

—

C₂₃

C₂×#490

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₂₃

yes

—

C₄²

C₂²⋊D₂₃

yes

—

C₄²

C₂²×D₂₃

yes

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

102

D₄

—

103

D₄

—

104

D₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

#67292

yes

—

109

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

110

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

111

C₂×C₄

C₂³×D₂₃

yes

characteristic

112

C₈

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

184

—

116

C₂²

#10625118

yes

—

117

C₂²

#10625120

yes

—

118

C₂²

C₂²×C₂²⋊D₂₃

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

#10625118

yes

—

122

C₄

C₂×#67292

yes

—

123

C₂

—

184

—

124

C₂

#13092783

yes

direct product factor

125

C₂

C₂×#10625118

yes

characteristic

126

C₂

C₂×#10625120

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂×C₂₃×#125	C₂
socle	C₂³×C₂₃	C₂×D₄
prime radical	C₂×#10625045	C₂
hypercenter	C₂×#125	D₂₃
hypocenter	C₂₃	C₂×#490
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃

commutator, cosocle

C₂×C₄×C₂₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₃

Fitting

C₂×C₂₃×#125

C₂

socle

C₂³×C₂₃

C₂×D₄

C₂

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #13092783	C₂	direct (N × H)
3. C₂×#10624991	C₂	semidirect
4. C₂×#10624985	C₂	semidirect
5. C₂×C₂₃×#125	C₂	semidirect
9. C₂₃×#125	C₂²	semidirect
16. #10624991	C₂²	semidirect
17. #10624985	C₂²	semidirect
61. C₂×#125	D₂₃	semidirect
65. C₂×C₂₃	#490	semidirect
75. #125	C₂×D₂₃	semidirect
81. C₂₃	C₂×#490	semidirect
124. C₂	#13092783	direct (N × H)

H ≅ G/N

type

2. #13092783

C₂

direct (N × H)

3. C₂×#10624991

C₂

semidirect

4. C₂×#10624985

C₂

semidirect

5. C₂×C₂₃×#125

C₂

semidirect

9. C₂₃×#125

C₂²

semidirect

16. #10624991

C₂²

semidirect

17. #10624985

C₂²

semidirect

61. C₂×#125

D₂₃

semidirect

65. C₂×C₂₃

#490

semidirect

75. #125

C₂×D₂₃

semidirect

81. C₂₃

C₂×#490

semidirect

124. C₂

#13092783

direct (N × H)

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

subgroup	index
C₂×#10625045	2
#13092783	2
C₂×#10624991	2
C₂×#10624985	2
C₂×C₂₃×#125	2
C₂×#490	23

subgroup

index

C₂×C₂₃×#125

Subgroups — C₂×#13092783 #15683733

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#13092783 #15683733

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13092783 #15683733