Group #13115772 — subgroups

Subgroups — #13115772

Order 1664.

292 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
4	#11718142	C₂	1	4	yes	maximal
5	C₂²×Dic₂₆	C₄	1	1	yes	characteristic
6	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
9	C₂×#6101	C₄	1	2	yes	—
10	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
11	C₂²×#1055	C₂²	1	2	yes	—
12	C₄×Dic₂₆	—	2	2	no	—
13	C₄×Dic₂₆	—	2	2	no	—
14	C₄×C₈×C₁₃	—	2	2	no	—
15	C₄⋊#1055	—	2	2	no	—
16	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
20	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
21	C₂×Dic₂₆	—	2	2	no	—
22	C₂×Dic₂₆	—	2	2	no	—
23	C₂×Dic₂₆	C₈	1	4	yes	complemented
24	C₂×C₈×C₁₃	—	2	2	no	—
25	C₂×C₈×C₁₃	—	2	2	no	—
26	C₂×C₈×C₁₃	Q₈	1	4	yes	complemented
27	C₄²×C₁₃	D₄	1	2	yes	—
28	C₄²×C₁₃	D₄	1	2	yes	—
29	C₄⋊Dic₁₃	—	2	2	no	—
30	#6101	—	2	4	no	—
31	C₄×Dic₁₃	—	2	2	no	—
32	C₄×Dic₁₃	—	2	2	no	—
33	C₂×#1055	—	2	2	no	—
34	C₂×#1055	—	2	4	no	—
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	C₂²⋊C₄	1	2	yes	—
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×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₁₃	#41	1	2	yes	—
46	C₂×Dic₁₃	—	2	2	no	—
47	C₂×Dic₁₃	—	2	2	no	—
48	C₂×Dic₁₃	—	2	2	no	—
49	C₂×Dic₁₃	—	2	4	no	—
50	C₂×Dic₁₃	C₂×C₈	1	4	yes	—
51	Dic₂₆	—	2	8	no	—
52	C₈×C₁₃	—	2	4	no	—
53	#1055	—	4	2	no	—
54	C₂×C₄×Q₈	—	13	1	no	—
55	C₂×C₄⋊C₈	—	13	1	no	—
56	C₂×C₄×C₈	D₁₃	1	1	yes	characteristic
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	—	2	2	no	—
64	C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
65	C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
66	C₄×C₁₃	#124	1	1	yes	characteristic
67	C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
68	Dic₁₃	—	2	4	no	—
69	Dic₁₃	—	4	2	no	—
70	C₂²×Q₈	—	13	1	no	—
71	C₂²×C₈	—	13	2	no	—
72	C₂²×C₈	C₂×D₁₃	1	2	yes	—
73	C₄×Q₈	—	26	2	no	—
74	C₄×Q₈	—	26	2	no	—
75	C₂×C₄⋊C₄	—	13	2	no	—
76	C₂×C₄⋊C₄	—	13	1	no	—
77	C₂×C₄²	—	13	2	no	—
78	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
79	C₄⋊C₈	—	26	2	no	—
80	C₄×C₈	—	2	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	2	no	—
90	C₂²×C₄	—	13	2	no	—
91	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
92	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	26	2	no	—
97	C₂×C₈	—	26	4	no	—
98	C₂×C₈	Dic₂₆	1	4	yes	complemented
99	C₄⋊C₄	—	26	4	no	—
100	C₄⋊C₄	—	26	2	no	—
101	C₄²	—	26	2	no	—
102	C₄²	—	26	2	no	—
103	C₄²	D₅₂	1	2	yes	—
104	C₄²	C₂²⋊D₁₃	1	2	yes	—
105	C₁₃	#1984	1	1	yes	characteristic, complemented, Sylow
106	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
107	Q₈	—	26	8	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	13	4	no	—
110	C₂×C₄	—	26	2	no	—
111	C₂×C₄	—	26	2	no	—
112	C₂×C₄	—	26	4	no	—
113	C₂×C₄	—	26	2	no	—
114	C₂×C₄	C₈×D₁₃	1	2	yes	—
115	C₂×C₄	#6094	1	2	yes	—
116	C₂×C₄	#6103	1	2	yes	—
117	C₂×C₄	#6103	1	2	yes	—
118	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
119	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
120	C₂×C₄	#6127	1	2	yes	—
121	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
122	C₈	—	2	4	no	—
123	C₈	—	52	2	no	—
124	C₂²	C₄×Dic₂₆	1	2	yes	—
125	C₂²	C₂×C₈×D₁₃	1	1	yes	characteristic
126	C₂²	C₂×#6094	1	1	yes	characteristic
127	C₂²	#88155	1	2	yes	—
128	C₂²	C₂×#6103	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	—	26	4	no	—
131	C₄	—	52	2	no	—
132	C₄	#88060	1	4	yes	—
133	C₄	#88123	1	1	yes	characteristic
134	C₄	#88129	1	1	yes	characteristic
135	C₄	#88178	1	2	yes	—
136	C₂	C₈×Dic₂₆	1	2	yes	—
137	C₂	#11718340	1	2	yes	—
138	C₂	#11718590	1	1	yes	characteristic
139	C₂	C₂×#88060	1	1	yes	characteristic
140	C₂	#11718771	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₂₆

