Group #13111001 — subgroups

Subgroups — C₁₃×#1943 #13111001

Order 1664.

240 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 128 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×#107	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#106	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#389	C₂	1	2	yes	maximal
5	C₁₃×#340	C₂	1	2	yes	maximal
6	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
9	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₁₃×#107	C₄	1	4	yes	complemented
13	C₁₃×#106	—	2	2	no	—
14	C₁₃×#97	—	2	2	no	—
15	C₁₃×#95	—	2	2	no	—
16	C₂²×C₄×C₁₃	—	2	2	no	—
17	C₂²×C₄×C₁₃	—	2	2	no	—
18	C₂²×C₄×C₁₃	D₄	1	2	yes	—
19	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂×C₈×C₁₃	—	2	1	no	—
21	C₂×C₈×C₁₃	—	2	1	no	—
22	C₂×C₈×C₁₃	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₁₃	D₄	1	2	yes	—
24	C₂×C₈×C₁₃	Q₈	1	2	yes	—
25	C₂×C₈×C₁₃	Q₈	1	2	yes	—
26	C₁₃×C₄⋊C₄	—	2	2	no	—
27	C₁₃×C₄⋊C₄	—	2	2	no	—
28	C₁₃×C₄⋊C₄	—	2	2	no	—
29	C₁₃×C₄⋊C₄	—	4	4	no	—
30	C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
31	C₄²×C₁₃	—	2	1	no	—
32	C₄²×C₁₃	—	2	1	no	—
33	#1943	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₃	—	2	4	no	—
36	C₂×C₄×C₁₃	—	2	2	no	—
37	C₂×C₄×C₁₃	—	2	2	no	—
38	C₂×C₄×C₁₃	—	2	2	no	—
39	C₂×C₄×C₁₃	—	4	2	no	—
40	C₂×C₄×C₁₃	—	4	2	no	—
41	C₂×C₄×C₁₃	—	4	2	no	—
42	C₂×C₄×C₁₃	—	4	2	no	—
43	C₂×C₄×C₁₃	—	4	2	no	—
44	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
49	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
50	C₈×C₁₃	—	2	2	no	—
51	C₈×C₁₃	C₄⋊C₄	1	4	yes	complemented
52	C₂×#107	C₂×C₁₃	1	1	yes	characteristic
53	C₂×#106	C₂×C₁₃	1	1	yes	characteristic
54	C₂×#97	C₂×C₁₃	1	1	yes	characteristic
55	#389	C₂×C₁₃	1	2	yes	—
56	#340	C₂×C₁₃	1	2	yes	—
57	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
58	C₂²×C₁₃	#121	1	1	yes	characteristic
59	C₂²×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
60	C₂²×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
61	C₂²×C₁₃	#137	1	2	yes	—
62	C₄×C₁₃	—	4	4	no	—
63	C₄×C₁₃	—	4	2	no	—
64	C₄×C₁₃	—	8	2	no	—
65	C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
67	C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
68	C₄×C₁₃	#125	1	1	yes	characteristic
69	C₂²×C₈	C₂²×C₁₃	1	1	yes	characteristic
70	C₂²×C₈	C₂²×C₁₃	1	1	yes	characteristic
71	C₂×C₄⋊C₄	—	2	2	no	—
72	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
73	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
74	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
75	#107	C₄×C₁₃	1	4	yes	complemented
76	#106	—	2	2	no	—
77	#97	—	2	2	no	—
78	#95	—	2	2	no	—
79	C₂×C₁₃	#389	1	1	yes	characteristic
80	C₂×C₁₃	#441	1	1	yes	characteristic
81	C₂×C₁₃	#442	1	1	yes	characteristic
82	C₂×C₁₃	#468	1	2	yes	—
83	C₂×C₁₃	#501	1	2	yes	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	D₄×C₁₃	1	2	yes	—
87	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	C₂×C₄×C₁₃	1	2	yes	—
91	C₂×C₈	D₄×C₁₃	1	2	yes	—
92	C₂×C₈	Q₈×C₁₃	1	2	yes	—
93	C₂×C₈	Q₈×C₁₃	1	2	yes	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	4	4	no	—
98	C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
99	C₄²	—	2	1	no	—
100	C₄²	—	2	1	no	—
101	C₁₃	#1943	1	1	yes	characteristic, direct product factor, Sylow
102	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
113	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
114	C₂×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
115	C₂×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
116	C₂×C₄	C₁₃×#41	1	1	yes	characteristic
117	C₂×C₄	C₁₃×#41	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	C₁₃×C₄⋊C₄	1	4	yes	complemented
120	C₂²	C₄×D₄×C₁₃	1	1	yes	characteristic
121	C₂²	C₁₃×#121	1	1	yes	characteristic
122	C₂²	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
123	C₂²	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
124	C₂²	C₁₃×#137	1	2	yes	—
125	C₄	—	4	4	no	—
126	C₄	—	4	2	no	—
127	C₄	—	8	2	no	—
128	C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
129	C₄	C₄×Q₈×C₁₃	1	1	yes	characteristic
130	C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
131	C₄	C₁₃×#125	1	1	yes	characteristic
132	C₂	C₁₃×#389	1	1	yes	characteristic
133	C₂	C₁₃×#441	1	1	yes	characteristic
134	C₂	C₁₃×#442	1	1	yes	characteristic
135	C₂	C₁₃×#468	1	2	yes	—
136	C₂	C₁₃×#501	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#107

