Group #13123199 — subgroups

Subgroups — #13123199

Order 1664.

472 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 213 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×#88124	C₂	1	2	yes	maximal
2	C₄³×C₁₃	C₂	1	1	yes	maximal, characteristic
3	#11718597	C₂	1	2	yes	maximal
4	#11718596	C₂	1	2	yes	maximal
5	#11718589	C₂	1	8	yes	maximal
6	C₂×C₄²×C₁₃	C₂²	1	4	yes	—
7	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₃	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₁₃	C₂²	1	8	yes	—
12	C₂×#6101	C₂²	1	8	yes	—
13	#88124	—	2	8	no	—
14	#88071	C₂²	1	8	yes	—
15	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
16	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
17	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
19	C₂²×Dic₁₃	C₂³	1	8	yes	—
20	C₄²×C₁₃	—	2	8	no	—
21	C₄²×C₁₃	D₄	1	4	yes	—
22	C₄²×C₁₃	Q₈	1	4	yes	—
23	C₄²×C₁₃	Q₈	1	4	yes	—
24	C₄⋊Dic₁₃	—	2	8	no	—
25	C₄⋊Dic₁₃	—	2	16	no	—
26	#6101	—	2	16	no	—
27	#2859	—	13	1	no	maximal, Sylow
28	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
29	C₂×C₄×C₁₃	—	2	4	no	—
30	C₂×C₄×C₁₃	—	2	4	no	—
31	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
32	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
33	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
34	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
35	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
36	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
37	C₂×C₄×C₁₃	#41	1	8	yes	—
38	C₂×Dic₁₃	—	2	8	no	—
39	C₂×Dic₁₃	—	2	16	no	—
40	C₂×#125	—	13	2	no	—
41	#389	—	13	8	no	—
42	#384	—	13	2	no	—
43	#382	—	13	2	no	—
44	C₄³	D₁₃	1	1	yes	characteristic
45	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
46	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
47	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
48	C₂²×C₁₃	C₂×#41	1	4	yes	—
49	C₄×C₁₃	—	2	4	no	—
50	C₄×C₁₃	—	2	4	no	—
51	C₄×C₁₃	#125	1	8	yes	complemented
52	C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
53	Dic₁₃	—	4	8	no	—
54	#125	—	26	8	no	—
55	C₂×C₄⋊C₄	—	13	8	no	—
56	C₂×C₄⋊C₄	—	13	4	no	—
57	C₂×C₄⋊C₄	—	13	8	no	—
58	C₂×C₄²	C₂×D₁₃	1	4	yes	—
59	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
60	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
61	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
62	#95	—	13	8	no	—
63	C₂×C₁₃	C₂×#125	1	2	yes	—
64	C₂×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₁₃	#532	1	2	yes	—
66	C₂×C₁₃	#533	1	2	yes	—
67	C₂²×C₄	—	13	8	no	—
68	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
69	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
70	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
71	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
72	C₄⋊C₄	—	26	16	no	—
73	C₄⋊C₄	—	26	8	no	—
74	C₄⋊C₄	—	26	16	no	—
75	C₄²	—	2	8	no	—
76	C₄²	Dic₂₆	1	4	yes	—
77	C₄²	Dic₂₆	1	4	yes	—
78	C₄²	D₅₂	1	4	yes	—
79	C₁₃	#2859	1	1	yes	characteristic, complemented, Sylow
80	C₂³	C₂³×D₁₃	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	26	8	no	—
84	C₂×C₄	—	26	16	no	—
85	C₂×C₄	C₂×Dic₂₆	1	4	yes	—
86	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
87	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
88	C₂×C₄	C₂×Dic₂₆	1	4	yes	—
89	C₂×C₄	C₂×D₅₂	1	4	yes	—
90	C₂×C₄	C₂×D₅₂	1	2	yes	—
91	C₂×C₄	#6127	1	8	yes	—
92	C₂²	C₂²×Dic₂₆	1	1	yes	characteristic
93	C₂²	C₂²×Dic₂₆	1	1	yes	characteristic
94	C₂²	C₂²×D₅₂	1	1	yes	characteristic
95	C₂²	C₂×#6127	1	4	yes	—
96	C₄	—	2	4	no	—
97	C₄	—	2	4	no	—
98	C₄	—	52	8	no	—
99	C₄	#88123	1	4	yes	—
100	C₄	#88124	1	8	yes	complemented
101	C₂	C₂×#88123	1	1	yes	characteristic
102	C₂	C₂×#88124	1	2	yes	—
103	C₂	#11719175	1	2	yes	—
104	C₂	#11719182	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88124

C₂

yes

maximal

C₄³×C₁₃

C₂

yes

maximal, characteristic

#11718597

C₂

yes

maximal

#11718596

C₂

yes

maximal

#11718589

C₂

yes

maximal

C₂×C₄²×C₁₃

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

#88124

—

#88071

C₂²

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₄²×C₁₃

—

C₄²×C₁₃

D₄

yes

—

C₄²×C₁₃

Q₈

yes

—

C₄²×C₁₃

Q₈

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

#2859

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

—

—

—

—

C₄³

D₁₃

yes

characteristic

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#125

yes

complemented

C₄×C₁₃

C₄⋊Q₈

yes

—

Dic₁₃

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₃

yes

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₂×C₄²

C₂×D₁₃

yes

characteristic

#95

—

C₂×C₁₃

C₂×#125

yes

—

C₂×C₁₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₃

#532

yes

—

C₂×C₁₃

#533

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂²×C₄

C₂²×D₁₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

Dic₂₆

yes

—

C₄²

Dic₂₆

yes

—

C₄²

D₅₂

yes

—

C₁₃

#2859

yes

characteristic, complemented, Sylow

C₂³

C₂³×D₁₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×Dic₂₆

yes

—

C₂×C₄

C₂×Dic₂₆

yes

—

C₂×C₄

C₂×Dic₂₆

yes

—

C₂×C₄

C₂×Dic₂₆

yes

—

C₂×C₄

C₂×D₅₂

yes

—

C₂×C₄

C₂×D₅₂

yes

—

C₂×C₄

#6127

yes

—

C₂²

C₂²×Dic₂₆

yes

characteristic

C₂²

C₂²×Dic₂₆

yes

characteristic

C₂²

C₂²×D₅₂

yes

characteristic

C₂²

C₂×#6127

yes

—

C₄

—

C₄

—

C₄

—

C₄

#88123

yes

—

100

C₄

#88124

yes

complemented

101

C₂

C₂×#88123

yes

characteristic

102

C₂

C₂×#88124

yes

—

103

C₂

#11719175

yes

—

104

C₂

#11719182

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₄³×C₁₃	C₂
hypercenter	C₄³	D₁₃
hypocenter	C₁₃	#2859
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₄³×C₁₃

C₂

hypercenter

C₄³

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₄³

N	H ≅ G/N	type
51. C₄×C₁₃	#125	semidirect
79. C₁₃	#2859	semidirect
100. C₄	#88124	semidirect

H ≅ G/N

type

51. C₄×C₁₃

#125

semidirect

79. C₁₃

#2859

semidirect

100. C₄

#88124

semidirect

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

subgroup	index
C₂×#88124	2
C₄³×C₁₃	2
#11718597	2
#11718596	2
#11718589	2
#2859	13

subgroup

index

C₄³×C₁₃

Subgroups — #13123199

Conjugacy classes of subgroups

Complements

Maximal subgroups