Group #13125172 — subgroups

Subgroups — C₂×#11718728 #13125172

Order 1664.

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×C₁₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88128	C₂	1	1	yes	maximal, characteristic
3	#11718728	C₂	1	8	yes	maximal, direct product factor
4	C₂×#88049	C₂	1	4	yes	maximal, complemented
5	C₂×#88043	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₅₂	—	2	2	no	—
7	C₂²×D₅₂	C₂²	1	2	yes	—
8	C₁₃×#125	—	2	2	no	—
9	C₁₃×#125	C₂²	1	4	yes	complemented
10	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
11	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	C₂²×#1055	C₂²	1	2	yes	—
14	#88128	—	2	2	no	—
15	#88128	C₂²	1	4	yes	—
16	#88049	C₂²	1	16	yes	complemented
17	#88043	C₂²	1	4	yes	complemented
18	C₂³×D₁₃	—	2	2	no	—
19	C₂²×C₄×C₁₃	—	2	2	no	—
20	C₂²×C₄×C₁₃	D₄	1	2	yes	—
21	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×D₅₂	—	2	8	no	—
23	C₂×D₅₂	—	2	2	no	—
24	C₂×D₅₂	—	2	8	no	—
25	C₂×D₅₂	—	4	8	no	—
26	C₂×D₅₂	C₂³	1	4	yes	—
27	C₁₃×C₄⋊C₄	—	2	8	no	—
28	C₁₃×C₄⋊C₄	—	2	2	no	—
29	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
30	C₄²×C₁₃	D₄	1	2	yes	—
31	C₄²×C₁₃	C₂³	1	2	yes	—
32	C₂×#1055	—	2	4	no	—
33	C₂×#1055	C₂³	1	4	yes	—
34	C₂×#490	—	13	1	no	maximal, Sylow
35	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×D₁₃	—	2	4	no	—
37	C₂²×D₁₃	—	4	4	no	—
38	C₂²×D₁₃	—	4	8	no	—
39	C₂×C₄×C₁₃	—	2	4	no	—
40	C₂×C₄×C₁₃	—	2	4	no	—
41	C₂×C₄×C₁₃	—	4	4	no	—
42	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₃	#41	1	4	yes	—
45	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
46	D₅₂	—	2	8	no	—
47	D₅₂	—	4	16	no	—
48	D₅₂	—	4	4	no	—
49	#1055	—	2	4	no	—
50	C₂×#127	—	13	1	no	—
51	C₂×#125	D₁₃	1	1	yes	characteristic, complemented
52	#490	—	13	8	no	—
53	C₂×#102	—	13	4	no	—
54	C₂×#97	—	13	1	no	—
55	C₂²×C₁₃	#124	1	2	yes	—
56	C₂²×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
57	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
58	C₂×D₁₃	—	4	4	no	—
59	C₂×D₁₃	—	4	8	no	—
60	C₂×D₁₃	—	8	8	no	—
61	C₄×C₁₃	—	2	4	no	—
62	C₄×C₁₃	—	4	4	no	—
63	C₄×C₁₃	#124	1	2	yes	—
64	C₄×C₁₃	C₂×#41	1	2	yes	—
65	C₂²×D₄	—	13	2	no	—
66	C₂²×D₄	—	26	2	no	—
67	C₂²×C₈	—	13	2	no	—
68	#127	—	13	4	no	—
69	#127	—	26	2	no	—
70	#125	—	2	2	no	—
71	#125	C₂×D₁₃	1	4	yes	complemented
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
74	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
75	#102	—	13	16	no	—
76	#97	—	13	4	no	—
77	C₂×C₁₃	#490	1	4	yes	complemented
78	C₂×C₁₃	C₂×#124	1	1	yes	characteristic
79	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
80	D₁₃	—	8	4	no	—
81	C₂⁴	—	26	2	no	—
82	C₂×D₄	—	13	4	no	—
83	C₂×D₄	—	26	8	no	—
84	C₂×D₄	—	26	8	no	—
85	C₂×D₄	—	26	2	no	—
86	C₂×D₄	—	52	8	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
90	C₂×C₈	—	13	4	no	—
91	C₂×C₈	—	26	4	no	—
92	C₄⋊C₄	—	2	8	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
95	C₄²	C₂²⋊D₁₃	1	2	yes	—
96	C₄²	C₂²×D₁₃	1	2	yes	—
97	C₁₃	C₂×#490	1	1	yes	characteristic, complemented, Sylow
98	C₂³	—	26	4	no	—
99	C₂³	—	52	4	no	—
100	C₂³	—	52	8	no	—
101	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
102	D₄	—	26	8	no	—
103	D₄	—	52	16	no	—
104	D₄	—	52	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	#6131	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
110	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
111	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
112	C₈	—	26	4	no	—
113	C₂²	—	52	4	no	—
114	C₂²	—	52	8	no	—
115	C₂²	—	104	8	no	—
116	C₂²	#88201	1	2	yes	—
117	C₂²	#88203	1	4	yes	—
118	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	—	4	4	no	—
121	C₄	#88201	1	2	yes	—
122	C₄	C₂×#6131	1	2	yes	—
123	C₂	—	104	4	no	—
124	C₂	#11718728	1	4	yes	direct product factor
125	C₂	C₂×#88201	1	1	yes	characteristic
126	C₂	C₂×#88203	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#125

