Group #13117199 — subgroups

Subgroups — #13117199

Order 1664.

628 conjugacy classes of subgroups; only the 325 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 325 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	C₂×C₄×Dic₂₆	C₂	1	3	yes	maximal
2	C₁₃×#376	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₁₃	C₂	1	3	yes	maximal
4	#11718310	C₂	1	6	yes	maximal
5	#11718309	C₂	1	2	yes	maximal
6	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
7	C₂×C₄²×C₁₃	C₂²	1	3	yes	—
8	C₂×C₄⋊Dic₁₃	C₂²	1	3	yes	—
9	C₂×#6101	C₂²	1	6	yes	—
10	C₂×C₄×Dic₁₃	C₂²	1	6	yes	—
11	C₂×C₄×Dic₁₃	C₂²	1	6	yes	—
12	C₄×Dic₂₆	C₄	1	24	yes	complemented
13	C₁₃×#95	C₂²	1	1	yes	characteristic
14	C₁₃×#95	C₂²	1	3	yes	—
15	#88071	C₂²	1	6	yes	—
16	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
17	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
18	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₁₃	C₂³	1	6	yes	—
20	C₂²×Dic₁₃	C₂³	1	2	yes	—
21	C₂×Dic₂₆	C₂×C₄	1	12	yes	—
22	C₄²×C₁₃	C₂×C₄	1	12	yes	—
23	C₄⋊Dic₁₃	C₂×C₄	1	12	yes	—
24	#6101	C₂×C₄	1	24	yes	—
25	C₄×Dic₁₃	—	4	6	no	—
26	C₄×Dic₁₃	C₂×C₄	1	24	yes	—
27	#2276	—	13	1	no	maximal, Sylow
28	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
29	C₂×C₄×C₁₃	—	2	3	no	—
30	C₂×C₄×C₁₃	—	2	6	no	—
31	C₂×C₄×C₁₃	—	2	3	no	—
32	C₂×C₄×C₁₃	C₂²×C₄	1	6	yes	—
33	C₂×C₄×C₁₃	C₂²×C₄	1	6	yes	—
34	C₂×C₄×C₁₃	C₂²×C₄	1	6	yes	—
35	C₂×Dic₁₃	—	2	6	no	—
36	C₂×Dic₁₃	—	2	6	no	—
37	C₂×Dic₁₃	C₂²×C₄	1	12	yes	—
38	C₂×Dic₁₃	C₂²×C₄	1	12	yes	—
39	Dic₂₆	C₄²	1	16	yes	complemented
40	C₂×C₄×Q₈	—	13	3	no	—
41	C₄×C₄⋊C₄	—	13	3	no	—
42	C₄×C₄⋊C₄	—	13	6	no	—
43	#376	—	13	2	no	—
44	#376	D₁₃	1	1	yes	characteristic
45	C₂²×C₁₃	C₂³×C₄	1	3	yes	—
46	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
47	C₂²×C₁₃	2₋¹⁺⁴	1	3	yes	—
48	C₄×C₁₃	—	2	6	no	—
49	C₄×C₁₃	—	2	6	no	—
50	C₄×C₁₃	C₂×C₄²	1	4	yes	—
51	Dic₁₃	—	2	12	no	—
52	Dic₁₃	C₂×C₄²	1	8	yes	—
53	C₂²×Q₈	—	13	1	no	—
54	C₄×Q₈	—	13	24	no	—
55	C₂×C₄⋊C₄	—	13	6	no	—
56	C₂×C₄⋊C₄	—	13	3	no	—
57	C₂×C₄²	—	13	6	no	—
58	C₂×C₄²	—	13	6	no	—
59	C₂×C₄²	C₂×D₁₃	1	3	yes	—
60	#95	—	13	6	no	—
61	#95	C₂×D₁₃	1	1	yes	characteristic
62	#95	C₂×D₁₃	1	3	yes	—
63	C₂×C₁₃	C₂²×C₄²	1	1	yes	characteristic
64	C₂×C₁₃	#519	1	3	yes	—
65	C₂×C₁₃	#520	1	3	yes	—
66	C₂×Q₈	—	13	12	no	—
67	C₂²×C₄	—	13	2	no	—
68	C₂²×C₄	—	13	6	no	—
69	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
70	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
71	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
72	C₄⋊C₄	—	13	24	no	—
73	C₄⋊C₄	—	13	12	no	—
74	C₄²	—	13	24	no	—
75	C₄²	—	52	6	no	—
76	C₄²	C₄×D₁₃	1	12	yes	—
77	C₁₃	#2276	1	1	yes	characteristic, complemented, Sylow
78	C₂³	C₂³×D₁₃	1	1	yes	characteristic
79	Q₈	—	13	16	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	—	2	3	no	—
83	C₂×C₄	—	13	12	no	—
84	C₂×C₄	—	13	12	no	—
85	C₂×C₄	—	26	6	no	—
86	C₂×C₄	—	26	6	no	—
87	C₂×C₄	C₂×C₄×D₁₃	1	6	yes	—
88	C₂×C₄	C₂×C₄×D₁₃	1	6	yes	—
89	C₂×C₄	C₂×C₄×D₁₃	1	6	yes	—
90	C₂²	C₂²×C₄×D₁₃	1	3	yes	—
91	C₂²	#88256	1	1	yes	characteristic
92	C₂²	#88257	1	3	yes	—
93	C₄	—	2	6	no	—
94	C₄	—	2	6	no	—
95	C₄	—	13	8	no	—
96	C₄	—	26	12	no	—
97	C₄	C₄²×D₁₃	1	4	yes	—
98	C₂	C₂×C₄²×D₁₃	1	1	yes	characteristic
99	C₂	#11719221	1	3	yes	—
100	C₂	#11719228	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₂₆

