Group #13125364 — subgroups

Subgroups — #13125364

Order 1664.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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	#11719205	C₂	1	2	yes	maximal
2	C₂×C₁₃×#106	C₂	1	1	yes	maximal, characteristic
3	#11718767	C₂	1	1	yes	maximal, characteristic
4	#11718762	C₂	1	1	yes	maximal, characteristic
5	#11718625	C₂	1	2	yes	maximal
6	#11718519	C₂	1	2	yes	maximal, complemented
7	#11718517	C₂	1	4	yes	maximal, complemented
8	#11718516	C₂	1	2	yes	maximal, complemented
9	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
11	#88180	C₂²	1	2	yes	—
12	#88180	C₂²	1	1	yes	characteristic
13	#88178	—	2	4	no	—
14	C₂×#6099	C₂²	1	1	yes	characteristic
15	#88144	—	2	2	no	—
16	#88144	C₂²	1	4	yes	—
17	#88142	—	2	2	no	—
18	#88142	C₂²	1	4	yes	—
19	C₁₃×#106	C₂²	1	2	yes	—
20	C₁₃×#106	C₂²	1	2	yes	—
21	#88058	C₂²	1	2	yes	—
22	#88057	C₂²	1	1	yes	characteristic
23	#88057	C₂²	1	1	yes	characteristic
24	#88055	C₂²	1	2	yes	—
25	C₈×Dic₁₃	C₂²	1	2	yes	—
26	#88048	C₂²	1	4	yes	—
27	#88047	C₂²	1	4	yes	—
28	C₂²×C₄×C₁₃	—	2	2	no	—
29	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
30	C₂×Dic₂₆	—	2	4	no	—
31	C₂×C₈×C₁₃	Q₈	1	4	yes	complemented
32	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
36	C₂²⋊Dic₁₃	—	2	2	no	—
37	C₂²⋊Dic₁₃	D₄	1	2	yes	—
38	C₄⋊Dic₁₃	—	2	4	no	—
39	C₄⋊Dic₁₃	D₄	1	2	yes	—
40	C₄⋊Dic₁₃	C₂³	1	2	yes	—
41	C₄⋊Dic₁₃	C₂³	1	2	yes	—
42	#6101	—	2	4	no	—
43	#6101	—	2	4	no	—
44	C₄×Dic₁₃	—	2	2	no	—
45	C₄×Dic₁₃	C₂³	1	2	yes	—
46	#6099	Q₈	1	4	yes	complemented
47	C₂×#1055	C₂³	1	2	yes	—
48	#3163	—	13	1	no	maximal, Sylow
49	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃	—	2	4	no	—
51	C₂×C₄×C₁₃	—	8	2	no	—
52	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₁₃	—	2	2	no	—
57	C₂×Dic₁₃	—	2	2	no	—
58	C₂×Dic₁₃	C₂×D₄	1	4	yes	—
59	Dic₂₆	—	8	4	no	—
60	C₈×C₁₃	C₂×Q₈	1	2	yes	—
61	C₈×C₁₃	C₂×Q₈	1	2	yes	—
62	#1055	C₂×Q₈	1	4	yes	—
63	#544	—	13	2	no	—
64	#501	—	13	4	no	—
65	#499	—	13	2	no	—
66	#498	—	13	2	no	—
67	#428	—	13	2	no	—
68	#427	—	13	1	no	—
69	C₂×#106	D₁₃	1	1	yes	characteristic
70	#405	—	13	1	no	—
71	C₂²×C₁₃	—	2	2	no	—
72	C₂²×C₁₃	C₄⋊Q₈	1	2	yes	—
73	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
74	C₄×C₁₃	—	4	4	no	—
75	C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
