Group #15767601 — subgroups

Subgroups — #15767601

Order 3328.

328 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 79 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	#13113627	C₂	1	3	yes	maximal, complemented
2	#13113505	C₂	1	3	yes	maximal
3	C₁₃×#1309	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#88128	C₂²	1	1	yes	characteristic
5	C₁₃×#423	C₂²	1	3	yes	—
6	#11718586	C₂²	1	3	yes	—
7	#11718077	—	2	3	no	—
8	#11718077	—	2	3	no	—
9	#11718072	—	2	3	no	—
10	#11718071	—	2	3	no	—
11	C₂²×D₅₂	—	2	3	no	—
12	C₂×C₁₃×M₁₆	—	2	3	no	—
13	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
14	#88128	—	2	3	no	—
15	#88128	Q₈	1	2	yes	—
16	C₁₃×C₄⋊C₈	—	2	3	no	—
17	C₁₃×C₄⋊C₈	—	2	3	no	—
18	C₂²⋊#1055	—	2	3	no	—
19	C₄⋊#1055	D₄	1	3	yes	—
20	C₄⋊#1055	D₄	1	3	yes	—
21	C₄×#1055	—	2	3	no	—
22	#10611	—	13	1	no	maximal, Sylow
23	C₂³×D₁₃	—	4	1	no	—
24	C₂²×C₄×C₁₃	#41	1	3	yes	—
25	C₂×D₅₂	—	2	6	no	—
26	C₂×D₅₂	—	4	3	no	—
27	C₂×D₅₂	—	8	3	no	—
28	C₂×D₅₂	—	8	3	no	—
29	C₁₃×M₁₆	—	8	3	no	—
30	C₂×C₈×C₁₃	—	4	3	no	—
31	C₄²×C₁₃	C₂×D₄	1	3	yes	—
32	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
33	C₂×#1055	—	2	3	no	—
34	C₂×#1055	—	2	3	no	—
35	#1574	—	13	3	no	—
36	#1545	—	13	3	no	—
37	#1309	D₁₃	1	1	yes	characteristic, complemented
38	C₂³×C₁₃	#126	1	1	yes	characteristic
39	C₂²×D₁₃	—	4	2	no	—
40	C₂²×D₁₃	—	8	6	no	—
41	C₂×C₄×C₁₃	—	2	3	no	—
42	C₂×C₄×C₁₃	—	2	3	no	—
43	C₂×C₄×C₁₃	#121	1	3	yes	—
44	C₂×C₄×C₁₃	C₂²⋊Q₈	1	3	yes	—
45	D₅₂	—	8	6	no	—
46	D₅₂	—	8	3	no	—
47	D₅₂	—	8	3	no	—
48	C₈×C₁₃	—	8	3	no	—
49	#1055	—	4	3	no	—
50	#1055	—	4	3	no	—
51	C₂×#127	—	13	1	no	—
52	#431	—	13	3	no	—
53	#423	C₂×D₁₃	1	3	yes	—
54	#335	—	26	3	no	—
55	#334	—	26	3	no	—
56	#331	—	26	3	no	—
57	#331	—	26	3	no	—
58	C₂²×C₁₃	—	2	3	no	—
59	C₂²×C₁₃	#396	1	1	yes	characteristic
60	C₂×D₁₃	—	8	6	no	—
61	C₂×D₁₃	—	16	4	no	—
62	C₄×C₁₃	—	2	3	no	—
63	C₄×C₁₃	#471	1	3	yes	—
64	C₄×C₁₃	#472	1	3	yes	—
65	C₂²×D₄	—	26	3	no	—
66	C₂×M₁₆	—	2	3	no	—
67	#127	—	13	2	no	—
68	#127	—	26	3	no	—
69	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
70	C₄⋊C₈	—	2	3	no	—
71	C₄⋊C₈	—	2	3	no	—
72	C₄⋊C₈	—	13	3	no	—
73	C₄⋊C₈	—	13	3	no	—
74	C₂²⋊C₈	—	26	3	no	—
75	C₄×C₈	—	26	3	no	—
76	C₂×C₁₃	—	4	1	no	—
77	C₂×C₁₃	#2063	1	3	yes	—
78	D₁₃	—	16	2	no	—
79	C₂⁴	—	52	1	no	—
80	C₂×D₄	—	26	6	no	—
81	C₂×D₄	—	52	3	no	—
82	C₂×D₄	—	104	3	no	—
83	C₂×D₄	—	104	3	no	—
84	C₂²×C₄	#6127	1	3	yes	—
85	M₁₆	—	8	3	no	—
86	C₂×C₈	—	4	3	no	—
87	C₂×C₈	—	26	3	no	—
88	C₂×C₈	—	26	3	no	—
89	C₄²	D₄×D₁₃	1	3	yes	—
90	C₄²	Q₈×D₁₃	1	1	yes	characteristic
91	C₁₃	#10611	1	1	yes	characteristic, complemented, Sylow
92	C₂³	—	52	2	no	—
93	C₂³	—	104	6	no	—
94	C₂³	#88130	1	1	yes	characteristic
95	D₄	—	104	6	no	—
96	D₄	—	104	3	no	—
97	D₄	—	104	3	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	#88138	1	3	yes	—
101	C₂×C₄	#88150	1	3	yes	—
102	C₈	—	8	3	no	—
103	C₈	—	52	3	no	—
104	C₈	—	52	3	no	—
105	C₂²	—	2	3	no	—
106	C₂²	—	104	6	no	—
107	C₂²	—	208	4	no	—
108	C₂²	#11718337	1	1	yes	characteristic
109	C₄	—	2	3	no	—
110	C₄	#11718410	1	3	yes	—
111	C₄	#11718415	1	3	yes	—
112	C₂	—	4	1	no	—
113	C₂	—	208	2	no	—
114	C₂	#13116197	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13113627

