Group #13116178 — subgroups

Subgroups — #13116178

Order 1664.

246 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 75 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	C₁₃×#544	C₂	1	1	yes	maximal, characteristic
2	#11718656	C₂	1	2	yes	maximal
3	#11718637	C₂	1	1	yes	maximal, characteristic
4	#11718124	C₂	1	2	yes	maximal
5	#11718122	C₂	1	1	yes	maximal, characteristic
6	C₁₃×C₄⋊Q₈	—	2	2	no	—
7	C₁₃×#125	—	2	2	no	—
8	C₁₃×C₂²⋊Q₈	—	2	2	no	—
9	C₁₃×#117	C₂²	1	2	yes	—
10	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂×#6101	—	2	2	no	—
13	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂×#6099	C₂²	1	2	yes	—
15	#88048	—	2	2	no	—
16	#88047	—	2	2	no	—
17	C₂×Q₈×C₁₃	—	2	2	no	—
18	C₂²×C₄×C₁₃	D₄	1	2	yes	—
19	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₁₃	—	2	1	no	—
21	C₂²×Dic₁₃	—	2	1	no	—
22	C₁₃×C₄⋊C₄	—	2	2	no	—
23	C₁₃×C₄⋊C₄	—	2	4	no	—
24	C₁₃×C₄⋊C₄	—	2	2	no	—
25	C₁₃×C₄⋊C₄	D₄	1	2	yes	—
26	C₁₃×C₄⋊C₄	D₄	1	2	yes	—
27	C₁₃×C₂²⋊C₄	—	2	2	no	—
28	C₄²×C₁₃	—	2	2	no	—
29	C₄⋊Dic₁₃	—	2	2	no	—
30	#6101	—	4	4	no	—
31	C₄×Dic₁₃	—	2	1	no	—
32	C₄×Dic₁₃	Q₈	1	2	yes	complemented
33	#6099	—	2	2	no	—
34	#6099	Q₈	1	4	yes	complemented
35	C₂×#1055	—	2	2	no	—
36	#2060	—	13	1	no	maximal, Sylow
37	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₁₃	—	8	2	no	—
39	C₂×C₄×C₁₃	—	2	2	no	—
40	C₂×C₄×C₁₃	—	2	2	no	—
41	C₂×C₄×C₁₃	—	2	2	no	—
42	C₂×C₄×C₁₃	—	4	2	no	—
43	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
47	C₂×Dic₁₃	—	2	2	no	—
48	C₂×Dic₁₃	—	2	2	no	—
49	C₂×Dic₁₃	—	2	2	no	—
50	C₂×Dic₁₃	—	4	2	no	—
51	C₂×Dic₁₃	—	4	1	no	—
52	#1055	—	2	4	no	—
53	#544	D₁₃	1	1	yes	characteristic
54	#428	—	13	2	no	—
55	#384	—	13	1	no	—
56	#343	—	13	1	no	—
57	#339	—	13	2	no	—
58	C₂²×C₁₃	—	2	1	no	—
59	C₂²×C₁₃	—	2	1	no	—
60	C₂²×C₁₃	#120	1	1	yes	characteristic
61	C₂²×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂²×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
63	C₄×C₁₃	—	4	2	no	—
64	C₄×C₁₃	—	4	2	no	—
65	C₄×C₁₃	—	4	2	no	—
66	C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
67	C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
68	Dic₁₃	—	2	4	no	—
69	Dic₁₃	—	8	1	no	—
70	C₂×M₁₆	—	13	2	no	—
71	C₄⋊Q₈	—	2	2	no	—
72	#125	—	2	2	no	—
73	C₂²⋊Q₈	—	2	2	no	—
74	#117	C₂×D₁₃	1	2	yes	—
75	C₂×C₄⋊C₄	—	13	1	no	—
76	C₂×C₄⋊C₄	—	26	2	no	—
77	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
78	C₂×C₄²	—	13	1	no	—
79	#107	—	26	2	no	—
80	#106	—	26	2	no	—
81	C₂×C₁₃	—	2	1	no	—
82	C₂×C₁₃	—	2	1	no	—
83	C₂×C₁₃	#395	1	1	yes	characteristic
84	C₂×C₁₃	#453	1	1	yes	characteristic
85	C₂×C₁₃	#456	1	1	yes	characteristic
86	C₂×Q₈	—	2	2	no	—
87	C₂²×C₄	—	26	1	no	—
88	C₂²×C₄	—	26	1	no	—
89	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
91	M₁₆	—	13	4	no	—
92	M₁₆	—	26	2	no	—
93	C₂×C₈	—	26	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	26	2	no	—
98	C₄⋊C₄	—	52	4	no	—
99	C₄⋊C₄	C₂²⋊D₁₃	1	2	yes	—
100	C₄⋊C₄	C₂²⋊D₁₃	1	2	yes	—
101	C₂²⋊C₄	—	2	2	no	—
102	C₄²	—	2	2	no	—
103	C₄²	—	13	2	no	—
104	C₄²	—	26	1	no	—
105	C₁₃	#2060	1	1	yes	characteristic, complemented, Sylow
106	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
107	Q₈	—	8	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	—	26	2	no	—
113	C₂×C₄	—	26	2	no	—
114	C₂×C₄	—	26	2	no	—
115	C₂×C₄	—	52	2	no	—
116	C₂×C₄	—	52	1	no	—
117	C₂×C₄	Q₈×D₁₃	1	1	yes	characteristic
118	C₂×C₄	Q₈×D₁₃	1	2	yes	—
119	C₂×C₄	#6131	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
121	C₈	—	26	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	#88198	1	1	yes	characteristic
125	C₂²	#88200	1	1	yes	characteristic
126	C₂²	#88207	1	1	yes	characteristic
127	C₄	—	4	2	no	—
128	C₄	—	4	2	no	—
129	C₄	—	4	2	no	—
130	C₄	—	26	4	no	—
131	C₄	—	104	1	no	—
132	C₄	#88198	1	2	yes	—
133	C₄	#88200	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#11718890	1	1	yes	characteristic
137	C₂	#11718902	1	1	yes	characteristic
138	C₂	#11718907	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#544

