Group #13122828 — subgroups

Subgroups — #13122828

Order 1664.

432 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 149 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
2	#11718890	C₂	1	2	yes	maximal
3	#11718640	C₂	1	3	yes	maximal
4	#11718635	C₂	1	3	yes	maximal
5	#11718307	C₂	1	6	yes	maximal
6	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
8	C₂²×Dic₂₆	C₂²	1	3	yes	—
9	C₂×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
10	C₂×#6101	C₂²	1	6	yes	—
11	C₂×C₄×Dic₁₃	C₂²	1	3	yes	—
12	C₁₃×#95	C₂²	1	3	yes	—
13	#88071	C₂²	1	6	yes	—
14	#88071	C₂²	1	6	yes	—
15	#88071	C₂²	1	3	yes	—
16	C₂×Q₈×C₁₃	—	4	3	no	—
17	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
19	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
20	C₂²×Dic₁₃	C₂³	1	2	yes	—
21	C₂²×Dic₁₃	C₂³	1	6	yes	—
22	C₂×Dic₂₆	—	4	3	no	—
23	C₂×Dic₂₆	—	4	3	no	—
24	C₂×Dic₂₆	—	4	3	no	—
25	C₂×Dic₂₆	Q₈	1	12	yes	complemented
26	C₁₃×C₄⋊C₄	—	4	3	no	—
27	#6101	—	4	6	no	—
28	C₄×Dic₁₃	—	4	3	no	—
29	#2811	—	13	1	no	maximal, Sylow
30	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
31	Q₈×C₁₃	—	4	4	no	—
32	C₂×C₄×C₁₃	—	2	3	no	—
33	C₂×C₄×C₁₃	—	2	3	no	—
34	C₂×C₄×C₁₃	—	2	6	no	—
35	C₂×C₄×C₁₃	—	2	3	no	—
36	C₂×C₄×C₁₃	—	2	3	no	—
37	C₂×C₄×C₁₃	C₂×Q₈	1	6	yes	—
38	C₂×Dic₁₃	—	2	6	no	—
39	C₂×Dic₁₃	—	2	6	no	—
40	C₂×Dic₁₃	—	2	6	no	—
41	C₂×Dic₁₃	C₂×Q₈	1	12	yes	—
42	Dic₂₆	—	4	12	no	—
43	Dic₂₆	—	4	4	no	—
44	#395	—	13	2	no	—
45	#395	—	13	3	no	—
46	#395	D₁₃	1	1	yes	characteristic
47	#391	—	13	3	no	—
48	#391	—	13	6	no	—
49	C₂²×C₁₃	C₂²×Q₈	1	3	yes	—
50	C₂²×C₁₃	H(5,𝔽₂)	1	3	yes	—
51	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
52	C₄×C₁₃	—	2	6	no	—
53	C₄×C₁₃	—	4	1	no	—
54	C₄×C₁₃	—	4	3	no	—
55	Dic₁₃	—	2	12	no	—
56	Dic₁₃	—	4	2	no	—
57	C₂²×Q₈	—	13	3	no	—
58	C₂²×Q₈	—	13	1	no	—
59	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
60	C₂×C₄⋊C₄	—	13	6	no	—
61	C₂×C₄⋊C₄	C₂×D₁₃	1	3	yes	—
62	C₂×C₄²	—	13	3	no	—
63	#95	—	13	3	no	—
64	#95	—	13	6	no	—
65	#95	—	13	6	no	—
66	#95	C₂×D₁₃	1	3	yes	—
67	C₂×C₁₃	#543	1	3	yes	—
68	C₂×C₁₃	Q₈²	1	3	yes	—
69	C₂×C₁₃	H(3,𝔽₄)	1	1	yes	characteristic
70	C₂×Q₈	—	4	3	no	—
71	C₂×Q₈	—	13	12	no	—
72	C₂×Q₈	—	52	3	no	—
73	C₂×Q₈	—	52	3	no	—
74	C₂×Q₈	—	52	3	no	—
75	C₂²×C₄	—	13	6	no	—
76	C₂²×C₄	—	13	2	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	6	no	—
82	C₄²	—	52	3	no	—
83	C₁₃	#2811	1	1	yes	characteristic, complemented, Sylow
84	C₂³	C₂³×D₁₃	1	1	yes	characteristic
85	Q₈	—	4	4	no	—
86	Q₈	—	52	12	no	—
87	Q₈	—	52	4	no	—
88	C₂×C₄	—	2	3	no	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	2	3	no	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	2	6	no	—
93	C₂×C₄	—	13	12	no	—
94	C₂×C₄	—	26	6	no	—
95	C₂×C₄	—	26	6	no	—
96	C₂×C₄	—	26	6	no	—
97	C₂×C₄	Q₈×D₁₃	1	6	yes	—
98	C₂²	#88251	1	3	yes	—
99	C₂²	C₂×Q₈×D₁₃	1	3	yes	—
100	C₂²	#88256	1	1	yes	characteristic
101	C₄	—	2	6	no	—
102	C₄	—	4	3	no	—
103	C₄	—	4	1	no	—
104	C₄	—	26	12	no	—
105	C₄	—	52	2	no	—
106	C₂	#11719299	1	1	yes	characteristic
107	C₂	#11719339	1	3	yes	—
108	C₂	#11719427	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#395

C₂

yes

maximal, characteristic

#11718890

C₂

yes

maximal

#11718640

C₂

yes

maximal

#11718635

C₂

yes

maximal

#11718307

C₂

yes

maximal

C₂²×Q₈×C₁₃

C₂²

yes

characteristic

C₂²×Dic₂₆

C₂²

yes

characteristic

C₂²×Dic₂₆

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₁₃×#95

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

Q₈

yes

complemented

C₁₃×C₄⋊C₄

—

#6101

—

C₄×Dic₁₃

—

#2811

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×Q₈

yes

—

Dic₂₆

—

Dic₂₆

—

#395

—

#395

—

#395

D₁₃

yes

characteristic

#391

—

#391

—

C₂²×C₁₃

C₂²×Q₈

yes

—

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₂²×C₁₃

H(5,𝔽₂)

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₁₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

—

—

—

—

C₂×D₁₃

yes

—

C₂×C₁₃

#543

yes

—

C₂×C₁₃

Q₈²

yes

—

C₂×C₁₃

H(3,𝔽₄)

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₁₃

#2811

yes

characteristic, complemented, Sylow

C₂³

C₂³×D₁₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Q₈×D₁₃

yes

—

C₂²

#88251

yes

—

C₂²

C₂×Q₈×D₁₃

yes

—

100

C₂²

#88256

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₂

#11719299

yes

characteristic

107

C₂

#11719339

yes

—

108

C₂

#11719427

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#395	C₂
hypercenter	#395	D₁₃
hypocenter	C₁₃	#2811
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #395

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#395

C₂

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #395

N	H ≅ G/N	type
25. C₂×Dic₂₆	Q₈	semidirect
83. C₁₃	#2811	semidirect

H ≅ G/N

type

25. C₂×Dic₂₆

Q₈

semidirect

83. C₁₃

#2811

semidirect

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

subgroup	index
C₁₃×#395	2
#11718890	2
#11718640	2
#11718635	2
#11718307	2
#2811	13

subgroup

index

C₁₃×#395

Subgroups — #13122828

Conjugacy classes of subgroups

Complements

Maximal subgroups