Group #13110682 — subgroups

Subgroups — C₁₃×#1624 #13110682

Order 1664.

240 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 68 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×#555	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#459	C₂	1	2	yes	maximal, complemented
4	C₁₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×D₄⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×SD₁₆	—	2	2	no	—
7	C₂×C₁₃×D₈	—	2	2	no	—
8	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₁₃×#127	C₂²	1	1	yes	characteristic
10	C₁₃×#124	—	2	2	no	—
11	C₁₃×#121	—	2	2	no	—
12	C₄×D₄×C₁₃	C₂²	1	2	yes	—
13	C₁₃×#117	—	2	2	no	—
14	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₁₃×#102	—	2	2	no	—
16	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₁₃	D₄	1	2	yes	—
18	C₂×D₄×C₁₃	—	2	2	no	—
19	C₂×D₄×C₁₃	—	2	2	no	—
20	C₂×D₄×C₁₃	—	2	1	no	—
21	C₂×D₄×C₁₃	D₄	1	2	yes	—
22	C₂²×C₄×C₁₃	—	2	2	no	—
23	C₁₃×SD₁₆	—	4	4	no	—
24	C₁₃×D₈	—	4	4	no	—
25	C₂×C₈×C₁₃	—	2	1	no	—
26	C₂×C₈×C₁₃	—	2	2	no	—
27	C₁₃×C₄⋊C₄	—	2	2	no	—
28	C₁₃×C₄⋊C₄	D₄	1	2	yes	—
29	C₁₃×C₂²⋊C₄	—	2	4	no	—
30	C₁₃×C₂²⋊C₄	—	2	2	no	—
31	C₄²×C₁₃	—	2	2	no	—
32	C₄²×C₁₃	C₂³	1	1	yes	characteristic
33	#1624	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂³×C₁₃	—	2	2	no	—
35	C₂³×C₁₃	—	4	1	no	—
36	Q₈×C₁₃	—	4	2	no	—
37	D₄×C₁₃	—	2	2	no	—
38	D₄×C₁₃	—	2	2	no	—
39	D₄×C₁₃	—	4	2	no	—
40	D₄×C₁₃	—	4	2	no	—
41	D₄×C₁₃	—	4	2	no	—
42	D₄×C₁₃	—	8	2	no	—
43	C₂×C₄×C₁₃	—	2	2	no	—
44	C₂×C₄×C₁₃	—	2	2	no	—
45	C₂×C₄×C₁₃	—	4	2	no	—
46	C₂×C₄×C₁₃	—	4	2	no	—
47	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
49	C₈×C₁₃	—	2	2	no	—
50	C₈×C₁₃	—	4	2	no	—
51	#555	C₂×C₁₃	1	1	yes	characteristic, complemented
52	#492	C₂×C₁₃	1	1	yes	characteristic, complemented
53	#459	C₂×C₁₃	1	2	yes	complemented
54	#333	C₂×C₁₃	1	1	yes	characteristic, complemented
55	D₄⋊C₈	C₂×C₁₃	1	2	yes	complemented
56	C₂²×C₁₃	—	4	2	no	—
57	C₂²×C₁₃	—	4	2	no	—
58	C₂²×C₁₃	—	4	2	no	—
59	C₂²×C₁₃	—	8	2	no	—
60	C₂²×C₁₃	—	8	1	no	—
61	C₂²×C₁₃	#120	1	1	yes	characteristic
62	C₄×C₁₃	—	2	1	no	—
63	C₄×C₁₃	—	4	2	no	—
64	C₄×C₁₃	—	4	2	no	—
65	C₄×C₁₃	#135	1	2	yes	—
66	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
67	C₂×SD₁₆	—	2	2	no	—
68	C₂×D₈	—	2	2	no	—
69	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
70	#127	C₂²×C₁₃	1	1	yes	characteristic
71	#124	—	2	2	no	—
72	#121	—	2	2	no	—
73	C₄×D₄	C₂²×C₁₃	1	2	yes	—
74	#117	—	2	2	no	—
75	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
76	#102	—	2	2	no	—
77	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
78	C₂×C₁₃	—	8	2	no	—
79	C₂×C₁₃	—	16	1	no	—
80	C₂×C₁₃	#449	1	2	yes	—
81	C₂×C₁₃	#453	1	1	yes	characteristic
82	C₂×Q₈	D₄×C₁₃	1	2	yes	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	D₄×C₁₃	1	2	yes	—
87	C₂²×C₄	—	2	2	no	—
88	SD₁₆	—	4	4	no	—
89	D₈	—	4	4	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	D₄×C₁₃	1	2	yes	—
94	C₂²⋊C₄	—	2	4	no	—
95	C₂²⋊C₄	—	2	2	no	—
96	C₄²	—	2	2	no	—
97	C₄²	C₂³×C₁₃	1	1	yes	characteristic
98	C₁₃	#1624	1	1	yes	characteristic, direct product factor, Sylow
99	C₂³	—	2	2	no	—
100	C₂³	—	4	1	no	—
101	Q₈	—	4	2	no	—
102	D₄	—	2	2	no	—
103	D₄	—	2	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	2	no	—
106	D₄	—	4	2	no	—
107	D₄	—	8	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
113	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	4	2	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	4	2	no	—
118	C₂²	—	4	2	no	—
119	C₂²	—	8	2	no	—
120	C₂²	—	8	1	no	—
121	C₂²	C₁₃×#120	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	4	2	no	—
124	C₄	—	4	2	no	—
125	C₄	C₁₃×#135	1	2	yes	—
126	C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
127	C₂	—	8	2	no	—
128	C₂	—	16	1	no	—
129	C₂	C₁₃×#449	1	2	yes	—
130	C₂	C₁₃×#453	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#555

