Group #26131018 — subgroups

Subgroups — #26131018

Order 12032.

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

normal subgroups only

This group has 238 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	#20245670	C₂	1	1	yes	maximal, characteristic
2	#20245661	C₂	1	2	yes	maximal
3	#20242997	C₂	1	2	yes	maximal
4	C₁₆×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#2254	C₂	1	1	yes	maximal, characteristic
6	#15689528	C₂²	1	2	yes	—
7	#15689475	—	2	2	no	—
8	#15689473	—	2	2	no	—
9	#15688951	—	2	2	no	—
10	#15688948	—	2	4	no	—
11	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₄₇×#500	C₂²	1	2	yes	—
14	C₄₇×#367	—	2	2	no	—
15	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
16	#13094024	—	2	2	no	—
17	C₂×C₄₇⋊C₁₆	D₄	1	2	yes	—
18	#13093958	—	2	2	no	—
19	#13093958	—	2	2	no	—
20	#13093948	—	4	4	no	—
21	#13093947	—	4	4	no	—
22	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
23	C₄⋊Q₈×C₄₇	—	2	2	no	—
24	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
25	#107×C₄₇	—	2	2	no	—
26	#107×C₄₇	—	2	2	no	—
27	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
28	C₂×Q₈×C₄₇	—	4	2	no	—
29	C₂×Dic₉₄	—	4	2	no	—
30	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
32	C₄⋊C₄×C₄₇	—	4	2	no	—
33	C₄⋊C₄×C₄₇	—	4	2	no	—
34	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₄⋊Dic₄₇	—	4	2	no	—
36	C₄⋊Dic₄₇	—	4	2	no	—
37	C₂×C₄₇⋊C₈	D₈	1	4	yes	—
38	C₁₆×C₄₇	Q₁₆	1	4	yes	—
39	C₄₇⋊C₁₆	Q₁₆	1	4	yes	—
40	Q₈×C₄₇	—	8	4	no	—
41	C₂×C₄×C₄₇	—	8	2	no	—
42	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
43	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
44	C₂×Dic₄₇	—	8	2	no	—
45	Dic₉₄	—	8	4	no	—
46	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
47	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
48	C₄₇⋊C₈	—	2	4	no	—
49	#12522	—	47	1	no	maximal, Sylow
50	C₂²×C₄₇	#493	1	1	yes	characteristic
51	C₄×C₄₇	—	16	2	no	—
52	C₄×C₄₇	#494	1	1	yes	characteristic
53	C₄×C₄₇	#494	1	1	yes	characteristic
54	C₄×C₄₇	#508	1	4	yes	—
55	Dic₄₇	—	16	2	no	—
56	#2254	—	47	1	no	—
57	#2254	D₄₇	1	1	yes	characteristic
58	#2252	—	47	2	no	—
59	#1710	—	47	2	no	—
60	C₈×C₁₆	—	47	1	no	—
61	C₂×C₄₇	#1713	1	1	yes	characteristic
62	C₂×C₄₇	#2249	1	2	yes	—
63	#500	—	47	2	no	—
64	#500	C₂×D₄₇	1	2	yes	—
65	#487	—	94	2	no	—
66	#487	—	94	2	no	—
67	#367	—	2	2	no	—
68	#367	—	94	2	no	—
69	#366	—	94	4	no	—
70	C₄×C₁₆	—	47	1	no	—
71	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
72	C₈²	—	47	1	no	—
73	C₄₇	#12522	1	1	yes	characteristic, complemented, Sylow
74	C₄⋊Q₈	—	2	2	no	—
75	C₄⋊Q₈	—	94	2	no	—
76	C₂×C₁₆	—	47	2	no	—
77	C₂×C₁₆	C₂²⋊D₄₇	1	2	yes	—
78	#107	—	2	2	no	—
79	#107	—	2	2	no	—
80	#107	—	94	2	no	—
81	#107	—	94	2	no	—
82	#103	—	188	4	no	—
83	#103	—	188	4	no	—
84	C₄×C₈	—	47	2	no	—
85	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
86	C₂×Q₈	—	4	2	no	—
87	C₂×Q₈	—	188	2	no	—
88	C₂×C₈	—	47	4	no	—
89	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
90	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
91	C₄⋊C₄	—	4	2	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	188	2	no	—
94	C₄⋊C₄	—	188	2	no	—
95	C₄²	D₄×D₄₇	1	1	yes	characteristic
96	C₁₆	—	47	4	no	—
97	C₁₆	#10625337	1	4	yes	—
98	Q₈	—	8	4	no	—
99	Q₈	—	376	4	no	—
100	C₂×C₄	—	8	2	no	—
101	C₂×C₄	—	376	2	no	—
102	C₂×C₄	#13094020	1	1	yes	characteristic
103	C₂×C₄	D₈×D₄₇	1	2	yes	—
104	C₈	—	94	4	no	—
105	C₈	Q₁₆×D₄₇	1	2	yes	—
106	C₈	C₂×#10625337	1	2	yes	—
107	C₂²	#15689498	1	1	yes	characteristic
108	C₄	—	16	2	no	—
109	C₄	—	752	2	no	—
110	C₄	#15689501	1	1	yes	characteristic
111	C₄	#15689502	1	1	yes	characteristic
112	C₄	#15689554	1	4	yes	—
113	C₂	#20243005	1	1	yes	characteristic
114	C₂	#20245648	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245670

