Group #13131008 — subgroups

Subgroups — #13131008

Order 1664.

280 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 90 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×#88141	C₂	1	1	yes	maximal, characteristic
2	C₂×#88112	C₂	1	1	yes	maximal, characteristic
3	C₂×#88110	C₂	1	1	yes	maximal, characteristic
4	#11718290	C₂	1	4	yes	maximal
5	C₂²×Dic₂₆	C₄	1	1	yes	characteristic
6	C₂²×#1057	C₂²	1	2	yes	—
7	C₂²×#1057	C₂²	1	2	yes	—
8	C₂×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
9	C₂×#6101	C₄	1	2	yes	—
10	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
12	#88141	—	2	2	no	—
13	#88141	—	2	2	no	—
14	#88112	—	2	2	no	—
15	#88110	—	2	2	no	—
16	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
20	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
21	C₂×Dic₂₆	—	2	2	no	—
22	C₂×Dic₂₆	—	2	4	no	—
23	C₂×#1057	—	2	2	no	—
24	C₂×#1057	—	2	2	no	—
25	C₂×#1057	—	2	2	no	—
26	C₂×#1057	—	2	4	no	—
27	C₂×#1057	Q₈	1	4	yes	complemented
28	C₁₃×C₄⋊C₄	—	2	2	no	—
29	#6101	—	2	4	no	—
30	C₄×Dic₁₃	—	2	2	no	—
31	C₄×Dic₁₃	—	2	2	no	—
32	C₄×Dic₁₃	D₄	1	2	yes	—
33	C₄×Dic₁₃	D₄	1	2	yes	—
34	#1985	—	13	1	no	maximal, Sylow
35	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₃	—	2	2	no	—
37	C₂×C₄×C₁₃	—	2	2	no	—
38	C₂×C₄×C₁₃	—	2	4	no	—
39	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
40	C₂×Dic₁₃	—	2	2	no	—
41	C₂×Dic₁₃	—	2	2	no	—
42	C₂×Dic₁₃	—	2	2	no	—
43	C₂×Dic₁₃	C₂²⋊C₄	1	2	yes	—
44	C₂×Dic₁₃	M₁₆	1	4	yes	—
45	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₁₃	C₂×Q₈	1	2	yes	—
48	C₂×Dic₁₃	#41	1	2	yes	—
49	Dic₂₆	—	2	8	no	—
50	#1057	—	2	4	no	—
51	#1057	—	4	2	no	—
52	C₂×C₄×Q₈	—	13	1	no	—
53	C₂×C₄⋊C₈	—	13	1	no	—
54	C₂×#97	—	13	1	no	—
55	#336	—	13	4	no	—
56	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₃	C₄×Q₈	1	2	yes	—
58	C₂²×C₁₃	C₂×M₁₆	1	2	yes	—
59	C₂²×C₁₃	#131	1	2	yes	—
60	C₄×C₁₃	—	2	2	no	—
61	C₄×C₁₃	—	4	2	no	—
62	Dic₁₃	—	2	4	no	—
63	Dic₁₃	—	2	2	no	—
64	Dic₁₃	C₂²⋊Q₈	1	2	yes	—
65	Dic₁₃	#124	1	1	yes	characteristic
66	Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂²×Q₈	—	13	1	no	—
68	C₂²×C₈	—	13	2	no	—
69	C₂²×C₈	—	13	2	no	—
70	C₄×Q₈	—	26	2	no	—
71	C₄×Q₈	—	26	2	no	—
72	C₂×C₄⋊C₄	—	13	2	no	—
73	C₂×C₄⋊C₄	#261	1	1	yes	characteristic
74	C₂×C₄²	—	13	1	no	—
75	C₂×C₄²	—	13	2	no	—
76	C₄⋊C₈	—	26	2	no	—
77	#97	—	26	2	no	—
78	C₂×C₁₃	#391	1	1	yes	characteristic
79	C₂×C₁₃	#407	1	1	yes	characteristic
80	C₂×C₁₃	#408	1	1	yes	characteristic
81	C₂×C₁₃	#446	1	4	yes	—
82	C₂×Q₈	—	26	4	no	—
83	C₂×Q₈	—	26	2	no	—
84	C₂²×C₄	—	13	1	no	—
85	C₂²×C₄	—	13	1	no	—
86	C₂²×C₄	—	13	2	no	—
87	C₂²×C₄	C₂×#261	1	2	yes	—
88	C₂²×C₄	C₂×#261	1	1	yes	characteristic
89	C₂×C₈	—	13	4	no	—
90	C₂×C₈	—	26	2	no	—
91	C₂×C₈	—	26	2	no	—
92	C₂×C₈	—	26	2	no	—
93	C₂×C₈	—	26	4	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	26	4	no	—
96	C₄²	—	13	2	no	—
97	C₄²	—	13	2	no	—
98	C₄²	—	26	2	no	—
99	C₄²	—	26	2	no	—
100	C₁₃	#1985	1	1	yes	characteristic, complemented, Sylow
101	C₂³	C₂²×#261	1	1	yes	characteristic
102	Q₈	—	26	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	13	2	no	—
107	C₂×C₄	—	13	2	no	—
108	C₂×C₄	—	13	1	no	—
109	C₂×C₄	—	13	1	no	—
110	C₂×C₄	—	13	4	no	—
111	C₂×C₄	—	13	2	no	—
112	C₂×C₄	—	26	2	no	—
113	C₂×C₄	—	26	2	no	—
114	C₂×C₄	—	26	2	no	—
115	C₂×C₄	C₂²⋊#261	1	2	yes	—
116	C₈	—	26	4	no	—
117	C₈	—	52	2	no	—
118	C₂²	D₁₃⋊M₁₆	1	2	yes	—
119	C₂²	#88238	1	2	yes	—
120	C₂²	Q₈×#261	1	2	yes	—
121	C₂²	C₂×C₂²⋊#261	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	4	2	no	—
124	C₄	—	13	2	no	—
125	C₄	—	13	1	no	—
126	C₄	—	13	1	no	—
127	C₄	—	26	4	no	—
128	C₄	—	26	2	no	—
129	C₂	#11719078	1	4	yes	—
130	C₂	#11719126	1	1	yes	characteristic
131	C₂	#11719147	1	1	yes	characteristic
132	C₂	(C₂×#261)⋊Q₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88141

