Group #13122879 — subgroups

Subgroups — #13122879

Order 1664.

630 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 133 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×#88130	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88128	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#383	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718333	C₂	1	6	yes	maximal, complemented
5	#11718332	C₂	1	6	yes	maximal, complemented
6	C₂²×D₅₂	C₂²	1	6	yes	—
7	C₂²×D₅₂	C₂²	1	4	yes	—
8	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#6103	C₂²	1	3	yes	—
10	C₂×#6103	C₂²	1	12	yes	—
11	C₂×#6101	C₂²	1	3	yes	—
12	#88130	—	2	4	no	—
13	#88128	—	2	4	no	—
14	C₁₃×#95	C₂²	1	6	yes	—
15	C₂³×D₁₃	C₂³	1	1	yes	characteristic
16	C₂³×D₁₃	C₂³	1	4	yes	—
17	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
18	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
19	C₂²×Dic₁₃	C₂³	1	3	yes	—
20	C₂×D₅₂	—	2	12	no	—
21	C₂×D₅₂	—	4	12	no	—
22	C₂×D₅₂	—	4	12	no	—
23	C₄²×C₁₃	D₄	1	4	yes	complemented
24	#6103	—	2	6	no	—
25	#6103	—	4	12	no	—
26	#6101	—	2	6	no	—
27	#2818	—	13	1	no	maximal, Sylow
28	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
29	C₂²×D₁₃	—	2	1	no	—
30	C₂²×D₁₃	—	2	4	no	—
31	C₂²×D₁₃	—	2	12	no	—
32	C₂²×D₁₃	—	2	12	no	—
33	C₂²×D₁₃	—	2	6	no	—
34	C₂×C₄×C₁₃	—	2	6	no	—
35	C₂×C₄×C₁₃	—	2	12	no	—
36	C₂×C₄×C₁₃	C₂×D₄	1	6	yes	—
37	C₂×Dic₁₃	—	2	3	no	—
38	C₂×Dic₁₃	—	2	6	no	—
39	D₅₂	—	4	24	no	—
40	D₅₂	—	4	16	no	—
41	C₂×#127	—	13	1	no	—
42	C₂×#126	—	13	1	no	—
43	#394	—	13	6	no	—
44	#392	—	13	6	no	—
45	#383	D₁₃	1	1	yes	characteristic, complemented
46	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
47	C₂²×C₁₃	H(5,𝔽₂)	1	6	yes	—
48	C₂×D₁₃	—	4	12	no	—
49	C₂×D₁₃	—	4	3	no	—
50	C₂×D₁₃	—	4	4	no	—
51	C₂×D₁₃	—	4	4	no	—
52	C₂×D₁₃	—	4	12	no	—
53	C₄×C₁₃	—	2	6	no	—
54	C₄×C₁₃	—	4	4	no	—
55	Dic₁₃	—	4	3	no	—
56	C₂²×D₄	—	13	6	no	—
57	C₂²×D₄	—	13	4	no	—
58	#127	—	26	4	no	—
59	#126	—	26	4	no	—
60	C₂×C₄⋊C₄	—	13	3	no	—
61	C₂×C₂²⋊C₄	—	13	3	no	—
62	C₂×C₂²⋊C₄	—	13	12	no	—
63	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
64	#95	C₂×D₁₃	1	6	yes	—
65	C₂×C₁₃	#535	1	3	yes	—
66	C₂×C₁₃	#560	1	4	yes	—
67	D₁₃	—	8	4	no	—
68	D₁₃	—	8	1	no	—
69	C₂⁴	—	13	1	no	—
70	C₂⁴	—	13	4	no	—
71	C₂×D₄	—	26	12	no	—
72	C₂×D₄	—	52	12	no	—
73	C₂×D₄	—	52	12	no	—
74	C₂²×C₄	—	13	3	no	—
75	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
76	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
77	C₄⋊C₄	—	26	6	no	—
78	C₂²⋊C₄	—	26	6	no	—
79	C₂²⋊C₄	—	52	12	no	—
80	C₄²	D₅₂	1	4	yes	complemented
81	C₁₃	#2818	1	1	yes	characteristic, complemented, Sylow
82	C₂³	—	26	1	no	—
83	C₂³	—	26	4	no	—
84	C₂³	—	26	12	no	—
85	C₂³	—	26	12	no	—
86	C₂³	—	26	6	no	—
87	C₂³	C₂³×D₁₃	1	1	yes	characteristic
88	D₄	—	52	24	no	—
89	D₄	—	52	16	no	—
90	C₂×C₄	—	2	6	no	—
91	C₂×C₄	—	2	12	no	—
92	C₂×C₄	—	26	3	no	—
93	C₂×C₄	—	26	6	no	—
94	C₂×C₄	C₂×D₅₂	1	6	yes	—
95	C₂²	—	52	12	no	—
96	C₂²	—	52	3	no	—
97	C₂²	—	52	4	no	—
98	C₂²	—	52	4	no	—
99	C₂²	—	52	12	no	—
100	C₂²	C₂²×D₅₂	1	1	yes	characteristic
101	C₂²	#88256	1	6	yes	—
102	C₄	—	2	6	no	—
103	C₄	—	4	4	no	—
104	C₄	—	52	3	no	—
105	C₂	—	104	4	no	—
106	C₂	—	104	1	no	—
107	C₂	#11719230	1	3	yes	—
108	C₂	#11719416	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88130

