Group #26103732 — subgroups

Subgroups — #26103732

Order 12032.

126 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 55 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	C₄×Dic₃₇₆	C₂	1	1	yes	maximal, characteristic
2	#20241460	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃₇₆	C₄	1	1	yes	characteristic
5	C₄×Dic₁₈₈	—	2	1	no	—
6	#15688949	C₄	1	1	yes	characteristic
7	#15688928	—	2	1	no	—
8	#15688876	—	2	1	no	—
9	C₂×C₃₂×C₄₇	—	2	1	no	—
10	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₈₈	—	2	1	no	—
12	C₄×Dic₉₄	—	4	1	no	—
13	Dic₃₇₆	—	2	1	no	—
14	Dic₃₇₆	C₈	1	2	yes	complemented
15	#13093958	—	2	1	no	—
16	C₄⋊(C₄₇⋊C₈)	—	4	1	no	—
17	#13093949	—	4	1	no	—
18	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
20	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
21	C₃₂×C₄₇	—	2	2	no	—
22	C₂×Dic₉₄	—	4	1	no	—
23	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
25	C₄²×C₄₇	D₈	1	1	yes	characteristic
26	C₄⋊Dic₄₇	—	4	1	no	—
27	Dic₄₇⋊C₄	—	8	1	no	—
28	C₄×Dic₄₇	—	8	1	no	—
29	C₂×C₄₇⋊C₈	—	8	1	no	—
30	Dic₁₈₈	—	2	2	no	—
31	Dic₁₈₈	—	4	1	no	—
32	C₁₆×C₄₇	—	2	1	no	—
33	C₁₆×C₄₇	C₂×C₈	1	1	yes	characteristic
34	C₁₆×C₄₇	M₁₆	1	1	yes	characteristic
35	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
36	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
37	C₂×C₄×C₄₇	SD₃₂	1	1	yes	characteristic
38	C₂×Dic₄₇	—	8	1	no	—
39	C₂×Dic₄₇	—	8	1	no	—
40	Dic₉₄	—	4	2	no	—
41	Dic₉₄	—	8	1	no	—
42	C₈×C₄₇	—	2	1	no	—
43	C₈×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
44	C₈×C₄₇	#104	1	1	yes	characteristic
45	C₄₇⋊C₈	—	16	1	no	—
46	#7384	—	47	1	no	maximal, Sylow
47	C₂²×C₄₇	#357	1	1	yes	characteristic
48	C₄×C₄₇	—	2	1	no	—
49	C₄×C₄₇	D₄⋊C₈	1	1	yes	characteristic
50	C₄×C₄₇	#359	1	1	yes	characteristic
51	C₄×C₄₇	SD₆₄	1	1	yes	characteristic
52	C₄×C₄₇	Q₆₄	1	1	yes	characteristic
53	Dic₄₇	—	8	2	no	—
54	Dic₄₇	—	16	1	no	—
55	C₄×Q₃₂	—	47	1	no	—
56	C₄×C₃₂	D₄₇	1	1	yes	characteristic
57	#1371	—	47	1	no	—
58	C₂×C₄₇	#1331	1	1	yes	characteristic
59	C₂×C₄₇	#1416	1	1	yes	characteristic
60	C₂×C₄₇	#1417	1	1	yes	characteristic
61	C₂×Q₃₂	—	47	1	no	—
62	C₄×Q₁₆	—	94	1	no	—
63	C₂×C₃₂	—	2	1	no	—
64	#366	—	47	1	no	—
65	#358	—	94	1	no	—
66	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
67	#335	—	94	1	no	—
68	C₄₇	#7384	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₁₆	—	94	1	no	—
70	C₄×Q₈	—	188	1	no	—
71	Q₃₂	—	47	2	no	—
72	Q₃₂	—	94	1	no	—
73	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
74	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
75	#107	—	94	1	no	—
76	C₄⋊C₈	—	188	1	no	—
77	#103	—	188	1	no	—
78	C₄×C₈	D₁₈₈	1	1	yes	characteristic
79	C₃₂	—	2	2	no	—
80	C₂×Q₈	—	188	1	no	—
81	Q₁₆	—	94	2	no	—
82	Q₁₆	—	188	1	no	—
83	C₂×C₈	—	376	1	no	—
84	C₂×C₈	#10625325	1	1	yes	characteristic
85	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
86	C₄⋊C₄	—	376	1	no	—
87	C₄⋊C₄	—	188	1	no	—
88	C₄²	—	376	1	no	—
89	C₄²	D₃₇₆	1	1	yes	characteristic
90	C₁₆	—	2	1	no	—
91	C₁₆	C₈×D₄₇	1	1	yes	characteristic
92	C₁₆	#10625324	1	1	yes	characteristic
93	Q₈	—	376	1	no	—
94	Q₈	—	188	2	no	—
95	C₂×C₄	—	376	1	no	—
96	C₂×C₄	—	376	1	no	—
97	C₂×C₄	#13093946	1	1	yes	characteristic
98	C₂×C₄	D₇₅₂	1	1	yes	characteristic
99	C₂×C₄	#13093969	1	1	yes	characteristic
100	C₈	—	2	1	no	—
101	C₈	—	752	1	no	—
102	C₈	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
103	C₈	#13093952	1	1	yes	characteristic
104	C₂²	#15688925	1	1	yes	characteristic
105	C₄	—	2	1	no	—
106	C₄	—	376	2	no	—
107	C₄	—	752	1	no	—
108	C₄	#15688852	1	1	yes	characteristic
109	C₄	#15688931	1	1	yes	characteristic
110	C₄	#15688962	1	1	yes	characteristic
111	C₄	Dic₇₅₂	1	1	yes	characteristic
112	C₂	#20241350	1	1	yes	characteristic
113	C₂	#20241570	1	1	yes	characteristic
114	C₂	#20241573	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₃₇₆

