Group #13113568 — subgroups

Subgroups — #13113568

Order 1664.

254 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 67 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	#11719421	C₂	1	1	yes	maximal, characteristic
2	#11718498	C₂	1	2	yes	maximal, complemented
3	C₁₃×#331	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718136	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718073	C₂	1	2	yes	maximal, complemented
6	C₂×#6129	C₄	1	2	yes	—
7	C₁₃×#127	C₂²	1	1	yes	characteristic
8	#88192	—	2	2	no	—
9	#88190	C₄	1	2	yes	—
10	D₄×Dic₁₃	—	2	2	no	—
11	C₂×#6094	—	2	2	no	—
12	C₄²×D₁₃	C₂²	1	1	yes	characteristic
13	#88123	C₂²	1	1	yes	characteristic
14	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
15	#88054	—	2	2	no	—
16	C₄⋊#1055	C₂²	1	2	yes	—
17	C₂×D₄×C₁₃	—	2	2	no	—
18	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
19	C₂×C₂²⋊D₁₃	—	2	2	no	—
20	C₂²×Dic₁₃	—	2	2	no	—
21	#6129	—	4	4	no	—
22	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₁₃	D₄	1	2	yes	—
24	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₁₃	—	2	2	no	—
26	C₄²×C₁₃	C₂³	1	1	yes	characteristic
27	C₂²⋊Dic₁₃	—	2	4	no	—
28	C₄⋊Dic₁₃	—	2	2	no	—
29	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₁₃	D₄	1	2	yes	—
31	C₂×#1055	—	2	2	no	—
32	#6094	—	4	4	no	—
33	#1560	—	13	1	no	maximal, Sylow
34	C₂³×C₁₃	—	2	2	no	—
35	C₂²×D₁₃	C₂²⋊C₄	1	1	yes	characteristic
36	D₄×C₁₃	—	4	2	no	—
37	D₄×C₁₃	—	4	2	no	—
38	D₄×C₁₃	—	4	2	no	—
39	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
41	C₂²⋊D₁₃	—	4	4	no	—
42	C₂×Dic₁₃	—	2	2	no	—
43	C₂×Dic₁₃	—	4	4	no	—
44	C₂×Dic₁₃	C₂²⋊C₄	1	2	yes	—
45	C₂×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₄×D₁₃	—	2	2	no	—
47	C₄×D₁₃	—	2	2	no	—
48	C₄×D₁₃	—	2	2	no	—
49	Dic₂₆	—	4	2	no	—
50	C₈×C₁₃	—	4	2	no	—
51	#1055	—	4	2	no	—
52	#532	—	13	1	no	—
53	#423	—	13	2	no	—
54	#333	—	13	2	no	—
55	#331	—	13	1	no	—
56	#331	D₁₃	1	1	yes	characteristic, complemented
57	C₂²×C₁₃	—	4	2	no	—
58	C₂²×C₁₃	—	4	4	no	—
59	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×D₁₃	—	2	1	no	—
61	C₂×D₁₃	—	2	2	no	—
62	C₄×C₁₃	—	2	1	no	—
63	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
64	C₄×C₁₃	#137	1	2	yes	—
65	Dic₁₃	—	2	2	no	—
66	Dic₁₃	—	4	2	no	—
67	Dic₁₃	#100	1	2	yes	—
68	C₂×#41	—	13	2	no	—
69	C₂×M₁₆	—	26	2	no	—
70	C₄⋊Q₈	—	13	1	no	—
71	#127	C₂×D₁₃	1	1	yes	characteristic
72	#124	—	13	2	no	—
73	#121	—	26	2	no	—
74	C₄×D₄	—	26	2	no	—
75	C₂×C₄²	—	13	1	no	—
76	C₄⋊C₈	—	13	2	no	—
77	C₄⋊C₈	—	26	2	no	—
78	C₄⋊C₈	C₂×D₁₃	1	2	yes	—
79	C₂×C₁₃	—	8	2	no	—
80	C₂×C₁₃	C₂×#100	1	1	yes	characteristic
81	C₂×C₁₃	#417	1	2	yes	—
82	D₁₃	—	4	1	no	—
83	#41	—	52	4	no	—
84	C₂×Q₈	—	13	2	no	—
85	C₂×D₄	—	2	2	no	—
86	C₂×D₄	—	26	2	no	—
87	C₂×D₄	C₄×D₁₃	1	2	yes	—
88	C₂²×C₄	—	13	1	no	—
89	C₂²×C₄	—	13	2	no	—
90	C₂²×C₄	—	26	2	no	—
91	M₁₆	—	52	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	26	2	no	—
94	C₄⋊C₄	—	26	2	no	—
95	C₂²⋊C₄	—	26	4	no	—
96	C₄²	—	13	1	no	—
97	C₄²	—	13	2	no	—
98	C₄²	C₂²×D₁₃	1	1	yes	characteristic
99	C₁₃	#1560	1	1	yes	characteristic, complemented, Sylow
100	C₂³	—	2	2	no	—
101	C₂³	—	13	1	no	—
102	Q₈	—	52	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	2	no	—
106	D₄	—	52	4	no	—
107	C₂×C₄	—	13	1	no	—
108	C₂×C₄	—	13	2	no	—
109	C₂×C₄	—	26	2	no	—
110	C₂×C₄	—	26	2	no	—
111	C₂×C₄	—	26	2	no	—
112	C₂×C₄	—	26	2	no	—
113	C₂×C₄	—	52	4	no	—
114	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
115	C₂×C₄	D₄×D₁₃	1	2	yes	—
116	C₈	—	4	2	no	—
117	C₈	—	52	2	no	—
118	C₂²	—	4	2	no	—
119	C₂²	—	4	4	no	—
120	C₂²	—	26	1	no	—
121	C₂²	—	26	2	no	—
122	C₂²	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	13	2	no	—
125	C₄	—	26	2	no	—
126	C₄	—	52	2	no	—
127	C₄	#88165	1	2	yes	—
128	C₄	#88169	1	2	yes	—
129	C₂	—	8	2	no	—
130	C₂	—	52	1	no	—
131	C₂	D₁₃×#100	1	1	yes	characteristic
132	C₂	#11718432	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11719421

