Group #15774800 — subgroups

Subgroups — #15774800

Order 3328.

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

normal subgroups only

This group has 214 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	#13116708	C₂	1	1	yes	maximal, characteristic
2	#13113656	C₂	1	2	yes	maximal
3	#13112902	C₂	1	1	yes	maximal, characteristic
4	#13112807	C₂	1	2	yes	maximal
5	C₁₃×#1313	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₀₄	C₄	1	1	yes	characteristic
7	C₄×Dic₅₂	C₂²	1	2	yes	—
8	C₈×Dic₂₆	—	2	2	no	—
9	C₁₃×#346	C₂²	1	1	yes	characteristic
10	C₁₃×#346	C₂²	1	1	yes	characteristic
11	C₁₃×#322	C₂²	1	1	yes	characteristic
12	#11718097	C₄	1	1	yes	characteristic
13	#11718095	C₄	1	2	yes	—
14	#11718049	C₂²	1	2	yes	—
15	#11718048	—	2	2	no	—
16	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
17	C₄×Dic₂₆	—	2	2	no	—
18	C₂×C₁₃×C₁₆	—	2	1	no	—
19	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₁₃	D₄	1	2	yes	—
22	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
23	#88058	C₂×C₄	1	2	yes	—
24	#88056	—	2	2	no	—
25	#88054	—	4	2	no	—
26	C₈×Dic₁₃	—	4	2	no	—
27	C₄⋊#1055	—	2	2	no	—
28	Dic₁₀₄	C₈	1	4	yes	complemented
29	#12066	—	13	1	no	maximal, Sylow
30	C₂×Dic₂₆	—	2	2	no	—
31	C₂×C₈×C₁₃	—	2	2	no	—
32	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
34	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
35	C₄⋊Dic₁₃	—	2	2	no	—
36	#6101	—	4	2	no	—
37	C₄×Dic₁₃	—	4	2	no	—
38	C₂×#1055	—	4	2	no	—
39	Dic₅₂	—	2	2	no	—
40	Dic₅₂	C₂×C₈	1	4	yes	—
41	C₁₃×C₁₆	—	2	1	no	—
42	C₁₃×C₁₆	—	4	1	no	—
43	C₁₃×C₁₆	C₂×C₈	1	2	yes	—
44	#2176	—	13	1	no	—
45	#1579	—	13	2	no	—
46	#1368	—	13	1	no	—
47	#1334	—	13	2	no	—
48	#1313	D₁₃	1	1	yes	characteristic
49	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
51	C₂×Dic₁₃	—	4	2	no	—
52	C₂×Dic₁₃	—	4	2	no	—
53	Dic₂₆	—	2	4	no	—
54	Dic₂₆	—	4	2	no	—
55	C₈×C₁₃	—	2	1	no	—
56	C₈×C₁₃	—	4	2	no	—
57	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
58	C₈×C₁₃	#131	1	1	yes	characteristic
59	#1055	—	8	2	no	—
60	C₂×Q₃₂	—	13	1	no	—
61	C₈×Q₈	—	26	2	no	—
62	C₄×Q₁₆	—	13	2	no	—
63	#367	—	13	1	no	—
64	#358	—	13	2	no	—
65	#346	C₂×D₁₃	1	1	yes	characteristic
66	#346	C₂×D₁₃	1	1	yes	characteristic
67	#334	—	13	2	no	—
68	Q₈⋊C₈	—	26	2	no	—
69	#322	C₂×D₁₃	1	1	yes	characteristic
70	C₂²×C₁₃	#442	1	1	yes	characteristic
71	C₄×C₁₃	—	2	2	no	—
72	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
73	C₄×C₁₃	#444	1	1	yes	characteristic
74	Dic₁₃	—	4	4	no	—
75	Dic₁₃	—	8	2	no	—
76	C₂×Q₁₆	—	13	2	no	—
77	C₄×Q₈	—	26	2	no	—
78	Q₃₂	—	13	4	no	—
79	C₂×C₁₆	—	2	1	no	—
80	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
81	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
82	#107	—	13	2	no	—
83	C₄⋊C₈	—	26	2	no	—
84	C₄⋊C₈	—	52	2	no	—
85	#103	—	26	2	no	—
86	C₄×C₈	—	52	2	no	—
87	C₄×C₈	D₅₂	1	2	yes	—
88	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
89	C₂×C₁₃	#1580	1	1	yes	characteristic
90	C₂×C₁₃	#2180	1	2	yes	—
91	C₂×Q₈	—	26	2	no	—
92	Q₁₆	—	13	4	no	—
93	Q₁₆	—	26	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	52	2	no	—
96	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
97	C₂×C₈	#6127	1	1	yes	characteristic
98	C₄⋊C₄	—	26	2	no	—
99	C₄⋊C₄	—	52	2	no	—
100	C₄²	—	52	2	no	—
101	C₄²	C₂×D₅₂	1	1	yes	characteristic
102	C₁₆	—	2	1	no	—
103	C₁₆	—	4	1	no	—
104	C₁₆	C₈×D₁₃	1	2	yes	—
105	C₁₃	#12066	1	1	yes	characteristic, complemented, Sylow
106	Q₈	—	26	4	no	—
107	Q₈	—	52	2	no	—
108	C₂×C₄	—	52	2	no	—
109	C₂×C₄	—	52	2	no	—
110	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
111	C₂×C₄	#88162	1	2	yes	—
112	C₈	—	2	1	no	—
113	C₈	—	4	2	no	—
114	C₈	—	104	2	no	—
115	C₈	C₂×C₈×D₁₃	1	1	yes	characteristic
116	C₈	#88155	1	1	yes	characteristic
117	C₂²	#11718372	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	52	4	no	—
120	C₄	—	104	2	no	—
121	C₄	C₈×D₅₂	1	1	yes	characteristic
122	C₄	#11718378	1	1	yes	characteristic
123	C₂	#13113661	1	1	yes	characteristic
124	C₂	#13116729	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13116708