C₂

yes

maximal, characteristic

C₂×#88112

C₂

yes

maximal, characteristic

C₂×#88110

C₂

yes

maximal, characteristic

#11718290

C₂

yes

maximal

C₂²×Dic₂₆

C₄

yes

characteristic

C₂²×#1057

C₂²

yes

—

C₂²×#1057

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₄

yes

characteristic

C₂×#6101

C₄

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

—

—

—

C₂²×C₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂²×Dic₁₃

C₂×C₄

yes

characteristic

C₂²×Dic₁₃

C₂×C₄

yes

—

C₂²×Dic₁₃

C₂³

yes

characteristic

C₂×Dic₂₆

—

C₂×Dic₂₆

—

—

—

—

—

Q₈

yes

complemented

C₁₃×C₄⋊C₄

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

D₄

yes

—

C₄×Dic₁₃

D₄

yes

—

#1985

—

maximal, Sylow

C₂³×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂²⋊C₄

yes

—

C₂×Dic₁₃

M₁₆

yes

—

C₂×Dic₁₃

C₂×D₄

yes

characteristic

C₂×Dic₁₃

C₂×D₄

yes

characteristic

C₂×Dic₁₃

C₂×Q₈

yes

—

C₂×Dic₁₃

#41

yes

—

Dic₂₆

—

#1057

—

#1057

—

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

#336

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃

C₄×Q₈

yes

—

C₂²×C₁₃

C₂×M₁₆

yes

—

C₂²×C₁₃

#131

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

C₂²⋊Q₈

yes

—

Dic₁₃

#124

yes

characteristic

Dic₁₃

C₄⋊Q₈

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

#261

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

#97

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#261

yes

—

C₂²×C₄

C₂×#261

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

100

C₁₃

#1985

yes

characteristic, complemented, Sylow

101

C₂³

C₂²×#261

yes

characteristic

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

C₂²⋊#261

yes

—

116

C₈

—

117

C₈

—

118

C₂²

D₁₃⋊M₁₆

yes

—

119

C₂²

#88238

yes

—

120

C₂²

Q₈×#261

yes

—

121

C₂²

C₂×C₂²⋊#261

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₂

#11719078

yes

—

130

C₂

#11719126

yes

characteristic

131

C₂

#11719147

yes

characteristic

132

C₂

(C₂×#261)⋊Q₈

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×#261
commutator, socle, prime radical	C₂³×C₁₃	C₂²×C₄
Fitting	C₂×C₁₃×C₄⋊C₄	C₄
cosocle	C₂²×Dic₁₃	C₂³
hypercenter	C₂×C₄⋊C₄	#261
hypocenter	C₁₃	#1985
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×#261

commutator, socle, prime radical

C₂³×C₁₃

C₂²×C₄

Fitting

C₂×C₁₃×C₄⋊C₄

C₄

cosocle

C₂²×Dic₁₃

C₂³

C₂×C₄⋊C₄

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
27. C₂×#1057	Q₈	semidirect
100. C₁₃	#1985	semidirect

H ≅ G/N

type

27. C₂×#1057

Q₈

semidirect

100. C₁₃

#1985

semidirect

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

subgroup	index
C₂×#88141	2
C₂×#88112	2
C₂×#88110	2
#11718290	2
#1985	13

subgroup

index

Subgroups — #13131008

Conjugacy classes of subgroups

Complements

Maximal subgroups