76	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
77	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
78	Dic₁₃	—	2	4	no	—
79	Dic₁₃	—	4	2	no	—
80	Dic₁₃	—	4	2	no	—
81	C₂×M₁₆	—	13	1	no	—
82	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
83	C₄⋊Q₈	—	13	4	no	—
84	C₄⋊Q₈	—	26	2	no	—
85	#125	—	13	4	no	—
86	#125	—	26	2	no	—
87	C₂²⋊Q₈	—	26	4	no	—
88	#117	—	13	1	no	—
89	#117	—	13	2	no	—
90	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
91	#107	—	13	2	no	—
92	#107	—	13	4	no	—
93	#106	—	13	1	no	—
94	#106	—	13	1	no	—
95	#106	—	13	4	no	—
96	#106	C₂×D₁₃	1	2	yes	—
97	#106	C₂×D₁₃	1	2	yes	—
98	#97	—	13	2	no	—
99	C₄×C₈	—	13	2	no	—
100	C₂×C₁₃	—	2	2	no	—
101	C₂×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₁₃	#577	1	2	yes	—
103	C₂×Q₈	—	26	4	no	—
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
106	M₁₆	—	13	4	no	—
107	C₂×C₈	—	13	2	no	—
108	C₂×C₈	Dic₂₆	1	4	yes	complemented
109	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
110	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	13	2	no	—
113	C₄⋊C₄	—	13	2	no	—
114	C₄⋊C₄	—	13	2	no	—
115	C₄⋊C₄	—	26	4	no	—
116	C₄⋊C₄	—	26	4	no	—
117	C₄⋊C₄	—	26	4	no	—
118	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
119	C₂²⋊C₄	—	13	2	no	—
120	C₂²⋊C₄	—	26	2	no	—
121	C₄²	—	13	2	no	—
122	C₄²	—	26	2	no	—
123	C₁₃	#3163	1	1	yes	characteristic, complemented, Sylow
124	C₂³	D₄×D₁₃	1	1	yes	characteristic
125	Q₈	—	104	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	8	2	no	—
128	C₂×C₄	—	13	4	no	—
129	C₂×C₄	—	26	2	no	—
130	C₂×C₄	—	26	2	no	—
131	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
132	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
133	C₂×C₄	Q₈×D₁₃	1	2	yes	—
134	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
135	C₈	—	13	4	no	—
136	C₈	C₂×Dic₂₆	1	2	yes	—
137	C₈	C₂×Dic₂₆	1	2	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	#88142	1	2	yes	—
140	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
141	C₄	—	4	4	no	—
142	C₄	—	26	4	no	—
143	C₄	—	52	2	no	—
144	C₄	—	52	2	no	—
145	C₄	#88142	1	2	yes	—
146	C₄	C₂²×Dic₂₆	1	1	yes	characteristic
147	C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
148	C₂	—	2	2	no	—
149	C₂	C₂×#88142	1	1	yes	characteristic
150	C₂	#11719467	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11719205