C₂

yes

maximal, characteristic

#13113656

C₂

yes

maximal

#13112902

C₂

yes

maximal, characteristic

#13112807

C₂

yes

maximal

C₁₃×#1313

C₂

yes

maximal, characteristic

C₂×Dic₁₀₄

C₄

yes

characteristic

C₄×Dic₅₂

C₂²

yes

—

C₈×Dic₂₆

—

C₁₃×#346

C₂²

yes

characteristic

C₁₃×#346

C₂²

yes

characteristic

C₁₃×#322

C₂²

yes

characteristic

#11718097

C₄

yes

characteristic

#11718095

C₄

yes

—

#11718049

C₂²

yes

—

#11718048

—

C₂×Dic₅₂

C₂×C₄

yes

—

C₄×Dic₂₆

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

C₂³

yes

characteristic

#88058

C₂×C₄

yes

—

#88056

—

#88054

—

C₈×Dic₁₃

—

C₄⋊#1055

—

Dic₁₀₄

C₈

yes

complemented

#12066

—

maximal, Sylow

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₂×#1055

—

Dic₅₂

—

Dic₅₂

C₂×C₈

yes

—

C₁₃×C₁₆

—

C₁₃×C₁₆

—

C₁₃×C₁₆

C₂×C₈

yes

—

—

—

—

—

D₁₃

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

C₂²×C₈

yes

C₈×C₁₃

yes

—

C₂×Q₃₂

—

C₈×Q₈

—

C₄×Q₁₆

—

#367

—

#358

—

#346

C₂×D₁₃

yes

characteristic

#346

C₂×D₁₃

yes

characteristic

#334

—

Q₈⋊C₈

—

#322

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

#442

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

C₈×D₄

yes

characteristic

C₄×C₁₃

#444

yes

characteristic

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₁₃

yes

characteristic

C₂×C₁₆

C₄×D₁₃

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

D₅₂

yes

—

C₄×C₈

C₂²×D₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×D₁₃

yes

characteristic

C₂×C₈

#6127

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

—

101

C₄²

C₂×D₅₂

yes

characteristic

102

C₁₆

—

103

C₁₆

—

104

C₁₆

C₈×D₁₃

yes

—

105

C₁₃

#12066

yes

characteristic, complemented, Sylow

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₄×D₅₂

yes

characteristic

111

C₂×C₄

#88162

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

104

—

115

C₈

C₂×C₈×D₁₃

yes

characteristic

116

C₈

#88155

yes

characteristic

117

C₂²

#11718372

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

104

—

121

C₄

C₈×D₅₂

yes

characteristic

122

C₄

#11718378

yes

characteristic

123

C₂

#13113661

yes

characteristic

124

C₂

#13116729

yes

—

	subgroup	G/H
center	C₂²	#11718372
commutator	C₈×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1313	C₂
socle, prime radical	C₂²×C₁₃	#442
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	#1313	D₁₃
hypocenter	C₁₃	#12066
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1313

subgroup

G/H

center

C₂²

#11718372

commutator

C₈×C₁₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1313

C₂

socle, prime radical

C₂²×C₁₃

#442

cosocle

C₄×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1313

N	H ≅ G/N	type
28. Dic₁₀₄	C₈	semidirect
105. C₁₃	#12066	semidirect

H ≅ G/N

type

28. Dic₁₀₄

C₈

semidirect

105. C₁₃

#12066

semidirect

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

subgroup	index
#13116708	2
#13113656	2
#13112902	2
#13112807	2
C₁₃×#1313	2
#12066	13

subgroup

index

C₁₃×#1313

Subgroups — #15774800

Conjugacy classes of subgroups

Complements

Maximal subgroups