Group #26128158 — subgroups

Subgroups — #26128158

Order 12032.

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	#20242207	C₂	1	2	yes	maximal
2	#20241454	C₂	1	1	yes	maximal, characteristic
3	#20241358	C₂	1	2	yes	maximal
4	#20241308	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#2176	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₄₇⋊C₈	—	2	2	no	—
7	#15688899	C₂²	1	1	yes	characteristic
8	#15688874	C₂²	1	2	yes	—
9	Q₈⋊(C₄₇⋊C₈)	—	2	2	no	—
10	#15688844	C₂²	1	1	yes	characteristic
11	C₂×Q₃₂×C₄₇	C₄	1	1	yes	characteristic
12	C₄×Q₁₆×C₄₇	C₂²	1	2	yes	—
13	C₄₇×#367	C₄	1	1	yes	characteristic
14	C₄₇×#358	C₄	1	2	yes	—
15	C₄₇×#346	C₂²	1	1	yes	characteristic
16	C₂×C₄₇⋊C₁₆	—	2	1	no	—
17	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
18	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
19	C₄×C₄₇⋊C₈	—	4	2	no	—
20	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
21	C₂×Q₁₆×C₄₇	C₂×C₄	1	2	yes	—
22	C₄×Q₈×C₄₇	—	2	2	no	—
23	Q₃₂×C₄₇	C₈	1	4	yes	complemented
24	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
26	#107×C₄₇	C₂×C₄	1	2	yes	—
27	#103×C₄₇	—	2	2	no	—
28	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₄₇	—	2	2	no	—
30	Q₁₆×C₄₇	—	2	2	no	—
31	Q₁₆×C₄₇	C₂×C₈	1	4	yes	—
32	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
34	C₄⋊C₄×C₄₇	—	2	2	no	—
35	C₄⋊C₄×C₄₇	—	4	2	no	—
36	C₄²×C₄₇	—	4	2	no	—
37	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄₇⋊C₈	—	2	2	no	—
39	C₂×C₄₇⋊C₈	—	4	2	no	—
40	C₁₆×C₄₇	—	2	1	no	—
41	C₁₆×C₄₇	C₂×C₈	1	2	yes	—
42	C₄₇⋊C₁₆	—	4	1	no	—
43	Q₈×C₄₇	—	2	4	no	—
44	Q₈×C₄₇	—	4	2	no	—
45	C₂×C₄×C₄₇	—	4	2	no	—
46	C₂×C₄×C₄₇	—	4	2	no	—
47	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
49	C₈×C₄₇	—	2	1	no	—
50	C₈×C₄₇	C₂²×C₈	1	1	yes	characteristic
51	C₈×C₄₇	#131	1	1	yes	characteristic
52	C₄₇⋊C₈	—	4	2	no	—
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	—	4	4	no	—
58	C₄×C₄₇	—	8	2	no	—
59	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
60	C₄×C₄₇	#444	1	1	yes	characteristic
61	#2176	D₄₇	1	1	yes	characteristic
62	#1579	—	47	2	no	—
63	#1368	—	47	1	no	—
64	#1334	—	47	2	no	—
65	#1313	—	47	1	no	—
66	C₂×C₄₇	#1580	1	1	yes	characteristic
67	C₂×C₄₇	#2180	1	2	yes	—
68	C₂×Q₃₂	Dic₄₇	1	1	yes	characteristic
69	C₈×Q₈	—	94	2	no	—
70	C₄×Q₁₆	C₂×D₄₇	1	2	yes	—
71	#367	Dic₄₇	1	1	yes	characteristic
72	#358	Dic₄₇	1	2	yes	—
73	#346	—	47	1	no	—
74	#346	C₂×D₄₇	1	1	yes	characteristic
75	#334	—	47	2	no	—
76	Q₈⋊C₈	—	94	2	no	—
77	#322	—	47	1	no	—
78	C₄₇	#12066	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₁₆	C₂×Dic₄₇	1	2	yes	—
80	C₄×Q₈	—	2	2	no	—
81	Q₃₂	C₄₇⋊C₈	1	4	yes	complemented
82	C₂×C₁₆	—	94	1	no	—
83	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
84	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
85	#107	C₂×Dic₄₇	1	2	yes	—
86	C₄⋊C₈	—	94	2	no	—
87	C₄⋊C₈	—	188	2	no	—
88	#103	—	2	2	no	—
89	C₄×C₈	—	47	2	no	—
90	C₄×C₈	—	188	2	no	—
91	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
92	C₂×Q₈	—	2	2	no	—
93	Q₁₆	—	2	2	no	—
94	Q₁₆	C₂×C₄₇⋊C₈	1	4	yes	—
95	C₂×C₈	—	94	2	no	—
96	C₂×C₈	—	188	2	no	—
97	C₂×C₈	#10625352	1	1	yes	characteristic
98	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄²	—	4	2	no	—
102	C₄²	D₄×D₄₇	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	—	188	1	no	—
105	C₁₆	C₂×C₄₇⋊C₈	1	2	yes	—
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	D₄×Dic₄₇	1	1	yes	characteristic
111	C₂×C₄	#13094048	1	2	yes	—
112	C₈	—	2	1	no	—
113	C₈	—	376	2	no	—
114	C₈	—	188	2	no	—
115	C₈	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
116	C₈	#13094031	1	1	yes	characteristic
117	C₂²	#15689235	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	4	4	no	—
120	C₄	—	8	2	no	—
121	C₄	D₄×C₄₇⋊C₈	1	1	yes	characteristic
122	C₄	#15689245	1	1	yes	characteristic
123	C₂	#20242212	1	1	yes	characteristic
124	C₂	#20245280	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20242207

