Group #26132671 — subgroups

Subgroups — #26132671

Order 12032.

274 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 147 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	#20243560	C₂	1	2	yes	maximal
2	#20241482	C₂	1	4	yes	maximal
3	C₂×C₄₇×#346	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093960	—	2	2	no	—
5	C₂×#13093960	C₂²	1	4	yes	—
6	#15689147	—	2	2	no	—
7	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
9	C₄₇×#346	C₄	1	4	yes	—
10	C₂×#10625329	—	2	4	no	—
11	#13093960	—	2	8	no	—
12	#13093960	—	4	4	no	—
13	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
14	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
17	C₂×C₁₆×C₄₇	—	2	4	no	—
18	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	—
19	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₄₇	Q₈	1	2	yes	—
21	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
22	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
23	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
24	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
25	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
27	C₂×C₈×C₄₇	#41	1	4	yes	—
28	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
29	C₄²×C₄₇	D₈	1	2	yes	—
30	#10625329	—	4	4	no	—
31	#10625329	—	4	4	no	—
32	C₂×C₄₇⋊C₈	—	4	4	no	—
33	C₁₆×C₄₇	—	2	4	no	—
34	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
35	C₂×C₄×C₄₇	—	4	2	no	—
36	C₂×C₄×C₄₇	#107	1	4	yes	—
37	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₄₇	#124	1	2	yes	—
39	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
40	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
41	C₈×C₄₇	#106	1	4	yes	—
42	C₈×C₄₇	#117	1	2	yes	—
43	C₈×C₄₇	#125	1	2	yes	—
44	C₄₇⋊C₈	—	8	4	no	—
45	#12892	—	47	1	no	maximal, Sylow
46	C₂²×C₄₇	—	2	2	no	—
47	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
48	C₂²×C₄₇	#487	1	2	yes	—
49	C₄×C₄₇	—	2	4	no	—
50	C₄×C₄₇	#382	1	1	yes	characteristic
51	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
52	C₄×C₄₇	#486	1	2	yes	—
53	C₂×#346	D₄₇	1	1	yes	characteristic
54	#1848	—	47	2	no	—
55	#1381	—	47	4	no	—
56	C₂×C₄₇	—	2	2	no	—
57	C₂×C₄₇	#1834	1	1	yes	characteristic
58	C₂×C₄₇	#2161	1	2	yes	—
59	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
60	C₂×#108	—	47	4	no	—
61	C₂×#108	—	94	2	no	—
62	#423	—	94	2	no	—
63	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
64	#346	Dic₄₇	1	4	yes	—
65	C₄₇	#12892	1	1	yes	characteristic, complemented, Sylow
66	C₂×M₁₆	—	94	4	no	—
67	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
68	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
69	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
70	C₂×C₁₆	—	2	4	no	—
71	C₂×C₁₆	C₂×Dic₄₇	1	4	yes	—
72	#108	—	94	8	no	—
73	#108	—	188	4	no	—
74	C₄⋊C₈	—	188	4	no	—
75	C₄×C₈	Dic₉₄	1	2	yes	—
76	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
77	C₂²×C₄	Dic₁₈₈	1	2	yes	—
78	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
79	M₁₆	—	188	4	no	—
80	M₁₆	—	188	4	no	—
81	C₂×C₈	—	188	4	no	—
82	C₂×C₈	#10625325	1	4	yes	—
83	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
84	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
85	C₂×C₈	#10625350	1	4	yes	—
86	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
87	C₄²	D₃₇₆	1	2	yes	—
88	C₄²	C₄⋊Dic₄₇	1	2	yes	—
89	C₁₆	—	2	4	no	—
90	C₂³	C₂×Dic₁₈₈	1	1	yes	characteristic
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	#13093958	1	4	yes	—
93	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
94	C₂×C₄	#13094012	1	2	yes	—
95	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
96	C₂×C₄	C₂×#10625325	1	2	yes	—
97	C₈	—	376	4	no	—
98	C₈	#13093956	1	4	yes	—
99	C₈	#13093983	1	2	yes	—
100	C₈	#13094016	1	2	yes	—
101	C₂²	—	2	2	no	—
102	C₂²	C₂×#13093958	1	1	yes	characteristic
103	C₂²	#15689474	1	2	yes	—
104	C₄	—	2	4	no	—
105	C₄	#15688989	1	1	yes	characteristic
106	C₄	C₂×#13093956	1	1	yes	characteristic
107	C₄	#15689470	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	#20243516	1	1	yes	characteristic
110	C₂	#20245183	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243560

