Group #13122544 — subgroups

Subgroups — #13122544

Order 1664.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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×#395	C₂	1	1	yes	maximal, characteristic, complemented
2	#11718892	C₂	1	2	yes	maximal, complemented
3	#11718652	C₂	1	3	yes	maximal
4	#11718638	C₂	1	3	yes	maximal, complemented
5	#11718331	C₂	1	6	yes	maximal, complemented
6	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×D₅₂	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
9	C₂×#6103	C₂²	1	6	yes	—
10	C₂×#6103	C₂²	1	6	yes	—
11	C₂×C₄⋊Dic₁₃	C₂²	1	3	yes	—
12	C₂×C₄×Dic₁₃	C₂²	1	3	yes	—
13	C₁₃×#95	C₂²	1	3	yes	—
14	#88071	C₂²	1	6	yes	—
15	#88071	C₂²	1	3	yes	—
16	C₂³×D₁₃	C₂³	1	2	yes	—
17	C₂×Q₈×C₁₃	—	4	3	no	—
18	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
20	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
21	C₂²×Dic₁₃	C₂³	1	6	yes	—
22	C₂×D₅₂	—	4	3	no	—
23	C₁₃×C₄⋊C₄	—	4	3	no	—
24	#6103	—	2	12	no	—
25	#6103	—	4	6	no	—
26	C₄⋊Dic₁₃	—	4	3	no	—
27	C₄×Dic₁₃	—	2	6	no	—
28	#2785	—	13	1	no	maximal, Sylow
29	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
30	C₂²×D₁₃	—	2	6	no	—
31	C₂²×D₁₃	—	2	6	no	—
32	C₂²×D₁₃	—	2	2	no	—
33	Q₈×C₁₃	—	4	4	no	—
34	C₂×C₄×C₁₃	—	2	3	no	—
35	C₂×C₄×C₁₃	—	2	3	no	—
36	C₂×C₄×C₁₃	—	2	6	no	—
37	C₂×C₄×C₁₃	—	2	3	no	—
38	C₂×C₄×C₁₃	—	2	3	no	—
39	C₂×C₄×C₁₃	#41	1	6	yes	—
40	C₂×Dic₁₃	—	2	6	no	—
41	C₂×Dic₁₃	—	2	6	no	—
42	C₂×Dic₁₃	—	2	6	no	—
43	D₅₂	—	4	4	no	—
44	#395	D₁₃	1	1	yes	characteristic, complemented
45	#394	—	13	3	no	—
46	#393	—	13	2	no	—
47	#388	—	13	6	no	—
48	#382	—	13	3	no	—
49	C₂²×C₁₃	C₂×#41	1	3	yes	—
50	C₂²×C₁₃	H(5,𝔽₂)	1	3	yes	—
51	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
52	C₂×D₁₃	—	4	6	no	—
53	C₂×D₁₃	—	4	6	no	—
54	C₂×D₁₃	—	4	2	no	—
55	C₄×C₁₃	—	2	6	no	—
56	C₄×C₁₃	—	4	1	no	—
57	C₄×C₁₃	—	4	3	no	—
58	Dic₁₃	—	4	6	no	—
59	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
60	C₂²×D₄	—	13	1	no	—
61	C₂×C₄⋊C₄	—	13	3	no	—
62	C₂×C₄⋊C₄	C₂×D₁₃	1	3	yes	—
63	C₂×C₂²⋊C₄	—	13	6	no	—
64	C₂×C₂²⋊C₄	—	13	6	no	—
65	C₂×C₄²	—	13	3	no	—
66	#95	—	13	3	no	—
67	#95	—	13	6	no	—
68	#95	C₂×D₁₃	1	3	yes	—
69	C₂×C₁₃	#539	1	3	yes	—
70	C₂×C₁₃	#555	1	3	yes	—
71	C₂×C₁₃	H(3,𝔽₄)	1	1	yes	characteristic
72	D₁₃	—	8	2	no	—
73	C₂⁴	—	13	2	no	—
74	C₂×Q₈	—	4	3	no	—
75	C₂×D₄	—	52	3	no	—
76	C₂²×C₄	—	13	6	no	—
77	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
78	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
79	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
80	C₄⋊C₄	—	4	3	no	—
81	C₄⋊C₄	—	52	3	no	—
82	C₂²⋊C₄	—	26	12	no	—
83	C₂²⋊C₄	—	52	6	no	—
84	C₄²	—	26	6	no	—
85	C₁₃	#2785	1	1	yes	characteristic, complemented, Sylow
86	C₂³	—	26	6	no	—
87	C₂³	—	26	6	no	—
88	C₂³	—	26	2	no	—
89	C₂³	C₂³×D₁₃	1	1	yes	characteristic
90	Q₈	—	4	4	no	—
91	D₄	—	52	4	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	2	3	no	—
94	C₂×C₄	—	2	6	no	—
95	C₂×C₄	—	2	3	no	—
96	C₂×C₄	—	2	3	no	—
97	C₂×C₄	—	26	6	no	—
98	C₂×C₄	—	26	6	no	—
99	C₂×C₄	—	26	6	no	—
100	C₂×C₄	#6131	1	6	yes	—
101	C₂²	—	52	6	no	—
102	C₂²	—	52	2	no	—
103	C₂²	—	52	6	no	—
104	C₂²	#88251	1	3	yes	—
105	C₂²	C₂×#6131	1	3	yes	—
106	C₂²	#88257	1	1	yes	characteristic
107	C₄	—	2	6	no	—
108	C₄	—	4	1	no	—
109	C₄	—	4	3	no	—
110	C₄	—	52	6	no	—
111	C₂	—	104	2	no	—
112	C₂	#11719299	1	1	yes	characteristic
113	C₂	#11719350	1	3	yes	—
114	C₂	#11719439	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#395