C₂

yes

maximal, complemented

#13113505

C₂

yes

maximal

C₁₃×#1309

C₂

yes

maximal, characteristic, complemented

C₂×#88128

C₂²

yes

characteristic

C₁₃×#423

C₂²

yes

—

#11718586

C₂²

yes

—

—

—

—

—

C₂²×D₅₂

—

C₂×C₁₃×M₁₆

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

#88128

—

#88128

Q₈

yes

—

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

—

C₂²⋊#1055

—

C₄⋊#1055

D₄

yes

—

C₄⋊#1055

D₄

yes

—

C₄×#1055

—

#10611

—

maximal, Sylow

C₂³×D₁₃

—

C₂²×C₄×C₁₃

#41

yes

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×Q₈

yes

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂³×C₁₃

#126

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#121

yes

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

D₅₂

—

D₅₂

—

D₅₂

—

C₈×C₁₃

—

—

—

—

—

C₂×D₁₃

yes

—

—

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

#396

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#471

yes

—

C₄×C₁₃

#472

yes

—

C₂²×D₄

—

C₂×M₁₆

—

#127

—

#127

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

#2063

yes

—

D₁₃

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

104

—

C₂×D₄

—

104

—

C₂²×C₄

#6127

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

D₄×D₁₃

yes

—

C₄²

Q₈×D₁₃

yes

characteristic

C₁₃

#10611

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

104

—

C₂³

#88130

yes

characteristic

D₄

—

104

—

D₄

—

104

—

D₄

—

104

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#88138

yes

—

101

C₂×C₄

#88150

yes

—

102

C₈

—

103

C₈

—

104

C₈

—

105

C₂²

—

106

C₂²

—

104

—

107

C₂²

—

208

—

108

C₂²

#11718337

yes

characteristic

109

C₄

—

110

C₄

#11718410

yes

—

111

C₄

#11718415

yes

—

112

C₂

—

113

C₂

—

208

—

114

C₂

#13116197

yes

—

	subgroup	G/H
center	C₂²	#11718337
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1309	C₂
socle	C₂²×C₁₃	#396
prime radical	C₂×#88128	C₂²
hypercenter	#1309	D₁₃
hypocenter	C₁₃	#10611
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#11718337

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1309

C₂

C₂²×C₁₃

C₂²

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
1. #13113627	C₂	semidirect
3. C₁₃×#1309	C₂	semidirect
37. #1309	D₁₃	semidirect
91. C₁₃	#10611	semidirect

H ≅ G/N

type

1. #13113627

C₂

semidirect

3. C₁₃×#1309

C₂

semidirect

37. #1309

D₁₃

semidirect

91. C₁₃

#10611

semidirect

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

subgroup	index
#13113627	2
#13113505	2
C₁₃×#1309	2
#10611	13

subgroup

index

#13113627

#13113505

C₁₃×#1309

#10611

Subgroups — #15767601

Conjugacy classes of subgroups

Complements

Maximal subgroups