Group #26102949 — subgroups

Subgroups — #26102949

Order 12032.

134 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 51 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	#20242286	C₂	1	1	yes	maximal, characteristic
2	#20241452	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1311	C₂	1	1	yes	maximal, characteristic
4	#15689253	—	2	1	no	—
5	C₄×Dic₁₈₈	C₄	1	1	yes	characteristic
6	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
7	#15688876	C₄	1	1	yes	characteristic
8	#15688855	—	2	1	no	—
9	C₄×C₁₆×C₄₇	—	2	1	no	—
10	C₈²×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₈₈	C₈	1	1	yes	characteristic
12	C₄×Dic₉₄	—	2	1	no	—
13	C₂×C₄₇⋊C₁₆	—	4	1	no	—
14	#13093958	C₈	1	1	yes	characteristic
15	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
16	Dic₄₇⋊C₈	—	4	1	no	—
17	#13093949	—	2	1	no	—
18	#13093934	—	4	1	no	—
19	C₂×C₁₆×C₄₇	—	2	2	no	—
20	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
22	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
23	C₂×Dic₉₄	—	2	1	no	—
24	C₂×C₈×C₄₇	—	2	1	no	—
25	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
26	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
27	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
28	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
29	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₄⋊Dic₄₇	—	2	1	no	—
31	Dic₄₇⋊C₄	—	4	1	no	—
32	C₄×Dic₄₇	—	4	1	no	—
33	C₂×C₄₇⋊C₈	—	4	1	no	—
34	Dic₁₈₈	—	4	1	no	—
35	C₁₆×C₄₇	—	2	4	no	—
36	C₄₇⋊C₁₆	—	8	1	no	—
37	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
38	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
39	C₂×C₄×C₄₇	#104	1	1	yes	characteristic
40	C₂×Dic₄₇	—	4	1	no	—
41	C₂×Dic₄₇	—	4	1	no	—
42	Dic₉₄	—	4	1	no	—
43	Dic₉₄	—	4	1	no	—
44	C₈×C₄₇	—	2	2	no	—
45	C₈×C₄₇	—	2	1	no	—
46	C₈×C₄₇	—	2	1	no	—
47	C₈×C₄₇	—	2	1	no	—
48	C₈×C₄₇	—	2	1	no	—
49	C₄₇⋊C₈	—	8	1	no	—
50	#7075	—	47	1	no	maximal, Sylow
51	C₂²×C₄₇	D₄⋊C₈	1	1	yes	characteristic
52	C₄×C₄₇	—	2	1	no	—
53	C₄×C₄₇	#349	1	1	yes	characteristic
54	C₄×C₄₇	#350	1	1	yes	characteristic
55	C₄×C₄₇	#360	1	1	yes	characteristic
56	C₄×C₄₇	#362	1	1	yes	characteristic
57	Dic₄₇	—	8	1	no	—
58	Dic₄₇	—	8	1	no	—
59	#1591	—	47	1	no	—
60	#1367	—	47	1	no	—
61	#1311	D₄₇	1	1	yes	characteristic
62	C₂×C₄₇	#1330	1	1	yes	characteristic
63	C₂×C₄₇	#1334	1	1	yes	characteristic
64	C₂×C₄₇	#1335	1	1	yes	characteristic
65	#446	—	94	1	no	—
66	C₄×Q₁₆	—	47	1	no	—
67	C₄⋊C₁₆	—	94	1	no	—
68	C₄×C₁₆	—	2	1	no	—
69	#335	—	47	1	no	—
70	Q₈⋊C₈	—	94	1	no	—
71	C₈²	C₂×D₄₇	1	1	yes	characteristic
72	C₄₇	#7075	1	1	yes	characteristic, complemented, Sylow
73	C₂×Q₁₆	—	47	1	no	—
74	C₄×Q₈	—	94	1	no	—
75	C₂×C₁₆	—	2	2	no	—
76	C₂×C₁₆	—	188	1	no	—
77	#107	—	47	1	no	—
78	C₄⋊C₈	—	94	1	no	—
79	C₄⋊C₈	—	188	1	no	—
80	#103	—	94	1	no	—
81	#97	—	188	1	no	—
82	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
83	C₄×C₈	D₁₈₈	1	1	yes	characteristic
84	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
85	C₂×Q₈	—	94	1	no	—
86	Q₁₆	—	188	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	188	1	no	—
89	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
90	C₂×C₈	#10625324	1	1	yes	characteristic
91	C₂×C₈	#10625325	1	1	yes	characteristic
92	C₂×C₈	D₃₇₆	1	1	yes	characteristic
93	C₄⋊C₄	—	94	1	no	—
94	C₄⋊C₄	—	188	1	no	—
95	C₄²	—	188	1	no	—
96	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
97	C₁₆	—	2	4	no	—
98	C₁₆	—	376	1	no	—
99	Q₈	—	188	1	no	—
100	Q₈	—	188	1	no	—
101	C₂×C₄	—	188	1	no	—
102	C₂×C₄	—	188	1	no	—
103	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
104	C₂×C₄	#13093946	1	1	yes	characteristic
105	C₂×C₄	#13093952	1	1	yes	characteristic
106	C₈	—	2	2	no	—
107	C₈	—	2	1	no	—
108	C₈	—	2	1	no	—
109	C₈	—	2	1	no	—
110	C₈	—	2	1	no	—
111	C₈	—	376	1	no	—
112	C₂²	#15688852	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	376	1	no	—
115	C₄	—	376	1	no	—
116	C₄	#15688904	1	1	yes	characteristic
117	C₄	#15688908	1	1	yes	characteristic
118	C₄	#15688934	1	1	yes	characteristic
119	C₄	#15688940	1	1	yes	characteristic
120	C₂	#20241347	1	1	yes	characteristic
121	C₂	#20241359	1	1	yes	characteristic
122	C₂	#20241362	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20242286