C₂

yes

maximal, characteristic

#20241460

C₂

yes

maximal, characteristic

C₄×C₃₂×C₄₇

C₂

yes

maximal, characteristic

C₂×Dic₃₇₆

C₄

yes

characteristic

C₄×Dic₁₈₈

—

#15688949

C₄

yes

characteristic

#15688928

—

#15688876

—

C₂×C₃₂×C₄₇

—

C₄×C₁₆×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

—

C₄×Dic₉₄

—

Dic₃₇₆

—

Dic₃₇₆

C₈

yes

complemented

#13093958

—

C₄⋊(C₄₇⋊C₈)

—

#13093949

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₃₂×C₄₇

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₇

SD₁₆

yes

characteristic

C₄²×C₄₇

D₈

yes

characteristic

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

Dic₁₈₈

—

Dic₁₈₈

—

C₁₆×C₄₇

—

C₁₆×C₄₇

C₂×C₈

yes

characteristic

C₁₆×C₄₇

M₁₆

yes

characteristic

C₂×C₄×C₄₇

#102

yes

characteristic

C₂×C₄×C₄₇

D₁₆

yes

characteristic

C₂×C₄×C₄₇

SD₃₂

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

C₂²⋊C₈

yes

characteristic

C₈×C₄₇

#104

yes

characteristic

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

D₄⋊C₈

yes

characteristic

C₄×C₄₇

#359

yes

characteristic

C₄×C₄₇

SD₆₄

yes

characteristic

C₄×C₄₇

Q₆₄

yes

characteristic

Dic₄₇

—

Dic₄₇

—

C₄×Q₃₂

—

C₄×C₃₂

D₄₇

yes

characteristic

#1371

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×Q₃₂

—

C₄×Q₁₆

—

C₂×C₃₂

—

#366

—

#358

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

#335

—

C₄₇

#7384

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

188

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

C₄×D₄₇

yes

characteristic

C₂×C₁₆

C₂²⋊D₄₇

yes

characteristic

#107

—

C₄⋊C₈

—

188

—

#103

—

188

—

C₄×C₈

D₁₈₈

yes

characteristic

C₃₂

—

C₂×Q₈

—

188

—

Q₁₆

—

Q₁₆

—

188

—

C₂×C₈

—

376

—

C₂×C₈

#10625325

yes

characteristic

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

characteristic

C₄⋊C₄

—

376

—

C₄⋊C₄

—

188

—

C₄²

—

376

—

C₄²

D₃₇₆

yes

characteristic

C₁₆

—

C₁₆

C₈×D₄₇

yes

characteristic

C₁₆

#10625324

yes

characteristic

Q₈

—

376

—

Q₈

—

188

—

C₂×C₄

—

376

—

C₂×C₄

—

376

—

C₂×C₄

#13093946

yes

characteristic

C₂×C₄

D₇₅₂

yes

characteristic

C₂×C₄

#13093969

yes

characteristic

100

C₈

—

101

C₈

—

752

—

102

C₈

(C₂×D₄₇)⋊C₈

yes

characteristic

103

C₈

#13093952

yes

characteristic

104

C₂²

#15688925

yes

characteristic

105

C₄

—

106

C₄

—

376

—

107

C₄

—

752

—

108

C₄

#15688852

yes

characteristic

109

C₄

#15688931

yes

characteristic

110

C₄

#15688962

yes

characteristic

111

C₄

Dic₇₅₂

yes

characteristic

112

C₂

#20241350

yes

characteristic

113

C₂

#20241570

yes

characteristic

114

C₂

#20241573

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₇₅₂
commutator	C₁₆×C₄₇	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₄₇
Fitting	C₄×C₃₂×C₄₇	C₂
socle, prime radical	C₂²×C₄₇	#357
cosocle	C₄×C₁₆×C₄₇	C₂²
hypercenter	C₄×C₃₂	D₄₇
hypocenter	C₄₇	#7384
LCS	G ▷ C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

subgroup

G/H

center

C₂×C₄

D₇₅₂

commutator

C₁₆×C₄₇

C₂×C₈

Frattini

C₄×C₁₆

C₂×D₄₇

Fitting

C₄×C₃₂×C₄₇

C₂

socle, prime radical

C₂²×C₄₇

#357

cosocle

C₄×C₁₆×C₄₇

C₂²

hypercenter

C₄×C₃₂

D₄₇

hypocenter

C₄₇

#7384

LCS

G ▷ C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

N	H ≅ G/N	type
14. Dic₃₇₆	C₈	semidirect
68. C₄₇	#7384	semidirect

H ≅ G/N

type

14. Dic₃₇₆

C₈

semidirect

68. C₄₇

#7384

semidirect

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

subgroup	index
C₄×Dic₃₇₆	2
#20241460	2
C₄×C₃₂×C₄₇	2
#7384	47

subgroup

index

C₄×Dic₃₇₆

#20241460

C₄×C₃₂×C₄₇

#7384

Subgroups — #26103732

Conjugacy classes of subgroups

Complements

Maximal subgroups