C₂

yes

maximal

#20241482

C₂

yes

maximal

C₂×C₄₇×#346

C₂

yes

maximal, characteristic

C₂×#13093960

—

C₂×#13093960

C₂²

yes

—

#15689147

—

C₂²×C₁₆×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#346

C₄

yes

—

C₂×#10625329

—

#13093960

—

#13093960

—

C₄⋊(C₄₇⋊C₈)

—

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

Q₁₆

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₄²×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

D₈

yes

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₂³×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#107

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#124

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₈×C₄₇

#106

yes

—

C₈×C₄₇

#117

yes

—

C₈×C₄₇

#125

yes

—

C₄₇⋊C₈

—

#12892

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×#107

yes

characteristic

C₂²×C₄₇

#487

yes

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₂×#346

D₄₇

yes

characteristic

#1848

—

#1381

—

C₂×C₄₇

—

C₂×C₄₇

#1834

yes

characteristic

C₂×C₄₇

#2161

yes

—

C₂²×C₁₆

C₂×D₄₇

yes

—

C₂×#108

—

C₂×#108

—

#423

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#346

Dic₄₇

yes

—

C₄₇

#12892

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

D₁₈₈

yes

characteristic

C₂²×C₈

C₂²×D₄₇

yes

characteristic

C₂×C₄²

D₁₈₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₄₇

yes

—

#108

—

#108

—

188

—

C₄⋊C₈

—

188

—

C₄×C₈

Dic₉₄

yes

—

C₄×C₈

C₂×Dic₄₇

yes

—

C₂²×C₄

Dic₁₈₈

yes

—

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

M₁₆

—

188

—

M₁₆

—

188

—

C₂×C₈

—

188

—

C₂×C₈

#10625325

yes

—

C₂×C₈

C₄⋊Dic₄₇

yes

—

C₂×C₈

C₂×Dic₉₄

yes

characteristic

C₂×C₈

#10625350

yes

—

C₂×C₈

C₂²×Dic₄₇

yes

characteristic

C₄²

D₃₇₆

yes

—

C₄²

C₄⋊Dic₄₇

yes

—

C₁₆

—

C₂³

C₂×Dic₁₈₈

yes

characteristic

C₂×C₄

—

C₂×C₄

#13093958

yes

—

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

characteristic

C₂×C₄

#13094012

yes

—

C₂×C₄

C₂×D₃₇₆

yes

characteristic

C₂×C₄

C₂×#10625325

yes

—

C₈

—

376

—

C₈

#13093956

yes

—

C₈

#13093983

yes

—

100

C₈

#13094016

yes

—

101

C₂²

—

102

C₂²

C₂×#13093958

yes

characteristic

103

C₂²

#15689474

yes

—

104

C₄

—

105

C₄

#15688989

yes

characteristic

106

C₄

C₂×#13093956

yes

characteristic

107

C₄

#15689470

yes

—

108

C₂

—

109

C₂

#20243516

yes

characteristic

110

C₂

#20245183

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₃₇₆
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#346	C₂
socle	C₂²×C₄₇	C₂×#107
prime radical	C₂³×C₄₇	C₂×Q₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#346	D₄₇
hypocenter	C₄₇	#12892
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₃₇₆

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₂×C₄₇×#346

C₂

socle

C₂²×C₄₇

C₂×#107

prime radical

C₂³×C₄₇

C₂×Q₁₆

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
65. C₄₇	#12892	semidirect

H ≅ G/N

type

65. C₄₇

#12892

semidirect

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

subgroup	index
#20243560	2
#20241482	2
C₂×C₄₇×#346	2
#12892	47

subgroup

index

#20243560

#20241482

C₂×C₄₇×#346

#12892

Subgroups — #26132671

Conjugacy classes of subgroups

Complements

Maximal subgroups