Group #15756573 — subgroups

Subgroups — #15756573

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	#13113584	C₂	1	2	yes	maximal
2	#13112681	C₂	1	2	yes	maximal
3	#13112665	C₂	1	2	yes	maximal
4	C₁₃×#1746	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₁₃×#376	C₂²	1	1	yes	characteristic
8	#11718586	C₂²	1	4	yes	—
9	C₁₃×#336	C₄	1	4	yes	—
10	#11718050	—	2	2	no	—
11	#11718049	—	2	2	no	—
12	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
13	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
16	C₁₃×#97	—	2	2	no	—
17	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
18	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
19	C₄×C₈×C₁₃	Q₈	1	2	yes	—
20	C₁₃×#95	—	2	2	no	—
21	C₂²⋊#1055	—	2	4	no	—
22	C₄⋊#1055	—	2	4	no	—
23	C₄×#1055	—	2	4	no	—
24	#8782	—	13	1	no	maximal, Sylow
25	C₂²×C₄×C₁₃	—	2	2	no	—
26	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₃	#41	1	2	yes	—
28	C₂×C₈×C₁₃	—	2	4	no	—
29	C₂×C₈×C₁₃	—	2	2	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	—	2	4	no	—
34	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
35	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
36	C₄²×C₁₃	#41	1	2	yes	—
37	C₂×#1055	—	4	4	no	—
38	#1746	D₁₃	1	1	yes	characteristic
39	#1564	—	13	2	no	—
40	#1281	—	13	2	no	—
41	#1274	—	13	2	no	—
42	C₂³×C₁₃	#117	1	1	yes	characteristic
43	C₂×C₄×C₁₃	—	4	2	no	—
44	C₂×C₄×C₁₃	—	4	4	no	—
45	C₂×C₄×C₁₃	—	4	2	no	—
46	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
48	C₂×C₄×C₁₃	#124	1	2	yes	—
49	C₂×C₄×C₁₃	#125	1	2	yes	—
50	C₂×C₄×C₁₃	#131	1	4	yes	—
51	C₈×C₁₃	—	2	2	no	—
52	C₈×C₁₃	—	2	2	no	—
53	C₈×C₁₃	—	4	2	no	—
54	#1055	—	8	4	no	—
55	#431	—	13	4	no	—
56	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
58	#376	C₂×D₁₃	1	1	yes	characteristic
59	#336	Dic₁₃	1	4	yes	—
60	#335	—	26	2	no	—
61	#334	—	26	2	no	—
62	C₂²×C₁₃	—	2	2	no	—
63	C₂²×C₁₃	#382	1	1	yes	characteristic
64	C₂²×C₁₃	#433	1	2	yes	—
65	C₄×C₁₃	—	2	2	no	—
66	C₄×C₁₃	—	2	2	no	—
67	C₄×C₁₃	—	8	2	no	—
68	C₄×C₁₃	#424	1	2	yes	—
69	C₄×C₁₃	#433	1	2	yes	—
70	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
71	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
72	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
73	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
74	C₄⋊C₈	—	26	4	no	—
75	C₂²⋊C₈	—	26	4	no	—
76	#97	—	2	2	no	—
77	C₄×C₈	—	26	4	no	—
78	C₄×C₈	Dic₂₆	1	2	yes	—
79	C₄×C₈	D₅₂	1	1	yes	characteristic
80	C₄×C₈	D₅₂	1	1	yes	characteristic
81	#95	—	2	2	no	—
82	C₂×C₁₃	—	2	2	no	—
83	C₂×C₁₃	#1832	1	1	yes	characteristic
84	C₂×C₁₃	#1833	1	2	yes	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	#6127	1	2	yes	—
87	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	4	no	—
91	C₂×C₈	—	52	4	no	—
92	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
93	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
94	C₄²	—	2	4	no	—
95	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
96	C₄²	C₂×D₅₂	1	1	yes	characteristic
97	C₄²	#6127	1	2	yes	—
98	C₁₃	#8782	1	1	yes	characteristic, complemented, Sylow
99	C₂³	#88180	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	4	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	#88124	1	2	yes	—
104	C₂×C₄	#88129	1	2	yes	—
105	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
106	C₂×C₄	#88180	1	1	yes	characteristic
107	C₂×C₄	#88202	1	4	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	2	2	no	—
110	C₈	—	4	2	no	—
111	C₈	—	104	4	no	—
112	C₂²	—	2	2	no	—
113	C₂²	#11718597	1	1	yes	characteristic
114	C₂²	#11718661	1	2	yes	—
115	C₄	—	2	2	no	—
116	C₄	—	2	2	no	—
117	C₄	—	8	2	no	—
118	C₄	#11718660	1	2	yes	—
119	C₄	#11718661	1	2	yes	—
120	C₂	—	2	2	no	—
121	C₂	#13114956	1	1	yes	characteristic
122	C₂	#13114960	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13113584

