Group #13111919 — subgroups

Subgroups — C₁₃×#2861 #13111919

Order 1664.

524 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 216 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×C₁₃×#124	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#391	C₂	1	2	yes	maximal, complemented
4	C₁₃×#390	C₂	1	2	yes	maximal
5	C₁₃×#388	C₂	1	8	yes	maximal, complemented
6	C₄³×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₁₃×#125	—	2	4	no	—
10	C₁₃×#124	—	2	4	no	—
11	C₂×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₁₃×C₂²⋊C₄	C₂²	1	8	yes	—
14	C₂×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
15	C₂×C₄²×C₁₃	C₂²	1	4	yes	—
16	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#95	C₂²	1	8	yes	—
19	C₂⁴×C₁₃	C₂³	1	2	yes	—
20	C₂×Q₈×C₁₃	—	2	2	no	—
21	C₂×Q₈×C₁₃	—	2	4	no	—
22	C₂×D₄×C₁₃	—	2	2	no	—
23	C₂×D₄×C₁₃	—	2	4	no	—
24	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
25	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
26	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
27	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
28	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₁₃×C₄⋊C₄	—	2	8	no	—
30	C₁₃×C₄⋊C₄	—	2	4	no	—
31	C₁₃×C₂²⋊C₄	—	2	8	no	—
32	C₁₃×C₂²⋊C₄	—	2	16	no	—
33	C₄²×C₁₃	—	2	8	no	—
34	C₄²×C₁₃	—	2	4	no	—
35	C₄²×C₁₃	D₄	1	4	yes	—
36	#2861	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂³×C₁₃	—	2	2	no	—
38	C₂³×C₁₃	—	2	8	no	—
39	C₂³×C₁₃	—	2	4	no	—
40	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
41	Q₈×C₁₃	—	4	4	no	—
42	D₄×C₁₃	—	4	4	no	—
43	C₂×C₄×C₁₃	—	2	4	no	—
44	C₂×C₄×C₁₃	—	2	4	no	—
45	C₂×C₄×C₁₃	—	2	4	no	—
46	C₂×C₄×C₁₃	—	2	2	no	—
47	C₂×C₄×C₁₃	—	2	4	no	—
48	C₂×C₄×C₁₃	—	2	8	no	—
49	C₂×C₄×C₁₃	—	2	4	no	—
50	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
51	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₃	#41	1	8	yes	—
53	C₂×C₄×C₁₃	#41	1	4	yes	—
54	C₂×#125	C₂×C₁₃	1	1	yes	characteristic, complemented
55	C₂×#124	C₂×C₁₃	1	1	yes	characteristic
56	#391	C₂×C₁₃	1	2	yes	complemented
57	#390	C₂×C₁₃	1	2	yes	—
58	#388	C₂×C₁₃	1	8	yes	complemented
59	C₄³	C₂×C₁₃	1	1	yes	characteristic, complemented
60	C₂²×C₁₃	—	4	2	no	—
61	C₂²×C₁₃	—	4	8	no	—
62	C₂²×C₁₃	—	4	4	no	—
63	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
64	C₂²×C₁₃	C₂×#41	1	2	yes	—
65	C₂²×C₁₃	C₂×#41	1	4	yes	—
66	C₄×C₁₃	—	2	8	no	—
67	C₄×C₁₃	—	2	4	no	—
68	C₄×C₁₃	—	4	2	no	—
69	C₄×C₁₃	—	4	4	no	—
70	C₄×C₁₃	#124	1	4	yes	—
71	C₂²×Q₈	C₂²×C₁₃	1	1	yes	characteristic
72	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
73	#125	—	2	4	no	—
74	#124	—	2	4	no	—
75	C₂×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
76	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
77	C₂×C₂²⋊C₄	C₂²×C₁₃	1	4	yes	—
78	C₂×C₂²⋊C₄	C₂²×C₁₃	1	8	yes	—
79	C₂×C₄²	C₂²×C₁₃	1	4	yes	—
80	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
81	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
82	#95	C₂²×C₁₃	1	8	yes	—
83	C₂×C₁₃	—	8	2	no	—
84	C₂×C₁₃	C₂×#124	1	1	yes	characteristic
85	C₂×C₁₃	#529	1	4	yes	—
86	C₂×C₁₃	#532	1	2	yes	—
87	C₂⁴	C₂³×C₁₃	1	2	yes	—
88	C₂×Q₈	—	2	2	no	—
89	C₂×Q₈	—	2	4	no	—
90	C₂×D₄	—	2	2	no	—
91	C₂×D₄	—	2	4	no	—
92	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
93	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
94	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
95	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
96	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
97	C₄⋊C₄	—	2	8	no	—
98	C₄⋊C₄	—	2	4	no	—
99	C₂²⋊C₄	—	2	8	no	—
100	C₂²⋊C₄	—	2	16	no	—
101	C₄²	—	2	8	no	—
102	C₄²	—	2	4	no	—
103	C₄²	D₄×C₁₃	1	4	yes	—
104	C₁₃	#2861	1	1	yes	characteristic, direct product factor, Sylow
105	C₂³	—	2	2	no	—
106	C₂³	—	2	8	no	—
107	C₂³	—	2	4	no	—
108	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
109	Q₈	—	4	4	no	—
110	D₄	—	4	4	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	8	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	C₂×D₄×C₁₃	1	4	yes	—
119	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
120	C₂×C₄	C₁₃×#41	1	8	yes	—
121	C₂×C₄	C₁₃×#41	1	4	yes	—
122	C₂²	—	4	2	no	—
123	C₂²	—	4	8	no	—
124	C₂²	—	4	4	no	—
125	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
126	C₂²	C₂×C₁₃×#41	1	2	yes	—
127	C₂²	C₂×C₁₃×#41	1	4	yes	—
128	C₄	—	2	8	no	—
129	C₄	—	2	4	no	—
130	C₄	—	4	2	no	—
131	C₄	—	4	4	no	—
132	C₄	C₁₃×#124	1	4	yes	—
133	C₂	—	8	2	no	—
134	C₂	C₂×C₁₃×#124	1	1	yes	characteristic
135	C₂	C₁₃×#529	1	4	yes	—
136	C₂	C₁₃×#532	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#125

