Group #13116451 — subgroups

Subgroups — #13116451

Order 1664.

228 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 93 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×#88155	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#6090	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718099	C₂	1	4	yes	maximal, complemented
5	C₂×#6127	C₄	1	1	yes	characteristic
6	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×#6099	C₄	1	1	yes	characteristic
8	#88155	—	2	2	no	—
9	#88155	—	2	2	no	—
10	C₂×#6094	C₄	1	2	yes	—
11	C₂×C₈×D₁₃	C₂²	1	2	yes	—
12	C₁₃×M₃₂	—	2	2	no	—
13	C₂×C₁₃×C₁₆	C₂²	1	2	yes	—
14	C₂×#6090	—	2	2	no	—
15	C₂×#6090	C₂²	1	2	yes	—
16	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₂²⋊D₁₃	C₈	1	2	yes	—
18	#6127	—	2	2	no	—
19	#6127	—	2	2	no	—
20	C₂×D₅₂	C₈	1	1	yes	characteristic
21	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
22	C₂×Dic₂₆	C₈	1	1	yes	characteristic
23	C₂×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₃	D₄	1	2	yes	—
25	C₂×C₈×C₁₃	D₄	1	2	yes	—
26	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	#6099	—	2	2	no	—
28	C₂×#1055	C₂×C₄	1	2	yes	—
29	#6094	—	2	4	no	—
30	C₈×D₁₃	—	2	2	no	—
31	C₈×D₁₃	—	2	2	no	—
32	C₁₃×C₁₆	—	2	2	no	—
33	#6090	—	2	2	no	—
34	#2113	—	13	1	no	maximal, Sylow
35	C₂³×C₁₃	C₂×C₈	1	1	yes	characteristic
36	C₂²×D₁₃	C₂×C₈	1	2	yes	—
37	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
39	C₂×C₄×C₁₃	M₁₆	1	2	yes	—
40	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²⋊D₁₃	—	2	4	no	—
42	C₂×Dic₁₃	C₂×C₈	1	2	yes	—
43	D₅₂	—	2	2	no	—
44	C₄×D₁₃	—	2	2	no	—
45	C₄×D₁₃	—	2	2	no	—
46	Dic₂₆	—	2	2	no	—
47	C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
48	C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
50	#1055	—	2	2	no	—
51	C₂×#131	—	13	1	no	—
52	C₂×M₃₂	D₁₃	1	1	yes	characteristic, complemented
53	C₂²×C₁₆	—	13	1	no	—
54	C₂²⋊C₁₆	—	13	4	no	—
55	C₂²×C₁₃	—	2	2	no	—
56	C₂²×C₁₃	C₂²⋊C₈	1	2	yes	—
57	C₂²×C₁₃	C₂²×C₈	1	1	yes	characteristic
58	C₂×D₁₃	—	2	2	no	—
59	C₂×D₁₃	—	2	4	no	—
60	C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
61	C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
63	Dic₁₃	—	2	2	no	—
64	C₂×#41	—	13	1	no	—
65	#131	—	26	2	no	—
66	#131	—	26	2	no	—
67	C₂×M₁₆	—	13	1	no	—
68	C₂×M₁₆	—	13	2	no	—
69	C₂²×C₈	—	13	2	no	—
70	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
71	M₃₂	—	2	2	no	—
72	C₂×C₁₆	—	13	2	no	—
73	C₂×C₁₆	—	26	2	no	—
74	C₂×C₁₆	C₂×D₁₃	1	2	yes	—
75	C₂×C₁₃	—	2	2	no	—
76	C₂×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
77	C₂×C₁₃	#504	1	2	yes	—
78	D₁₃	—	4	2	no	—
79	#41	—	26	2	no	—
80	#41	—	26	2	no	—
81	C₂×Q₈	—	13	1	no	—
82	C₂×D₄	—	13	2	no	—
83	C₂×D₄	—	13	1	no	—
84	C₂²×C₄	—	13	2	no	—
85	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
86	M₁₆	—	26	2	no	—
87	M₁₆	—	26	4	no	—
88	C₂×C₈	—	13	2	no	—
89	C₂×C₈	—	26	2	no	—
90	C₂×C₈	—	26	2	no	—
91	C₂×C₈	C₄×D₁₃	1	1	yes	characteristic
92	C₂×C₈	D₅₂	1	2	yes	—
93	C₂×C₈	C₂²⋊D₁₃	1	2	yes	—
94	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
95	C₁₆	—	2	2	no	—
96	C₁₆	—	26	2	no	—
97	C₁₃	#2113	1	1	yes	characteristic, complemented, Sylow
98	C₂³	—	13	2	no	—
99	C₂³	C₈×D₁₃	1	1	yes	characteristic
100	Q₈	—	26	2	no	—
101	D₄	—	26	4	no	—
102	D₄	—	26	2	no	—
103	C₂×C₄	—	13	2	no	—
104	C₂×C₄	—	26	2	no	—
105	C₂×C₄	—	26	2	no	—
106	C₂×C₄	C₈×D₁₃	1	1	yes	characteristic
107	C₂×C₄	#6094	1	2	yes	—
108	C₂×C₄	#6103	1	2	yes	—
109	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
110	C₈	—	26	2	no	—
111	C₈	#6103	1	2	yes	—
112	C₈	C₂×D₅₂	1	1	yes	characteristic
113	C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
114	C₂²	—	2	2	no	—
115	C₂²	—	26	2	no	—
116	C₂²	—	26	4	no	—
117	C₂²	#88060	1	2	yes	—
118	C₂²	C₂×C₈×D₁₃	1	1	yes	characteristic
119	C₄	—	26	2	no	—
120	C₄	#88060	1	2	yes	—
121	C₄	C₂×#6094	1	1	yes	characteristic
122	C₄	C₂×#6103	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	52	2	no	—
125	C₂	#11718568	1	2	yes	—
126	C₂	C₂×#88060	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88155

