Group #13120779 — subgroups

Subgroups — #13120779

Order 1664.

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

normal subgroups only

This group has 584 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	C₂×#88128	C₂	1	1	yes	maximal, characteristic
2	C₂×#88124	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718600	C₂	1	4	yes	maximal, complemented
5	#11718333	C₂	1	8	yes	maximal, complemented
6	C₂²×D₅₂	C₂²	1	2	yes	—
7	C₂²×D₅₂	C₂²	1	4	yes	—
8	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
9	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#6103	C₂²	1	16	yes	—
11	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
12	C₂×#6101	C₂²	1	4	yes	—
13	#88128	—	2	4	no	—
14	#88124	—	2	4	no	—
15	C₁₃×#95	C₂²	1	4	yes	—
16	C₂³×D₁₃	C₂³	1	4	yes	—
17	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
18	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
19	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₁₃	C₂³	1	4	yes	—
21	C₂×D₅₂	—	2	8	no	—
22	C₂×D₅₂	—	2	4	no	—
23	C₂×D₅₂	—	2	8	no	—
24	C₂×D₅₂	—	4	8	no	—
25	C₄²×C₁₃	—	2	4	no	—
26	C₄²×C₁₃	D₄	1	4	yes	—
27	#6103	—	4	16	no	—
28	C₄⋊Dic₁₃	—	2	4	no	—
29	#6101	—	2	8	no	—
30	#2641	—	13	1	no	maximal, Sylow
31	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
32	C₂²×D₁₃	—	2	4	no	—
33	C₂²×D₁₃	—	2	8	no	—
34	C₂²×D₁₃	—	2	16	no	—
35	C₂×C₄×C₁₃	—	2	4	no	—
36	C₂×C₄×C₁₃	—	2	4	no	—
37	C₂×C₄×C₁₃	—	2	8	no	—
38	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
39	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₃	#41	1	4	yes	—
41	C₂×Dic₁₃	—	2	4	no	—
42	C₂×Dic₁₃	—	2	8	no	—
43	D₅₂	—	4	16	no	—
44	D₅₂	—	4	8	no	—
45	C₂×#127	—	13	1	no	—
46	C₂×#125	—	13	1	no	—
47	#394	—	13	8	no	—
48	#390	—	13	4	no	—
49	#376	D₁₃	1	1	yes	characteristic, complemented
50	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
51	C₂²×C₁₃	C₂×#41	1	2	yes	—
52	C₂²×C₁₃	H(5,𝔽₂)	1	4	yes	—
53	C₂×D₁₃	—	4	4	no	—
54	C₂×D₁₃	—	4	8	no	—
55	C₂×D₁₃	—	4	16	no	—
56	C₄×C₁₃	—	2	4	no	—
57	C₄×C₁₃	—	4	4	no	—
58	C₄×C₁₃	#124	1	4	yes	—
59	Dic₁₃	—	4	4	no	—
60	C₂²×D₄	—	13	2	no	—
61	C₂²×D₄	—	13	4	no	—
62	#127	—	26	4	no	—
63	#125	—	26	4	no	—
64	C₂×C₄⋊C₄	—	13	4	no	—
65	C₂×C₄⋊C₄	—	13	2	no	—
66	C₂×C₂²⋊C₄	—	13	16	no	—
67	C₂×C₄²	C₂×D₁₃	1	2	yes	—
68	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
69	#95	C₂×D₁₃	1	4	yes	—
70	C₂×C₁₃	C₂×#124	1	1	yes	characteristic
71	C₂×C₁₃	#535	1	2	yes	—
72	C₂×C₁₃	#540	1	4	yes	—
73	D₁₃	—	8	4	no	—
74	C₂⁴	—	13	4	no	—
75	C₂×D₄	—	26	8	no	—
76	C₂×D₄	—	26	4	no	—
77	C₂×D₄	—	26	8	no	—
78	C₂×D₄	—	52	8	no	—
79	C₂²×C₄	—	13	4	no	—
80	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
81	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
82	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
83	C₄⋊C₄	—	26	8	no	—
84	C₄⋊C₄	—	26	4	no	—
85	C₂²⋊C₄	—	52	16	no	—
86	C₄²	—	2	4	no	—
87	C₄²	D₅₂	1	4	yes	—
88	C₁₃	#2641	1	1	yes	characteristic, complemented, Sylow
89	C₂³	—	26	4	no	—
90	C₂³	—	26	8	no	—
91	C₂³	—	26	16	no	—
92	C₂³	C₂³×D₁₃	1	1	yes	characteristic
93	D₄	—	52	16	no	—
94	D₄	—	52	8	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	8	no	—
98	C₂×C₄	—	26	4	no	—
99	C₂×C₄	—	26	8	no	—
100	C₂×C₄	C₂×D₅₂	1	4	yes	—
101	C₂×C₄	C₂×D₅₂	1	2	yes	—
102	C₂×C₄	#6127	1	4	yes	—
103	C₂²	—	52	4	no	—
104	C₂²	—	52	8	no	—
105	C₂²	—	52	16	no	—
106	C₂²	C₂²×D₅₂	1	1	yes	characteristic
107	C₂²	C₂×#6127	1	2	yes	—
108	C₂²	#88256	1	4	yes	—
109	C₄	—	2	4	no	—
110	C₄	—	4	4	no	—
111	C₄	—	52	4	no	—
112	C₄	#88129	1	4	yes	—
113	C₂	—	104	4	no	—
114	C₂	C₂×#88129	1	1	yes	characteristic
115	C₂	#11719230	1	2	yes	—
116	C₂	#11719240	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88128

