Group #15699986 — subgroups

Subgroups — C₁₃×#13647 #15699986

Order 3328.

186 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 118 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	C₁₃×#2258	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2170	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1422	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1401	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₃×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#504	C₂²	1	2	yes	—
7	C₁₃×#443	C₄	1	1	yes	characteristic
8	C₁₃×M₆₄	—	2	2	no	—
9	C₁₃×M₆₄	C₂²	1	2	yes	—
10	C₂×C₁₃×C₃₂	—	2	2	no	—
11	C₂×C₁₃×C₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×C₃₂	C₂²	1	1	yes	characteristic
13	C₁₃×#364	C₄	1	1	yes	characteristic
14	C₁₃×#350	C₄	1	2	yes	—
15	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
16	C₁₃×#135	C₈	1	1	yes	characteristic
17	C₁₃×#131	C₂×C₄	1	2	yes	—
18	C₁₃×M₃₂	—	2	2	no	—
19	C₁₃×M₃₂	C₂×C₄	1	2	yes	—
20	C₂×C₁₃×C₁₆	—	2	2	no	—
21	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
23	C₁₃×#108	C₈	1	1	yes	characteristic
24	C₁₃×#104	C₈	1	2	yes	—
25	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₁₃×C₃₂	—	2	2	no	—
27	C₁₃×C₃₂	—	2	1	no	—
28	C₁₃×C₃₂	D₄	1	1	yes	characteristic
29	C₁₃×C₃₂	D₄	1	1	yes	characteristic
30	#13647	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₁₃×#41	C₂×C₈	1	2	yes	—
32	C₁₃×Q₁₆	C₁₆	1	1	yes	characteristic
33	C₁₃×SD₁₆	C₁₆	1	2	yes	—
34	C₁₃×D₈	C₁₆	1	1	yes	characteristic
35	C₁₃×M₁₆	—	2	2	no	—
36	C₁₃×M₁₆	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₁₃	—	2	2	no	—
38	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
41	C₁₃×C₁₆	—	2	2	no	—
42	C₁₃×C₁₆	—	2	1	no	—
43	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
44	C₁₃×C₁₆	#41	1	1	yes	characteristic
45	#2258	C₂×C₁₃	1	2	yes	complemented
46	#2170	C₂×C₁₃	1	1	yes	characteristic
47	#1422	C₂×C₁₃	1	1	yes	characteristic, complemented
48	#1401	C₂×C₁₃	1	2	yes	complemented
49	C₄×C₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
50	Q₈×C₁₃	C₂×C₁₆	1	2	yes	—
51	D₄×C₁₃	—	2	2	no	—
52	D₄×C₁₃	C₂×C₁₆	1	2	yes	—
53	C₂×C₄×C₁₃	—	2	1	no	—
54	C₂×C₄×C₁₃	—	2	2	no	—
55	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₁₃	—	2	2	no	—
57	C₈×C₁₃	C₂×C₁₆	1	1	yes	characteristic
58	C₈×C₁₃	C₂×C₁₆	1	1	yes	characteristic
59	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
60	C₈×C₁₃	#131	1	1	yes	characteristic
61	#504	C₂²×C₁₃	1	2	yes	—
62	#443	C₄×C₁₃	1	1	yes	characteristic
63	M₆₄	—	2	2	no	—
64	M₆₄	C₂²×C₁₃	1	2	yes	—
65	C₂×C₃₂	—	2	2	no	—
66	C₂×C₃₂	C₂²×C₁₃	1	1	yes	characteristic
67	C₂×C₃₂	C₂²×C₁₃	1	1	yes	characteristic
68	#364	C₄×C₁₃	1	1	yes	characteristic
69	#350	C₄×C₁₃	1	2	yes	—
70	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
71	C₂²×C₁₃	—	2	2	no	—
72	C₂²×C₁₃	#504	1	1	yes	characteristic
73	C₄×C₁₃	—	2	2	no	—
74	C₄×C₁₃	—	2	2	no	—
75	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
76	C₄×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
77	#135	C₈×C₁₃	1	1	yes	characteristic
78	#131	C₂×C₄×C₁₃	1	2	yes	—
79	M₃₂	—	2	2	no	—
80	M₃₂	C₂×C₄×C₁₃	1	2	yes	—
81	C₂×C₁₆	—	2	2	no	—
82	C₂×C₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
83	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
84	#108	C₈×C₁₃	1	1	yes	characteristic
85	#104	C₈×C₁₃	1	2	yes	—
86	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
87	C₃₂	—	2	2	no	—
88	C₃₂	—	2	1	no	—
89	C₃₂	D₄×C₁₃	1	1	yes	characteristic
90	C₃₂	D₄×C₁₃	1	1	yes	characteristic
91	C₂×C₁₃	—	2	1	no	—
92	C₂×C₁₃	—	4	2	no	—
93	C₂×C₁₃	D₄×C₁₆	1	1	yes	characteristic
94	#41	C₂×C₈×C₁₃	1	2	yes	—
95	Q₁₆	C₁₃×C₁₆	1	1	yes	characteristic
96	SD₁₆	C₁₃×C₁₆	1	2	yes	—
97	D₈	C₁₃×C₁₆	1	1	yes	characteristic
98	M₁₆	—	2	2	no	—
99	M₁₆	C₂×C₈×C₁₃	1	2	yes	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
102	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
103	C₄²	C₂×C₈×C₁₃	1	1	yes	characteristic
104	C₁₆	—	2	2	no	—
105	C₁₆	—	2	1	no	—
106	C₁₆	C₂×D₄×C₁₃	1	1	yes	characteristic
107	C₁₆	C₁₃×#41	1	1	yes	characteristic
108	C₁₃	#13647	1	1	yes	characteristic, direct product factor, Sylow
109	Q₈	C₂×C₁₃×C₁₆	1	2	yes	—
110	D₄	—	2	2	no	—
111	D₄	C₂×C₁₃×C₁₆	1	2	yes	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
115	C₈	—	2	2	no	—
116	C₈	C₂×C₁₃×C₁₆	1	1	yes	characteristic
117	C₈	C₂×C₁₃×C₁₆	1	1	yes	characteristic
118	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
119	C₈	C₁₃×#131	1	1	yes	characteristic
120	C₂²	—	2	2	no	—
121	C₂²	C₁₃×#504	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	C₈×D₄×C₁₃	1	1	yes	characteristic
125	C₄	C₂²×C₁₃×C₁₆	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	4	2	no	—
128	C₂	D₄×C₁₃×C₁₆	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#2258