C₂

yes

maximal, characteristic

C₂×C₁₃×#106

C₂

yes

maximal, characteristic

C₂×C₁₃×#97

C₂

yes

maximal, characteristic

C₁₃×#389

C₂

yes

maximal

C₁₃×#340

C₂

yes

maximal

C₂²×C₈×C₁₃

C₂²

yes

characteristic

C₂²×C₈×C₁₃

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₁₃×#107

C₄

yes

complemented

C₁₃×#106

—

C₁₃×#97

—

C₁₃×#95

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₈×C₁₃

D₄

yes

—

C₂×C₈×C₁₃

Q₈

yes

—

C₂×C₈×C₁₃

Q₈

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₄²×C₁₃

—

C₄²×C₁₃

—

#1943

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

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×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₈

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₈×C₁₃

—

C₈×C₁₃

C₄⋊C₄

yes

complemented

C₂×#107

C₂×C₁₃

yes

characteristic

C₂×#106

C₂×C₁₃

yes

characteristic

C₂×#97

C₂×C₁₃

yes

characteristic

#389

C₂×C₁₃

yes

—

#340

C₂×C₁₃

yes

—

C₂²×C₁₃

C₄×D₄

yes

characteristic

C₂²×C₁₃

#121

yes

characteristic

C₂²×C₁₃

C₄⋊Q₈

yes

characteristic

C₂²×C₁₃

AGL(1,ℤ/8)

yes

—

C₂²×C₁₃

#137

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₃

C₄×Q₈

yes

characteristic

C₄×C₁₃

C₂²⋊Q₈

yes

characteristic

C₄×C₁₃

#125

yes

characteristic

C₂²×C₈

C₂²×C₁₃

yes

characteristic

C₂²×C₈

C₂²×C₁₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₁₃

yes

—

C₂×C₄⋊C₄

C₂²×C₁₃

yes

—

C₂×C₄²

C₂²×C₁₃

yes

characteristic

#107

C₄×C₁₃

yes

—

—

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

C₂×C₁₃

#501

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₁₃

yes

—

C₂²×C₄

C₂³×C₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₁₃

yes

—

C₂×C₈

D₄×C₁₃

yes

—

C₂×C₈

Q₈×C₁₃

yes

—

C₂×C₈

Q₈×C₁₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₄×C₁₃

yes

—

C₄²

—

100

C₄²

—

101

C₁₃

#1943

yes

characteristic, direct product factor, Sylow

102

C₂³

C₂×D₄×C₁₃

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂²×C₄×C₁₃

yes

characteristic

113

C₂×C₄

C₂×D₄×C₁₃

yes

characteristic

114

C₂×C₄

C₂×Q₈×C₁₃

yes

characteristic

115

C₂×C₄

C₂×Q₈×C₁₃

yes

characteristic

116

C₂×C₄

C₁₃×#41

yes

characteristic

117

C₂×C₄

C₁₃×#41

yes

characteristic

118

C₈

—

119

C₈

C₁₃×C₄⋊C₄

yes

complemented

120

C₂²

C₄×D₄×C₁₃

yes

characteristic

121

C₂²

C₁₃×#121

yes

characteristic

122

C₂²

C₁₃×C₄⋊Q₈

yes

characteristic

123

C₂²

C₁₃×AGL(1,ℤ/8)

yes

—

124

C₂²

C₁₃×#137

yes

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

C₂×C₁₃×C₄⋊C₄

yes

characteristic

129

C₄

C₄×Q₈×C₁₃

yes

characteristic

130

C₄

C₁₃×C₂²⋊Q₈

yes

characteristic

131

C₄

C₁₃×#125

yes

characteristic

132

C₂

C₁₃×#389

yes

characteristic

133

C₂

C₁₃×#441

yes

characteristic

134

C₂

C₁₃×#442

yes

characteristic

135

C₂

C₁₃×#468

yes

—

136

C₂

C₁₃×#501

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₄	C₂³×C₁₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₃

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₁₃

Frattini, cosocle

C₂²×C₄

C₂³×C₁₃

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

N	H ≅ G/N	type
12. C₁₃×#107	C₄	semidirect
33. #1943	C₁₃	direct (N × H)
51. C₈×C₁₃	C₄⋊C₄	semidirect
75. #107	C₄×C₁₃	semidirect
101. C₁₃	#1943	direct (N × H)
119. C₈	C₁₃×C₄⋊C₄	semidirect

H ≅ G/N

type

12. C₁₃×#107

C₄

semidirect

33. #1943

C₁₃

direct (N × H)

51. C₈×C₁₃

C₄⋊C₄

semidirect

75. #107

C₄×C₁₃

semidirect

101. C₁₃

#1943

direct (N × H)

119. C₈

C₁₃×C₄⋊C₄

semidirect

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

subgroup	index
C₂×C₁₃×#107	2
C₂×C₁₃×#106	2
C₂×C₁₃×#97	2
C₁₃×#389	2
C₁₃×#340	2
#1943	13

subgroup

index

C₂×C₁₃×#107

C₂×C₁₃×#106

C₂×C₁₃×#97

C₁₃×#389

C₁₃×#340

#1943

Subgroups — C₁₃×#1943 #13111001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C13×#1943 #13111001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#1943 #13111001