Group #13110618 — subgroups

Subgroups — C₁₃×#1560 #13110618

Order 1664.

254 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 96 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×#532	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#423	C₂	1	2	yes	maximal, complemented
3	C₁₃×#333	C₂	1	2	yes	maximal, complemented
4	C₁₃×#331	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#41	C₄	1	2	yes	—
6	C₂×C₁₃×M₁₆	—	2	2	no	—
7	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₁₃×#127	C₂²	1	1	yes	characteristic
9	C₁₃×#124	C₄	1	2	yes	—
10	C₁₃×#121	—	2	2	no	—
11	C₄×D₄×C₁₃	—	2	2	no	—
12	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	C₁₃×C₄⋊C₈	—	2	2	no	—
14	C₁₃×C₄⋊C₈	C₂²	1	4	yes	—
15	C₁₃×#41	—	4	4	no	—
16	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
17	C₂×D₄×C₁₃	—	2	2	no	—
18	C₂×D₄×C₁₃	—	2	2	no	—
19	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₁₃	—	2	2	no	—
21	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₃	D₄	1	2	yes	—
23	C₁₃×M₁₆	—	4	4	no	—
24	C₂×C₈×C₁₃	—	2	4	no	—
25	C₁₃×C₄⋊C₄	—	2	2	no	—
26	C₁₃×C₂²⋊C₄	—	2	4	no	—
27	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₁₃	D₄	1	2	yes	—
29	C₄²×C₁₃	C₂³	1	1	yes	characteristic
30	#1560	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂³×C₁₃	—	2	2	no	—
32	C₂³×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
33	Q₈×C₁₃	—	4	2	no	—
34	D₄×C₁₃	—	4	2	no	—
35	D₄×C₁₃	—	4	4	no	—
36	D₄×C₁₃	—	4	2	no	—
37	D₄×C₁₃	—	4	2	no	—
38	C₂×C₄×C₁₃	—	2	2	no	—
39	C₂×C₄×C₁₃	—	2	2	no	—
40	C₂×C₄×C₁₃	—	2	2	no	—
41	C₂×C₄×C₁₃	—	2	2	no	—
42	C₂×C₄×C₁₃	—	4	4	no	—
43	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
47	C₈×C₁₃	—	4	4	no	—
48	#532	C₂×C₁₃	1	1	yes	characteristic
49	#423	C₂×C₁₃	1	2	yes	complemented
50	#333	C₂×C₁₃	1	2	yes	complemented
51	#331	C₂×C₁₃	1	2	yes	complemented
52	C₂²×C₁₃	—	2	1	no	—
53	C₂²×C₁₃	—	2	2	no	—
54	C₂²×C₁₃	—	4	2	no	—
55	C₂²×C₁₃	—	4	4	no	—
56	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₄×C₁₃	—	2	1	no	—
58	C₄×C₁₃	—	2	2	no	—
59	C₄×C₁₃	—	4	2	no	—
60	C₄×C₁₃	#100	1	2	yes	—
61	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₁₃	#137	1	2	yes	—
63	C₂×#41	C₄×C₁₃	1	2	yes	—
64	C₂×M₁₆	—	2	2	no	—
65	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
66	#127	C₂²×C₁₃	1	1	yes	characteristic
67	#124	C₄×C₁₃	1	2	yes	—
68	#121	—	2	2	no	—
69	C₄×D₄	—	2	2	no	—
70	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
71	C₄⋊C₈	—	2	2	no	—
72	C₄⋊C₈	C₂²×C₁₃	1	4	yes	—
73	C₂×C₁₃	—	4	1	no	—
74	C₂×C₁₃	—	8	2	no	—
75	C₂×C₁₃	C₂×#100	1	1	yes	characteristic
76	C₂×C₁₃	#417	1	2	yes	—
77	#41	—	4	4	no	—
78	C₂×Q₈	C₂×C₄×C₁₃	1	2	yes	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	C₂×C₄×C₁₃	1	2	yes	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
84	C₂²×C₄	D₄×C₁₃	1	2	yes	—
85	M₁₆	—	4	4	no	—
86	C₂×C₈	—	2	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₂²⋊C₄	—	2	4	no	—
89	C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
90	C₄²	D₄×C₁₃	1	2	yes	—
91	C₄²	C₂³×C₁₃	1	1	yes	characteristic
92	C₁₃	#1560	1	1	yes	characteristic, direct product factor, Sylow
93	C₂³	—	2	2	no	—
94	C₂³	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
95	Q₈	—	4	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	4	no	—
98	D₄	—	4	2	no	—
99	D₄	—	4	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
106	C₂×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
108	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
109	C₈	—	4	4	no	—
110	C₂²	—	2	1	no	—
111	C₂²	—	2	2	no	—
112	C₂²	—	4	2	no	—
113	C₂²	—	4	4	no	—
114	C₂²	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	2	no	—
117	C₄	—	4	2	no	—
118	C₄	C₁₃×#100	1	2	yes	—
119	C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
120	C₄	C₁₃×#137	1	2	yes	—
121	C₂	—	4	1	no	—
122	C₂	—	8	2	no	—
123	C₂	C₂×C₁₃×#100	1	1	yes	characteristic
124	C₂	C₁₃×#417	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#532