C₂

yes

maximal, characteristic

#20245661

C₂

yes

maximal

#20242997

C₂

yes

maximal

C₁₆×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₄₇×#2254

C₂

yes

maximal, characteristic

#15689528

C₂²

yes

—

—

—

—

—

C₄×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₈×C₄₇⋊C₈

C₂²

yes

characteristic

C₄₇×#500

C₂²

yes

—

C₄₇×#367

—

C₄×C₁₆×C₄₇

C₂²

yes

characteristic

#13094024

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

—

—

—

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₄⋊Q₈×C₄₇

—

C₂×C₁₆×C₄₇

D₄

yes

—

#107×C₄₇

—

#107×C₄₇

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

D₈

yes

—

C₁₆×C₄₇

Q₁₆

yes

—

C₄₇⋊C₁₆

Q₁₆

yes

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

C₂×Q₁₆

yes

—

C₈×C₄₇

C₂×Q₁₆

yes

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

#2254

—

#2254

D₄₇

yes

characteristic

#2252

—

#1710

—

C₈×C₁₆

—

C₂×C₄₇

#1713

yes

characteristic

C₂×C₄₇

#2249

yes

—

#500

—

#500

C₂×D₄₇

yes

—

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

C₈²

—

C₄₇

#12522

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₄₇

yes

—

—

—

—

—

—

188

—

#103

—

188

—

C₄×C₈

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

188

—

C₂×C₈

—

C₂×C₈

D₄×D₄₇

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

188

—

C₄⋊C₄

—

188

—

C₄²

D₄×D₄₇

yes

characteristic

C₁₆

—

C₁₆

#10625337

yes

—

Q₈

—

Q₈

—

376

—

100

C₂×C₄

—

101

C₂×C₄

—

376

—

102

C₂×C₄

#13094020

yes

characteristic

103

C₂×C₄

D₈×D₄₇

yes

—

104

C₈

—

105

C₈

Q₁₆×D₄₇

yes

—

106

C₈

C₂×#10625337

yes

—

107

C₂²

#15689498

yes

characteristic

108

C₄

—

109

C₄

—

752

—

110

C₄

#15689501

yes

characteristic

111

C₄

#15689502

yes

characteristic

112

C₄

#15689554

yes

—

113

C₂

#20243005

yes

characteristic

114

C₂

#20245648

yes

—

	subgroup	G/H
center	C₂²	#15689498
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#2254	C₂
socle, prime radical	C₂²×C₄₇	#493
hypercenter	#2254	D₄₇
hypocenter	C₄₇	#12522
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

subgroup

G/H

center

C₂²

#15689498

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#2254

C₂

C₂²×C₄₇

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

N	H ≅ G/N	type
73. C₄₇	#12522	semidirect

H ≅ G/N

type

73. C₄₇

#12522

semidirect

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

subgroup	index
#20245670	2
#20245661	2
#20242997	2
C₁₆×C₄₇⋊C₈	2
C₄₇×#2254	2
#12522	47

subgroup

index

#20245670

#20245661

#20242997

C₁₆×C₄₇⋊C₈

C₄₇×#2254

#12522

Subgroups — #26131018

Conjugacy classes of subgroups

Complements

Maximal subgroups