Group #15756723 — subgroups

Subgroups — #15756723

Order 3328.

354 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 203 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	#13115008	C₂	1	2	yes	maximal
2	#13113595	C₂	1	4	yes	maximal
3	C₂×C₈²×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
5	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂×#88059	—	2	4	no	—
7	#11718585	C₂²	1	4	yes	—
8	C₈²×C₁₃	C₄	1	1	yes	characteristic
9	C₈²×C₁₃	C₄	1	2	yes	—
10	C₈²×C₁₃	C₄	1	1	yes	characteristic
11	#11718050	—	2	8	no	—
12	C₂²×C₈×C₁₃	D₄	1	2	yes	—
13	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
14	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
15	C₂×#6099	—	2	4	no	—
16	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
17	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₁₃	D₄	1	4	yes	—
21	C₄×C₈×C₁₃	Q₈	1	2	yes	—
22	#88059	—	4	8	no	—
23	C₄⋊#1055	—	2	8	no	—
24	#8823	—	13	1	no	maximal, Sylow
25	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
27	C₂×C₈×C₁₃	—	2	8	no	—
28	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
29	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
32	C₂×C₈×C₁₃	D₈	1	4	yes	—
33	C₂×C₈×C₁₃	Q₁₆	1	4	yes	—
34	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
35	C₄²×C₁₃	C₂×D₄	1	2	yes	—
36	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
37	#6099	—	4	8	no	—
38	C₂×#1055	—	4	4	no	—
39	#1848	—	13	2	no	—
40	#1567	—	13	4	no	—
41	C₂×C₈²	D₁₃	1	1	yes	characteristic
42	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₃	—	2	4	no	—
44	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
47	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
48	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
49	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
50	C₈×C₁₃	—	2	4	no	—
51	C₈×C₁₃	#107	1	4	yes	—
52	C₈×C₁₃	#107	1	4	yes	—
53	C₈×C₁₃	#108	1	8	yes	complemented
54	#1055	—	8	4	no	—
55	C₂×#108	—	26	4	no	—
56	#423	—	13	4	no	—
57	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
59	#335	—	26	8	no	—
60	C₈²	Dic₁₃	1	1	yes	characteristic
61	C₈²	Dic₁₃	1	2	yes	—
62	C₈²	Dic₁₃	1	1	yes	characteristic
63	C₂²×C₁₃	—	2	2	no	—
64	C₂²×C₁₃	#384	1	1	yes	characteristic
65	C₂²×C₁₃	#500	1	2	yes	—
66	C₄×C₁₃	—	2	2	no	—
67	C₄×C₁₃	C₂×#107	1	2	yes	—
68	C₄×C₁₃	C₂×#108	1	4	yes	—
69	C₄×C₁₃	#493	1	1	yes	characteristic
70	C₄×C₁₃	#494	1	1	yes	characteristic
71	C₂×M₁₆	—	26	4	no	—
72	C₂²×C₈	Dic₂₆	1	4	yes	—
73	C₂²×C₈	D₅₂	1	2	yes	—
74	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
75	#108	—	52	8	no	—
76	C₄⋊C₈	—	26	8	no	—
77	C₄×C₈	Dic₂₆	1	2	yes	—
78	C₄×C₈	D₅₂	1	4	yes	—
79	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
80	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
81	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
82	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
83	C₂×C₁₃	—	2	2	no	—
84	C₂×C₁₃	#1845	1	1	yes	characteristic
85	C₂×C₁₃	#1848	1	2	yes	—
86	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
87	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
88	M₁₆	—	52	8	no	—
89	C₂×C₈	—	2	8	no	—
90	C₂×C₈	—	52	4	no	—
91	C₂×C₈	D₁₀₄	1	4	yes	—
92	C₂×C₈	Dic₅₂	1	4	yes	—
93	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
94	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
95	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
96	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
97	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
98	C₄²	C₂×D₅₂	1	2	yes	—
99	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
100	C₁₃	#8823	1	1	yes	characteristic, complemented, Sylow
101	C₂³	#88123	1	1	yes	characteristic
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	#88123	1	2	yes	—
104	C₂×C₄	#88128	1	1	yes	characteristic
105	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
106	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
107	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
108	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₈	—	104	4	no	—
111	C₈	#88058	1	4	yes	—
112	C₈	#88058	1	4	yes	—
113	C₈	#88059	1	8	yes	complemented
114	C₂²	—	2	2	no	—
115	C₂²	#11718343	1	2	yes	—
116	C₂²	#11718596	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	#11718356	1	1	yes	characteristic
119	C₄	#11718360	1	1	yes	characteristic
120	C₄	C₂×#88058	1	2	yes	—
121	C₄	C₂×#88059	1	4	yes	—
122	C₂	—	2	2	no	—
123	C₂	#13114996	1	1	yes	characteristic
124	C₂	#13115008	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13115008

