Group #26103389 — subgroups

Subgroups — #26103389

Order 12032.

184 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 83 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	#20242151	C₂	1	2	yes	maximal
2	C₄₇×#2150	C₂	1	1	yes	maximal, characteristic
3	#15689180	—	2	2	no	—
4	#15688876	—	2	2	no	—
5	#15688874	C₄	1	4	yes	—
6	C₂×M₃₂×C₄₇	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
8	C₄₇×C₄⋊C₁₆	—	2	2	no	—
9	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
10	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
11	#13093935	—	4	2	no	—
12	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
15	M₃₂×C₄₇	—	4	2	no	—
16	C₂×C₁₆×C₄₇	—	2	2	no	—
17	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₄₇	D₄	1	2	yes	—
19	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
21	C₂×C₈×C₄₇	—	2	1	no	—
22	C₂×C₈×C₄₇	—	2	1	no	—
23	C₂×C₈×C₄₇	—	4	1	no	—
24	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
25	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
27	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
28	C₄²×C₄₇	D₈	1	1	yes	characteristic
29	C₄²×C₄₇	Q₁₆	1	1	yes	characteristic
30	C₂×C₄₇⋊C₈	—	4	4	no	—
31	C₁₆×C₄₇	—	4	2	no	—
32	C₂³×C₄₇	#106	1	1	yes	characteristic
33	C₂×C₄×C₄₇	—	2	1	no	—
34	C₂×C₄×C₄₇	—	2	1	no	—
35	C₂×C₄×C₄₇	—	4	2	no	—
36	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
37	C₂×C₄×C₄₇	#102	1	2	yes	—
38	C₂×C₄×C₄₇	#103	1	2	yes	—
39	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
40	C₈×C₄₇	—	2	2	no	—
41	C₈×C₄₇	—	2	1	no	—
42	C₈×C₄₇	#100	1	1	yes	characteristic
43	C₈×C₄₇	#101	1	1	yes	characteristic
44	C₄₇⋊C₈	—	8	4	no	—
45	#7246	—	47	1	no	maximal, Sylow
46	C₂²×C₄₇	—	2	1	no	—
47	C₂²×C₄₇	—	2	2	no	—
48	C₂²×C₄₇	#340	1	1	yes	characteristic
49	C₄×C₄₇	—	2	1	no	—
50	C₄×C₄₇	—	2	2	no	—
51	C₄×C₄₇	—	2	2	no	—
52	C₄×C₄₇	#338	1	1	yes	characteristic
53	C₄×C₄₇	#343	1	1	yes	characteristic
54	#2150	D₄₇	1	1	yes	characteristic
55	#1568	—	47	2	no	—
56	C₂×C₄₇	—	4	1	no	—
57	C₂×C₄₇	#1292	1	1	yes	characteristic
58	C₂×C₄₇	#1389	1	2	yes	—
59	C₂×M₃₂	Dic₄₇	1	2	yes	—
60	#432	—	94	2	no	—
61	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
62	C₄⋊C₁₆	—	2	2	no	—
63	#335	—	94	2	no	—
64	#334	—	47	4	no	—
65	C₄₇	#7246	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
67	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
68	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
69	M₃₂	—	4	2	no	—
70	C₂×C₁₆	—	2	2	no	—
71	C₄⋊C₈	—	94	4	no	—
72	C₂²⋊C₈	—	188	2	no	—
73	#97	—	188	2	no	—
74	C₄×C₈	C₄×D₄₇	1	2	yes	—
75	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
76	C₂²×C₄	#10625325	1	2	yes	—
77	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	4	1	no	—
81	C₂×C₈	—	188	4	no	—
82	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
83	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
84	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
85	C₄²	D₃₇₆	1	1	yes	characteristic
86	C₄²	Dic₁₈₈	1	1	yes	characteristic
87	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
88	C₁₆	—	4	2	no	—
89	C₂³	#13093956	1	1	yes	characteristic
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
94	C₂×C₄	#13093946	1	2	yes	—
95	C₂×C₄	#13093949	1	2	yes	—
96	C₂×C₄	#13093958	1	1	yes	characteristic
97	C₈	—	2	2	no	—
98	C₈	—	2	1	no	—
99	C₈	—	376	4	no	—
100	C₈	#13093941	1	1	yes	characteristic
101	C₈	#13093943	1	1	yes	characteristic
102	C₂²	—	2	2	no	—
103	C₂²	—	2	1	no	—
104	C₂²	#15688886	1	1	yes	characteristic
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	2	1	no	—
108	C₄	C₄₇⋊#338	1	1	yes	characteristic
109	C₄	#15688892	1	1	yes	characteristic
110	C₂	—	4	1	no	—
111	C₂	#20241263	1	1	yes	characteristic
112	C₂	#20241503	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20242151