C₂

yes

maximal, complemented

C₁₃×#2170

C₂

yes

maximal, characteristic

C₁₃×#1422

C₂

yes

maximal, characteristic, complemented

C₁₃×#1401

C₂

yes

maximal, complemented

C₄×C₁₃×C₃₂

C₂

yes

maximal, characteristic, complemented

C₁₃×#504

C₂²

yes

—

C₁₃×#443

C₄

yes

characteristic

C₁₃×M₆₄

—

C₁₃×M₆₄

C₂²

yes

—

C₂×C₁₃×C₃₂

—

C₂×C₁₃×C₃₂

C₂²

yes

characteristic

C₂×C₁₃×C₃₂

C₂²

yes

characteristic

C₁₃×#364

C₄

yes

characteristic

C₁₃×#350

C₄

yes

—

C₄×C₁₃×C₁₆

C₂²

yes

characteristic

C₁₃×#135

C₈

yes

characteristic

C₁₃×#131

C₂×C₄

yes

—

C₁₃×M₃₂

—

C₁₃×M₃₂

C₂×C₄

yes

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₂×C₁₃×C₁₆

C₂³

yes

characteristic

C₁₃×#108

C₈

yes

characteristic

C₁₃×#104

C₈

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₁₃×C₃₂

—

C₁₃×C₃₂

—

C₁₃×C₃₂

D₄

yes

characteristic

C₁₃×C₃₂

D₄

yes

characteristic

#13647

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₁₃×#41

C₂×C₈

yes

—

C₁₃×Q₁₆

C₁₆

yes

characteristic

C₁₃×SD₁₆

C₁₆

yes

—

C₁₃×D₈

C₁₆

yes

characteristic

C₁₃×M₁₆

—

C₁₃×M₁₆

C₂×C₈

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂×C₈

yes

characteristic

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₁₃×C₁₆

—

C₁₃×C₁₆

—

C₁₃×C₁₆

C₂×D₄

yes

C₁₃×C₁₆

yes

C₂×C₁₃

yes

complemented

#2170

C₂×C₁₃

yes

characteristic

#1422

C₂×C₁₃

yes

characteristic, complemented

#1401

C₂×C₁₃

yes

complemented

C₄×C₃₂

C₂×C₁₃

yes

characteristic, complemented

Q₈×C₁₃

C₂×C₁₆

yes

—

D₄×C₁₃

—

D₄×C₁₃

C₂×C₁₆

yes

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₈×C₁₃

—

C₈×C₁₃

C₂×C₁₆

yes

characteristic

C₈×C₁₃

C₂×C₁₆

yes

characteristic

C₈×C₁₃

C₄×D₄

yes

C₈×C₁₃

yes

C₂²×C₁₃

yes

—

#443

C₄×C₁₃

yes

characteristic

M₆₄

—

M₆₄

C₂²×C₁₃

yes

—

C₂×C₃₂

—

C₂×C₃₂

C₂²×C₁₃

yes

characteristic

C₂×C₃₂

C₂²×C₁₃

yes

characteristic

#364

C₄×C₁₃

yes

characteristic

#350

C₄×C₁₃

yes

—

C₄×C₁₆

C₂²×C₁₃

yes

characteristic

C₂²×C₁₃

—

C₂²×C₁₃

#504

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₈×D₄

yes

characteristic

C₄×C₁₃

C₂²×C₁₆

yes

characteristic

#135

C₈×C₁₃

yes

characteristic

#131

C₂×C₄×C₁₃

yes

—

M₃₂

—

M₃₂

C₂×C₄×C₁₃

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₁₃

yes

characteristic

C₂×C₁₆

C₂³×C₁₃

yes

characteristic

#108

C₈×C₁₃

yes

characteristic

#104

C₈×C₁₃

yes

—

C₄×C₈

C₂×C₄×C₁₃

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

D₄×C₁₃

yes

characteristic

C₃₂

D₄×C₁₃

yes

characteristic

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