C₂

yes

maximal

#13113595

C₂

yes

maximal

C₂×C₈²×C₁₃

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88059

—

#11718585

C₂²

yes

—

C₈²×C₁₃

C₄

yes

characteristic

C₈²×C₁₃

C₄

yes

—

C₈²×C₁₃

C₄

yes

characteristic

#11718050

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂²×C₈×C₁₃

Q₈

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×#6099

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

Q₈

yes

—

—

—

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

Q₁₆

yes

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×Q₈

yes

—

—

—

—

C₂×C₈²

D₁₃

yes

characteristic

C₂³×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×C₄×C₁₃

C₂×Q₁₆

yes

—

C₈×C₁₃

—

C₈×C₁₃

#107

yes

—

C₈×C₁₃

#107

yes

—

C₈×C₁₃

yes

—

—

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

C₂×C₄×C₈

C₂×D₁₃

yes

—

#335

—

C₈²

Dic₁₃

yes

characteristic

C₈²

Dic₁₃

yes

—

C₈²

Dic₁₃

yes

characteristic

C₂²×C₁₃

—

C₂²×C₁₃

#384

yes

characteristic

C₂²×C₁₃

#500

yes

—

C₄×C₁₃

—

C₄×C₁₃

C₂×#107

yes

—

C₄×C₁₃

C₂×#108

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₂×M₁₆

—

C₂²×C₈

Dic₂₆

yes

—

C₂²×C₈

D₅₂

yes

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

#108

—

C₄⋊C₈

—

C₄×C₈

Dic₂₆

yes

—

C₄×C₈

D₅₂

yes

—

C₄×C₈

C₂×Dic₁₃

yes

—

C₄×C₈

C₂×Dic₁₃

yes

characteristic

C₄×C₈

C₂×Dic₁₃

yes

characteristic

C₄×C₈

C₂×Dic₁₃

yes

—

C₂×C₁₃

—

C₂×C₁₃

#1845

yes

characteristic

C₂×C₁₃

#1848

yes

—

C₂²×C₄

C₂×Dic₂₆

yes

—

C₂²×C₄

C₂×D₅₂

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₀₄

yes

—

C₂×C₈

Dic₅₂

yes

—

C₂×C₈

C₄⋊Dic₁₃

yes

—

C₂×C₈

C₄⋊Dic₁₃

yes

—

C₂×C₈

C₄⋊Dic₁₃

yes

—

C₂×C₈

C₄⋊Dic₁₃

yes

—

C₄²

C₂×Dic₂₆

yes

characteristic

C₄²

C₂×D₅₂

yes

—

C₄²

C₂²×Dic₁₃

yes

characteristic

100

C₁₃

#8823

yes

characteristic, complemented, Sylow

101

C₂³

#88123

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

#88123

yes

—

104

C₂×C₄

#88128

yes

characteristic

105

C₂×C₄

C₂×D₁₀₄

yes

—

106

C₂×C₄

C₂×Dic₅₂

yes

—

107

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

characteristic

108

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

—

109

C₈

—

110

C₈

—

104

—

111

C₈

#88058

yes

—

112

C₈

#88058

yes

—

113

C₈

#88059

yes

complemented

114

C₂²

—

115

C₂²

#11718343

yes

—

116

C₂²

#11718596

yes

characteristic

117

C₄

—

118

C₄

#11718356

yes

characteristic

119

C₄

#11718360

yes

characteristic

120

C₄

C₂×#88058

yes

—

121

C₄

C₂×#88059

yes

—

122

C₂

—

123

C₂

#13114996

yes

characteristic

124

C₂

#13115008

yes

—

	subgroup	G/H
center	C₂×C₄	#88128
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₈²×C₁₃	C₂
socle	C₂²×C₁₃	#384
prime radical	C₂³×C₁₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×C₈²	D₁₃
hypocenter	C₁₃	#8823
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂×C₄

#88128

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₂×C₈²×C₁₃

C₂

socle

C₂²×C₁₃

#384

prime radical

C₂³×C₁₃

C₄⋊Q₈

cosocle

C₂×C₄²×C₁₃

C₂³

hypercenter

C₂×C₈²

D₁₃

hypocenter

C₁₃

#8823

LCS

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
53. C₈×C₁₃	#108	semidirect
100. C₁₃	#8823	semidirect
113. C₈	#88059	semidirect

H ≅ G/N

type

53. C₈×C₁₃

#108

semidirect

100. C₁₃

#8823

semidirect

113. C₈

#88059

semidirect

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

subgroup	index
#13115008	2
#13113595	2
C₂×C₈²×C₁₃	2
#8823	13

subgroup

index

#13115008

#13113595

C₂×C₈²×C₁₃

#8823

Subgroups — #15756723

Conjugacy classes of subgroups

Complements

Maximal subgroups