Group #13116188 — subgroups

Subgroups — #13116188

Order 1664.

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

normal subgroups only

This group has 314 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×#543	C₂	1	1	yes	maximal, characteristic
2	#11718607	C₂	1	3	yes	maximal
3	#11718118	C₂	1	3	yes	maximal
4	C₂³×Dic₁₃	C₂²	1	1	yes	characteristic
5	C₁₃×C₂²⋊Q₈	—	2	3	no	—
6	C₁₃×C₂²⋊Q₈	—	2	3	no	—
7	C₂×C₁₃×C₂²⋊C₄	C₂²	1	3	yes	—
8	C₂×C₂²⋊Dic₁₃	C₂²	1	3	yes	—
9	C₂×#6101	—	2	3	no	—
10	#88071	—	2	3	no	—
11	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
12	C₂×Q₈×C₁₃	—	2	1	no	—
13	C₂×Q₈×C₁₃	—	2	1	no	—
14	C₂²×C₄×C₁₃	D₄	1	3	yes	—
15	C₂²×C₄×C₁₃	D₄	1	3	yes	—
16	C₂²×Dic₁₃	—	2	3	no	—
17	C₂²×Dic₁₃	—	2	3	no	—
18	C₂²×Dic₁₃	—	4	2	no	—
19	C₁₃×C₄⋊C₄	—	2	3	no	—
20	C₁₃×C₄⋊C₄	—	2	3	no	—
21	C₁₃×C₂²⋊C₄	—	2	6	no	—
22	C₂²⋊Dic₁₃	—	2	3	no	—
23	C₂²⋊Dic₁₃	Q₈	1	6	yes	complemented
24	#6101	—	4	6	no	—
25	#2062	—	13	1	no	maximal, Sylow
26	C₂³×C₁₃	—	4	1	no	—
27	C₂³×C₁₃	—	4	1	no	—
28	C₂³×C₁₃	C₂×D₄	1	3	yes	—
29	C₂³×C₁₃	C₂×Q₈	1	3	yes	—
30	C₂³×C₁₃	#41	1	1	yes	characteristic
31	Q₈×C₁₃	—	8	1	no	—
32	Q₈×C₁₃	—	8	1	no	—
33	C₂×C₄×C₁₃	—	2	3	no	—
34	C₂×C₄×C₁₃	—	2	3	no	—
35	C₂×C₄×C₁₃	—	4	3	no	—
36	C₂×C₄×C₁₃	—	4	3	no	—
37	C₂×Dic₁₃	—	2	6	no	—
38	C₂×Dic₁₃	—	2	6	no	—
39	C₂×Dic₁₃	—	4	3	no	—
40	C₂×Dic₁₃	—	4	3	no	—
41	C₂×Dic₁₃	—	4	3	no	—
42	C₂×Dic₁₃	—	4	1	no	—
43	#543	D₁₃	1	1	yes	characteristic
44	#385	—	13	3	no	—
45	#342	—	13	3	no	—
46	C₂²×C₁₃	—	2	3	no	—
47	C₂²×C₁₃	—	2	3	no	—
48	C₂²×C₁₃	—	4	3	no	—
49	C₂²×C₁₃	—	4	1	no	—
50	C₂²×C₁₃	#120	1	1	yes	characteristic
51	C₂²×C₁₃	C₂²⋊Q₈	1	3	yes	—
52	C₂²×C₁₃	C₄⋊Q₈	1	3	yes	—
53	C₄×C₁₃	—	4	3	no	—
54	C₄×C₁₃	—	4	3	no	—
55	Dic₁₃	—	4	6	no	—
56	Dic₁₃	—	4	2	no	—
57	C₂³×C₄	—	13	1	no	—
58	C₂²⋊Q₈	—	2	3	no	—
59	C₂²⋊Q₈	—	2	3	no	—
60	C₂×C₄⋊C₄	—	26	3	no	—
61	C₂×C₂²⋊C₄	—	13	3	no	—
62	C₂×C₂²⋊C₄	C₂×D₁₃	1	3	yes	—
63	#95	—	26	3	no	—
64	C₂×C₁₃	—	2	3	no	—
65	C₂×C₁₃	—	2	3	no	—
66	C₂×C₁₃	#395	1	1	yes	characteristic
67	C₂×C₁₃	B(4,𝔽₂)	1	1	yes	characteristic
68	C₂×C₁₃	#458	1	1	yes	characteristic
69	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
70	C₂×Q₈	—	2	1	no	—
71	C₂×Q₈	—	2	1	no	—
72	C₂²×C₄	—	26	3	no	—
73	C₂²×C₄	—	26	3	no	—
74	C₂²×C₄	—	52	2	no	—
75	C₂²×C₄	C₂²⋊D₁₃	1	3	yes	—
76	C₂²×C₄	C₂²⋊D₁₃	1	3	yes	—
77	C₄⋊C₄	—	2	3	no	—
78	C₄⋊C₄	—	2	3	no	—
79	C₄⋊C₄	—	52	6	no	—
80	C₂²⋊C₄	—	2	6	no	—
81	C₂²⋊C₄	—	13	6	no	—
82	C₂²⋊C₄	—	26	3	no	—
83	C₁₃	#2062	1	1	yes	characteristic, complemented, Sylow
84	C₂³	—	4	1	no	—
85	C₂³	—	4	1	no	—
86	C₂³	Q₈×D₁₃	1	3	yes	—
87	C₂³	#6131	1	1	yes	characteristic
88	C₂³	C₂×C₂²⋊D₁₃	1	3	yes	—
89	Q₈	—	8	1	no	—
90	Q₈	—	8	1	no	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	4	3	no	—
94	C₂×C₄	—	4	3	no	—
95	C₂×C₄	—	26	6	no	—
96	C₂×C₄	—	26	6	no	—
97	C₂×C₄	—	52	3	no	—
98	C₂×C₄	—	52	3	no	—
99	C₂×C₄	—	52	3	no	—
100	C₂×C₄	—	52	1	no	—
101	C₂²	—	2	3	no	—
102	C₂²	—	2	3	no	—
103	C₂²	—	4	3	no	—
104	C₂²	—	4	1	no	—
105	C₂²	#88198	1	3	yes	—
106	C₂²	#88200	1	3	yes	—
107	C₂²	#88207	1	1	yes	characteristic
108	C₄	—	4	3	no	—
109	C₄	—	4	3	no	—
110	C₄	—	52	6	no	—
111	C₄	—	52	2	no	—
112	C₂	—	2	3	no	—
113	C₂	—	2	3	no	—
114	C₂	#11718890	1	1	yes	characteristic
115	C₂	#11718904	1	1	yes	characteristic
116	C₂	#11718905	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#543

