Group #13131006 — subgroups

Subgroups — #13131006

Order 1664.

292 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 110 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×C₄⋊#1057	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#1057	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	C₄⋊#1057	—	2	2	no	—
15	C₄×#1057	—	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	2	no	—
23	C₂×Dic₂₆	C₈	1	4	yes	complemented
24	C₂×#1057	—	2	2	no	—
25	C₂×#1057	—	2	2	no	—
26	C₂×#1057	—	2	2	no	—
27	C₂×#1057	—	2	4	no	—
28	C₂×#1057	Q₈	1	4	yes	complemented
29	C₁₃×C₄⋊C₄	—	2	2	no	—
30	#6101	—	2	4	no	—
31	C₄×Dic₁₃	—	2	2	no	—
32	C₄×Dic₁₃	—	2	2	no	—
33	C₄×Dic₁₃	D₄	1	2	yes	—
34	C₄×Dic₁₃	D₄	1	2	yes	—
35	#1984	—	13	1	no	maximal, Sylow
36	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
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×C₄×C₁₃	C₂×C₈	1	2	yes	—
41	C₂×C₄×C₁₃	M₁₆	1	2	yes	—
42	C₂×Dic₁₃	—	2	2	no	—
43	C₂×Dic₁₃	—	2	2	no	—
44	C₂×Dic₁₃	—	2	2	no	—
45	C₂×Dic₁₃	C₂²⋊C₄	1	2	yes	—
46	C₂×Dic₁₃	C₂×C₈	1	4	yes	—
47	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₁₃	C₂×Q₈	1	2	yes	—
50	C₂×Dic₁₃	#41	1	2	yes	—
51	Dic₂₆	—	2	8	no	—
52	#1057	—	2	4	no	—
53	#1057	—	4	2	no	—
54	C₂×C₄×Q₈	—	13	1	no	—
55	C₂×C₄⋊C₈	—	13	1	no	—
56	C₂×C₄×C₈	—	13	1	no	—
57	#336	—	13	4	no	—
58	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₃	C₄×Q₈	1	2	yes	—
60	C₂²×C₁₃	C₂²×C₈	1	1	yes	characteristic
61	C₂²×C₁₃	C₂×M₁₆	1	1	yes	characteristic
62	C₂²×C₁₃	#131	1	2	yes	—
63	C₄×C₁₃	—	4	2	no	—
64	C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
65	Dic₁₃	—	2	4	no	—
66	Dic₁₃	—	2	2	no	—
67	Dic₁₃	C₂²⋊Q₈	1	2	yes	—
68	Dic₁₃	#124	1	1	yes	characteristic
69	Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
70	C₂²×Q₈	—	13	1	no	—
71	C₂²×C₈	—	13	2	no	—
72	C₂²×C₈	—	13	2	no	—
73	C₄×Q₈	—	26	2	no	—
74	C₄×Q₈	—	26	2	no	—
75	C₂×C₄⋊C₄	—	13	2	no	—
76	C₂×C₄⋊C₄	#261	1	1	yes	characteristic
77	C₂×C₄²	—	13	1	no	—
78	C₂×C₄²	—	13	2	no	—
79	C₄⋊C₈	—	26	2	no	—
80	C₄×C₈	—	26	2	no	—
81	C₂×C₁₃	#391	1	1	yes	characteristic
82	C₂×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
83	C₂×C₁₃	#408	1	1	yes	characteristic
84	C₂×C₁₃	C₈×Q₈	1	2	yes	—
85	C₂×C₁₃	#446	1	2	yes	—
86	C₂×Q₈	—	13	4	no	—
87	C₂×Q₈	—	26	2	no	—
88	C₂×Q₈	—	26	2	no	—
89	C₂²×C₄	—	13	1	no	—
90	C₂²×C₄	—	13	1	no	—
91	C₂²×C₄	—	13	2	no	—
92	C₂²×C₄	C₂×#261	1	2	yes	—
93	C₂²×C₄	C₂×#261	1	1	yes	characteristic
94	C₂×C₈	—	13	4	no	—
95	C₂×C₈	—	26	2	no	—
96	C₂×C₈	—	26	2	no	—
97	C₂×C₈	—	26	2	no	—
98	C₂×C₈	—	26	4	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	26	4	no	—
101	C₄²	—	13	2	no	—
102	C₄²	—	13	2	no	—
103	C₄²	—	26	2	no	—
104	C₄²	—	26	2	no	—
105	C₁₃	#1984	1	1	yes	characteristic, complemented, Sylow
106	C₂³	C₂²×#261	1	1	yes	characteristic
107	Q₈	—	26	8	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	13	2	no	—
111	C₂×C₄	—	13	2	no	—
112	C₂×C₄	—	13	1	no	—
113	C₂×C₄	—	13	1	no	—
114	C₂×C₄	—	13	4	no	—
115	C₂×C₄	—	13	2	no	—
116	C₂×C₄	—	26	2	no	—
117	C₂×C₄	—	26	2	no	—
118	C₂×C₄	—	26	2	no	—
119	C₂×C₄	D₁₃⋊C₈	1	2	yes	—
120	C₂×C₄	#6118	1	2	yes	—
121	C₂×C₄	C₂²⋊#261	1	2	yes	—
122	C₈	—	26	4	no	—
123	C₈	—	52	2	no	—
124	C₂²	C₂×D₁₃⋊C₈	1	1	yes	characteristic
125	C₂²	C₂×#6118	1	1	yes	characteristic
126	C₂²	#88238	1	2	yes	—
127	C₂²	Q₈×#261	1	2	yes	—
128	C₂²	C₂×C₂²⋊#261	1	1	yes	characteristic
129	C₄	—	4	2	no	—
130	C₄	—	13	2	no	—
131	C₄	—	13	1	no	—
132	C₄	—	13	1	no	—
133	C₄	—	26	4	no	—
134	C₄	—	26	2	no	—
135	C₄	#88111	1	4	yes	—
136	C₂	#11719074	1	2	yes	—
137	C₂	#11719080	1	2	yes	—
138	C₂	C₂×#88111	1	1	yes	characteristic
139	C₂	#11719147	1	1	yes	characteristic
140	C₂	(C₂×#261)⋊Q₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88141