C₂

yes

maximal

C₄₇×#2150

C₂

yes

maximal, characteristic

#15689180

—

#15688876

—

#15688874

C₄

yes

—

C₂×M₃₂×C₄₇

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×C₄⋊C₁₆

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

Q₈

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

M₃₂×C₄₇

—

C₂×C₁₆×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄²

yes

characteristic

C₂×C₈×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

C₂²⋊C₄

yes

—

C₄²×C₄₇

D₈

yes

characteristic

C₄²×C₄₇

Q₁₆

yes

characteristic

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₂³×C₄₇

#106

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#95

yes

characteristic

C₂×C₄×C₄₇

#102

yes

—

C₂×C₄×C₄₇

#103

yes

—

C₂×C₄×C₄₇

#107

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

yes

C₈×C₄₇

yes

C₄₇⋊C₈

—

#7246

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#340

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

D₄₇

yes

characteristic

#1568

—

C₂×C₄₇

—

C₂×C₄₇

#1292

yes

characteristic

C₂×C₄₇

#1389

yes

—

C₂×M₃₂

Dic₄₇

yes

—

#432

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₄⋊C₁₆

—

#335

—

#334

—

C₄₇

#7246

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₉₄

yes

characteristic

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

C₂×C₄²

D₁₈₈

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₄⋊C₈

—

C₂²⋊C₈

—

188

—

#97

—

188

—

C₄×C₈

C₄×D₄₇

yes

—

C₄×C₈

C₂²⋊D₄₇

yes

—

C₂²×C₄

#10625325

yes

—

C₂²×C₄

C₄⋊Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

C₄×Dic₄₇

yes

characteristic

C₂×C₈

Dic₄₇⋊C₄

yes

—

C₂×C₈

C₂²⋊Dic₄₇

yes

characteristic

C₄²

D₃₇₆

yes

characteristic

C₄²

Dic₁₈₈

yes

characteristic

C₄²

(C₂×D₄₇)⋊C₄

yes

—

C₁₆

—

C₂³

#13093956

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄₇⋊#95

yes

characteristic

C₂×C₄

#13093946

yes

—

C₂×C₄

#13093949

yes

—

C₂×C₄

#13093958

yes

characteristic

C₈

—

C₈

—

C₈

—

376

—

100

C₈

#13093941

yes

characteristic

101

C₈

#13093943

yes

characteristic

102

C₂²

—

103

C₂²

—

104

C₂²

#15688886

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

C₄₇⋊#338

yes

characteristic

109

C₄

#15688892

yes

characteristic

110

C₂

—

111

C₂

#20241263

yes

characteristic

112

C₂

#20241503

yes

—

	subgroup	G/H
center	C₂²	#15688886
commutator	C₂×C₈×C₄₇	C₄²
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₄₇×#2150	C₂
socle	C₂²×C₄₇	#340
prime radical	C₂³×C₄₇	#106
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	#2150	D₄₇
hypocenter	C₄₇	#7246
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂²

#15688886

commutator

C₂×C₈×C₄₇

C₄²

Frattini

C₂×C₄×C₈

C₂×D₄₇

Fitting

C₄₇×#2150

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄×C₈×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
65. C₄₇	#7246	semidirect

H ≅ G/N

type

65. C₄₇

#7246

semidirect

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

subgroup	index
#20242151	2
C₄₇×#2150	2
#7246	47

subgroup

index

#20242151

C₄₇×#2150

#7246

Subgroups — #26103389

Conjugacy classes of subgroups

Complements

Maximal subgroups