C₂

yes

maximal, characteristic

C₂×C₁₃×M₃₂

C₂

yes

maximal, characteristic, complemented

C₂²×#6090

C₂

yes

maximal, characteristic, complemented

#11718099

C₂

yes

maximal, complemented

C₂×#6127

C₄

yes

characteristic

C₂²×C₈×C₁₃

C₂²

yes

characteristic

C₂×#6099

C₄

yes

—

—

C₄

yes

—

C₂×C₈×D₁₃

C₂²

yes

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

C₂²

yes

—

C₂×#6090

—

C₂×#6090

C₂²

yes

—

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₁₃

C₈

yes

—

#6127

—

#6127

—

C₂×D₅₂

C₈

yes

characteristic

C₂×C₄×D₁₃

C₂×C₄

yes

—

C₂×Dic₂₆

C₈

yes

characteristic

C₂×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃

D₄

yes

—

C₂×C₈×C₁₃

D₄

yes

—

C₂×C₈×C₁₃

C₂³

yes

characteristic

#6099

—

C₂×#1055

C₂×C₄

yes

—

#6094

—

C₈×D₁₃

—

C₈×D₁₃

—

C₁₃×C₁₆

—

#6090

—

#2113

—

maximal, Sylow

C₂³×C₁₃

C₂×C₈

yes

characteristic

C₂²×D₁₃

C₂×C₈

yes

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃

M₁₆

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²⋊D₁₃

—

C₂×Dic₁₃

C₂×C₈

yes

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

Dic₂₆

—

C₈×C₁₃

C₂²⋊C₄

yes

—

C₈×C₁₃

C₂×D₄

yes

characteristic

C₈×C₁₃

C₂×D₄

yes

characteristic

#1055

—

C₂×#131

—

C₂×M₃₂

D₁₃

yes

characteristic, complemented

C₂²×C₁₆

—

C₂²⋊C₁₆

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂²⋊C₈

yes

—

C₂²×C₁₃

C₂²×C₈

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

C₂²⋊C₈

yes

—

C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₃

C₂×M₁₆

yes

characteristic

Dic₁₃

—

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₁₃

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₁₃

yes

—

C₂×C₁₃

—

C₂×C₁₃

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₁₃

#504

yes

—

D₁₃

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₁₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₁₃

yes

characteristic

C₂×C₈

D₅₂

yes

—

C₂×C₈

C₂²⋊D₁₃

yes

—

C₂×C₈

C₂²×D₁₃

yes

characteristic

C₁₆

—

C₁₆

—

C₁₃

#2113

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

C₈×D₁₃

yes

characteristic

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₈×D₁₃

yes

characteristic

107

C₂×C₄

#6094

yes

—

108

C₂×C₄

#6103

yes

—

109

C₂×C₄

C₂×C₄×D₁₃

yes

characteristic

110

C₈

—

111

C₈

#6103

yes

—

112

C₈

C₂×D₅₂

yes

characteristic

113

C₈

C₂×C₂²⋊D₁₃

yes

characteristic

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

#88060

yes

—

118

C₂²

C₂×C₈×D₁₃

yes

characteristic

119

C₄

—

120

C₄

#88060

yes

—

121

C₄

C₂×#6094

yes

characteristic

122

C₄

C₂×#6103

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#11718568

yes

—

126

C₂

C₂×#88060

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₃
commutator, socle	C₂²×C₁₃	C₂²×C₈
Fitting	C₂×C₁₃×M₃₂	C₂
prime radical	C₂×#6127	C₄
cosocle	C₂×C₈×C₁₃	C₂³
hypercenter	C₂×M₃₂	D₁₃
hypocenter	C₁₃	#2113
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₃

commutator, socle

C₂²×C₁₃

C₂²×C₈

Fitting

C₂×C₁₃×M₃₂

C₂

prime radical

C₂×#6127

C₄

cosocle

C₂×C₈×C₁₃

C₂³

hypercenter

C₂×M₃₂

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
2. C₂×C₁₃×M₃₂	C₂	semidirect
3. C₂²×#6090	C₂	semidirect
4. #11718099	C₂	semidirect
52. C₂×M₃₂	D₁₃	semidirect
97. C₁₃	#2113	semidirect

H ≅ G/N

type

2. C₂×C₁₃×M₃₂

C₂

semidirect

3. C₂²×#6090

C₂

semidirect

4. #11718099

C₂

semidirect

52. C₂×M₃₂

D₁₃

semidirect

97. C₁₃

#2113

semidirect

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

subgroup	index
C₂×#88155	2
C₂×C₁₃×M₃₂	2
C₂²×#6090	2
#11718099	2
#2113	13

subgroup

index

C₂×#88155

C₂×C₁₃×M₃₂

C₂²×#6090

#11718099

#2113

Subgroups — #13116451

Conjugacy classes of subgroups

Complements

Maximal subgroups