C₂

yes

maximal, characteristic

#11718656

C₂

yes

maximal

#11718637

C₂

yes

maximal, characteristic

#11718124

C₂

yes

maximal

#11718122

C₂

yes

maximal, characteristic

C₁₃×C₄⋊Q₈

—

C₁₃×#125

—

C₁₃×C₂²⋊Q₈

—

C₁₃×#117

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×#6101

—

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

C₂×#6099

C₂²

yes

—

#88048

—

#88047

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

D₄

yes

—

C₁₃×C₄⋊C₄

D₄

yes

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

Q₈

yes

complemented

#6099

—

#6099

Q₈

yes

—

—

C₂³×C₁₃

C₂×D₄

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃

#41

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

#1055

—

#544

D₁₃

yes

—

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#120

yes

characteristic

C₂²×C₁₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₃

C₄⋊Q₈

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂²⋊Q₈

yes

—

C₄×C₁₃

C₄⋊Q₈

yes

—

Dic₁₃

—

Dic₁₃

—

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

C₂×D₁₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

characteristic

C₂×C₄²

—

#107

—

#106

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₁₃

yes

—

100

C₄⋊C₄

C₂²⋊D₁₃

yes

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₁₃

#2060

yes

characteristic, complemented, Sylow

106

C₂³

C₂×C₂²⋊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

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

Q₈×D₁₃

yes

characteristic

118

C₂×C₄

Q₈×D₁₃

yes

—

119

C₂×C₄

#6131

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

#88198

yes

characteristic

125

C₂²

#88200

yes

characteristic

126

C₂²

#88207

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

104

—

132

C₄

#88198

yes

—

133

C₄

#88200

yes

—

134

C₂

—

135

C₂

—

136

C₂

#11718890

yes

characteristic

137

C₂

#11718902

yes

characteristic

138

C₂

#11718907

yes

characteristic

	subgroup	G/H
center	C₂²	#88207
commutator, cosocle	C₂²×C₄×C₁₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#544	C₂
socle	C₂²×C₁₃	#120
prime radical	C₂³×C₁₃	C₂×D₄
hypercenter	#544	D₁₃
hypocenter	C₁₃	#2060
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

subgroup

G/H

center

C₂²

#88207

commutator, cosocle

C₂²×C₄×C₁₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₁₃×#544

C₂

socle

C₂²×C₁₃

#120

prime radical

C₂³×C₁₃

C₂×D₄

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

N	H ≅ G/N	type
32. C₄×Dic₁₃	Q₈	semidirect
34. #6099	Q₈	semidirect
105. C₁₃	#2060	semidirect

H ≅ G/N

type

32. C₄×Dic₁₃

Q₈

semidirect

34. #6099

Q₈

semidirect

105. C₁₃

#2060

semidirect

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

subgroup	index
C₁₃×#544	2
#11718656	2
#11718637	2
#11718124	2
#11718122	2
#2060	13

subgroup

index

C₁₃×#544

Subgroups — #13116178

Conjugacy classes of subgroups

Complements

Maximal subgroups