Group #15774799 — subgroups

Subgroups — #15774799

Order 3328.

214 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 93 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	#13113655	C₂	1	2	yes	maximal
2	#13112902	C₂	1	1	yes	maximal, characteristic
3	#13112806	C₂	1	2	yes	maximal
4	#13112756	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#2176	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×Q₃₂	C₄	1	1	yes	characteristic
7	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
8	Q₈×#1055	—	2	2	no	—
9	C₁₃×#367	C₄	1	1	yes	characteristic
10	C₁₃×#358	C₄	1	2	yes	—
11	C₁₃×#346	C₂²	1	1	yes	characteristic
12	Q₈⋊#1055	—	2	2	no	—
13	#11718053	C₂²	1	1	yes	characteristic
14	#11718049	C₂²	1	2	yes	—
15	#11718047	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
17	C₄×Q₈×C₁₃	—	2	2	no	—
18	C₁₃×Q₃₂	C₈	1	4	yes	complemented
19	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₁₃×#107	C₂×C₄	1	2	yes	—
22	C₁₃×#103	—	2	2	no	—
23	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×#6090	—	2	1	no	—
25	C₄⋊#1055	—	2	2	no	—
26	C₄⋊#1055	—	4	2	no	—
27	C₄×#1055	—	4	2	no	—
28	C₄×#1055	D₄	1	2	yes	—
29	#12066	—	13	1	no	maximal, Sylow
30	C₂×Q₈×C₁₃	—	2	2	no	—
31	C₁₃×Q₁₆	—	2	2	no	—
32	C₁₃×Q₁₆	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
35	C₁₃×C₄⋊C₄	—	2	2	no	—
36	C₁₃×C₄⋊C₄	—	4	2	no	—
37	C₄²×C₁₃	—	4	2	no	—
38	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂×#1055	—	2	2	no	—
40	C₂×#1055	—	4	2	no	—
41	C₁₃×C₁₆	—	2	1	no	—
42	C₁₃×C₁₆	C₂×C₈	1	2	yes	—
43	#6090	—	4	1	no	—
44	#2176	D₁₃	1	1	yes	characteristic
45	#1579	—	13	2	no	—
46	#1368	—	13	1	no	—
47	#1334	—	13	2	no	—
48	#1313	—	13	1	no	—
49	Q₈×C₁₃	—	2	4	no	—
50	Q₈×C₁₃	—	4	2	no	—
51	C₂×C₄×C₁₃	—	4	2	no	—
52	C₂×C₄×C₁₃	—	4	2	no	—
53	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
55	C₈×C₁₃	—	2	1	no	—
56	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
57	C₈×C₁₃	#131	1	1	yes	characteristic
58	#1055	—	4	2	no	—
59	#1055	—	8	2	no	—
60	C₂×Q₃₂	Dic₁₃	1	1	yes	characteristic
61	C₈×Q₈	—	26	2	no	—
62	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
63	#367	Dic₁₃	1	1	yes	characteristic
64	#358	Dic₁₃	1	2	yes	—
65	#346	—	13	1	no	—
66	#346	C₂×D₁₃	1	1	yes	characteristic
67	#334	—	13	2	no	—
68	Q₈⋊C₈	—	26	2	no	—
69	#322	—	13	1	no	—
70	C₂²×C₁₃	#442	1	1	yes	characteristic
71	C₄×C₁₃	—	2	2	no	—
72	C₄×C₁₃	—	4	4	no	—
73	C₄×C₁₃	—	8	2	no	—
74	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
75	C₄×C₁₃	#444	1	1	yes	characteristic
76	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
77	C₄×Q₈	—	2	2	no	—
78	Q₃₂	#1055	1	4	yes	complemented
79	C₂×C₁₆	—	26	1	no	—
80	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
81	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
82	#107	C₂×Dic₁₃	1	2	yes	—
83	C₄⋊C₈	—	26	2	no	—
84	C₄⋊C₈	—	52	2	no	—
85	#103	—	2	2	no	—
86	C₄×C₈	—	13	2	no	—
87	C₄×C₈	—	52	2	no	—
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	—	2	2	no	—
92	Q₁₆	—	2	2	no	—
93	Q₁₆	C₂×#1055	1	4	yes	—
94	C₂×C₈	—	26	2	no	—
95	C₂×C₈	—	52	2	no	—
96	C₂×C₈	#6129	1	1	yes	characteristic
97	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	—	4	2	no	—
100	C₄²	—	4	2	no	—
101	C₄²	D₄×D₁₃	1	1	yes	characteristic
102	C₁₆	—	2	1	no	—
103	C₁₆	—	52	1	no	—
104	C₁₆	C₂×#1055	1	2	yes	—
105	C₁₃	#12066	1	1	yes	characteristic, complemented, Sylow
106	Q₈	—	2	4	no	—
107	Q₈	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	#88165	1	2	yes	—
111	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
112	C₈	—	2	1	no	—
113	C₈	—	52	2	no	—
114	C₈	—	104	2	no	—
115	C₈	C₂²×#1055	1	1	yes	characteristic
116	C₈	#88202	1	1	yes	characteristic
117	C₂²	#11718813	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	4	4	no	—
120	C₄	—	8	2	no	—
121	C₄	D₄×#1055	1	1	yes	characteristic
122	C₄	#11718858	1	1	yes	characteristic
123	C₂	#13113660	1	1	yes	characteristic
124	C₂	#13116728	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13113655

