Group #13131723 — subgroups

Subgroups — #13131723

Order 1664.

394 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 144 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	#11719430	C₂	1	1	yes	maximal, characteristic
2	#11719159	C₂	1	4	yes	maximal, complemented
3	#11719075	C₂	1	8	yes	maximal, complemented
4	#11719055	C₂	1	2	yes	maximal, complemented
5	C₂×#6131	C₂²	1	2	yes	—
6	C₂×#6118	C₂²	1	4	yes	—
7	C₂×D₁₃⋊C₈	C₂²	1	4	yes	—
8	C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
9	#88201	C₄	1	2	yes	—
10	#88149	C₄	1	4	yes	—
11	#88147	C₂²	1	4	yes	—
12	#88128	C₄	1	1	yes	characteristic
13	C₄²×D₁₃	C₂²	1	1	yes	characteristic
14	#88112	C₂²	1	4	yes	—
15	#88111	C₂²	1	4	yes	—
16	#88111	C₂²	1	8	yes	—
17	#88110	C₂²	1	4	yes	—
18	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
19	#6131	—	4	2	no	—
20	#6131	—	4	2	no	—
21	C₂×D₅₂	C₂×C₄	1	4	yes	—
22	C₂×D₅₂	C₂×C₄	1	2	yes	—
23	C₂×D₅₂	C₂×C₄	1	4	yes	—
24	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
25	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₄×D₁₃	C₂³	1	4	yes	—
27	C₂×#1057	C₂³	1	8	yes	—
28	#6118	—	4	4	no	—
29	D₁₃⋊C₈	—	4	4	no	—
30	C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
31	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
32	#6103	C₂×C₄	1	8	yes	—
33	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₁₃	C₂³	1	2	yes	—
35	#2980	—	13	1	no	maximal, Sylow
36	C₂²×D₁₃	C₂²×C₄	1	4	yes	—
37	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
38	Q₈×C₁₃	—	4	2	no	—
39	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
41	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
42	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
43	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
44	D₅₂	—	4	4	no	—
45	D₅₂	—	4	2	no	—
46	D₅₂	—	4	4	no	—
47	C₄×D₁₃	—	2	4	no	—
48	C₄×D₁₃	—	2	4	no	—
49	C₄×D₁₃	—	2	2	no	—
50	C₄×D₁₃	—	4	2	no	—
51	#1057	—	2	8	no	—
52	#532	—	13	1	no	—
53	#435	—	13	8	no	—
54	#432	—	13	2	no	—
55	#408	—	13	4	no	—
56	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
57	C₂×D₁₃	—	2	1	no	—
58	C₂×D₁₃	—	2	4	no	—
59	C₂×D₁₃	—	2	2	no	—
60	C₂×D₁₃	—	2	8	no	—
61	C₄×C₁₃	—	2	1	no	—
62	C₄×C₁₃	—	2	4	no	—
63	C₄×C₁₃	—	2	2	no	—
64	Dic₁₃	—	2	2	no	—
65	Dic₁₃	H(5,𝔽₂)	1	2	yes	—
66	C₂×#41	—	13	2	no	—
67	C₂×M₁₆	—	13	4	no	—
68	C₂²×C₈	—	13	4	no	—
69	C₄⋊Q₈	#261	1	1	yes	characteristic
70	#127	—	13	1	no	—
71	#124	—	13	2	no	—
72	#121	—	13	4	no	—
73	C₄×D₄	—	13	4	no	—
74	C₂×C₄²	—	13	1	no	—
75	C₄⋊C₈	—	13	4	no	—
76	C₂²⋊C₈	—	13	4	no	—
77	C₂²⋊C₈	—	13	8	no	—
78	#97	—	13	4	no	—
79	C₂×C₁₃	#518	1	1	yes	characteristic
80	C₂×C₁₃	#568	1	2	yes	—
81	D₁₃	—	4	1	no	—
82	D₁₃	—	4	4	no	—
83	#41	—	52	2	no	—
84	#41	—	52	2	no	—
85	C₂×Q₈	C₂×#261	1	2	yes	—
86	C₂×D₄	—	13	4	no	—
87	C₂×D₄	—	13	4	no	—
88	C₂×D₄	—	13	2	no	—
89	C₂²×C₄	—	13	1	no	—
90	C₂²×C₄	—	13	4	no	—
91	C₂²×C₄	—	13	2	no	—
92	M₁₆	—	52	4	no	—
93	C₂×C₈	—	13	8	no	—
94	C₂×C₈	—	52	4	no	—
95	C₄⋊C₄	C₂×#261	1	4	yes	—
96	C₂²⋊C₄	—	13	8	no	—
97	C₄²	—	13	2	no	—
98	C₄²	—	13	1	no	—
99	C₄²	C₂×#261	1	1	yes	characteristic
100	C₁₃	#2980	1	1	yes	characteristic, complemented, Sylow
101	C₂³	—	13	4	no	—
102	C₂³	—	13	1	no	—
103	Q₈	—	4	2	no	—
104	D₄	—	52	4	no	—
105	D₄	—	52	2	no	—
106	D₄	—	52	4	no	—
107	C₂×C₄	—	13	1	no	—
108	C₂×C₄	—	13	2	no	—
109	C₂×C₄	—	26	4	no	—
110	C₂×C₄	—	26	4	no	—
111	C₂×C₄	—	26	2	no	—
112	C₂×C₄	—	52	2	no	—
113	C₂×C₄	C₂²×#261	1	2	yes	—
114	C₂×C₄	C₂²×#261	1	1	yes	characteristic
115	C₂×C₄	C₂²×#261	1	4	yes	—
116	C₈	—	26	8	no	—
117	C₂²	—	26	1	no	—
118	C₂²	—	26	4	no	—
119	C₂²	—	26	2	no	—
120	C₂²	—	26	8	no	—
121	C₂²	C₂³×#261	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	2	4	no	—
124	C₄	—	2	2	no	—
125	C₄	—	13	2	no	—
126	C₄	—	26	2	no	—
127	C₂	—	52	1	no	—
128	C₂	—	52	4	no	—
129	C₂	#11719623	1	1	yes	characteristic
130	C₂	#11719625	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11719430

