Group #13123209 — subgroups

Subgroups — #13123209

Order 1664.

524 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 157 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×#88129	C₂	1	1	yes	maximal, characteristic
3	#11718651	C₂	1	8	yes	maximal, complemented
4	#11718649	C₂	1	2	yes	maximal
5	#11718635	C₂	1	2	yes	maximal, complemented
6	C₄²×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₅₂	C₂²	1	1	yes	characteristic
8	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
9	C₁₃×#125	—	2	4	no	—
10	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#6103	C₂²	1	4	yes	—
14	C₂×#6103	C₂²	1	8	yes	—
15	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
16	C₂×C₄×Dic₁₃	C₂²	1	4	yes	—
17	#88129	—	2	4	no	—
18	#88071	C₂²	1	8	yes	—
19	C₂³×D₁₃	C₂³	1	2	yes	—
20	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
21	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
22	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₁₃	C₂³	1	2	yes	—
24	C₂²×Dic₁₃	C₂³	1	4	yes	—
25	C₂×D₅₂	—	2	2	no	—
26	C₂×D₅₂	—	2	4	no	—
27	C₂×Dic₂₆	—	2	2	no	—
28	C₂×Dic₂₆	—	2	4	no	—
29	C₁₃×C₄⋊C₄	—	2	8	no	—
30	C₁₃×C₄⋊C₄	—	2	4	no	—
31	C₄²×C₁₃	D₄	1	4	yes	—
32	#6103	—	2	8	no	—
33	#6103	—	2	16	no	—
34	C₄×Dic₁₃	—	2	4	no	—
35	C₄×Dic₁₃	—	2	8	no	—
36	#2861	—	13	1	no	maximal, Sylow
37	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
38	C₂²×D₁₃	—	2	2	no	—
39	C₂²×D₁₃	—	2	8	no	—
40	C₂²×D₁₃	—	2	4	no	—
41	C₂×C₄×C₁₃	—	2	4	no	—
42	C₂×C₄×C₁₃	—	2	4	no	—
43	C₂×C₄×C₁₃	—	2	8	no	—
44	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
45	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₃	#41	1	4	yes	—
47	C₂×Dic₁₃	—	2	4	no	—
48	C₂×Dic₁₃	—	2	4	no	—
49	C₂×Dic₁₃	—	2	2	no	—
50	C₂×Dic₁₃	—	2	4	no	—
51	C₂×Dic₁₃	#41	1	8	yes	—
52	D₅₂	—	4	4	no	—
53	Dic₂₆	—	4	4	no	—
54	C₂×#125	D₁₃	1	1	yes	characteristic, complemented
55	C₂×#124	—	13	1	no	—
56	#391	—	13	2	no	—
57	#390	—	13	2	no	—
58	#388	—	13	8	no	—
59	C₄³	—	13	1	no	—
60	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
61	C₂²×C₁₃	C₂×#41	1	2	yes	—
62	C₂²×C₁₃	C₂×#41	1	4	yes	—
63	C₂×D₁₃	—	4	2	no	—
64	C₂×D₁₃	—	4	8	no	—
65	C₂×D₁₃	—	4	4	no	—
66	C₄×C₁₃	—	2	4	no	—
67	C₄×C₁₃	—	4	4	no	—
68	C₄×C₁₃	#124	1	4	yes	—
69	Dic₁₃	—	2	8	no	—
70	Dic₁₃	—	4	2	no	—
71	C₂²×Q₈	—	13	1	no	—
72	C₂²×D₄	—	13	1	no	—
73	#125	—	2	4	no	—
74	#124	—	26	4	no	—
75	C₂×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
76	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
77	C₂×C₂²⋊C₄	—	13	4	no	—
78	C₂×C₂²⋊C₄	—	13	8	no	—
79	C₂×C₄²	—	13	4	no	—
80	C₂×C₄²	—	13	2	no	—
81	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
82	#95	—	13	8	no	—
83	C₂×C₁₃	C₂×#124	1	1	yes	characteristic
84	C₂×C₁₃	#529	1	4	yes	—
85	C₂×C₁₃	#532	1	2	yes	—
86	D₁₃	—	8	2	no	—
87	C₂⁴	—	13	2	no	—
88	C₂×Q₈	—	26	4	no	—
89	C₂×Q₈	—	26	2	no	—
90	C₂×D₄	—	26	4	no	—
91	C₂×D₄	—	26	2	no	—
92	C₂²×C₄	—	13	4	no	—
93	C₂²×C₄	—	13	2	no	—
94	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
95	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
97	C₄⋊C₄	—	2	8	no	—
98	C₄⋊C₄	—	2	4	no	—
99	C₂²⋊C₄	—	26	16	no	—
100	C₂²⋊C₄	—	26	8	no	—
101	C₄²	—	26	8	no	—
102	C₄²	—	26	4	no	—
103	C₄²	C₂²⋊D₁₃	1	4	yes	—
104	C₁₃	#2861	1	1	yes	characteristic, complemented, Sylow
105	C₂³	—	26	4	no	—
106	C₂³	—	26	8	no	—
107	C₂³	—	26	2	no	—
108	C₂³	C₂³×D₁₃	1	1	yes	characteristic
109	Q₈	—	52	4	no	—
110	D₄	—	52	4	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	2	8	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	13	8	no	—
115	C₂×C₄	—	26	4	no	—
116	C₂×C₄	—	26	4	no	—
117	C₂×C₄	—	26	4	no	—
118	C₂×C₄	—	26	2	no	—
119	C₂×C₄	#6131	1	4	yes	—
120	C₂×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
121	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
122	C₂²	—	52	2	no	—
123	C₂²	—	52	4	no	—
124	C₂²	—	52	8	no	—
125	C₂²	C₂×#6131	1	2	yes	—
126	C₂²	#41×D₁₃	1	4	yes	—
127	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
128	C₄	—	2	4	no	—
129	C₄	—	4	4	no	—
130	C₄	—	26	8	no	—
131	C₄	—	52	2	no	—
132	C₄	#88201	1	4	yes	—
133	C₂	—	104	2	no	—
134	C₂	#11719396	1	4	yes	—
135	C₂	C₂×#88201	1	1	yes	characteristic
136	C₂	#11719537	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#125

