Group #13110871 — subgroups

Subgroups — C₁₃×#1813 #13110871

Order 1664.

252 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 116 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×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#108	C₂	1	2	yes	maximal, complemented
3	C₁₃×#413	C₂	1	2	yes	maximal
4	C₁₃×#338	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
6	C₁₃×#131	—	2	4	no	—
7	C₂×C₁₃×M₁₆	C₄	1	2	yes	—
8	C₂×C₁₃×M₁₆	C₂²	1	4	yes	—
9	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₃	C₄	1	2	yes	—
11	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₁₃×#108	—	2	4	no	—
13	C₁₃×#101	—	2	2	no	—
14	C₁₃×#100	—	2	2	no	—
15	C₁₃×#41	—	2	2	no	—
16	C₁₃×#41	—	2	2	no	—
17	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
18	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
19	C₂×D₄×C₁₃	Q₈	1	2	yes	—
20	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₁₃×M₁₆	—	2	2	no	—
23	C₁₃×M₁₆	—	2	4	no	—
24	C₁₃×M₁₆	—	2	4	no	—
25	C₁₃×M₁₆	—	2	4	no	—
26	C₂×C₈×C₁₃	—	2	4	no	—
27	C₂×C₈×C₁₃	—	2	4	no	—
28	C₂×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₃	D₄	1	4	yes	—
32	#1813	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂³×C₁₃	C₄⋊C₄	1	2	yes	—
34	C₂³×C₁₃	C₂×Q₈	1	1	yes	characteristic
35	Q₈×C₁₃	—	2	2	no	—
36	D₄×C₁₃	—	2	4	no	—
37	D₄×C₁₃	—	2	2	no	—
38	C₂×C₄×C₁₃	—	2	2	no	—
39	C₂×C₄×C₁₃	—	2	2	no	—
40	C₂×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₃	#41	1	2	yes	—
45	C₈×C₁₃	—	2	2	no	—
46	C₈×C₁₃	—	4	4	no	—
47	C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
48	C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
49	C₂×#131	C₂×C₁₃	1	1	yes	characteristic
50	C₂×#108	C₂×C₁₃	1	2	yes	complemented
51	#413	C₂×C₁₃	1	2	yes	—
52	#338	C₂×C₁₃	1	2	yes	complemented
53	C₂²×C₁₃	—	2	2	no	—
54	C₂²×C₁₃	—	4	1	no	—
55	C₂²×C₁₃	—	4	2	no	—
56	C₂²×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₃	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₁₃	—	2	2	no	—
59	C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₁₃	#117	1	1	yes	characteristic
61	C₄×C₁₃	#121	1	2	yes	—
62	C₂×#41	C₂²×C₁₃	1	1	yes	characteristic
63	#131	—	2	4	no	—
64	C₂×M₁₆	C₄×C₁₃	1	2	yes	—
65	C₂×M₁₆	C₂²×C₁₃	1	4	yes	—
66	C₂×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
67	C₂²×C₈	C₄×C₁₃	1	2	yes	—
68	C₂²×C₈	C₂²×C₁₃	1	1	yes	characteristic
69	#108	—	2	4	no	—
70	#101	—	2	2	no	—
71	#100	—	2	2	no	—
72	C₂×C₁₃	—	2	1	no	—
73	C₂×C₁₃	—	2	2	no	—
74	C₂×C₁₃	—	4	2	no	—
75	C₂×C₁₃	#380	1	1	yes	characteristic
76	#41	—	2	2	no	—
77	#41	—	2	2	no	—
78	C₂×Q₈	D₄×C₁₃	1	1	yes	characteristic
79	C₂×D₄	D₄×C₁₃	1	1	yes	characteristic
80	C₂×D₄	Q₈×C₁₃	1	2	yes	—
81	C₂²×C₄	C₂×C₄×C₁₃	1	2	yes	—
82	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
83	M₁₆	—	2	2	no	—
84	M₁₆	—	2	4	no	—
85	M₁₆	—	2	4	no	—
86	M₁₆	—	2	4	no	—
87	C₂×C₈	—	2	4	no	—
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	C₂×C₄×C₁₃	1	2	yes	—
90	C₂×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
91	C₂×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
92	C₂×C₈	D₄×C₁₃	1	4	yes	—
93	C₁₃	#1813	1	1	yes	characteristic, direct product factor, Sylow
94	C₂³	C₁₃×C₄⋊C₄	1	2	yes	—
95	C₂³	C₂×Q₈×C₁₃	1	1	yes	characteristic
96	Q₈	—	2	2	no	—
97	D₄	—	2	4	no	—
98	D₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	C₁₃×C₄⋊C₄	1	2	yes	—
102	C₂×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
103	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
104	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
105	C₂×C₄	C₁₃×#41	1	2	yes	—
106	C₈	—	2	2	no	—
107	C₈	—	4	4	no	—
108	C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
109	C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
110	C₂²	—	2	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	4	2	no	—
113	C₂²	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
114	C₂²	C₁₃×C₂²⋊Q₈	1	2	yes	—
115	C₄	—	2	2	no	—
116	C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
117	C₄	C₁₃×#117	1	1	yes	characteristic
118	C₄	C₁₃×#121	1	2	yes	—
119	C₂	—	2	1	no	—
120	C₂	—	2	2	no	—
121	C₂	—	4	2	no	—
122	C₂	C₁₃×#380	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#131

