Group #26103387 — subgroups

Subgroups — #26103387

Order 12032.

200 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 91 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	#20242147	C₂	1	2	yes	maximal
2	C₄₇×#2150	C₂	1	1	yes	maximal, characteristic
3	#15689147	—	2	2	no	—
4	#15688876	—	2	2	no	—
5	#15688876	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	C₂×#10625329	—	4	2	no	—
10	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
11	C₄⋊(C₄₇⋊C₈)	—	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	#10625329	—	8	4	no	—
31	C₂×C₄₇⋊C₈	—	4	4	no	—
32	C₁₆×C₄₇	—	4	2	no	—
33	C₂³×C₄₇	#106	1	1	yes	characteristic
34	C₂×C₄×C₄₇	—	2	1	no	—
35	C₂×C₄×C₄₇	—	2	1	no	—
36	C₂×C₄×C₄₇	—	2	2	no	—
37	C₂×C₄×C₄₇	—	2	2	no	—
38	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
39	C₂×C₄×C₄₇	#102	1	2	yes	—
40	C₂×C₄×C₄₇	#103	1	2	yes	—
41	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
42	C₈×C₄₇	—	2	2	no	—
43	C₈×C₄₇	—	2	1	no	—
44	C₈×C₄₇	#100	1	1	yes	characteristic
45	C₈×C₄₇	#101	1	1	yes	characteristic
46	C₄₇⋊C₈	—	8	4	no	—
47	#7245	—	47	1	no	maximal, Sylow
48	C₂²×C₄₇	—	2	1	no	—
49	C₂²×C₄₇	—	2	2	no	—
50	C₂²×C₄₇	#340	1	1	yes	characteristic
51	C₄×C₄₇	—	2	1	no	—
52	C₄×C₄₇	—	2	2	no	—
53	C₄×C₄₇	#338	1	1	yes	characteristic
54	C₄×C₄₇	#343	1	1	yes	characteristic
55	C₄×C₄₇	#361	1	2	yes	—
56	C₄×C₄₇	#362	1	2	yes	—
57	#2150	D₄₇	1	1	yes	characteristic
58	#1567	—	47	2	no	—
59	C₂×C₄₇	—	4	1	no	—
60	C₂×C₄₇	#1292	1	1	yes	characteristic
61	C₂×C₄₇	#1387	1	2	yes	—
62	C₂×M₃₂	Dic₄₇	1	2	yes	—
63	#423	—	94	2	no	—
64	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
65	C₄⋊C₁₆	—	2	2	no	—
66	#335	—	47	4	no	—
67	#335	—	94	2	no	—
68	C₄₇	#7245	1	1	yes	characteristic, complemented, Sylow
69	C₂×M₁₆	—	188	2	no	—
70	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
71	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
72	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
73	M₃₂	—	4	2	no	—
74	C₂×C₁₆	—	2	2	no	—
75	C₄⋊C₈	—	94	4	no	—
76	C₄⋊C₈	—	188	2	no	—
77	C₄×C₈	C₄×D₄₇	1	2	yes	—
78	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
79	C₂²×C₄	#10625325	1	2	yes	—
80	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
81	M₁₆	—	376	4	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	188	4	no	—
86	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
87	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
88	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
89	C₄²	D₃₇₆	1	1	yes	characteristic
90	C₄²	Dic₁₈₈	1	1	yes	characteristic
91	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
92	C₁₆	—	4	2	no	—
93	C₂³	#13093956	1	1	yes	characteristic
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
99	C₂×C₄	#13093946	1	2	yes	—
100	C₂×C₄	#13093949	1	2	yes	—
101	C₂×C₄	#13093958	1	1	yes	characteristic
102	C₈	—	2	2	no	—
103	C₈	—	2	1	no	—
104	C₈	—	376	4	no	—
105	C₈	#13093941	1	1	yes	characteristic
106	C₈	#13093943	1	1	yes	characteristic
107	C₂²	—	2	2	no	—
108	C₂²	—	2	1	no	—
109	C₂²	#15688886	1	1	yes	characteristic
110	C₄	—	2	2	no	—
111	C₄	—	2	1	no	—
112	C₄	C₄₇⋊#338	1	1	yes	characteristic
113	C₄	#15688892	1	1	yes	characteristic
114	C₄	#15688937	1	2	yes	—
115	C₄	#15688940	1	2	yes	—
116	C₂	—	4	1	no	—
117	C₂	#20241263	1	1	yes	characteristic
118	C₂	#20241497	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20242147

C₂

yes

maximal

C₄₇×#2150

C₂

yes

maximal, characteristic

#15689147

—

#15688876

—

#15688876

C₄

yes

—

C₂×M₃₂×C₄₇

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×C₄⋊C₁₆

—

C₂×#10625329

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

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

#10625329

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₂³×C₄₇

#106

yes

characteristic

C₂×C₄×C₄₇

—

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

—

#7245

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#340

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₄×C₄₇

#362

yes

—

#2150

D₄₇

yes

characteristic

#1567

—

C₂×C₄₇

—

C₂×C₄₇

#1292

yes

characteristic

C₂×C₄₇

#1387

yes

—

C₂×M₃₂

Dic₄₇

yes

—

#423

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₄⋊C₁₆

—

#335

—

#335

—

C₄₇

#7245

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

188

—

C₂²×C₈

Dic₉₄

yes

characteristic

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

C₂×C₄²

D₁₈₈

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

188

—

C₄×C₈

C₄×D₄₇

yes

—

C₄×C₈

C₂²⋊D₄₇

yes

—

C₂²×C₄

#10625325

yes

—

C₂²×C₄

C₄⋊Dic₄₇

yes

characteristic

M₁₆

—

376

—

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

—

C₂×C₄

C₄₇⋊#95

yes

characteristic

C₂×C₄

#13093946

yes

—

100

C₂×C₄

#13093949

yes

—

101

C₂×C₄

#13093958

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

376

—

105

C₈

#13093941

yes

characteristic

106

C₈

#13093943

yes

characteristic

107

C₂²

—

108

C₂²

—

109

C₂²

#15688886

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

C₄₇⋊#338

yes

characteristic

113

C₄

#15688892

yes

characteristic

114

C₄

#15688937

yes

—

115

C₄

#15688940

yes

—

116

C₂

—

117

C₂

#20241263

yes

characteristic

118

C₂

#20241497

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	#7245
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
68. C₄₇	#7245	semidirect

H ≅ G/N

type

68. C₄₇

#7245

semidirect

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

subgroup	index
#20242147	2
C₄₇×#2150	2
#7245	47

subgroup

index

#20242147

C₄₇×#2150

#7245

Subgroups — #26103387

Conjugacy classes of subgroups

Complements

Maximal subgroups