Group #13130967 — subgroups

Subgroups — #13130967

Order 1664.

308 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 78 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	D₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#88117	C₂	1	4	yes	maximal
3	#11719060	C₂	1	1	yes	maximal, characteristic
4	#11719047	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈×D₁₃	C₂²	1	2	yes	—
6	C₂×C₄⋊#261	C₂²	1	2	yes	—
7	C₂×D₁₃⋊C₈	C₂²	1	2	yes	—
8	C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
9	#88198	C₄	1	2	yes	complemented
10	C₄⋊C₄×D₁₃	—	2	2	no	—
11	#88142	—	2	2	no	—
12	C₄²×D₁₃	C₂²	1	1	yes	characteristic
13	#88123	C₄	1	1	yes	characteristic, complemented
14	#88117	—	2	8	no	—
15	(C₂×#261)⋊C₄	—	2	2	no	—
16	#88110	—	2	2	no	—
17	C₂²×#261	—	2	2	no	—
18	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
19	Q₈×D₁₃	—	2	4	no	—
20	Q₈×D₁₃	—	2	4	no	—
21	C₂×C₄×D₁₃	—	2	2	no	—
22	C₂×C₄×D₁₃	D₄	1	2	yes	—
23	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
24	C₂×Dic₂₆	—	2	2	no	—
25	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
26	C₂×#1057	—	2	2	no	—
27	C₄⋊#261	—	2	4	no	—
28	D₁₃⋊C₈	—	2	2	no	—
29	D₁₃⋊C₈	D₄	1	4	yes	—
30	C₁₃×C₄⋊C₄	—	2	2	no	—
31	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₄⋊Dic₁₃	—	2	2	no	—
33	#6101	—	2	4	no	—
34	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
35	C₄×Dic₁₃	D₄	1	2	yes	—
36	#1960	—	13	1	no	maximal, Sylow
37	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂×#261	—	4	2	no	—
39	C₂×#261	—	4	4	no	—
40	Q₈×C₁₃	—	2	4	no	—
41	C₂×C₄×C₁₃	—	2	2	no	—
42	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×Dic₁₃	—	2	2	no	—
45	C₂×Dic₁₃	C₂²⋊C₄	1	2	yes	—
46	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
47	C₄×D₁₃	—	2	4	no	—
48	C₄×D₁₃	—	4	2	no	—
49	C₄×D₁₃	—	4	2	no	—
50	C₄×D₁₃	C₂×D₄	1	2	yes	—
51	C₄×D₁₃	#41	1	2	yes	—
52	Dic₂₆	—	2	4	no	—
53	Dic₂₆	—	4	4	no	—
54	#1057	—	2	4	no	—
55	C₂×C₄⋊Q₈	—	13	1	no	—
56	C₂×#103	—	13	4	no	—
57	C₂×#97	—	13	1	no	—
58	#382	—	13	1	no	—
59	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×D₁₃	#124	1	1	yes	characteristic
61	C₂×D₁₃	#127	1	1	yes	characteristic
62	C₂×D₁₃	#137	1	4	yes	—
63	#261	—	8	2	no	—
64	C₄×C₁₃	—	2	2	no	—
65	C₄×C₁₃	—	4	2	no	—
66	C₄×C₁₃	C₄×D₄	1	2	yes	—
67	Dic₁₃	—	2	2	no	—
68	Dic₁₃	—	4	2	no	—
69	Dic₁₃	#121	1	2	yes	—
70	C₂²×Q₈	—	13	2	no	—
71	C₂²×C₈	—	13	2	no	—
72	C₄⋊Q₈	—	13	2	no	—
73	C₄⋊Q₈	—	13	1	no	—
74	C₄⋊Q₈	—	26	2	no	—
75	C₄⋊Q₈	#261	1	1	yes	characteristic, complemented
76	C₂×C₄⋊C₄	—	13	2	no	—
77	C₂×C₄⋊C₄	—	26	2	no	—
78	C₂×C₄²	—	13	1	no	—
79	#103	—	26	8	no	—
80	#97	—	26	2	no	—
81	#95	—	26	2	no	—
82	C₂×C₁₃	#390	1	1	yes	characteristic
83	C₂×C₁₃	#419	1	2	yes	—
84	D₁₃	#491	1	2	yes	—
85	D₁₃	#497	1	2	yes	—
86	C₂×Q₈	—	13	2	no	—
87	C₂×Q₈	—	26	2	no	—
88	C₂×Q₈	—	26	4	no	—
89	C₂×Q₈	—	26	4	no	—
90	C₂×Q₈	C₂×#261	1	2	yes	—
91	C₂²×C₄	—	13	1	no	—
92	C₂²×C₄	—	13	2	no	—
93	C₂²×C₄	—	26	2	no	—
94	C₂²×C₄	—	26	2	no	—
95	C₂×C₈	—	13	4	no	—
96	C₂×C₈	—	26	2	no	—
97	C₂×C₈	—	26	2	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	—	26	4	no	—
100	C₄⋊C₄	—	26	4	no	—
101	C₄⋊C₄	—	26	2	no	—
102	C₄²	—	13	2	no	—
103	C₄²	—	13	1	no	—
104	C₄²	C₂×#261	1	1	yes	characteristic
105	C₁₃	#1960	1	1	yes	characteristic, complemented, Sylow
106	C₂³	—	13	1	no	—
107	Q₈	—	2	4	no	—
108	Q₈	—	26	4	no	—
109	Q₈	—	52	4	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	13	1	no	—
112	C₂×C₄	—	13	2	no	—
113	C₂×C₄	—	13	2	no	—
114	C₂×C₄	—	13	2	no	—
115	C₂×C₄	—	26	4	no	—
116	C₂×C₄	—	26	2	no	—
117	C₂×C₄	—	52	2	no	—
118	C₂×C₄	—	52	2	no	—
119	C₂×C₄	—	52	2	no	—
120	C₂×C₄	—	52	4	no	—
121	C₂×C₄	C₂²⋊#261	1	2	yes	—
122	C₂×C₄	C₂²×#261	1	1	yes	characteristic
123	C₈	—	26	4	no	—
124	C₂²	—	13	1	no	—
125	C₂²	—	13	1	no	—
126	C₂²	—	13	4	no	—
127	C₂²	C₂×C₂²⋊#261	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	4	2	no	—
130	C₄	—	13	2	no	—
131	C₄	—	26	2	no	—
132	C₄	—	52	2	no	—
133	C₄	—	104	2	no	—
134	C₄	D₄×#261	1	2	yes	—
135	C₂	—	13	2	no	—
136	C₂	—	13	2	no	—
137	C₂	#11719151	1	1	yes	characteristic
138	C₂	#11719153	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