C₂

yes

maximal

#13112681

C₂

yes

maximal

#13112665

C₂

yes

maximal

C₁₃×#1746

C₂

yes

maximal, characteristic

C₂×C₁₃×#97

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₁₃×#376

C₂²

yes

characteristic

#11718586

C₂²

yes

—

C₁₃×#336

C₄

yes

—

#11718050

—

#11718049

—

C₂²×C₈×C₁₃

C₂×C₄

yes

—

C₂²×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₁₃×#97

—

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

Q₈

yes

—

C₁₃×#95

—

C₂²⋊#1055

—

—

—

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

characteristic

C₄²×C₁₃

#41

yes

—

C₂×#1055

—

#1746

D₁₃

yes

—

—

—

C₂³×C₁₃

#117

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

—

C₂×C₄×C₁₃

#125

yes

—

C₂×C₄×C₁₃

#131

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

#1055

—

#431

—

C₂×#97

C₂×D₁₃

yes

characteristic

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

#376

C₂×D₁₃

yes

characteristic

#336

Dic₁₃

yes

—

#335

—

#334

—

C₂²×C₁₃

—

C₂²×C₁₃

#382

yes

characteristic

C₂²×C₁₃

#433

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#424

yes

—

C₄×C₁₃

#433

yes

—

C₂²×C₈

C₂×Dic₁₃

yes

—

C₂²×C₈

C₂×Dic₁₃

yes

—

C₂×C₄²

C₂×Dic₁₃

yes

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

Dic₂₆

yes

—

C₄×C₈

D₅₂

yes

characteristic

C₄×C₈

D₅₂

yes

characteristic

#95

—

C₂×C₁₃

—

C₂×C₁₃

#1832

yes

characteristic

C₂×C₁₃

#1833

yes

—

C₂²×C₄

—

C₂²×C₄

#6127

yes

—

C₂²×C₄

C₂²×Dic₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊Dic₁₃

yes

—

C₂×C₈

C₄⋊Dic₁₃

yes

—

C₄²

—

C₄²

C₂×Dic₂₆

yes

characteristic

C₄²

C₂×D₅₂

yes

characteristic

C₄²

#6127

yes

—

C₁₃

#8782

yes

characteristic, complemented, Sylow

C₂³

#88180

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

#88124

yes

—

104

C₂×C₄

#88129

yes

—

105

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

characteristic

106

C₂×C₄

#88180

yes

characteristic

107

C₂×C₄

#88202

yes

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

104

—

112

C₂²

—

113

C₂²

#11718597

yes

characteristic

114

C₂²

#11718661

yes

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

#11718660

yes

—

119

C₄

#11718661

yes

—

120

C₂

—

121

C₂

#13114956

yes

characteristic

122

C₂

#13114960

yes

—

	subgroup	G/H
center	C₂×C₄	#88180
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1746	C₂
socle	C₂²×C₁₃	#382
prime radical	C₂³×C₁₃	#117
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1746	D₁₃
hypocenter	C₁₃	#8782
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ #1746

subgroup

G/H

center

C₂×C₄

#88180

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1746

C₂

C₂²×C₁₃

C₂³×C₁₃

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ #376 ◁ #1746

N	H ≅ G/N	type
98. C₁₃	#8782	semidirect

H ≅ G/N

type

98. C₁₃

#8782

semidirect

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

subgroup	index
#13113584	2
#13112681	2
#13112665	2
C₁₃×#1746	2
#8782	13

subgroup

index

C₁₃×#1746

Subgroups — #15756573

Conjugacy classes of subgroups

Complements

Maximal subgroups