C₂

yes

maximal

C₁₃×#376

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₁₃

C₂

yes

maximal

#11718310

C₂

yes

maximal

#11718309

C₂

yes

maximal

C₂²×Dic₂₆

C₂²

yes

characteristic

C₂×C₄²×C₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₄×Dic₂₆

C₄

yes

complemented

C₁₃×#95

C₂²

yes

characteristic

C₁₃×#95

C₂²

yes

—

#88071

C₂²

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×Dic₂₆

C₂×C₄

yes

—

C₄²×C₁₃

C₂×C₄

yes

—

C₄⋊Dic₁₃

C₂×C₄

yes

—

#6101

C₂×C₄

yes

—

C₄×Dic₁₃

—

C₄×Dic₁₃

C₂×C₄

yes

—

#2276

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂²×C₄

yes

—

C₂×Dic₁₃

C₂²×C₄

yes

—

Dic₂₆

C₄²

yes

complemented

C₂×C₄×Q₈

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

#376

—

#376

D₁₃

yes

characteristic

C₂²×C₁₃

C₂³×C₄

yes

—

C₂²×C₁₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂×C₄²

yes

—

Dic₁₃

—

Dic₁₃

C₂×C₄²

yes

—

C₂²×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

—

#95

—

#95

C₂×D₁₃

yes

characteristic

#95

C₂×D₁₃

yes

—

C₂×C₁₃

C₂²×C₄²

yes

characteristic

C₂×C₁₃

#519

yes

—

C₂×C₁₃

#520

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₄×D₁₃

yes

—

C₁₃

#2276

yes

characteristic, complemented, Sylow

C₂³

C₂³×D₁₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄×D₁₃

yes

—

C₂×C₄

C₂×C₄×D₁₃

yes

—

C₂×C₄

C₂×C₄×D₁₃

yes

—

C₂²

C₂²×C₄×D₁₃

yes

—

C₂²

#88256

yes

characteristic

C₂²

#88257

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

C₄²×D₁₃

yes

—

C₂

C₂×C₄²×D₁₃

yes

characteristic

C₂

#11719221

yes

—

100

C₂

#11719228

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator	C₂×C₁₃	C₂²×C₄²
Fitting	C₁₃×#376	C₂
socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
hypercenter	#376	D₁₃
hypocenter	C₁₃	#2276
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #376

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator

C₂×C₁₃

C₂²×C₄²

Fitting

C₁₃×#376

C₂

socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

D₁₃

C₁₃

G ▷ C₂×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #376

N	H ≅ G/N	type
12. C₄×Dic₂₆	C₄	semidirect
39. Dic₂₆	C₄²	semidirect
77. C₁₃	#2276	semidirect

H ≅ G/N

type

12. C₄×Dic₂₆

C₄

semidirect

39. Dic₂₆

C₄²

semidirect

77. C₁₃

#2276

semidirect

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

subgroup	index
C₂×C₄×Dic₂₆	2
C₁₃×#376	2
C₄×C₄⋊Dic₁₃	2
#11718310	2
#11718309	2
#2276	13

subgroup

index

C₂×C₄×Dic₂₆

C₁₃×#376

C₄×C₄⋊Dic₁₃

#11718310

#11718309

#2276

Subgroups — #13117199

Conjugacy classes of subgroups

Complements

Maximal subgroups