Group #26128159 — subgroups

Subgroups — #26128159

Order 12032.

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	#20245260	C₂	1	1	yes	maximal, characteristic
2	#20242208	C₂	1	2	yes	maximal
3	#20241454	C₂	1	1	yes	maximal, characteristic
4	#20241359	C₂	1	2	yes	maximal
5	C₄₇×#1313	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₇₆	C₄	1	1	yes	characteristic
7	C₈×Dic₉₄	—	2	2	no	—
8	C₄×Dic₁₈₈	C₂²	1	2	yes	—
9	#15688951	C₄	1	1	yes	characteristic
10	#15688928	C₄	1	2	yes	—
11	#15688874	C₂²	1	2	yes	—
12	#15688855	—	2	2	no	—
13	C₄₇×#346	C₂²	1	1	yes	characteristic
14	C₄₇×#346	C₂²	1	1	yes	characteristic
15	C₄₇×#322	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₈₈	C₂×C₄	1	2	yes	—
17	C₄×Dic₉₄	—	2	2	no	—
18	Dic₃₇₆	C₈	1	4	yes	complemented
19	#13093958	C₂×C₄	1	2	yes	—
20	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
21	Dic₄₇⋊C₈	—	4	2	no	—
22	#13093949	—	2	2	no	—
23	C₈×Dic₄₇	—	4	2	no	—
24	C₂×C₁₆×C₄₇	—	2	1	no	—
25	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₄₇	D₄	1	2	yes	—
28	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	C₂×Dic₉₄	—	2	2	no	—
30	C₂×C₈×C₄₇	—	2	2	no	—
31	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
33	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
34	C₄⋊Dic₄₇	—	2	2	no	—
35	Dic₄₇⋊C₄	—	4	2	no	—
36	C₄×Dic₄₇	—	4	2	no	—
37	C₂×C₄₇⋊C₈	—	4	2	no	—
38	Dic₁₈₈	—	2	2	no	—
39	Dic₁₈₈	C₂×C₈	1	4	yes	—
40	C₁₆×C₄₇	—	2	1	no	—
41	C₁₆×C₄₇	—	4	1	no	—
42	C₁₆×C₄₇	C₂×C₈	1	2	yes	—
43	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
44	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
45	C₂×Dic₄₇	—	4	2	no	—
46	C₂×Dic₄₇	—	4	2	no	—
47	Dic₉₄	—	2	4	no	—
48	Dic₉₄	—	4	2	no	—
49	C₈×C₄₇	—	2	1	no	—
50	C₈×C₄₇	—	4	2	no	—
51	C₈×C₄₇	C₂²×C₈	1	1	yes	characteristic
52	C₈×C₄₇	#131	1	1	yes	characteristic
53	C₄₇⋊C₈	—	8	2	no	—
54	#12066	—	47	1	no	maximal, Sylow
55	C₂²×C₄₇	#442	1	1	yes	characteristic
56	C₄×C₄₇	—	2	2	no	—
57	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
58	C₄×C₄₇	#444	1	1	yes	characteristic
59	Dic₄₇	—	4	4	no	—
60	Dic₄₇	—	8	2	no	—
61	#2176	—	47	1	no	—
62	#1579	—	47	2	no	—
63	#1368	—	47	1	no	—
64	#1334	—	47	2	no	—
65	#1313	D₄₇	1	1	yes	characteristic
66	C₂×C₄₇	#1580	1	1	yes	characteristic
67	C₂×C₄₇	#2180	1	2	yes	—
68	C₂×Q₃₂	—	47	1	no	—
69	C₈×Q₈	—	94	2	no	—
70	C₄×Q₁₆	—	47	2	no	—
71	#367	—	47	1	no	—
72	#358	—	47	2	no	—
73	#346	C₂×D₄₇	1	1	yes	characteristic
74	#346	C₂×D₄₇	1	1	yes	characteristic
75	#334	—	47	2	no	—
76	Q₈⋊C₈	—	94	2	no	—
77	#322	C₂×D₄₇	1	1	yes	characteristic
78	C₄₇	#12066	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₁₆	—	47	2	no	—
80	C₄×Q₈	—	94	2	no	—
81	Q₃₂	—	47	4	no	—
82	C₂×C₁₆	—	2	1	no	—
83	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
84	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
85	#107	—	47	2	no	—
86	C₄⋊C₈	—	94	2	no	—
87	C₄⋊C₈	—	188	2	no	—
88	#103	—	94	2	no	—
89	C₄×C₈	—	188	2	no	—
90	C₄×C₈	D₁₈₈	1	2	yes	—
91	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
92	C₂×Q₈	—	94	2	no	—
93	Q₁₆	—	47	4	no	—
94	Q₁₆	—	94	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	188	2	no	—
97	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
98	C₂×C₈	#10625350	1	1	yes	characteristic
99	C₄⋊C₄	—	94	2	no	—
100	C₄⋊C₄	—	188	2	no	—
101	C₄²	—	188	2	no	—
102	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	—	4	1	no	—
105	C₁₆	C₈×D₄₇	1	2	yes	—
106	Q₈	—	94	4	no	—
107	Q₈	—	188	2	no	—
108	C₂×C₄	—	188	2	no	—
109	C₂×C₄	—	188	2	no	—
110	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
111	C₂×C₄	#13094052	1	2	yes	—
112	C₈	—	2	1	no	—
113	C₈	—	4	2	no	—
114	C₈	—	376	2	no	—
115	C₈	C₂×C₈×D₄₇	1	1	yes	characteristic
116	C₈	#13094030	1	1	yes	characteristic
117	C₂²	#15689236	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	376	2	no	—
120	C₄	—	188	4	no	—
121	C₄	C₈×D₁₈₈	1	1	yes	characteristic
122	C₄	#15689246	1	1	yes	characteristic
123	C₂	#20242213	1	1	yes	characteristic
124	C₂	#20245281	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245260