C₂

yes

maximal, characteristic, complemented

#11718892

C₂

yes

maximal, complemented

#11718652

C₂

yes

maximal

#11718638

C₂

yes

maximal, complemented

#11718331

C₂

yes

maximal, complemented

C₂²×Q₈×C₁₃

C₂²

yes

characteristic

C₂²×D₅₂

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×#6103

C₂²

yes

—

C₂×#6103

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₁₃×#95

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂³×D₁₃

C₂³

yes

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×D₅₂

—

C₁₃×C₄⋊C₄

—

#6103

—

#6103

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

#2785

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

D₅₂

—

#395

D₁₃

yes

characteristic, complemented

—

—

—

—

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₂²×C₁₃

2₋¹⁺⁴

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

C₂²×Q₈

C₂×D₁₃

yes

characteristic

C₂²×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

#95

—

#95

C₂×D₁₃

yes

—

C₂×C₁₃

#539

yes

—

C₂×C₁₃

#555

yes

—

C₂×C₁₃

H(3,𝔽₄)

yes

characteristic

D₁₃

—

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₃

#2785

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₁₃

yes

characteristic

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#6131

yes

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

#88251

yes

—

105

C₂²

C₂×#6131

yes

—

106

C₂²

#88257

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₂

—

104

—

112

C₂

#11719299

yes

characteristic

113

C₂

#11719350

yes

—

114

C₂

#11719439

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#395	C₂
prime radical	C₂²×D₅₂	C₂²
hypercenter	#395	D₁₃
hypocenter	C₁₃	#2785
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #395

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#395

C₂

prime radical

C₂²×D₅₂

C₂²

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #395

N	H ≅ G/N	type
1. C₁₃×#395	C₂	semidirect
2. #11718892	C₂	semidirect
4. #11718638	C₂	semidirect
5. #11718331	C₂	semidirect
44. #395	D₁₃	semidirect
85. C₁₃	#2785	semidirect

H ≅ G/N

type

1. C₁₃×#395

C₂

semidirect

2. #11718892

C₂

semidirect

4. #11718638

C₂

semidirect

5. #11718331

C₂

semidirect

44. #395

D₁₃

semidirect

85. C₁₃

#2785

semidirect

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

subgroup	index
C₁₃×#395	2
#11718892	2
#11718652	2
#11718638	2
#11718331	2
#2785	13

subgroup

index

C₁₃×#395

Subgroups — #13122544

Conjugacy classes of subgroups

Complements

Maximal subgroups