Group #15756604 — subgroups

Subgroups — #15756604

Order 3328.

262 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 115 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	#13114974	C₂	1	1	yes	maximal, characteristic
2	#13114967	C₂	1	1	yes	maximal, characteristic
3	#13112695	C₂	1	4	yes	maximal
4	C₁₃×#1450	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#431	C₂²	1	2	yes	—
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	#11718597	C₂²	1	2	yes	—
8	#11718585	C₂²	1	2	yes	—
9	C₁₃×#322	C₄	1	2	yes	complemented
10	C₈²×C₁₃	C₄	1	2	yes	complemented
11	#11718144	—	2	2	no	—
12	#11718143	—	2	2	no	—
13	C₂²×C₈×C₁₃	D₄	1	2	yes	—
14	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₃	—	2	2	no	—
16	C₂×#6099	—	2	2	no	—
17	C₁₃×C₄⋊C₈	—	2	2	no	—
18	C₁₃×C₂²⋊C₈	—	2	2	no	—
19	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
20	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₁₃	Q₈	1	2	yes	—
23	#88071	—	2	4	no	—
24	C₄⋊#1055	—	2	4	no	—
25	#8791	—	13	1	no	maximal, Sylow
26	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₃	#41	1	2	yes	—
28	C₂²×Dic₁₃	—	4	2	no	—
29	C₂×C₈×C₁₃	—	2	4	no	—
30	C₂×C₈×C₁₃	—	2	4	no	—
31	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
33	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
35	C₄²×C₁₃	#41	1	2	yes	—
36	C₄⋊Dic₁₃	—	4	4	no	—
37	#6099	—	4	4	no	—
38	C₂×#1055	—	4	2	no	—
39	#1837	—	13	1	no	—
40	#1835	—	13	1	no	—
41	#1450	D₁₃	1	1	yes	characteristic
42	#1286	—	13	4	no	—
43	C₂³×C₁₃	#124	1	1	yes	characteristic
44	C₂×C₄×C₁₃	—	4	2	no	—
45	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃	#117	1	2	yes	—
47	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
48	C₂×C₄×C₁₃	#125	1	2	yes	—
49	C₂×C₄×C₁₃	#135	1	4	yes	—
50	C₂×Dic₁₃	—	8	4	no	—
51	C₂×Dic₁₃	—	8	2	no	—
52	C₈×C₁₃	—	2	2	no	—
53	C₈×C₁₃	—	2	2	no	—
54	C₈×C₁₃	—	4	4	no	—
55	#1055	—	8	2	no	—
56	#431	C₂×D₁₃	1	2	yes	—
57	#423	—	13	2	no	—
58	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
59	#382	—	13	2	no	—
60	#341	—	26	2	no	—
61	#340	—	26	2	no	—
62	#322	Dic₁₃	1	2	yes	complemented
63	C₈²	Dic₁₃	1	2	yes	complemented
64	C₂²×C₁₃	—	2	2	no	—
65	C₂²×C₁₃	#382	1	1	yes	characteristic
66	C₂²×C₁₃	#488	1	2	yes	—
67	C₄×C₁₃	—	2	2	no	—
68	C₄×C₁₃	—	2	2	no	—
69	C₄×C₁₃	#427	1	2	yes	—
70	C₄×C₁₃	#488	1	2	yes	—
71	Dic₁₃	—	16	2	no	—
72	C₂×M₁₆	—	26	2	no	—
73	C₂²×C₈	D₅₂	1	2	yes	—
74	C₂×C₄⋊C₄	—	26	2	no	—
75	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
76	C₄⋊C₈	—	2	2	no	—
77	C₄⋊C₈	—	26	4	no	—
78	C₂²⋊C₈	—	2	2	no	—
79	C₄×C₈	Dic₂₆	1	2	yes	—
80	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
81	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
82	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
83	#95	—	26	4	no	—
84	C₂×C₁₃	—	2	2	no	—
85	C₂×C₁₃	#1833	1	2	yes	—
86	C₂×C₁₃	#1835	1	1	yes	characteristic
87	C₂²×C₄	—	52	2	no	—
88	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
89	C₂²×C₄	#6127	1	2	yes	—
90	M₁₆	—	52	4	no	—
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	2	4	no	—
93	C₂×C₈	—	52	2	no	—
94	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
95	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
96	C₄⋊C₄	—	52	4	no	—
97	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
98	C₄²	#6127	1	2	yes	—
99	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
100	C₁₃	#8791	1	1	yes	characteristic, complemented, Sylow
101	C₂³	#88129	1	1	yes	characteristic
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	104	4	no	—
104	C₂×C₄	—	104	2	no	—
105	C₂×C₄	#88124	1	2	yes	—
106	C₂×C₄	#88129	1	1	yes	characteristic
107	C₂×C₄	#88158	1	4	yes	—
108	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
109	C₂×C₄	#88180	1	2	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	2	2	no	—
112	C₈	—	4	4	no	—
113	C₈	—	104	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#11718358	1	2	yes	—
116	C₂²	#11718597	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	208	2	no	—
120	C₄	#11718358	1	2	yes	—
121	C₄	#11718767	1	2	yes	—
122	C₂	—	2	2	no	—
123	C₂	#13114960	1	2	yes	—
124	C₂	#13114967	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13114974

