Group #26103379 — subgroups

Subgroups — #26103379

Order 12032.

190 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 87 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	C₂×C₄₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#15689180	—	2	2	no	—
4	#15688876	—	2	2	no	—
5	#15688874	C₄	1	4	yes	—
6	C₂²×C₁₆×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	C₂×C₁₆×C₄₇	—	2	2	no	—
16	C₂×C₁₆×C₄₇	—	4	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	D₈	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
22	C₂×C₈×C₄₇	—	2	1	no	—
23	C₂×C₈×C₄₇	—	2	1	no	—
24	C₂×C₈×C₄₇	—	4	1	no	—
25	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
26	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
28	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
29	C₄²×C₄₇	SD₁₆	1	2	yes	—
30	C₂×C₄₇⋊C₈	—	4	4	no	—
31	C₁₆×C₄₇	—	4	2	no	—
32	C₂³×C₄₇	#107	1	1	yes	characteristic
33	C₂×C₄×C₄₇	—	2	2	no	—
34	C₂×C₄×C₄₇	—	2	1	no	—
35	C₂×C₄×C₄₇	—	2	1	no	—
36	C₂×C₄×C₄₇	—	4	1	no	—
37	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
38	C₂×C₄×C₄₇	#102	1	2	yes	—
39	C₂×C₄×C₄₇	#103	1	2	yes	—
40	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
41	C₈×C₄₇	—	2	2	no	—
42	C₈×C₄₇	—	2	1	no	—
43	C₈×C₄₇	#100	1	1	yes	characteristic
44	C₈×C₄₇	#101	1	1	yes	characteristic
45	C₄₇⋊C₈	—	8	4	no	—
46	#7242	—	47	1	no	maximal, Sylow
47	C₂²×C₄₇	—	2	1	no	—
48	C₂²×C₄₇	—	2	1	no	—
49	C₂²×C₄₇	#340	1	1	yes	characteristic
50	C₂²×C₄₇	#368	1	2	yes	—
51	C₄×C₄₇	—	2	1	no	—
52	C₄×C₄₇	—	2	2	no	—
53	C₄×C₄₇	—	2	2	no	—
54	C₄×C₄₇	#338	1	1	yes	characteristic
55	C₄×C₄₇	#343	1	1	yes	characteristic
56	C₂×C₄⋊C₁₆	D₄₇	1	1	yes	characteristic
57	#1568	—	47	2	no	—
58	C₂×C₄₇	—	2	2	no	—
59	C₂×C₄₇	#1292	1	1	yes	characteristic
60	C₂×C₄₇	#1381	1	1	yes	characteristic
61	C₂×C₄₇	#1384	1	1	yes	characteristic
62	C₂²×C₁₆	Dic₄₇	1	2	yes	—
63	#432	—	94	2	no	—
64	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
65	C₄⋊C₁₆	—	2	2	no	—
66	#335	—	94	2	no	—
67	#334	—	47	4	no	—
68	C₄₇	#7242	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
70	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
71	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	—	4	2	no	—
74	C₄⋊C₈	—	94	4	no	—
75	C₂²⋊C₈	—	188	2	no	—
76	#97	—	188	2	no	—
77	C₄×C₈	C₄×D₄₇	1	2	yes	—
78	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
79	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
80	C₂²×C₄	Dic₁₈₈	1	1	yes	characteristic
81	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
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	#10625325	1	2	yes	—
90	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
91	C₁₆	—	4	2	no	—
92	C₂³	#13093958	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
98	C₂×C₄	#13093946	1	2	yes	—
99	C₂×C₄	#13093949	1	2	yes	—
100	C₂×C₄	#13093956	1	1	yes	characteristic
101	C₈	—	2	2	no	—
102	C₈	—	2	1	no	—
103	C₈	—	376	4	no	—
104	C₈	#13093941	1	1	yes	characteristic
105	C₈	#13093943	1	1	yes	characteristic
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	#15688886	1	1	yes	characteristic
109	C₂²	#15688953	1	2	yes	—
110	C₄	—	2	2	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	1	no	—
113	C₄	C₄₇⋊#338	1	1	yes	characteristic
114	C₄	#15688892	1	1	yes	characteristic
115	C₂	—	2	2	no	—
116	C₂	#20241263	1	1	yes	characteristic
117	C₂	#20241482	1	1	yes	characteristic
118	C₂	#20241490	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20242151

C₂

yes

maximal

C₂×C₄₇×C₄⋊C₁₆

C₂

yes

maximal, characteristic

#15689180

—

#15688876

—

#15688874

C₄

yes

—

C₂²×C₁₆×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

C₂×C₁₆×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

D₈

yes

characteristic

C₂²×C₄×C₄₇

Q₁₆

yes

characteristic

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

SD₁₆

yes

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₂³×C₄₇

#107

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

#106

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

yes

C₈×C₄₇

yes

C₄₇⋊C₈

—

#7242

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#340

yes

characteristic

C₂²×C₄₇

#368

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₂×C₄⋊C₁₆

D₄₇

yes

characteristic

#1568

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂²×C₁₆

Dic₄₇

yes

—

#432

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₄⋊C₁₆

—

#335

—

#334

—

C₄₇

#7242

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₉₄

yes

characteristic

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

C₂×C₄²

D₁₈₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₂²⋊C₈

—

188

—

#97

—

188

—

C₄×C₈

C₄×D₄₇

yes

—

C₄×C₈

C₂²⋊D₄₇

yes

—

C₂²×C₄

D₃₇₆

yes

characteristic

C₂²×C₄

Dic₁₈₈

yes

characteristic

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

#10625325

yes

—

C₄²

(C₂×D₄₇)⋊C₄

yes

—

C₁₆

—

C₂³

#13093958

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄₇⋊#95

yes

characteristic

C₂×C₄

#13093946

yes

—

C₂×C₄

#13093949

yes

—

100

C₂×C₄

#13093956

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

376

—

104

C₈

#13093941

yes

characteristic

105

C₈

#13093943

yes

characteristic

106

C₂²

—

107

C₂²

—

108

C₂²

#15688886

yes

characteristic

109

C₂²

#15688953

yes

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

C₄₇⋊#338

yes

characteristic

114

C₄

#15688892

yes

characteristic

115

C₂

—

116

C₂

#20241263

yes

characteristic

117

C₂

#20241482

yes

characteristic

118

C₂

#20241490

yes

characteristic

	subgroup	G/H
center	C₂²	#15688886
commutator	C₂×C₈×C₄₇	C₄²
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₂×C₄₇×C₄⋊C₁₆	C₂
socle	C₂²×C₄₇	#340
prime radical	C₂³×C₄₇	#107
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₄₇
hypocenter	C₄₇	#7242
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center

C₂²

#15688886

commutator

C₂×C₈×C₄₇

C₄²

Frattini

C₂×C₄×C₈

C₂×D₄₇

Fitting

C₂×C₄₇×C₄⋊C₁₆

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄×C₈×C₄₇

C₂²

hypercenter

C₂×C₄⋊C₁₆

D₄₇

hypocenter

C₄₇

#7242

LCS

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
68. C₄₇	#7242	semidirect

H ≅ G/N

type

68. C₄₇

#7242

semidirect

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

subgroup	index
#20242151	2
C₂×C₄₇×C₄⋊C₁₆	2
#7242	47

subgroup

index

#20242151

C₂×C₄₇×C₄⋊C₁₆

#7242

Subgroups — #26103379

Conjugacy classes of subgroups

Complements

Maximal subgroups