D₄×C₁₆

yes

characteristic

#41

C₂×C₈×C₁₃

yes

—

Q₁₆

C₁₃×C₁₆

yes

characteristic

SD₁₆

C₁₃×C₁₆

yes

—

D₈

C₁₃×C₁₆

yes

characteristic

M₁₆

—

M₁₆

C₂×C₈×C₁₃

yes

—

100

C₂×C₈

—

101

C₂×C₈

C₂×C₈×C₁₃

yes

characteristic

102

C₂×C₈

C₂²×C₄×C₁₃

yes

characteristic

103

C₄²

C₂×C₈×C₁₃

yes

characteristic

104

C₁₆

—

105

C₁₆

—

106

C₁₆

C₂×D₄×C₁₃

yes

characteristic

107

C₁₆

C₁₃×#41

yes

characteristic

108

C₁₃

#13647

yes

characteristic, direct product factor, Sylow

109

Q₈

C₂×C₁₃×C₁₆

yes

—

110

D₄

—

111

D₄

C₂×C₁₃×C₁₆

yes

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂²×C₈×C₁₃

yes

characteristic

115

C₈

—

116

C₈

C₂×C₁₃×C₁₆

yes

characteristic

117

C₈

C₂×C₁₃×C₁₆

yes

characteristic

118

C₈

C₄×D₄×C₁₃

yes

characteristic

119

C₈

C₁₃×#131

yes

characteristic

120

C₂²

—

121

C₂²

C₁₃×#504

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

C₈×D₄×C₁₃

yes

characteristic

125

C₄

C₂²×C₁₃×C₁₆

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

D₄×C₁₃×C₁₆

yes

characteristic

	subgroup	G/H
center	C₁₃×C₃₂	D₄
commutator	C₄	C₂²×C₁₃×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₃
socle	C₂×C₁₃	D₄×C₁₆
prime radical	C₁₃×#135	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₃×C₃₂ ◁ C₂×C₁₃×C₃₂ ◁ G

subgroup

G/H

center

C₁₃×C₃₂

D₄

commutator

C₄

C₂²×C₁₃×C₁₆

Frattini, cosocle

C₂×C₁₆

C₂³×C₁₃

socle

C₂×C₁₃

D₄×C₁₆

prime radical

C₁₃×#135

C₈

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₁₃×C₃₂ ◁ C₂×C₁₃×C₃₂ ◁ G

N	H ≅ G/N	type
1. C₁₃×#2258	C₂	semidirect
3. C₁₃×#1422	C₂	semidirect
4. C₁₃×#1401	C₂	semidirect
5. C₄×C₁₃×C₃₂	C₂	semidirect
30. #13647	C₁₃	direct (N × H)
45. #2258	C₂×C₁₃	semidirect
47. #1422	C₂×C₁₃	semidirect
48. #1401	C₂×C₁₃	semidirect
49. C₄×C₃₂	C₂×C₁₃	semidirect
108. C₁₃	#13647	direct (N × H)

H ≅ G/N

type

1. C₁₃×#2258

C₂

semidirect

3. C₁₃×#1422

C₂

semidirect

4. C₁₃×#1401

C₂

semidirect

5. C₄×C₁₃×C₃₂

C₂

semidirect

30. #13647

C₁₃

direct (N × H)

45. #2258

C₂×C₁₃

semidirect

47. #1422

C₂×C₁₃

semidirect

48. #1401

C₂×C₁₃

semidirect

49. C₄×C₃₂

C₂×C₁₃

semidirect

108. C₁₃

#13647

direct (N × H)

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

subgroup	index
C₁₃×#2258	2
C₁₃×#2170	2
C₁₃×#1422	2
C₁₃×#1401	2
C₄×C₁₃×C₃₂	2
#13647	13

subgroup

index

C₁₃×#2258

C₁₃×#2170

C₁₃×#1422

C₁₃×#1401

C₄×C₁₃×C₃₂

#13647

Subgroups — C₁₃×#13647 #15699986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C13×#13647 #15699986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13647 #15699986