C₂

yes

maximal, characteristic

#13114967

C₂

yes

maximal, characteristic

#13112695

C₂

yes

maximal

C₁₃×#1450

C₂

yes

maximal, characteristic

C₁₃×#431

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

#11718597

C₂²

yes

—

#11718585

C₂²

yes

—

C₁₃×#322

C₄

yes

complemented

C₈²×C₁₃

C₄

yes

complemented

#11718144

—

#11718143

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×#6099

—

C₁₃×C₄⋊C₈

—

C₁₃×C₂²⋊C₈

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

Q₈

yes

—

—

—

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

—

C₂²×Dic₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

characteristic

C₄²×C₁₃

#41

yes

—

C₄⋊Dic₁₃

—

—

—

—

—

D₁₃

yes

characteristic

#1286

—

C₂³×C₁₃

#124

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

#124

yes

characteristic

C₂×C₄×C₁₃

#125

yes

—

C₂×C₄×C₁₃

#135

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

#1055

—

#431

C₂×D₁₃

yes

—

#423

—

C₂×C₄×C₈

C₂×D₁₃

yes

—

—

—

Dic₁₃

yes

complemented

C₈²

Dic₁₃

yes

complemented

C₂²×C₁₃

—

C₂²×C₁₃

#382

yes

characteristic

C₂²×C₁₃

#488

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#427

yes

—

C₄×C₁₃

#488

yes

—

Dic₁₃

—

C₂×M₁₆

—

C₂²×C₈

D₅₂

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

Dic₂₆

yes

—

C₄×C₈

C₂×Dic₁₃

yes

—

C₄×C₈

C₂×Dic₁₃

yes

characteristic

C₄×C₈

C₂×Dic₁₃

yes

characteristic

#95

—

C₂×C₁₃

—

C₂×C₁₃

#1833

yes

—

C₂×C₁₃

#1835

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₅₂

yes

characteristic

C₂²×C₄

#6127

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊Dic₁₃

yes

—

C₂×C₈

C₄⋊Dic₁₃

yes

—

C₄⋊C₄

—

C₄²

C₂×Dic₂₆

yes

characteristic

C₄²

#6127

yes

—

C₄²

C₂²×Dic₁₃

yes

characteristic

100

C₁₃

#8791

yes

characteristic, complemented, Sylow

101

C₂³

#88129

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

—

104

C₂×C₄

—

104

—

105

C₂×C₄

#88124

yes

—

106

C₂×C₄

#88129

yes

characteristic

107

C₂×C₄

#88158

yes

—

108

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

characteristic

109

C₂×C₄

#88180

yes

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

104

—

114

C₂²

—

115

C₂²

#11718358

yes

—

116

C₂²

#11718597

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

208

—

120

C₄

#11718358

yes

—

121

C₄

#11718767

yes

—

122

C₂

—

123

C₂

#13114960

yes

—

124

C₂

#13114967

yes

characteristic

	subgroup	G/H
center	C₂²	#11718597
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1450	C₂
socle	C₂²×C₁₃	#382
prime radical	C₂³×C₁₃	#124
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1450	D₁₃
hypocenter	C₁₃	#8791
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

subgroup

G/H

center

C₂²

#11718597

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1450

C₂

C₂²×C₁₃

C₂³×C₁₃

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

N	H ≅ G/N	type
9. C₁₃×#322	C₄	semidirect
10. C₈²×C₁₃	C₄	semidirect
62. #322	Dic₁₃	semidirect
63. C₈²	Dic₁₃	semidirect
100. C₁₃	#8791	semidirect

H ≅ G/N

type

9. C₁₃×#322

C₄

semidirect

10. C₈²×C₁₃

C₄

semidirect

62. #322

Dic₁₃

semidirect

63. C₈²

Dic₁₃

semidirect

100. C₁₃

#8791

semidirect

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

subgroup	index
#13114974	2
#13114967	2
#13112695	2
C₁₃×#1450	2
#8791	13

subgroup

index

C₁₃×#1450

Subgroups — #15756604

Conjugacy classes of subgroups

Complements

Maximal subgroups