C₂

yes

maximal, characteristic

#11719159

C₂

yes

maximal, complemented

#11719075

C₂

yes

maximal, complemented

#11719055

C₂

yes

maximal, complemented

C₂×#6131

C₂²

yes

—

C₂×#6118

C₂²

yes

—

C₂×D₁₃⋊C₈

C₂²

yes

—

C₁₃×C₄⋊Q₈

C₄

yes

characteristic

#88201

C₄

yes

—

#88149

C₄

yes

—

#88147

C₂²

yes

—

#88128

C₄

yes

characteristic

C₄²×D₁₃

C₂²

yes

characteristic

#88112

C₂²

yes

—

#88111

C₂²

yes

—

#88111

C₂²

yes

—

#88110

C₂²

yes

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

#6131

—

#6131

—

C₂×D₅₂

C₂×C₄

yes

—

C₂×D₅₂

C₂×C₄

yes

—

C₂×D₅₂

C₂×C₄

yes

—

C₂×C₄×D₁₃

C₂×C₄

yes

—

C₂×C₄×D₁₃

C₂³

yes

characteristic

C₂×C₄×D₁₃

C₂³

yes

—

C₂×#1057

C₂³

yes

—

#6118

—

D₁₃⋊C₈

—

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

#6103

C₂×C₄

yes

—

C₄×Dic₁₃

C₂×C₄

yes

characteristic

C₄×Dic₁₃

C₂³

yes

—

#2980

—

maximal, Sylow

C₂²×D₁₃

C₂²×C₄

yes

—

C₂²×D₁₃

C₂²×C₄

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×Dic₁₃

C₂²×C₄

yes

—

C₂×Dic₁₃

C₂⁴

yes

characteristic

D₅₂

—

D₅₂

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

—

—

—

—

—

C₂²×C₁₃

C₂³×C₄

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

H(5,𝔽₂)

yes

—

C₂×#41

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

yes

—

—

—

C₄×D₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₂×C₁₃

#518

yes

characteristic

C₂×C₁₃

#568

yes

—

D₁₃

—

D₁₃

—

#41

—

#41

—

C₂×Q₈

C₂×#261

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×#261

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×#261

yes

characteristic

100

C₁₃

#2980

yes

characteristic, complemented, Sylow

101

C₂³

—

102

C₂³

—

103

Q₈

—

104

D₄

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²×#261

yes

—

114

C₂×C₄

C₂²×#261

yes

characteristic

115

C₂×C₄

C₂²×#261

yes

—

116

C₈

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂³×#261

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₂

—

128

C₂

—

129

C₂

#11719623

yes

characteristic

130

C₂

#11719625

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×C₄⋊Q₈	C₄
prime radical	#11719430	C₂
cosocle	C₂×Dic₁₃	C₂⁴
hypercenter	C₄⋊Q₈	#261
hypocenter	C₁₃	#2980
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#261

commutator, socle

C₂²×C₁₃

C₂³×C₄

Fitting

C₁₃×C₄⋊Q₈

C₄

prime radical

#11719430

C₂

cosocle

C₂×Dic₁₃

C₂⁴

C₄⋊Q₈

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. #11719159	C₂	semidirect
3. #11719075	C₂	semidirect
4. #11719055	C₂	semidirect
100. C₁₃	#2980	semidirect

H ≅ G/N

type

2. #11719159

C₂

semidirect

3. #11719075

C₂

semidirect

4. #11719055

C₂

semidirect

100. C₁₃

#2980

semidirect

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

subgroup	index
#11719430	2
#11719159	2
#11719075	2
#11719055	2
#2980	13

subgroup

index

Subgroups — #13131723

Conjugacy classes of subgroups

Complements

Maximal subgroups