C₂

yes

maximal, characteristic

C₂×C₄⋊#1057

C₂

yes

maximal, characteristic

C₂×C₄×#1057

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

—

C₂×Dic₂₆

C₈

yes

—

—

—

—

Q₈

yes

complemented

C₁₃×C₄⋊C₄

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

D₄

yes

—

C₄×Dic₁₃

D₄

yes

—

#1984

—

maximal, Sylow

C₂³×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂×C₈

yes

—

C₂×C₄×C₁₃

M₁₆

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂²⋊C₄

yes

—

C₂×Dic₁₃

C₂×C₈

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×C₄×C₈

—

#336

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃

C₄×Q₈

yes

—

C₂²×C₁₃

C₂²×C₈

yes

characteristic

C₂²×C₁₃

C₂×M₁₆

yes

characteristic

C₂²×C₁₃

#131

yes

—

C₄×C₁₃

—

C₄×C₁₃

C₂²⋊C₈

yes

—

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

—

C₄×C₈

—

C₂×C₁₃

#391

yes

characteristic

C₂×C₁₃

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₁₃

#408

yes

characteristic

C₂×C₁₃

C₈×Q₈

yes

—

C₂×C₁₃

#446

yes

—

C₂×Q₈

—

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

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₁₃

#1984

yes

characteristic, complemented, Sylow

106

C₂³

C₂²×#261

yes

characteristic

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

D₁₃⋊C₈

yes

—

120

C₂×C₄

#6118

yes

—

121

C₂×C₄

C₂²⋊#261

yes

—

122

C₈

—

123

C₈

—

124

C₂²

C₂×D₁₃⋊C₈

yes

characteristic

125

C₂²

C₂×#6118

yes

characteristic

126

C₂²

#88238

yes

—

127

C₂²

Q₈×#261

yes

—

128

C₂²

C₂×C₂²⋊#261

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#88111

yes

—

136

C₂

#11719074

yes

—

137

C₂

#11719080

yes

—

138

C₂

C₂×#88111

yes

characteristic

139

C₂

#11719147

yes

characteristic

140

C₂

(C₂×#261)⋊Q₈

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×#261
commutator	C₂²×C₁₃	C₂²×C₈
Fitting	C₂×C₁₃×C₄⋊C₄	C₄
socle, prime radical	C₂³×C₁₃	C₂²×C₄
cosocle	C₂²×Dic₁₃	C₂³
hypercenter	C₂×C₄⋊C₄	#261
hypocenter	C₁₃	#1984
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×#261

commutator

C₂²×C₁₃

C₂²×C₈

Fitting

C₂×C₁₃×C₄⋊C₄

C₄

socle, prime radical

C₂³×C₁₃

C₂²×C₄

cosocle

C₂²×Dic₁₃

C₂³

C₂×C₄⋊C₄

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
23. C₂×Dic₂₆	C₈	semidirect
28. C₂×#1057	Q₈	semidirect
105. C₁₃	#1984	semidirect

H ≅ G/N

type

23. C₂×Dic₂₆

C₈

semidirect

28. C₂×#1057

Q₈

semidirect

105. C₁₃

#1984

semidirect

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

subgroup	index
C₂×#88141	2
C₂×C₄⋊#1057	2
C₂×C₄×#1057	2
#11718290	2
#1984	13

subgroup

index

C₂×#88141

C₂×C₄⋊#1057

C₂×C₄×#1057

#11718290

#1984

Subgroups — #13131006

Conjugacy classes of subgroups

Complements

Maximal subgroups