C₂

yes

maximal

#13112902

C₂

yes

maximal, characteristic

#13112806

C₂

yes

maximal

#13112756

C₂

yes

maximal, characteristic

C₁₃×#2176

C₂

yes

maximal, characteristic

C₂×C₁₃×Q₃₂

C₄

yes

characteristic

C₄×C₁₃×Q₁₆

C₂²

yes

—

Q₈×#1055

—

C₁₃×#367

C₄

yes

characteristic

C₁₃×#358

C₄

yes

—

C₁₃×#346

C₂²

yes

characteristic

Q₈⋊#1055

—

#11718053

C₂²

yes

characteristic

#11718049

C₂²

yes

—

#11718047

C₂²

yes

characteristic

C₂×C₁₃×Q₁₆

C₂×C₄

yes

—

C₄×Q₈×C₁₃

—

C₁₃×Q₃₂

C₈

yes

complemented

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₁₃×#107

C₂×C₄

yes

—

C₁₃×#103

—

C₄×C₈×C₁₃

C₂³

yes

—

—

—

—

D₄

yes

—

#12066

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₁₃×Q₁₆

—

C₁₃×Q₁₆

C₂×C₈

yes

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₂×#1055

—

C₂×#1055

—

C₁₃×C₁₆

—

C₁₃×C₁₆

C₂×C₈

yes

—

#6090

—

#2176

D₁₃

yes

—

—

—

—

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₈×C₁₃

—

C₈×C₁₃

C₂²×C₈

yes

C₈×C₁₃

yes

—

—

C₂×Q₃₂

Dic₁₃

yes

characteristic

C₈×Q₈

—

C₄×Q₁₆

C₂×D₁₃

yes

—

#367

Dic₁₃

yes

characteristic

#358

Dic₁₃

yes

—

#346

—

#346

C₂×D₁₃

yes

characteristic

#334

—

Q₈⋊C₈

—

#322

—

C₂²×C₁₃

#442

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₈×D₄

yes

characteristic

C₄×C₁₃

#444

yes

characteristic

C₂×Q₁₆

C₂×Dic₁₃

yes

—

C₄×Q₈

—

Q₃₂

#1055

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₁₃

yes

characteristic

C₂×C₁₆

C₂×Dic₁₃

yes

characteristic

#107

C₂×Dic₁₃

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

C₂×#1055

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#6129

yes

characteristic

C₂×C₈

C₂²×Dic₁₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

—

101

C₄²

D₄×D₁₃

yes

characteristic

102

C₁₆

—

103

C₁₆

—

104

C₁₆

C₂×#1055

yes

—

105

C₁₃

#12066

yes

characteristic, complemented, Sylow

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

#88165

yes

—

111

C₂×C₄

D₄×Dic₁₃

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

104

—

115

C₈

C₂²×#1055

yes

characteristic

116

C₈

#88202

yes

characteristic

117

C₂²

#11718813

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

D₄×#1055

yes

characteristic

122

C₄

#11718858

yes

characteristic

123

C₂

#13113660

yes

characteristic

124

C₂

#13116728

yes

—

	subgroup	G/H
center	C₂²	#11718813
commutator	C₈×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#2176	C₂
socle, prime radical	C₂²×C₁₃	#442
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	#2176	D₁₃
hypocenter	C₁₃	#12066
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

subgroup

G/H

center

C₂²

#11718813

commutator

C₈×C₁₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#2176

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 ◁ C₄×C₈ ◁ #2176

N	H ≅ G/N	type
18. C₁₃×Q₃₂	C₈	semidirect
78. Q₃₂	#1055	semidirect
105. C₁₃	#12066	semidirect

H ≅ G/N

type

18. C₁₃×Q₃₂

C₈

semidirect

78. Q₃₂

#1055

semidirect

105. C₁₃

#12066

semidirect

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

subgroup	index
#13113655	2
#13112902	2
#13112806	2
#13112756	2
C₁₃×#2176	2
#12066	13

subgroup

index

C₁₃×#2176

Subgroups — #15774799

Conjugacy classes of subgroups

Complements

Maximal subgroups