C₂

yes

maximal, characteristic

#11718607

C₂

yes

maximal

#11718118

C₂

yes

maximal

C₂³×Dic₁₃

C₂²

yes

characteristic

C₁₃×C₂²⋊Q₈

—

C₁₃×C₂²⋊Q₈

—

C₂×C₁₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

C₂×#6101

—

#88071

—

C₂⁴×C₁₃

C₂³

yes

characteristic

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₂²⋊Dic₁₃

—

C₂²⋊Dic₁₃

Q₈

yes

—

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂×D₄

yes

—

C₂³×C₁₃

C₂×Q₈

yes

—

C₂³×C₁₃

#41

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

#543

D₁₃

yes

characteristic

#385

—

#342

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#120

yes

characteristic

C₂²×C₁₃

C₂²⋊Q₈

yes

—

C₂²×C₁₃

C₄⋊Q₈

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂³×C₄

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₁₃

yes

—

#95

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂⁴

C₂²×D₁₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₃

yes

—

C₂²×C₄

C₂²⋊D₁₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₁₃

#2062

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

C₂³

Q₈×D₁₃

yes

—

C₂³

#6131

yes

characteristic

C₂³

C₂×C₂²⋊D₁₃

yes

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

#88198

yes

—

106

C₂²

#88200

yes

—

107

C₂²

#88207

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₂

—

113

C₂

—

114

C₂

#11718890

yes

characteristic

115

C₂

#11718904

yes

characteristic

116

C₂

#11718905

yes

characteristic

	subgroup	G/H
center	C₂²	#88207
commutator, prime radical, cosocle	C₂⁴×C₁₃	C₂³
Frattini	C₂⁴	C₂²×D₁₃
Fitting	C₁₃×#543	C₂
socle	C₂²×C₁₃	#120
hypercenter	#543	D₁₃
hypocenter	C₁₃	#2062
LCS	G ▷ C₂⁴×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #543

subgroup

G/H

center

C₂²

#88207

commutator, prime radical, cosocle

C₂⁴×C₁₃

C₂³

Frattini

C₂⁴

C₂²×D₁₃

Fitting

C₁₃×#543

C₂

C₂²×C₁₃

D₁₃

C₁₃

G ▷ C₂⁴×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂⁴ ◁ #543

N	H ≅ G/N	type
23. C₂²⋊Dic₁₃	Q₈	semidirect
83. C₁₃	#2062	semidirect

H ≅ G/N

type

23. C₂²⋊Dic₁₃

Q₈

semidirect

83. C₁₃

#2062

semidirect

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

subgroup	index
C₁₃×#543	2
#11718607	2
#11718118	2
#2062	13

subgroup

index

C₁₃×#543

#11718607

#11718118

#2062

Subgroups — #13116188

Conjugacy classes of subgroups

Complements

Maximal subgroups