C₂

yes

maximal

#20241454

C₂

yes

maximal, characteristic

#20241358

C₂

yes

maximal

#20241308

C₂

yes

maximal, characteristic

C₄₇×#2176

C₂

yes

maximal, characteristic

Q₈×C₄₇⋊C₈

—

#15688899

C₂²

yes

characteristic

#15688874

C₂²

yes

—

Q₈⋊(C₄₇⋊C₈)

—

#15688844

C₂²

yes

characteristic

C₂×Q₃₂×C₄₇

C₄

yes

characteristic

C₄×Q₁₆×C₄₇

C₂²

yes

—

C₄₇×#367

C₄

yes

characteristic

C₄₇×#358

C₄

yes

—

C₄₇×#346

C₂²

yes

characteristic

C₂×C₄₇⋊C₁₆

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₂×Q₁₆×C₄₇

C₂×C₄

yes

—

C₄×Q₈×C₄₇

—

Q₃₂×C₄₇

C₈

yes

complemented

C₂×C₁₆×C₄₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₇

C₂×C₄

yes

characteristic

#107×C₄₇

C₂×C₄

yes

—

#103×C₄₇

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

—

Q₁₆×C₄₇

—

Q₁₆×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₁₆×C₄₇

C₂×C₈

yes

—

C₄₇⋊C₁₆

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄×D₄

yes

characteristic

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₈×C₄₇

—

C₈×C₄₇

C₂²×C₈

yes

characteristic

C₈×C₄₇

#131

yes

characteristic

C₄₇⋊C₈

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₈×D₄

yes

C₄×C₄₇

yes

D₄₇

yes

—

—

—

—

C₂×C₄₇

#1580

yes

characteristic

C₂×C₄₇

#2180

yes

—

C₂×Q₃₂

Dic₄₇

yes

characteristic

C₈×Q₈

—

C₄×Q₁₆

C₂×D₄₇

yes

—

#367

Dic₄₇

yes

characteristic

#358

Dic₄₇

yes

—

#346

—

#346

C₂×D₄₇

yes

characteristic

#334

—

Q₈⋊C₈

—

#322

—

C₄₇

#12066

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×Dic₄₇

yes

—

C₄×Q₈

—

Q₃₂

C₄₇⋊C₈

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₄₇

yes

characteristic

C₂×C₁₆

C₂×Dic₄₇

yes

characteristic

#107

C₂×Dic₄₇

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

188

—

#103

—

C₄×C₈

—

C₄×C₈

—

188

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

Q₁₆

—

Q₁₆

C₂×C₄₇⋊C₈

yes

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

#10625352

yes

characteristic

C₂×C₈

C₂²×Dic₄₇

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

D₄×D₄₇

yes

characteristic

103

C₁₆

—

104

C₁₆

—

188

—

105

C₁₆

C₂×C₄₇⋊C₈

yes

—

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

D₄×Dic₄₇

yes

characteristic

111

C₂×C₄

#13094048

yes

—

112

C₈

—

113

C₈

—

376

—

114

C₈

—

188

—

115

C₈

C₂²×C₄₇⋊C₈

yes

characteristic

116

C₈

#13094031

yes

characteristic

117

C₂²

#15689235

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

D₄×C₄₇⋊C₈

yes

characteristic

122

C₄

#15689245

yes

characteristic

123

C₂

#20242212

yes

characteristic

124

C₂

#20245280

yes

—

	subgroup	G/H
center	C₂²	#15689235
commutator	C₈×C₄₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#2176	C₂
socle, prime radical	C₂²×C₄₇	#442
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#2176	D₄₇
hypocenter	C₄₇	#12066
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

subgroup

G/H

center

C₂²

#15689235

commutator

C₈×C₄₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#2176

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 ◁ C₄×C₈ ◁ #2176

N	H ≅ G/N	type
23. Q₃₂×C₄₇	C₈	semidirect
78. C₄₇	#12066	semidirect
81. Q₃₂	C₄₇⋊C₈	semidirect

H ≅ G/N

type

23. Q₃₂×C₄₇

C₈

semidirect

78. C₄₇

#12066

semidirect

81. Q₃₂

C₄₇⋊C₈

semidirect

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

subgroup	index
#20242207	2
#20241454	2
#20241358	2
#20241308	2
C₄₇×#2176	2
#12066	47

subgroup

index

C₄₇×#2176

Subgroups — #26128158

Conjugacy classes of subgroups

Complements

Maximal subgroups