C₂

yes

maximal, characteristic, complemented

C₁₃×#492

C₂

yes

maximal, characteristic, complemented

C₁₃×#459

C₂

yes

maximal, complemented

C₁₃×#333

C₂

yes

maximal, characteristic, complemented

C₁₃×D₄⋊C₈

C₂

yes

maximal, complemented

C₂×C₁₃×SD₁₆

—

C₂×C₁₃×D₈

—

C₁₃×C₄⋊Q₈

C₂²

yes

characteristic

C₁₃×#127

C₂²

yes

characteristic

C₁₃×#124

—

C₁₃×#121

—

C₄×D₄×C₁₃

C₂²

yes

—

C₁₃×#117

—

C₁₃×C₄⋊C₈

C₂²

yes

—

C₁₃×#102

—

C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×Q₈×C₁₃

D₄

yes

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

—

C₁₃×SD₁₆

—

C₁₃×D₈

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

D₄

yes

—

C₁₃×C₂²⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂³

yes

characteristic

#1624

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×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×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

#555

C₂×C₁₃

yes

characteristic, complemented

#492

C₂×C₁₃

yes

characteristic, complemented

#459

C₂×C₁₃

yes

complemented

#333

C₂×C₁₃

yes

characteristic, complemented

D₄⋊C₈

C₂×C₁₃

yes

complemented

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#120

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#135

yes

—

C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

C₂²×C₁₃

yes

characteristic

#127

C₂²×C₁₃

yes

characteristic

#124

—

#121

—

C₄×D₄

C₂²×C₁₃

yes

—

#117

—

C₄⋊C₈

C₂²×C₁₃

yes

—

#102

—

C₄×C₈

C₂²×C₁₃

yes

characteristic

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

#449

yes

—

C₂×C₁₃

#453

yes

characteristic

C₂×Q₈

D₄×C₁₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₄×C₁₃

yes

—

C₂²×C₄

—

SD₁₆

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₁₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂³×C₁₃

yes

characteristic

C₁₃

#1624

yes

characteristic, direct product factor, Sylow

C₂³

—

100

C₂³

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

106

D₄

—

107

D₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂×D₄×C₁₃

yes

characteristic

113

C₂×C₄

C₂×D₄×C₁₃

yes

—

114

C₈

—

115

C₈

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₁₃×#120

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

C₁₃×#135

yes

—

126

C₄

C₁₃×AGL(1,ℤ/8)

yes

—

127

C₂

—

128

C₂

—

129

C₂

C₁₃×#449

yes

—

130

C₂

C₁₃×#453

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₁₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄²×C₁₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₃

#120

commutator, Frattini, cosocle

C₄²

C₂³×C₁₃

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₃ ◁ C₄²×C₁₃ ◁ G

N	H ≅ G/N	type
1. C₁₃×#555	C₂	semidirect
2. C₁₃×#492	C₂	semidirect
3. C₁₃×#459	C₂	semidirect
4. C₁₃×#333	C₂	semidirect
5. C₁₃×D₄⋊C₈	C₂	semidirect
33. #1624	C₁₃	direct (N × H)
51. #555	C₂×C₁₃	semidirect
52. #492	C₂×C₁₃	semidirect
53. #459	C₂×C₁₃	semidirect
54. #333	C₂×C₁₃	semidirect
55. D₄⋊C₈	C₂×C₁₃	semidirect
98. C₁₃	#1624	direct (N × H)

H ≅ G/N

type

1. C₁₃×#555

C₂

semidirect

2. C₁₃×#492

C₂

semidirect

3. C₁₃×#459

C₂

semidirect

4. C₁₃×#333

C₂

semidirect

5. C₁₃×D₄⋊C₈

C₂

semidirect

33. #1624

C₁₃

direct (N × H)

51. #555

C₂×C₁₃

semidirect

52. #492

C₂×C₁₃

semidirect

53. #459

C₂×C₁₃

semidirect

54. #333

C₂×C₁₃

semidirect

55. D₄⋊C₈

C₂×C₁₃

semidirect

98. C₁₃

#1624

direct (N × H)

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

subgroup	index
C₁₃×#555	2
C₁₃×#492	2
C₁₃×#459	2
C₁₃×#333	2
C₁₃×D₄⋊C₈	2
#1624	13

subgroup

index

C₁₃×#555

C₁₃×#492

C₁₃×#459

C₁₃×#333

C₁₃×D₄⋊C₈

#1624

Subgroups — C₁₃×#1624 #13110682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C13×#1624 #13110682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#1624 #13110682