Group #13131068 — subgroups

Subgroups — #13131068

Order 1664.

228 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 82 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×#88161	C₂	1	1	yes	maximal, characteristic
2	C₂×#88106	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#6092	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718259	C₂	1	4	yes	maximal, complemented
5	C₂×#6127	C₄	1	1	yes	characteristic
6	C₂×C₁₃×M₁₆	C₄	1	1	yes	characteristic
7	C₂²×#1055	C₂²	1	1	yes	characteristic
8	#88161	—	2	2	no	—
9	#88161	—	2	2	no	—
10	C₂×#6094	C₄	1	2	yes	—
11	C₂×C₈×D₁₃	C₂²	1	2	yes	—
12	#88106	—	2	2	no	—
13	C₂×#6092	—	2	2	no	—
14	C₂×#6092	C₂²	1	2	yes	—
15	C₂×#6092	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	C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₂×#1055	C₂×C₄	1	1	yes	characteristic
26	C₂×#1055	D₄	1	2	yes	—
27	C₂×#1055	D₄	1	2	yes	—
28	C₂×#1055	C₂³	1	1	yes	characteristic
29	#6094	—	2	4	no	—
30	C₈×D₁₃	—	2	2	no	—
31	C₈×D₁₃	—	2	2	no	—
32	#6092	—	2	2	no	—
33	#6092	—	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	—	2	2	no	—
48	#1055	C₂²⋊C₄	1	2	yes	—
49	#1055	C₂×D₄	1	1	yes	characteristic
50	#1055	C₂×D₄	1	1	yes	characteristic
51	C₂×#131	—	13	1	no	—
52	C₂×M₃₂	—	13	1	no	—
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	2	no	—
68	C₂×M₁₆	#261	1	1	yes	characteristic
69	C₂²×C₈	—	13	1	no	—
70	C₂²×C₈	—	13	2	no	—
71	M₃₂	—	26	2	no	—
72	C₂×C₁₆	—	13	2	no	—
73	C₂×C₁₆	—	13	2	no	—
74	C₂×C₁₆	—	26	2	no	—
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	C₂×#261	1	1	yes	characteristic
86	M₁₆	—	2	2	no	—
87	M₁₆	—	26	4	no	—
88	C₂×C₈	—	13	2	no	—
89	C₂×C₈	—	13	2	no	—
90	C₂×C₈	—	13	1	no	—
91	C₂×C₈	—	13	1	no	—
92	C₂×C₈	—	26	2	no	—
93	C₂×C₈	—	26	2	no	—
94	C₂×C₈	C₂×#261	1	2	yes	—
95	C₁₆	—	26	2	no	—
96	C₁₆	—	26	2	no	—
97	C₁₃	#2113	1	1	yes	characteristic, complemented, Sylow
98	C₂³	—	13	2	no	—
99	C₂³	D₁₃⋊C₈	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	D₁₃⋊C₈	1	1	yes	characteristic
107	C₂×C₄	#6118	1	2	yes	—
108	C₂×C₄	C₂²⋊#261	1	2	yes	—
109	C₂×C₄	C₂²×#261	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	—	13	1	no	—
112	C₈	—	13	1	no	—
113	C₈	—	13	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	—	26	4	no	—
116	C₂²	—	26	2	no	—
117	C₂²	#88111	1	2	yes	—
118	C₂²	C₂×D₁₃⋊C₈	1	1	yes	characteristic
119	C₄	—	26	2	no	—
120	C₄	#88111	1	2	yes	—
121	C₄	C₂×#6118	1	1	yes	characteristic
122	C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	52	2	no	—
125	C₂	#11719096	1	2	yes	—
126	C₂	C₂×#88111	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88161

C₂

yes

maximal, characteristic

C₂×#88106

C₂

yes

maximal, characteristic, complemented

C₂²×#6092

C₂

yes

maximal, characteristic, complemented

#11718259

C₂

yes

maximal, complemented

C₂×#6127

C₄

yes

characteristic

C₂×C₁₃×M₁₆

C₄

yes

characteristic

C₂²×#1055

C₂²

yes

—

—

C₄

yes

—

C₂×C₈×D₁₃

C₂²

yes

—

#88106

—

C₂×#6092

—

C₂×#6092

C₂²

yes

—

C₂×#6092

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

C₁₃×M₁₆

—

C₂×C₈×C₁₃

C₂×C₄

yes

—

C₂×#1055

C₂×C₄

yes

characteristic

C₂×#1055

D₄

yes

—

C₂×#1055

D₄

yes

—

C₂×#1055

C₂³

yes

characteristic

#6094

—

C₈×D₁₃

—

C₈×D₁₃

—

—

—

—

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

—

#1055

C₂²⋊C₄

yes

—

#1055

C₂×D₄

yes

characteristic

#1055

C₂×D₄

yes

characteristic

C₂×#131

—

C₂×M₃₂

—

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

#261

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

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

C₂×#261

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#261

yes

—

C₁₆

—

C₁₆

—

C₁₃

#2113

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

D₁₃⋊C₈

yes

characteristic

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

D₁₃⋊C₈

yes

characteristic

107

C₂×C₄

#6118

yes

—

108

C₂×C₄

C₂²⋊#261

yes

—

109

C₂×C₄

C₂²×#261

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

#88111

yes

—

118

C₂²

C₂×D₁₃⋊C₈

yes

characteristic

119

C₄

—

120

C₄

#88111

yes

—

121

C₄

C₂×#6118

yes

characteristic

122

C₄

C₂×C₂²⋊#261

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#11719096

yes

—

126

C₂

C₂×#88111

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#261
commutator, socle	C₂²×C₁₃	C₂²×C₈
Fitting	C₂×C₁₃×M₁₆	C₄
prime radical	C₂×#6127	C₄
cosocle	C₂×#1055	C₂³
hypercenter	C₂×M₁₆	#261
hypocenter	C₁₃	#2113
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#261

commutator, socle

C₂²×C₁₃

C₂²×C₈

Fitting

C₂×C₁₃×M₁₆

C₄

C₄

C₂³

C₂×M₁₆

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
2. C₂×#88106	C₂	semidirect
3. C₂²×#6092	C₂	semidirect
4. #11718259	C₂	semidirect
97. C₁₃	#2113	semidirect

H ≅ G/N

type

2. C₂×#88106

C₂

semidirect

3. C₂²×#6092

C₂

semidirect

4. #11718259

C₂

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×#88161	2
C₂×#88106	2
C₂²×#6092	2
#11718259	2
#2113	13

subgroup

index

C₂²×#6092

Subgroups — #13131068

Conjugacy classes of subgroups

Complements

Maximal subgroups