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₃

C₂

yes

maximal, characteristic

C₂×C₄⋊#1055

C₂

yes

maximal, characteristic

#11718142

C₂

yes

maximal

C₂²×Dic₂₆

C₄

yes

characteristic

C₂²×C₈×C₁₃

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₄

yes

characteristic

C₂×#6101

C₄

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂²×#1055

C₂²

yes

—

C₄×Dic₂₆

—

C₄×Dic₂₆

—

C₄×C₈×C₁₃

—

C₄⋊#1055

—

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂×C₄

yes

—

C₂²×Dic₁₃

C₂×C₄

yes

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

C₈

yes

complemented

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

Q₈

yes

complemented

C₄²×C₁₃

D₄

yes

—

C₄²×C₁₃

D₄

yes

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

—

—

—

C₂³×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂×C₈

yes

—

C₂×C₄×C₁₃

M₁₆

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×C₈

yes

—

Dic₂₆

—

C₈×C₁₃

—

#1055

—

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

D₁₃

yes

characteristic

#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

—

C₄×C₁₃

C₂²⋊Q₈

yes

—

C₄×C₁₃

#124

yes

characteristic

C₄×C₁₃

C₄⋊Q₈

yes

characteristic

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₁₃

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

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

C₄×D₁₃

yes

characteristic

C₂²×C₄

C₄×D₁₃

yes

characteristic

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₂₆

yes

complemented

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

—

103

C₄²

D₅₂

yes

—

104

C₄²

C₂²⋊D₁₃

yes

—

105

C₁₃

#1984

yes

characteristic, complemented, Sylow

106

C₂³

C₂×C₄×D₁₃

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

C₈×D₁₃

yes

—

115

C₂×C₄

#6094

yes

—

116

C₂×C₄

#6103

yes

—

117

C₂×C₄

#6103

yes

—

118

C₂×C₄

C₂×Dic₂₆

yes

—

119

C₂×C₄

C₂×D₅₂

yes

characteristic

120

C₂×C₄

#6127

yes

—

121

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

122

C₈

—

123

C₈

—

124

C₂²

C₄×Dic₂₆

yes

—

125

C₂²

C₂×C₈×D₁₃

yes

characteristic

126

C₂²

C₂×#6094

yes

characteristic

127

C₂²

#88155

yes

—

128

C₂²

C₂×#6103

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#88060

yes

—

133

C₄

#88123

yes

characteristic

134

C₄

#88129

yes

characteristic

135

C₄

#88178

yes

—

136

C₂

C₈×Dic₂₆

yes

—

137

C₂

#11718340

yes

—

138

C₂

#11718590

yes

characteristic

139

C₂

C₂×#88060

yes

characteristic

140

C₂

#11718771

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂²×C₈
Fitting	C₂×C₄×C₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	C₂×C₄×C₈	D₁₃
hypocenter	C₁₃	#1984
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₃

commutator

C₂²×C₁₃

C₂²×C₈

Fitting

C₂×C₄×C₈×C₁₃

C₂

socle, prime radical

C₂³×C₁₃

C₂²×C₄

cosocle

C₂²×C₄×C₁₃

C₂³

hypercenter

C₂×C₄×C₈

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

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

H ≅ G/N

type

23. C₂×Dic₂₆

C₈

semidirect

26. C₂×C₈×C₁₃

Q₈

semidirect

98. C₂×C₈

Dic₂₆

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×C₄×Dic₂₆	2
C₂×C₄×C₈×C₁₃	2
C₂×C₄⋊#1055	2
#11718142	2
#1984	13

subgroup

index

C₂×C₄×Dic₂₆

C₂×C₄×C₈×C₁₃

C₂×C₄⋊#1055

#11718142

#1984

Subgroups — #13115772

Conjugacy classes of subgroups

Complements

Maximal subgroups