C₂

yes

maximal, characteristic

C₂×#88124

C₂

yes

maximal, characteristic, complemented

C₁₃×#376

C₂

yes

maximal, characteristic, complemented

#11718600

C₂

yes

maximal, complemented

#11718333

C₂

yes

maximal, complemented

C₂²×D₅₂

C₂²

yes

—

C₂²×D₅₂

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×#6103

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

#88128

—

#88124

—

C₁₃×#95

C₂²

yes

—

C₂³×D₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₄²×C₁₃

—

C₄²×C₁₃

D₄

yes

—

#6103

—

C₄⋊Dic₁₃

—

#6101

—

#2641

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#124

yes

—

Dic₁₃

—

C₂²×D₄

—

C₂²×D₄

—

#127

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₃

yes

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#95

C₂×D₁₃

yes

—

C₂×C₁₃

C₂×#124

yes

characteristic

C₂×C₁₃

#535

yes

—

C₂×C₁₃

#540

yes

—

D₁₃

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

D₅₂

yes

—

C₁₃

#2641

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₁₃

yes

characteristic

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₂×D₅₂

yes

—

101

C₂×C₄

C₂×D₅₂

yes

—

102

C₂×C₄

#6127

yes

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

C₂²×D₅₂

yes

characteristic

107

C₂²

C₂×#6127

yes

—

108

C₂²

#88256

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

#88129

yes

—

113

C₂

—

104

—

114

C₂

C₂×#88129

yes

characteristic

115

C₂

#11719230

yes

—

116

C₂

#11719240

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#376	C₂
prime radical	C₂×#88128	C₂
hypercenter	#376	D₁₃
hypocenter	C₁₃	#2641
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #376

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#376

C₂

C₂

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #376

N	H ≅ G/N	type
2. C₂×#88124	C₂	semidirect
3. C₁₃×#376	C₂	semidirect
4. #11718600	C₂	semidirect
5. #11718333	C₂	semidirect
49. #376	D₁₃	semidirect
88. C₁₃	#2641	semidirect

H ≅ G/N

type

2. C₂×#88124

C₂

semidirect

3. C₁₃×#376

C₂

semidirect

4. #11718600

C₂

semidirect

5. #11718333

C₂

semidirect

49. #376

D₁₃

semidirect

88. C₁₃

#2641

semidirect

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

subgroup	index
C₂×#88128	2
C₂×#88124	2
C₁₃×#376	2
#11718600	2
#11718333	2
#2641	13

subgroup

index

C₁₃×#376

Subgroups — #13120779

Conjugacy classes of subgroups

Complements

Maximal subgroups