C₂

yes

maximal

C₂×C₁₃×#106

C₂

yes

maximal, characteristic

#11718767

C₂

yes

maximal, characteristic

#11718762

C₂

yes

maximal, characteristic

#11718625

C₂

yes

maximal

#11718519

C₂

yes

maximal, complemented

#11718517

C₂

yes

maximal, complemented

#11718516

C₂

yes

maximal, complemented

C₂²×C₈×C₁₃

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

#88180

C₂²

yes

—

#88180

C₂²

yes

characteristic

#88178

—

C₂×#6099

C₂²

yes

characteristic

#88144

—

#88144

C₂²

yes

—

#88142

—

#88142

C₂²

yes

—

C₁₃×#106

C₂²

yes

—

C₁₃×#106

C₂²

yes

—

#88058

C₂²

yes

—

#88057

C₂²

yes

characteristic

#88057

C₂²

yes

characteristic

#88055

C₂²

yes

—

C₈×Dic₁₃

C₂²

yes

—

#88048

C₂²

yes

—

#88047

C₂²

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂×Dic₂₆

—

C₂×C₈×C₁₃

Q₈

yes

complemented

C₂×C₈×C₁₃

C₂³

yes

characteristic

C₂×C₈×C₁₃

C₂³

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

C₂³

yes

—

C₂²⋊Dic₁₃

—

C₂²⋊Dic₁₃

D₄

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

D₄

yes

—

C₄⋊Dic₁₃

C₂³

yes

—

C₄⋊Dic₁₃

C₂³

yes

—

#6101

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

C₂³

yes

—

#6099

Q₈

yes

complemented

C₂×#1055

C₂³

yes

—

#3163

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×D₄

yes

—

Dic₂₆

—

C₈×C₁₃

C₂×Q₈

yes

—

C₈×C₁₃

C₂×Q₈

yes

—

#1055

C₂×Q₈

yes

—

—

—

—

—

—

—

D₁₃

yes

characteristic

#405

—

C₂²×C₁₃

—

C₂²×C₁₃

C₄⋊Q₈

yes

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

C₄⋊Q₈

yes

—

C₄×C₁₃

C₂²×Q₈

yes

characteristic

C₄×C₁₃

C₂²×Q₈

yes

characteristic

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₁₃

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

#117

—

#117

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

—

—

—

—

—

C₂×D₁₃

yes

—

#106

C₂×D₁₃

yes

—

#97

—

C₄×C₈

—

100

C₂×C₁₃

—

101

C₂×C₁₃

C₂×C₄⋊Q₈

yes

characteristic

102

C₂×C₁₃

#577

yes

—

103

C₂×Q₈

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂²×D₁₃

yes

characteristic

106

M₁₆

—

107

C₂×C₈

—

108

C₂×C₈

Dic₂₆

yes

complemented

109

C₂×C₈

C₂²×D₁₃

yes

characteristic

110

C₂×C₈

C₂²×D₁₃

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

C₂²×D₁₃

yes

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₁₃

#3163

yes

characteristic, complemented, Sylow

124

C₂³

D₄×D₁₃

yes

characteristic

125

Q₈

—

104

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₂×Dic₂₆

yes

—

132

C₂×C₄

D₄×D₁₃

yes

characteristic

133

C₂×C₄

Q₈×D₁₃

yes

—

134

C₂×C₄

C₂³×D₁₃

yes

characteristic

135

C₈

—

136

C₈

C₂×Dic₂₆

yes

—

137

C₈

C₂×Dic₂₆

yes

—

138

C₂²

—

139

C₂²

#88142

yes

—

140

C₂²

C₂×D₄×D₁₃

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

#88142

yes

—

146

C₄

C₂²×Dic₂₆

yes

characteristic

147

C₄

C₂×Q₈×D₁₃

yes

characteristic

148

C₂

—

149

C₂

C₂×#88142

yes

characteristic

150

C₂

#11719467

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#106	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂³×C₁₃	C₂×D₄
hypercenter	C₂×#106	D₁₃
hypocenter	C₁₃	#3163
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂²

C₂×D₄×D₁₃

commutator, cosocle

C₂×C₄×C₁₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃

Fitting

C₂×C₁₃×#106

C₂

socle

C₂²×C₁₃

C₂²×D₄

prime radical

C₂³×C₁₃

C₂×D₄

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
6. #11718519	C₂	semidirect
7. #11718517	C₂	semidirect
8. #11718516	C₂	semidirect
31. C₂×C₈×C₁₃	Q₈	semidirect
46. #6099	Q₈	semidirect
108. C₂×C₈	Dic₂₆	semidirect
123. C₁₃	#3163	semidirect

H ≅ G/N

type

6. #11718519

C₂

semidirect

7. #11718517

C₂

semidirect

8. #11718516

C₂

semidirect

31. C₂×C₈×C₁₃

Q₈

semidirect

46. #6099

Q₈

semidirect

108. C₂×C₈

Dic₂₆

semidirect

123. C₁₃

#3163

semidirect

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

subgroup	index
#11719205	2
C₂×C₁₃×#106	2
#11718767	2
#11718762	2
#11718625	2
#11718519	2
#11718517	2
#11718516	2
#3163	13

subgroup

index

C₂×C₁₃×#106

Subgroups — #13125364

Conjugacy classes of subgroups

Complements

Maximal subgroups