C₂

yes

maximal, characteristic, complemented

C₂×#88128

C₂

yes

maximal, characteristic, complemented

C₁₃×#383

C₂

yes

maximal, characteristic, complemented

#11718333

C₂

yes

maximal, complemented

#11718332

C₂

yes

maximal, complemented

C₂²×D₅₂

C₂²

yes

—

C₂²×D₅₂

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×#6103

C₂²

yes

—

C₂×#6103

C₂²

yes

—

C₂×#6101

C₂²

yes

—

#88130

—

#88128

—

C₁₃×#95

C₂²

yes

—

C₂³×D₁₃

C₂³

yes

characteristic

C₂³×D₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₄²×C₁₃

D₄

yes

—

—

—

—

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

C₂²×D₄

—

C₂²×D₄

—

#127

—

#126

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#95

C₂×D₁₃

yes

—

C₂×C₁₃

#535

yes

—

C₂×C₁₃

#560

yes

—

D₁₃

—

D₁₃

—

C₂⁴

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

D₅₂

yes

complemented

C₁₃

#2818

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₁₃

yes

characteristic

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₅₂

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

C₂²×D₅₂

yes

characteristic

101

C₂²

#88256

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₂

—

104

—

106

C₂

—

104

—

107

C₂

#11719230

yes

—

108

C₂

#11719416

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#383	C₂
hypercenter	#383	D₁₃
hypocenter	C₁₃	#2818
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #383

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#383

C₂

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #383

N	H ≅ G/N	type
1. C₂×#88130	C₂	semidirect
2. C₂×#88128	C₂	semidirect
3. C₁₃×#383	C₂	semidirect
4. #11718333	C₂	semidirect
5. #11718332	C₂	semidirect
23. C₄²×C₁₃	D₄	semidirect
45. #383	D₁₃	semidirect
80. C₄²	D₅₂	semidirect
81. C₁₃	#2818	semidirect

H ≅ G/N

type

1. C₂×#88130

C₂

semidirect

2. C₂×#88128

C₂

semidirect

3. C₁₃×#383

C₂

semidirect

4. #11718333

C₂

semidirect

5. #11718332

C₂

semidirect

23. C₄²×C₁₃

D₄

semidirect

45. #383

D₁₃

semidirect

80. C₄²

D₅₂

semidirect

81. C₁₃

#2818

semidirect

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

subgroup	index
C₂×#88130	2
C₂×#88128	2
C₁₃×#383	2
#11718333	2
#11718332	2
#2818	13

subgroup

index

C₁₃×#383

Subgroups — #13122879

Conjugacy classes of subgroups

Complements

Maximal subgroups