C₂

yes

maximal, characteristic

C₁₃×#423

C₂

yes

maximal, complemented

C₁₃×#333

C₂

yes

maximal, complemented

C₁₃×#331

C₂

yes

maximal, complemented

C₂×C₁₃×#41

C₄

yes

—

C₂×C₁₃×M₁₆

—

C₁₃×C₄⋊Q₈

C₂²

yes

characteristic

C₁₃×#127

C₂²

yes

characteristic

C₁₃×#124

C₄

yes

—

C₁₃×#121

—

C₄×D₄×C₁₃

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

C₂²

yes

—

C₁₃×#41

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

D₄

yes

—

C₄²×C₁₃

C₂³

yes

characteristic

#1560

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

C₂²⋊C₄

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₈×C₁₃

—

#532

C₂×C₁₃

yes

characteristic

#423

C₂×C₁₃

yes

complemented

#333

C₂×C₁₃

yes

complemented

#331

C₂×C₁₃

yes

complemented

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#100

yes

—

C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₄×C₁₃

#137

yes

—

C₂×#41

C₄×C₁₃

yes

—

C₂×M₁₆

—

C₄⋊Q₈

C₂²×C₁₃

yes

characteristic

#127

C₂²×C₁₃

yes

characteristic

#124

C₄×C₁₃

yes

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂²×C₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₁₃

yes

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

C₂×#100

yes

characteristic

C₂×C₁₃

#417

yes

—

#41

—

C₂×Q₈

C₂×C₄×C₁₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₄×C₁₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₁₃

yes

characteristic

C₂²×C₄

D₄×C₁₃

yes

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×C₄×C₁₃

yes

characteristic

C₄²

D₄×C₁₃

yes

—

C₄²

C₂³×C₁₃

yes

characteristic

C₁₃

#1560

yes

characteristic, direct product factor, Sylow

C₂³

—

C₂³

C₁₃×C₂²⋊C₄

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₁₃×C₂²⋊C₄

yes

—

106

C₂×C₄

C₁₃×C₂²⋊C₄

yes

characteristic

107

C₂×C₄

C₂²×C₄×C₁₃

yes

characteristic

108

C₂×C₄

C₂×D₄×C₁₃

yes

—

109

C₈

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×C₁₃×C₂²⋊C₄

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

C₁₃×#100

yes

—

119

C₄

C₁₃×AGL(1,ℤ/8)

yes

—

120

C₄

C₁₃×#137

yes

—

121

C₂

—

122

C₂

—

123

C₂

C₂×C₁₃×#100

yes

characteristic

124

C₂

C₁₃×#417

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₄²	C₂³×C₁₃
prime radical	C₁₃×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₃

C₂×C₂²⋊C₄

commutator

C₂×C₄

C₂²×C₄×C₁₃

Frattini, cosocle

C₄²

C₂³×C₁₃

prime radical

C₁₃×#532

C₂

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

N	H ≅ G/N	type
2. C₁₃×#423	C₂	semidirect
3. C₁₃×#333	C₂	semidirect
4. C₁₃×#331	C₂	semidirect
30. #1560	C₁₃	direct (N × H)
49. #423	C₂×C₁₃	semidirect
50. #333	C₂×C₁₃	semidirect
51. #331	C₂×C₁₃	semidirect
92. C₁₃	#1560	direct (N × H)

H ≅ G/N

type

2. C₁₃×#423

C₂

semidirect

3. C₁₃×#333

C₂

semidirect

4. C₁₃×#331

C₂

semidirect

30. #1560

C₁₃

direct (N × H)

49. #423

C₂×C₁₃

semidirect

50. #333

C₂×C₁₃

semidirect

51. #331

C₂×C₁₃

semidirect

92. C₁₃

#1560

direct (N × H)

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

subgroup	index
C₁₃×#532	2
C₁₃×#423	2
C₁₃×#333	2
C₁₃×#331	2
#1560	13

subgroup

index

C₁₃×#532

C₁₃×#423

C₁₃×#333

C₁₃×#331

#1560

Subgroups — C₁₃×#1560 #13110618

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C13×#1560 #13110618

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#1560 #13110618