Group #26137264 — subgroups

Subgroups — #26137264

Order 12032.

194 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 67 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	#20245667	C₂	1	1	yes	maximal, characteristic, complemented
2	#20241566	C₂	1	4	yes	maximal, complemented
3	#20241522	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#2246	C₂	1	1	yes	maximal, characteristic
5	#15689525	—	2	1	no	—
6	C₂×C₄₇⋊C₃₂	C₂²	1	2	yes	—
7	#15688951	—	2	1	no	—
8	#15688949	C₂²	1	2	yes	—
9	C₂×D₁₆×C₄₇	—	2	1	no	—
10	C₂×D₁₆×C₄₇	C₂²	1	2	yes	—
11	C₄₇×#493	—	2	1	no	—
12	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
13	#13094016	—	4	1	no	—
14	#13093958	—	2	2	no	—
15	#13093956	—	4	1	no	—
16	C₄₇⋊C₃₂	—	2	2	no	—
17	C₂×D₈×C₄₇	—	2	2	no	—
18	C₂×D₈×C₄₇	—	4	1	no	—
19	#127×C₄₇	—	4	1	no	—
20	D₁₆×C₄₇	—	2	4	no	—
21	D₁₆×C₄₇	—	4	2	no	—
22	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
25	C₂×D₄×C₄₇	—	4	2	no	—
26	C₂×D₄×C₄₇	—	8	2	no	—
27	D₈×C₄₇	—	4	2	no	—
28	D₈×C₄₇	—	4	2	no	—
29	D₈×C₄₇	—	8	2	no	—
30	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
31	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
32	C₄²×C₄₇	D₈	1	1	yes	characteristic
33	C₄⋊Dic₄₇	—	4	2	no	—
34	Dic₄₇⋊C₄	—	8	2	no	—
35	C₁₆×C₄₇	SD₁₆	1	2	yes	—
36	C₁₆×C₄₇	#41	1	2	yes	—
37	C₂³×C₄₇	—	8	2	no	—
38	D₄×C₄₇	—	8	2	no	—
39	D₄×C₄₇	—	8	2	no	—
40	D₄×C₄₇	—	16	4	no	—
41	C₂×C₄×C₄₇	D₁₆	1	2	yes	—
42	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
43	C₂×Dic₄₇	—	8	2	no	—
44	C₈×C₄₇	SD₃₂	1	2	yes	—
45	C₈×C₄₇	#124	1	1	yes	characteristic
46	C₈×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
47	#13723	—	47	1	no	maximal, Sylow
48	C₂²×C₄₇	—	16	2	no	—
49	C₂²×C₄₇	—	16	4	no	—
50	C₂²×C₄₇	C₂×D₁₆	1	1	yes	characteristic
51	C₄×C₄₇	—	2	2	no	—
52	C₄×C₄₇	#486	1	1	yes	characteristic
53	C₄×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
54	Dic₄₇	—	16	2	no	—
55	#2253	—	47	1	no	—
56	#2246	D₄₇	1	1	yes	characteristic
57	#1415	—	47	4	no	—
58	#1397	—	47	1	no	—
59	C₂×C₄₇	—	32	2	no	—
60	C₂×C₄₇	#2240	1	1	yes	characteristic
61	C₂×C₄₇	#2263	1	2	yes	—
62	C₂×D₁₆	—	2	1	no	—
63	C₂×D₁₆	C₂×D₄₇	1	2	yes	—
64	#499	—	94	1	no	—
65	#493	—	2	1	no	—
66	C₂×C₃₂	—	47	2	no	—
67	#367	—	94	1	no	—
68	#366	—	47	2	no	—
69	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
70	C₄₇	#13723	1	1	yes	characteristic, complemented, Sylow
71	C₂×D₈	—	2	2	no	—
72	C₂×D₈	—	4	1	no	—
73	#127	—	4	1	no	—
74	#125	—	188	1	no	—
75	D₁₆	—	2	4	no	—
76	D₁₆	—	4	2	no	—
77	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
78	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
79	#107	—	94	2	no	—
80	#106	—	188	1	no	—
81	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
82	C₃₂	—	94	2	no	—
83	C₂×D₄	—	4	2	no	—
84	C₂×D₄	—	8	2	no	—
85	D₈	—	4	2	no	—
86	D₈	—	4	2	no	—
87	D₈	—	8	2	no	—
88	C₂×C₈	#10625334	1	1	yes	characteristic
89	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
90	C₄⋊C₄	—	376	2	no	—
91	C₄⋊C₄	—	188	2	no	—
92	C₄²	#10625334	1	1	yes	characteristic
93	C₁₆	#10625335	1	2	yes	—
94	C₁₆	#10625352	1	2	yes	—
95	C₂³	—	8	2	no	—
96	D₄	—	8	2	no	—
97	D₄	—	8	2	no	—
98	D₄	—	16	4	no	—
99	C₂×C₄	—	376	2	no	—
100	C₂×C₄	#13093965	1	2	yes	—
101	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
102	C₈	#13093968	1	2	yes	—
103	C₈	#13094011	1	1	yes	characteristic
104	C₈	C₂×#10625335	1	1	yes	characteristic
105	C₂²	—	16	2	no	—
106	C₂²	—	16	4	no	—
107	C₂²	C₂×#13093965	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	752	2	no	—
110	C₄	#15689469	1	1	yes	characteristic
111	C₄	C₂×#13093968	1	1	yes	characteristic
112	C₂	—	32	2	no	—
113	C₂	#20245611	1	1	yes	characteristic
114	C₂	#20245701	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245667