C₂

yes

maximal, characteristic, complemented

C₂×#88128

C₂

yes

maximal, characteristic

#11718728

C₂

yes

maximal, direct product factor

C₂×#88049

C₂

yes

maximal, complemented

C₂×#88043

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₂

—

C₂²×D₅₂

C₂²

yes

—

C₁₃×#125

—

C₁₃×#125

C₂²

yes

complemented

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂²×#1055

C₂²

yes

—

#88128

—

#88128

C₂²

yes

—

#88049

C₂²

yes

complemented

#88043

C₂²

yes

complemented

C₂³×D₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

C₂³

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

C₂³

yes

—

C₄²×C₁₃

D₄

yes

—

C₄²×C₁₃

C₂³

yes

—

C₂×#1055

—

C₂×#1055

C₂³

yes

—

C₂×#490

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

D₅₂

—

D₅₂

—

D₅₂

—

#1055

—

C₂×#127

—

C₂×#125

D₁₃

yes

characteristic, complemented

#490

—

C₂×#102

—

C₂×#97

—

C₂²×C₁₃

#124

yes

—

C₂²×C₁₃

AGL(1,ℤ/8)

yes

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#124

yes

—

C₄×C₁₃

C₂×#41

yes

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×C₈

—

—

—

—

C₂×D₁₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#102

—

#97

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

D₁₃

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₁₃

yes

—

C₄²

C₂²⋊D₁₃

yes

—

C₄²

C₂²×D₁₃

yes

—

C₁₃

C₂×#490

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

102

D₄

—

103

D₄

—

104

D₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

#6131

yes

—

109

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

110

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

111

C₂×C₄

C₂³×D₁₃

yes

characteristic

112

C₈

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

104

—

116

C₂²

#88201

yes

—

117

C₂²

#88203

yes

—

118

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

#88201

yes

—

122

C₄

C₂×#6131

yes

—

123

C₂

—

104

—

124

C₂

#11718728

yes

direct product factor

125

C₂

C₂×#88201

yes

characteristic

126

C₂

C₂×#88203

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#125	C₂
socle	C₂³×C₁₃	C₂×D₄
prime radical	C₂×#88128	C₂
hypercenter	C₂×#125	D₁₃
hypocenter	C₁₃	C₂×#490
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃

commutator, cosocle

C₂×C₄×C₁₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃

Fitting

C₂×C₁₃×#125

C₂

socle

C₂³×C₁₃

C₂×D₄

C₂

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
1. C₂×C₁₃×#125	C₂	semidirect
3. #11718728	C₂	direct (N × H)
4. C₂×#88049	C₂	semidirect
5. C₂×#88043	C₂	semidirect
9. C₁₃×#125	C₂²	semidirect
16. #88049	C₂²	semidirect
17. #88043	C₂²	semidirect
51. C₂×#125	D₁₃	semidirect
71. #125	C₂×D₁₃	semidirect
77. C₂×C₁₃	#490	semidirect
97. C₁₃	C₂×#490	semidirect
124. C₂	#11718728	direct (N × H)

H ≅ G/N

type

1. C₂×C₁₃×#125

C₂

semidirect

3. #11718728

C₂

direct (N × H)

4. C₂×#88049

C₂

semidirect

5. C₂×#88043

C₂

semidirect

9. C₁₃×#125

C₂²

semidirect

16. #88049

C₂²

semidirect

17. #88043

C₂²

semidirect

51. C₂×#125

D₁₃

semidirect

71. #125

C₂×D₁₃

semidirect

77. C₂×C₁₃

#490

semidirect

97. C₁₃

C₂×#490

semidirect

124. C₂

#11718728

direct (N × H)

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

subgroup	index
C₂×C₁₃×#125	2
C₂×#88128	2
#11718728	2
C₂×#88049	2
C₂×#88043	2
C₂×#490	13

subgroup

index

C₂×C₁₃×#125

Subgroups — C₂×#11718728 #13125172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#11718728 #13125172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11718728 #13125172