C₂

yes

maximal, characteristic, complemented

C₂×#88129

C₂

yes

maximal, characteristic

#11718651

C₂

yes

maximal, complemented

#11718649

C₂

yes

maximal

#11718635

C₂

yes

maximal, complemented

C₄²×Dic₁₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₂

C₂²

yes

characteristic

C₂²×Dic₂₆

C₂²

yes

characteristic

C₁₃×#125

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×#6103

C₂²

yes

—

C₂×#6103

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

#88129

—

#88071

C₂²

yes

—

C₂³×D₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

D₄

yes

—

#6103

—

#6103

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

#2861

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

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

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

#41

yes

—

D₅₂

—

Dic₂₆

—

C₂×#125

D₁₃

yes

characteristic, complemented

—

—

—

—

C₄³

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

C₂×#41

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#124

yes

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×D₄

—

#125

—

#124

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#95

—

C₂×C₁₃

C₂×#124

yes

characteristic

C₂×C₁₃

#529

yes

—

C₂×C₁₃

#532

yes

—

D₁₃

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₄²

—

102

C₄²

—

103

C₄²

C₂²⋊D₁₃

yes

—

104

C₁₃

#2861

yes

characteristic, complemented, Sylow

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

C₂³×D₁₃

yes

characteristic

109

Q₈

—

110

D₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

#6131

yes

—

120

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

121

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂×#6131

yes

—

126

C₂²

#41×D₁₃

yes

—

127

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#88201

yes

—

133

C₂

—

104

—

134

C₂

#11719396

yes

—

135

C₂

C₂×#88201

yes

characteristic

136

C₂

#11719537

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂×C₁₃×#125	C₂
prime radical	C₂²×D₅₂	C₂²
hypercenter	C₂×#125	D₁₃
hypocenter	C₁₃	#2861
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₂×C₁₃×#125

C₂

prime radical

C₂²×D₅₂

C₂²

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
1. C₂×C₁₃×#125	C₂	semidirect
3. #11718651	C₂	semidirect
5. #11718635	C₂	semidirect
6. C₄²×Dic₁₃	C₂	semidirect
54. C₂×#125	D₁₃	semidirect
104. C₁₃	#2861	semidirect

H ≅ G/N

type

1. C₂×C₁₃×#125

C₂

semidirect

3. #11718651

C₂

semidirect

5. #11718635

C₂

semidirect

6. C₄²×Dic₁₃

C₂

semidirect

54. C₂×#125

D₁₃

semidirect

104. C₁₃

#2861

semidirect

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

subgroup	index
C₂×C₁₃×#125	2
C₂×#88129	2
#11718651	2
#11718649	2
#11718635	2
C₄²×Dic₁₃	2
#2861	13

subgroup

index

C₂×C₁₃×#125

C₄²×Dic₁₃

Subgroups — #13123209

Conjugacy classes of subgroups

Complements

Maximal subgroups