C₂

yes

maximal, characteristic

C₂×C₁₃×#108

C₂

yes

maximal, complemented

C₁₃×#413

C₂

yes

maximal

C₁₃×#338

C₂

yes

maximal, complemented

C₂×C₁₃×#41

C₂²

yes

characteristic

C₁₃×#131

—

C₂×C₁₃×M₁₆

C₄

yes

—

C₂×C₁₃×M₁₆

C₂²

yes

—

C₂×C₁₃×M₁₆

C₂²

yes

characteristic

C₂²×C₈×C₁₃

C₄

yes

—

C₂²×C₈×C₁₃

C₂²

yes

characteristic

C₁₃×#108

—

C₁₃×#101

—

C₁₃×#100

—

C₁₃×#41

—

C₁₃×#41

—

C₂×Q₈×C₁₃

D₄

yes

characteristic

C₂×D₄×C₁₃

D₄

yes

characteristic

C₂×D₄×C₁₃

Q₈

yes

—

C₂²×C₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃

D₄

yes

—

#1813

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₃

C₄⋊C₄

yes

—

C₂³×C₁₃

C₂×Q₈

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

#41

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

C₂²⋊C₄

yes

—

C₈×C₁₃

C₂²⋊C₄

yes

—

C₂×#131

C₂×C₁₃

yes

characteristic

C₂×#108

C₂×C₁₃

yes

complemented

#413

C₂×C₁₃

yes

—

#338

C₂×C₁₃

yes

complemented

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃

C₂²⋊Q₈

yes

—

C₄×C₁₃

—

C₄×C₁₃

C₂×C₂²⋊C₄

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

C₂×#41

C₂²×C₁₃

yes

characteristic

#131

—

C₂×M₁₆

C₄×C₁₃

yes

—

C₂×M₁₆

C₂²×C₁₃

yes

—

C₂×M₁₆

C₂²×C₁₃

yes

characteristic

C₂²×C₈

C₄×C₁₃

yes

—

C₂²×C₈

C₂²×C₁₃

yes

—

—

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

yes

—

—

C₂×Q₈

D₄×C₁₃

yes

characteristic

C₂×D₄

D₄×C₁₃

yes

characteristic

C₂×D₄

Q₈×C₁₃

yes

—

C₂²×C₄

C₂×C₄×C₁₃

yes

—

C₂²×C₄

C₂³×C₁₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₁₃

yes

—

C₂×C₈

C₂×C₄×C₁₃

yes

characteristic

C₂×C₈

C₂×C₄×C₁₃

yes

characteristic

C₂×C₈

D₄×C₁₃

yes

—

C₁₃

#1813

yes

characteristic, direct product factor, Sylow

C₂³

C₁₃×C₄⋊C₄

yes

—

C₂³

C₂×Q₈×C₁₃

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₁₃×C₄⋊C₄

yes

—

102

C₂×C₄

C₂²×C₄×C₁₃

yes

characteristic

103

C₂×C₄

C₂×D₄×C₁₃

yes

—

104

C₂×C₄

C₂×D₄×C₁₃

yes

characteristic

105

C₂×C₄

C₁₃×#41

yes

—

106

C₈

—

107

C₈

—

108

C₈

C₁₃×C₂²⋊C₄

yes

—

109

C₈

C₁₃×C₂²⋊C₄

yes

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

C₂×C₁₃×C₄⋊C₄

yes

characteristic

114

C₂²

C₁₃×C₂²⋊Q₈

yes

—

115

C₄

—

116

C₄

C₂×C₁₃×C₂²⋊C₄

yes

characteristic

117

C₄

C₁₃×#117

yes

characteristic

118

C₄

C₁₃×#121

yes

—

119

C₂

—

120

C₂

—

121

C₂

—

122

C₂

C₁₃×#380

yes

characteristic

	subgroup	G/H
center	C₄×C₁₃	#117
commutator	C₂×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₄	C₂³×C₁₃
socle	C₂×C₁₃	#380
prime radical	C₂×C₁₃×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂×C₁₃×M₁₆ ◁ G

subgroup

G/H

center

C₄×C₁₃

#117

commutator

C₂×C₄

C₂²×C₄×C₁₃

Frattini, cosocle

C₂²×C₄

C₂³×C₁₃

socle

C₂×C₁₃

#380

prime radical

C₂×C₁₃×#41

C₂²

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₁₃ ◁ C₂×C₁₃×M₁₆ ◁ G

N	H ≅ G/N	type
2. C₂×C₁₃×#108	C₂	semidirect
4. C₁₃×#338	C₂	semidirect
32. #1813	C₁₃	direct (N × H)
50. C₂×#108	C₂×C₁₃	semidirect
52. #338	C₂×C₁₃	semidirect
93. C₁₃	#1813	direct (N × H)

H ≅ G/N

type

2. C₂×C₁₃×#108

C₂

semidirect

4. C₁₃×#338

C₂

semidirect

32. #1813

C₁₃

direct (N × H)

50. C₂×#108

C₂×C₁₃

semidirect

52. #338

C₂×C₁₃

semidirect

93. C₁₃

#1813

direct (N × H)

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

subgroup	index
C₂×C₁₃×#131	2
C₂×C₁₃×#108	2
C₁₃×#413	2
C₁₃×#338	2
#1813	13

subgroup

index

C₂×C₁₃×#131

C₂×C₁₃×#108

C₁₃×#413

C₁₃×#338

#1813

Subgroups — C₁₃×#1813 #13110871

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C13×#1813 #13110871

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#1813 #13110871