C₂

yes

maximal, characteristic

#20242208

C₂

yes

maximal

#20241454

C₂

yes

maximal, characteristic

#20241359

C₂

yes

maximal

C₄₇×#1313

C₂

yes

maximal, characteristic

C₂×Dic₃₇₆

C₄

yes

characteristic

C₈×Dic₉₄

—

C₄×Dic₁₈₈

C₂²

yes

—

#15688951

C₄

yes

characteristic

#15688928

C₄

yes

—

#15688874

C₂²

yes

—

#15688855

—

C₄₇×#346

C₂²

yes

characteristic

C₄₇×#346

C₂²

yes

characteristic

C₄₇×#322

C₂²

yes

characteristic

C₂×Dic₁₈₈

C₂×C₄

yes

—

C₄×Dic₉₄

—

Dic₃₇₆

C₈

yes

complemented

#13093958

C₂×C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

Dic₄₇⋊C₈

—

#13093949

—

C₈×Dic₄₇

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

Dic₁₈₈

—

Dic₁₈₈

C₂×C₈

yes

—

C₁₆×C₄₇

—

C₁₆×C₄₇

—

C₁₆×C₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

C₄×D₄

yes

characteristic

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂²×C₈

yes

characteristic

C₈×C₄₇

#131

yes

characteristic

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

C₈×D₄

yes

characteristic

C₄×C₄₇

#444

yes

characteristic

Dic₄₇

—

Dic₄₇

—

—

—

—

—

D₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×Q₃₂

—

C₈×Q₈

—

C₄×Q₁₆

—

#367

—

#358

—

#346

C₂×D₄₇

yes

characteristic

#346

C₂×D₄₇

yes

characteristic

#334

—

Q₈⋊C₈

—

#322

C₂×D₄₇

yes

characteristic

C₄₇

#12066

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₄₇

yes

characteristic

C₂×C₁₆

C₄×D₄₇

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

188

—

#103

—

C₄×C₈

—

188

—

C₄×C₈

D₁₈₈

yes

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

C₂×C₈

#10625350

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

188

—

101

C₄²

—

188

—

102

C₄²

C₂×D₁₈₈

yes

characteristic

103

C₁₆

—

104

C₁₆

—

105

C₁₆

C₈×D₄₇

yes

—

106

Q₈

—

107

Q₈

—

188

—

108

C₂×C₄

—

188

—

109

C₂×C₄

—

188

—

110

C₂×C₄

C₄×D₁₈₈

yes

characteristic

111

C₂×C₄

#13094052

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

376

—

115

C₈

C₂×C₈×D₄₇

yes

characteristic

116

C₈

#13094030

yes

characteristic

117

C₂²

#15689236

yes

characteristic

118

C₄

—

119

C₄

—

376

—

120

C₄

—

188

—

121

C₄

C₈×D₁₈₈

yes

characteristic

122

C₄

#15689246

yes

characteristic

123

C₂

#20242213

yes

characteristic

124

C₂

#20245281

yes

—

	subgroup	G/H
center	C₂²	#15689236
commutator	C₈×C₄₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1313	C₂
socle, prime radical	C₂²×C₄₇	#442
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#1313	D₄₇
hypocenter	C₄₇	#12066
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1313

subgroup

G/H

center

C₂²

#15689236

commutator

C₈×C₄₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1313

C₂

socle, prime radical

C₂²×C₄₇

#442

cosocle

C₄×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1313

N	H ≅ G/N	type
18. Dic₃₇₆	C₈	semidirect
78. C₄₇	#12066	semidirect

H ≅ G/N

type

18. Dic₃₇₆

C₈

semidirect

78. C₄₇

#12066

semidirect

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

subgroup	index
#20245260	2
#20242208	2
#20241454	2
#20241359	2
C₄₇×#1313	2
#12066	47

subgroup

index

C₄₇×#1313

Subgroups — #26128159

Conjugacy classes of subgroups

Complements

Maximal subgroups