C₂

yes

maximal, characteristic, complemented

#20241566

C₂

yes

maximal, complemented

#20241522

C₂

yes

maximal, characteristic, complemented

C₄₇×#2246

C₂

yes

maximal, characteristic

#15689525

—

C₂×C₄₇⋊C₃₂

C₂²

yes

—

#15688951

—

#15688949

C₂²

yes

—

C₂×D₁₆×C₄₇

—

C₂×D₁₆×C₄₇

C₂²

yes

—

C₄₇×#493

—

C₄×C₁₆×C₄₇

C₂²

yes

—

—

—

C₄₇⋊C₃₂

—

C₂×D₈×C₄₇

—

C₂×D₈×C₄₇

—

#127×C₄₇

—

D₁₆×C₄₇

—

D₁₆×C₄₇

—

C₂×C₁₆×C₄₇

D₄

yes

characteristic

C₂×C₁₆×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

D₈×C₄₇

—

D₈×C₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

D₈

yes

characteristic

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₁₆×C₄₇

SD₁₆

yes

—

C₁₆×C₄₇

#41

yes

—

C₂³×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

D₁₆

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×Dic₄₇

—

C₈×C₄₇

SD₃₂

yes

—

C₈×C₄₇

#124

yes

characteristic

C₈×C₄₇

C₂×SD₁₆

yes

characteristic

#13723

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×D₁₆

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

#486

yes

characteristic

C₄×C₄₇

C₂×SD₃₂

yes

characteristic

Dic₄₇

—

#2253

—

#2246

D₄₇

yes

characteristic

#1415

—

#1397

—

C₂×C₄₇

—

C₂×C₄₇

#2240

yes

characteristic

C₂×C₄₇

#2263

yes

—

C₂×D₁₆

—

C₂×D₁₆

C₂×D₄₇

yes

—

#499

—

#493

—

C₂×C₃₂

—

#367

—

#366

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

C₄₇

#13723

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

#127

—

#125

—

188

—

D₁₆

—

D₁₆

—

C₂×C₁₆

C₂²⋊D₄₇

yes

characteristic

C₂×C₁₆

C₂²×D₄₇

yes

characteristic

#107

—

#106

—

188

—

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

C₃₂

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

D₈

—

C₂×C₈

#10625334

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

C₄⋊C₄

—

376

—

C₄⋊C₄

—

188

—

C₄²

#10625334

yes

characteristic

C₁₆

#10625335

yes

—

C₁₆

#10625352

yes

—

C₂³

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

376

—

100

C₂×C₄

#13093965

yes

—

101

C₂×C₄

C₂×#10625334

yes

characteristic

102

C₈

#13093968

yes

—

103

C₈

#13094011

yes

characteristic

104

C₈

C₂×#10625335

yes

characteristic

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×#13093965

yes

characteristic

108

C₄

—

109

C₄

—

752

—

110

C₄

#15689469

yes

characteristic

111

C₄

C₂×#13093968

yes

characteristic

112

C₂

—

113

C₂

#20245611

yes

characteristic

114

C₂

#20245701

yes

—

	subgroup	G/H
center	C₂²	C₂×#13093965
commutator, cosocle	C₂×C₁₆×C₄₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₄₇
Fitting, prime radical	C₄₇×#2246	C₂
socle	C₂²×C₄₇	C₂×D₁₆
hypercenter	#2246	D₄₇
hypocenter	C₄₇	#13723
LCS	G ▷ C₂×C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2246

subgroup

G/H

center

C₂²

C₂×#13093965

commutator, cosocle

C₂×C₁₆×C₄₇

C₂³

Frattini

C₂×C₁₆

C₂²×D₄₇

Fitting, prime radical

C₄₇×#2246

C₂

socle

C₂²×C₄₇

C₂×D₁₆

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2246

N	H ≅ G/N	type
1. #20245667	C₂	semidirect
2. #20241566	C₂	semidirect
3. #20241522	C₂	semidirect
70. C₄₇	#13723	semidirect

H ≅ G/N

type

1. #20245667

C₂

semidirect

2. #20241566

C₂

semidirect

3. #20241522

C₂

semidirect

70. C₄₇

#13723

semidirect

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

subgroup	index
#20245667	2
#20241566	2
#20241522	2
C₄₇×#2246	2
#13723	47

subgroup

index

C₄₇×#2246

Subgroups — #26137264

Conjugacy classes of subgroups

Complements

Maximal subgroups