Group #13131717 — subgroups

Subgroups — #13131717

Order 1664.

410 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 152 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	#11719421	C₂	1	1	yes	maximal, characteristic
2	#11719146	C₂	1	4	yes	maximal, complemented
3	#11719145	C₂	1	4	yes	maximal, complemented
4	D₄×#1057	C₂	1	4	yes	maximal, complemented
5	#11719051	C₂	1	2	yes	maximal, complemented
6	C₂×#6129	C₂²	1	2	yes	—
7	C₂×#6122	C₂²	1	4	yes	—
8	C₂²×#1057	C₂²	1	4	yes	—
9	C₁₃×#127	C₄	1	1	yes	characteristic
10	#88192	C₄	1	4	yes	—
11	#88190	C₄	1	2	yes	—
12	D₄×Dic₁₃	C₂²	1	4	yes	—
13	C₄²×D₁₃	C₂²	1	1	yes	characteristic
14	#88123	C₄	1	1	yes	characteristic
15	C₂²⋊#1057	C₂²	1	8	yes	—
16	#88111	C₂²	1	4	yes	—
17	C₄⋊#1057	C₂²	1	2	yes	—
18	C₄⋊#1057	C₂²	1	2	yes	—
19	C₄×#1057	C₂²	1	4	yes	—
20	C₂×D₄×C₁₃	C₂×C₄	1	4	yes	—
21	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
22	C₂×C₂²⋊D₁₃	C₂×C₄	1	4	yes	—
23	C₂²×Dic₁₃	C₂³	1	4	yes	—
24	#6129	—	4	2	no	—
25	#6129	—	4	2	no	—
26	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
27	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
28	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
29	#6122	—	4	4	no	—
30	C₂×#1057	—	4	4	no	—
31	C₂×#1057	C₂³	1	4	yes	—
32	C₂×#1057	C₂³	1	4	yes	—
33	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²⋊Dic₁₃	C₂×C₄	1	8	yes	—
35	C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
36	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
37	C₄×Dic₁₃	C₂³	1	2	yes	—
38	#2979	—	13	1	no	maximal, Sylow
39	C₂³×C₁₃	C₂²×C₄	1	4	yes	—
40	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
41	D₄×C₁₃	—	2	8	no	—
42	D₄×C₁₃	—	4	2	no	—
43	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
45	C₂²⋊D₁₃	—	4	4	no	—
46	C₂×Dic₁₃	—	2	4	no	—
47	C₂×Dic₁₃	—	2	4	no	—
48	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
49	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
50	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
51	C₄×D₁₃	—	2	4	no	—
52	C₄×D₁₃	—	2	1	no	—
53	C₄×D₁₃	—	2	1	no	—
54	Dic₂₆	—	4	2	no	—
55	#1057	—	2	4	no	—
56	#1057	—	2	4	no	—
57	#532	—	13	1	no	—
58	#436	—	13	4	no	—
59	C₈×D₄	—	13	4	no	—
60	#431	—	13	2	no	—
61	#408	—	13	4	no	—
62	C₂²×C₁₃	—	2	4	no	—
63	C₂²×C₁₃	—	2	8	no	—
64	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₁₃	—	2	1	no	—
66	C₂×D₁₃	—	2	2	no	—
67	C₄×C₁₃	—	2	1	no	—
68	C₄×C₁₃	#131	1	4	yes	—
69	Dic₁₃	—	2	4	no	—
70	Dic₁₃	—	2	2	no	—
71	Dic₁₃	H(5,𝔽₂)	1	1	yes	characteristic
72	Dic₁₃	H(5,𝔽₂)	1	1	yes	characteristic
73	C₂×#41	—	13	2	no	—
74	C₂×M₁₆	—	13	4	no	—
75	C₂²×C₈	—	13	4	no	—
76	C₄⋊Q₈	—	13	1	no	—
77	#127	#261	1	1	yes	characteristic
78	#124	—	13	2	no	—
79	#121	—	13	4	no	—
80	C₄×D₄	—	13	4	no	—
81	C₂×C₄²	—	13	1	no	—
82	C₄⋊C₈	—	13	2	no	—
83	C₄⋊C₈	—	13	2	no	—
84	C₂²⋊C₈	—	13	4	no	—
85	C₂²⋊C₈	—	13	8	no	—
86	C₄×C₈	—	13	4	no	—
87	C₂×C₁₃	—	4	4	no	—
88	C₂×C₁₃	#518	1	1	yes	characteristic
89	C₂×C₁₃	C₂×#131	1	2	yes	—
90	D₁₃	—	4	1	no	—
91	#41	—	52	2	no	—
92	#41	—	52	2	no	—
93	C₂×Q₈	—	13	2	no	—
94	C₂×D₄	—	13	4	no	—
95	C₂×D₄	C₂×#261	1	4	yes	—
96	C₂×D₄	C₂×#261	1	2	yes	—
97	C₂²×C₄	—	13	4	no	—
98	C₂²×C₄	—	13	1	no	—
99	C₂²×C₄	—	13	2	no	—
100	M₁₆	—	52	4	no	—
101	C₂×C₈	—	13	4	no	—
102	C₂×C₈	—	13	4	no	—
103	C₂×C₈	—	52	4	no	—
104	C₄⋊C₄	—	13	4	no	—
105	C₂²⋊C₄	—	13	8	no	—
106	C₄²	—	13	2	no	—
107	C₄²	—	13	1	no	—
108	C₄²	C₂×#261	1	1	yes	characteristic
109	C₁₃	#2979	1	1	yes	characteristic, complemented, Sylow
110	C₂³	—	13	1	no	—
111	C₂³	C₂²×#261	1	4	yes	—
112	Q₈	—	52	2	no	—
113	D₄	—	2	8	no	—
114	D₄	—	4	2	no	—
115	D₄	—	52	4	no	—
116	C₂×C₄	—	13	1	no	—
117	C₂×C₄	—	13	2	no	—
118	C₂×C₄	—	13	4	no	—
119	C₂×C₄	—	26	1	no	—
120	C₂×C₄	—	26	1	no	—
121	C₂×C₄	—	26	4	no	—
122	C₂×C₄	—	26	4	no	—
123	C₂×C₄	—	26	4	no	—
124	C₂×C₄	C₂²×#261	1	1	yes	characteristic
125	C₂×C₄	C₂²×#261	1	2	yes	—
126	C₈	—	26	4	no	—
127	C₈	—	26	4	no	—
128	C₂²	—	2	8	no	—
129	C₂²	—	2	4	no	—
130	C₂²	—	26	2	no	—
131	C₂²	—	26	1	no	—
132	C₂²	C₂³×#261	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	13	1	no	—
135	C₄	—	13	1	no	—
136	C₄	—	26	2	no	—
137	C₄	—	26	4	no	—
138	C₄	#88238	1	4	yes	—
139	C₂	—	4	4	no	—
140	C₂	—	52	1	no	—
141	C₂	C₂×#88238	1	2	yes	—
142	C₂	#11719623	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11719421