C₂

yes

maximal, characteristic, complemented

C₂×C₁₃×#124

C₂

yes

maximal, characteristic

C₁₃×#391

C₂

yes

maximal, complemented

C₁₃×#390

C₂

yes

maximal

C₁₃×#388

C₂

yes

maximal, complemented

C₄³×C₁₃

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₃

C₂²

yes

characteristic

C₂²×D₄×C₁₃

C₂²

yes

characteristic

C₁₃×#125

—

C₁₃×#124

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₁₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₁₃×#95

C₂²

yes

—

C₂⁴×C₁₃

C₂³

yes

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

D₄

yes

—

#2861

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂⁴

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

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×C₄×C₁₃

#41

yes

—

C₂×#125

C₂×C₁₃

yes

characteristic, complemented

C₂×#124

C₂×C₁₃

yes

characteristic

#391

C₂×C₁₃

yes

complemented

#390

C₂×C₁₃

yes

—

#388

C₂×C₁₃

yes

complemented

C₄³

C₂×C₁₃

yes

characteristic, complemented

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

C₂×#41

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#124

yes

—

C₂²×Q₈

C₂²×C₁₃

yes

characteristic

C₂²×D₄

C₂²×C₁₃

yes

characteristic

#125

—

#124

—

C₂×C₄⋊C₄

C₂²×C₁₃

yes

—

C₂×C₄⋊C₄

C₂²×C₁₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₁₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₁₃

yes

—

C₂×C₄²

C₂²×C₁₃

yes

—

C₂×C₄²

C₂²×C₁₃

yes

—

C₂×C₄²

C₂²×C₁₃

yes

characteristic

#95

C₂²×C₁₃

yes

—

C₂×C₁₃

—

C₂×C₁₃

C₂×#124

yes

characteristic

C₂×C₁₃

#529

yes

—

C₂×C₁₃

#532

yes

—

C₂⁴

C₂³×C₁₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂³×C₁₃

yes

—

C₂²×C₄

C₂³×C₁₃

yes

—

C₂²×C₄

C₂³×C₁₃

yes

—

C₂²×C₄

C₂³×C₁₃

yes

—

C₂²×C₄

C₂³×C₁₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₄²

—

102

C₄²

—

103

C₄²

D₄×C₁₃

yes

—

104

C₁₃

#2861

yes

characteristic, direct product factor, Sylow

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

C₂⁴×C₁₃

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

C₂×D₄×C₁₃

yes

—

119

C₂×C₄

C₂×D₄×C₁₃

yes

—

120

C₂×C₄

C₁₃×#41

yes

—

121

C₂×C₄

C₁₃×#41

yes

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂²×D₄×C₁₃

yes

characteristic

126

C₂²

C₂×C₁₃×#41

yes

—

127

C₂²

C₂×C₁₃×#41

yes

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

C₁₃×#124

yes

—

133

C₂

—

134

C₂

C₂×C₁₃×#124

yes

characteristic

135

C₂

C₁₃×#529

yes

—

136

C₂

C₁₃×#532

yes

—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
prime radical	C₂²×D₄×C₁₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₃

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₁₃

prime radical

C₂²×D₄×C₁₃

C₂²

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₃ ◁ G

N	H ≅ G/N	type
1. C₂×C₁₃×#125	C₂	semidirect
3. C₁₃×#391	C₂	semidirect
5. C₁₃×#388	C₂	semidirect
6. C₄³×C₁₃	C₂	semidirect
36. #2861	C₁₃	direct (N × H)
54. C₂×#125	C₂×C₁₃	semidirect
56. #391	C₂×C₁₃	semidirect
58. #388	C₂×C₁₃	semidirect
59. C₄³	C₂×C₁₃	semidirect
104. C₁₃	#2861	direct (N × H)

H ≅ G/N

type

1. C₂×C₁₃×#125

C₂

semidirect

3. C₁₃×#391

C₂

semidirect

5. C₁₃×#388

C₂

semidirect

6. C₄³×C₁₃

C₂

semidirect

36. #2861

C₁₃

direct (N × H)

54. C₂×#125

C₂×C₁₃

semidirect

56. #391

C₂×C₁₃

semidirect

58. #388

C₂×C₁₃

semidirect

59. C₄³

C₂×C₁₃

semidirect

104. C₁₃

#2861

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×C₁₃×#124	2
C₁₃×#391	2
C₁₃×#390	2
C₁₃×#388	2
C₄³×C₁₃	2
#2861	13

subgroup

index

C₂×C₁₃×#125

C₂×C₁₃×#124

C₁₃×#391

C₁₃×#390

C₁₃×#388

C₄³×C₁₃

#2861

Subgroups — C₁₃×#2861 #13111919

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C13×#2861 #13111919

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2861 #13111919