C₂

yes

maximal, characteristic

#11718498

C₂

yes

maximal, complemented

C₁₃×#331

C₂

yes

maximal, characteristic, complemented

#11718136

C₂

yes

maximal, characteristic, complemented

#11718073

C₂

yes

maximal, complemented

C₂×#6129

C₄

yes

—

C₁₃×#127

C₂²

yes

characteristic

#88192

—

#88190

C₄

yes

—

D₄×Dic₁₃

—

C₂×#6094

—

C₄²×D₁₃

C₂²

yes

characteristic

#88123

C₂²

yes

characteristic

C₁₃×C₄⋊C₈

C₂²

yes

—

#88054

—

C₄⋊#1055

C₂²

yes

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

C₂×C₄

yes

—

C₂×C₂²⋊D₁₃

—

C₂²×Dic₁₃

—

#6129

—

C₂×C₄×D₁₃

C₂×C₄

yes

characteristic

C₂×C₄×D₁₃

D₄

yes

—

C₂×Dic₂₆

C₂×C₄

yes

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

C₂³

yes

characteristic

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

C₂×C₄

yes

characteristic

C₄×Dic₁₃

D₄

yes

—

—

—

—

C₂³×C₁₃

—

C₂²×D₁₃

C₂²⋊C₄

yes

characteristic

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂²⋊C₄

yes

—

C₂×Dic₁₃

C₂²⋊C₄

yes

characteristic

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

Dic₂₆

—

C₈×C₁₃

—

—

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₄×C₁₃

#137

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

#100

yes

—

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

C₂×D₁₃

yes

characteristic

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₁₃

yes

—

C₂×C₁₃

—

C₂×C₁₃

C₂×#100

yes

characteristic

C₂×C₁₃

#417

yes

—

D₁₃

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₄×D₁₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₁₃

yes

characteristic

C₁₃

#1560

yes

characteristic, complemented, Sylow

100

C₂³

—

101

C₂³

—

102

Q₈

—

103

D₄

—

104

D₄

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×C₄×D₁₃

yes

characteristic

115

C₂×C₄

D₄×D₁₃

yes

—

116

C₈

—

117

C₈

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂²⋊C₄×D₁₃

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

#88165

yes

—

128

C₄

#88169

yes

—

129

C₂

—

130

C₂

—

131

C₂

D₁₃×#100

yes

characteristic

132

C₂

#11718432

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₄²	C₂²×D₁₃
Fitting	C₁₃×#331	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₄
prime radical	#11719421	C₂
cosocle	C₄²×C₁₃	C₂³
hypercenter	#331	D₁₃
hypocenter	C₁₃	#1560
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₃

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₄²

C₂²×D₁₃

Fitting

C₁₃×#331

C₂

socle

C₂²×C₁₃

C₂×C₂²⋊C₄

prime radical

#11719421

C₂

cosocle

C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #11718498	C₂	semidirect
3. C₁₃×#331	C₂	semidirect
4. #11718136	C₂	semidirect
5. #11718073	C₂	semidirect
56. #331	D₁₃	semidirect
99. C₁₃	#1560	semidirect

H ≅ G/N

type

2. #11718498

C₂

semidirect

3. C₁₃×#331

C₂

semidirect

4. #11718136

C₂

semidirect

5. #11718073

C₂

semidirect

56. #331

D₁₃

semidirect

99. C₁₃

#1560

semidirect

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

subgroup	index
#11719421	2
#11718498	2
C₁₃×#331	2
#11718136	2
#11718073	2
#1560	13

subgroup

index

C₁₃×#331

Subgroups — #13113568

Conjugacy classes of subgroups

Complements

Maximal subgroups