D₁₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×#88117

C₂

yes

maximal

#11719060

C₂

yes

maximal, characteristic

#11719047

C₂

yes

maximal, characteristic

C₂×Q₈×D₁₃

C₂²

yes

—

C₂×C₄⋊#261

C₂²

yes

—

C₂×D₁₃⋊C₈

C₂²

yes

—

C₁₃×C₄⋊Q₈

C₄

yes

characteristic, complemented

#88198

C₄

yes

complemented

C₄⋊C₄×D₁₃

—

#88142

—

C₄²×D₁₃

C₂²

yes

characteristic

#88123

C₄

yes

characteristic, complemented

#88117

—

(C₂×#261)⋊C₄

—

#88110

—

C₂²×#261

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

Q₈×D₁₃

—

Q₈×D₁₃

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

D₄

yes

—

C₂×C₄×D₁₃

C₂³

yes

characteristic

C₂×Dic₂₆

—

C₂×Dic₂₆

C₂×C₄

yes

—

C₂×#1057

—

C₄⋊#261

—

D₁₃⋊C₈

—

D₁₃⋊C₈

D₄

yes

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

C₂×C₄

yes

characteristic

C₄×Dic₁₃

D₄

yes

—

#1960

—

maximal, Sylow

C₂²×D₁₃

C₂×D₄

yes

characteristic

C₂×#261

—

C₂×#261

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂²⋊C₄

yes

—

C₂×Dic₁₃

C₂×D₄

yes

characteristic

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

C₂×D₄

yes

—

C₄×D₁₃

#41

yes

—

Dic₂₆

—

Dic₂₆

—

#1057

—

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×#97

—

#382

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

C₂×D₁₃

yes

C₂×D₁₃

yes

C₂×D₁₃

yes

—

#261

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄×D₄

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

#121

yes

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#261

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

#103

—

#97

—

#95

—

C₂×C₁₃

#390

yes

characteristic

C₂×C₁₃

#419

yes

—

D₁₃

#491

yes

—

D₁₃

#497

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×#261

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₄²

C₂×#261

yes

characteristic

105

C₁₃

#1960

yes

characteristic, complemented, Sylow

106

C₂³

—

107

Q₈

—

108

Q₈

—

109

Q₈

—

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

—

120

C₂×C₄

—

121

C₂×C₄

C₂²⋊#261

yes

—

122

C₂×C₄

C₂²×#261

yes

characteristic

123

C₈

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂×C₂²⋊#261

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

104

—

134

C₄

D₄×#261

yes

—

135

C₂

—

136

C₂

—

137

C₂

#11719151

yes

characteristic

138

C₂

#11719153

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#261
Fitting	C₁₃×C₄⋊Q₈	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₃	C₂×D₄
cosocle	C₂×C₄×D₁₃	C₂³
hypercenter	C₄⋊Q₈	#261
hypocenter	C₁₃	#1960
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#261

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#261

Fitting

C₁₃×C₄⋊Q₈

C₄

socle

C₂²×C₁₃

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₃

C₂×D₄

cosocle

C₂×C₄×D₁₃

C₂³

C₄⋊Q₈

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
8. C₁₃×C₄⋊Q₈	C₄	semidirect
9. #88198	C₄	semidirect
13. #88123	C₄	semidirect
75. C₄⋊Q₈	#261	semidirect
105. C₁₃	#1960	semidirect

H ≅ G/N

type

8. C₁₃×C₄⋊Q₈

C₄

semidirect

9. #88198

C₄

semidirect

13. #88123

C₄

semidirect

75. C₄⋊Q₈

#261

semidirect

105. C₁₃

#1960

semidirect

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

subgroup	index
D₁₃×C₄⋊Q₈	2
C₂×#88117	2
#11719060	2
#11719047	2
#1960	13

subgroup

index

D₁₃×C₄⋊Q₈

Subgroups — #13130967

Conjugacy classes of subgroups

Complements

Maximal subgroups