C₂

yes

maximal, characteristic

#11719146

C₂

yes

maximal, complemented

#11719145

C₂

yes

maximal, complemented

D₄×#1057

C₂

yes

maximal, complemented

#11719051

C₂

yes

maximal, complemented

C₂×#6129

C₂²

yes

—

C₂×#6122

C₂²

yes

—

C₂²×#1057

C₂²

yes

—

C₁₃×#127

C₄

yes

characteristic

#88192

C₄

yes

—

#88190

C₄

yes

—

D₄×Dic₁₃

C₂²

yes

—

C₄²×D₁₃

C₂²

yes

characteristic

#88123

C₄

yes

characteristic

C₂²⋊#1057

C₂²

yes

—

#88111

C₂²

yes

—

C₄⋊#1057

C₂²

yes

—

C₄⋊#1057

C₂²

yes

—

C₄×#1057

C₂²

yes

—

C₂×D₄×C₁₃

C₂×C₄

yes

—

C₂×D₄×C₁₃

C₂×C₄

yes

—

C₂×C₂²⋊D₁₃

C₂×C₄

yes

—

C₂²×Dic₁₃

C₂³

yes

—

#6129

—

#6129

—

C₂×C₄×D₁₃

C₂×C₄

yes

—

C₂×C₄×D₁₃

C₂³

yes

characteristic

C₂×Dic₂₆

C₂×C₄

yes

—

#6122

—

C₂×#1057

—

C₂×#1057

C₂³

yes

—

C₂×#1057

C₂³

yes

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂²⋊Dic₁₃

C₂×C₄

yes

—

C₄⋊Dic₁₃

C₂×C₄

yes

—

C₄×Dic₁₃

C₂×C₄

yes

characteristic

C₄×Dic₁₃

C₂³

yes

—

#2979

—

maximal, Sylow

C₂³×C₁₃

C₂²×C₄

yes

—

C₂²×D₁₃

C₂²×C₄

yes

characteristic

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂²×C₄

yes

—

C₂×Dic₁₃

C₂²×C₄

yes

—

C₂×Dic₁₃

C₂⁴

yes

characteristic

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

Dic₂₆

—

—

—

—

—

C₈×D₄

—

#431

—

#408

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂³×C₄

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#131

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

yes

Dic₁₃

yes

C₂×#41

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

yes

—

—

C₄×D₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

#518

yes

characteristic

C₂×C₁₃

C₂×#131

yes

—

D₁₃

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

C₂×#261

yes

—

C₂×D₄

C₂×#261

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₄⋊C₄

—

105

C₂²⋊C₄

—

106

C₄²

—

107

C₄²

—

108

C₄²

C₂×#261

yes

characteristic

109

C₁₃

#2979

yes

characteristic, complemented, Sylow

110

C₂³

—

111

C₂³

C₂²×#261

yes

—

112

Q₈

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₂²×#261

yes

characteristic

125

C₂×C₄

C₂²×#261

yes

—

126

C₈

—

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂³×#261

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#88238

yes

—

139

C₂

—

140

C₂

—

141

C₂

C₂×#88238

yes

—

142

C₂

#11719623

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#127	C₄
prime radical	#11719421	C₂
cosocle	C₂×Dic₁₃	C₂⁴
hypercenter	#127	#261
hypocenter	C₁₃	#2979
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #127

subgroup

G/H

center, Frattini

C₂²

C₂³×#261

commutator, socle

C₂²×C₁₃

C₂³×C₄

Fitting

C₁₃×#127

C₄

prime radical

#11719421

C₂

cosocle

C₂×Dic₁₃

C₂⁴

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂² ◁ #127

N	H ≅ G/N	type
2. #11719146	C₂	semidirect
3. #11719145	C₂	semidirect
4. D₄×#1057	C₂	semidirect
5. #11719051	C₂	semidirect
109. C₁₃	#2979	semidirect

H ≅ G/N

type

2. #11719146

C₂

semidirect

3. #11719145

C₂

semidirect

4. D₄×#1057

C₂

semidirect

5. #11719051

C₂

semidirect

109. C₁₃

#2979

semidirect

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

subgroup	index
#11719421	2
#11719146	2
#11719145	2
D₄×#1057	2
#11719051	2
#2979	13

subgroup

index

Subgroups — #13131717

Conjugacy classes of subgroups

Complements

Maximal subgroups