C₂

yes

maximal, characteristic

#20241452

C₂

yes

maximal, characteristic

C₄₇×#1311

C₂

yes

maximal, characteristic

#15689253

—

C₄×Dic₁₈₈

C₄

yes

characteristic

C₄⋊(C₄₇⋊C₁₆)

—

#15688876

C₄

yes

characteristic

#15688855

—

C₄×C₁₆×C₄₇

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

C₈

yes

characteristic

C₄×Dic₉₄

—

C₂×C₄₇⋊C₁₆

—

#13093958

C₈

yes

characteristic

C₄⋊(C₄₇⋊C₈)

—

Dic₄₇⋊C₈

—

#13093949

—

#13093934

—

C₂×C₁₆×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₈

yes

characteristic

C₂×C₈×C₄₇

M₁₆

yes

characteristic

C₂×C₈×C₄₇

D₈

yes

characteristic

C₂×C₈×C₄₇

SD₁₆

yes

characteristic

C₄²×C₄₇

C₂²⋊C₄

yes

characteristic

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

Dic₁₈₈

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₂×C₄×C₄₇

C₂²⋊C₈

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#7075

—

maximal, Sylow

C₂²×C₄₇

D₄⋊C₈

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

#1591

—

#1367

—

#1311

D₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#7075

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

188

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

188

—

#103

—

#97

—

188

—

C₄×C₈

C₄×D₄₇

yes

characteristic

C₄×C₈

D₁₈₈

yes

characteristic

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

C₂×Q₈

—

Q₁₆

—

188

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

C₈×D₄₇

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

D₃₇₆

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

188

—

C₄²

—

188

—

C₄²

(C₂×D₄₇)⋊C₄

yes

characteristic

C₁₆

—

C₁₆

—

376

—

Q₈

—

188

—

100

Q₈

—

188

—

101

C₂×C₄

—

188

—

102

C₂×C₄

—

188

—

103

C₂×C₄

(C₂×D₄₇)⋊C₈

yes

characteristic

104

C₂×C₄

#13093946

yes

characteristic

105

C₂×C₄

#13093952

yes

characteristic

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

376

—

112

C₂²

#15688852

yes

characteristic

113

C₄

—

114

C₄

—

376

—

115

C₄

—

376

—

116

C₄

#15688904

yes

characteristic

117

C₄

#15688908

yes

characteristic

118

C₄

#15688934

yes

characteristic

119

C₄

#15688940

yes

characteristic

120

C₂

#20241347

yes

characteristic

121

C₂

#20241359

yes

characteristic

122

C₂

#20241362

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13093946
commutator	C₂×C₈×C₄₇	C₂×C₈
Frattini	C₈²	C₂×D₄₇
Fitting	C₄₇×#1311	C₂
socle, prime radical	C₂²×C₄₇	D₄⋊C₈
cosocle	C₈²×C₄₇	C₂²
hypercenter	#1311	D₄₇
hypocenter	C₄₇	#7075
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂×C₄

#13093946

commutator

C₂×C₈×C₄₇

C₂×C₈

Frattini

C₈²

C₂×D₄₇

Fitting

C₄₇×#1311

C₂

socle, prime radical

C₂²×C₄₇

D₄⋊C₈

cosocle

C₈²×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

N	H ≅ G/N	type
72. C₄₇	#7075	semidirect

H ≅ G/N

type

72. C₄₇

#7075

semidirect

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

subgroup	index
#20242286	2
#20241452	2
C₄₇×#1311	2
#7075	47

subgroup

index

#20242286

#20241452

C₄₇×#1311

#7075

Subgroups — #26102949

Conjugacy classes of subgroups

Complements

Maximal subgroups