Group #26132661 — subgroups

Subgroups — #26132661

Order 12032.

310 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 163 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	#20243552	C₂	1	2	yes	maximal
2	#20241480	C₂	1	4	yes	maximal
3	C₂×C₄×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093958	C₂²	1	4	yes	—
5	C₂×#13093956	—	2	2	no	—
6	#15688989	—	2	2	no	—
7	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
9	C₄×C₁₆×C₄₇	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₄₇	—	2	4	no	—
11	#13093958	—	2	8	no	—
12	#13093956	—	4	4	no	—
13	C₄₇⋊#95	—	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	D₈	1	2	yes	—
22	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
23	C₂²×Dic₄₇	—	4	4	no	—
24	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
25	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
26	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₈×C₄₇	#41	1	4	yes	—
29	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
30	C₄²×C₄₇	Q₁₆	1	2	yes	—
31	C₄⋊Dic₄₇	—	4	4	no	—
32	C₄⋊Dic₄₇	—	4	4	no	—
33	C₁₆×C₄₇	—	2	4	no	—
34	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
35	C₂×C₄×C₄₇	—	2	4	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×SD₁₆	1	2	yes	—
40	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
41	C₂×Dic₄₇	—	8	4	no	—
42	C₂×Dic₄₇	—	8	8	no	—
43	C₈×C₄₇	#106	1	4	yes	—
44	C₈×C₄₇	#117	1	2	yes	—
45	C₈×C₄₇	#125	1	2	yes	—
46	#12889	—	47	1	no	maximal, Sylow
47	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
48	C₂²×C₄₇	#486	1	2	yes	—
49	C₂²×C₄₇	#508	1	4	yes	—
50	C₄×C₄₇	—	2	4	no	—
51	C₄×C₄₇	#382	1	1	yes	characteristic
52	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
53	C₄×C₄₇	#487	1	2	yes	—
54	Dic₄₇	—	16	4	no	—
55	C₂×C₄×C₁₆	D₄₇	1	1	yes	characteristic
56	#1846	—	47	2	no	—
57	#1380	—	47	4	no	—
58	C₂×C₄₇	#1834	1	1	yes	characteristic
59	C₂×C₄₇	#2158	1	2	yes	—
60	C₂×C₄₇	#2242	1	4	yes	—
61	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
62	C₂×#107	—	47	4	no	—
63	C₂×#106	—	94	2	no	—
64	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
65	#382	—	94	2	no	—
66	C₄×C₁₆	Dic₄₇	1	4	yes	complemented
67	C₄₇	#12889	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
69	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
70	C₂×C₄⋊C₄	—	94	4	no	—
71	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
72	C₂×C₁₆	—	2	4	no	—
73	C₂×C₁₆	C₂×Dic₄₇	1	4	yes	—
74	#107	—	94	8	no	—
75	#106	—	188	4	no	—
76	C₄×C₈	Dic₉₄	1	2	yes	—
77	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
78	#95	—	188	4	no	—
79	C₂²×C₄	—	188	4	no	—
80	C₂²×C₄	D₃₇₆	1	2	yes	—
81	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
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	C₄⋊C₄	—	188	4	no	—
88	C₄⋊C₄	—	188	4	no	—
89	C₄²	Dic₁₈₈	1	2	yes	—
90	C₄²	C₄⋊Dic₄₇	1	2	yes	—
91	C₁₆	—	2	4	no	—
92	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	376	8	no	—
95	C₂×C₄	—	376	4	no	—
96	C₂×C₄	#13093958	1	4	yes	—
97	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
98	C₂×C₄	#13094012	1	2	yes	—
99	C₂×C₄	C₂×#10625325	1	2	yes	—
100	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
101	C₈	#13093956	1	4	yes	—
102	C₈	#13093983	1	2	yes	—
103	C₈	#13094016	1	2	yes	—
104	C₂²	C₂×#13093958	1	1	yes	characteristic
105	C₂²	#15689470	1	2	yes	—
106	C₂²	#15689555	1	4	yes	—
107	C₄	—	2	4	no	—
108	C₄	—	752	4	no	—
109	C₄	#15688989	1	1	yes	characteristic
110	C₄	C₂×#13093956	1	1	yes	characteristic
111	C₄	#15689474	1	2	yes	—
112	C₂	#20243516	1	1	yes	characteristic
113	C₂	#20245172	1	2	yes	—
114	C₂	#20245620	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243552

C₂

yes

maximal

#20241480

C₂

yes

maximal

C₂×C₄×C₁₆×C₄₇

C₂

yes

maximal, characteristic

C₂×#13093958

C₂²

yes

—

C₂×#13093956

—

#15688989

—

C₂²×C₁₆×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄×C₁₆×C₄₇

C₄

yes

complemented

C₂×C₄⋊Dic₄₇

—

#13093958

—

#13093956

—

C₄₇⋊#95

—

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

D₈

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

—

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

Q₁₆

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₁₆×C₄₇

—

C₂³×C₄₇

C₂×D₈

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×SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

#106

yes

—

C₈×C₄₇

#117

yes

—

C₈×C₄₇

#125

yes

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#508

yes

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

C₂×C₄×C₁₆

D₄₇

yes

characteristic

#1846

—

#1380

—

C₂×C₄₇

#1834

yes

characteristic

C₂×C₄₇

#2158

yes

—

C₂×C₄₇

#2242

yes

—

C₂²×C₁₆

C₂×D₄₇

yes

—

C₂×#107

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#382

—

C₄×C₁₆

Dic₄₇

yes

complemented

C₄₇

#12889

yes

characteristic, complemented, Sylow

C₂²×C₈

D₁₈₈

yes

characteristic

C₂²×C₈

C₂²×D₄₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

D₁₈₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₄₇

yes

—

#107

—

#106

—

188

—

C₄×C₈

Dic₉₄

yes

—

C₄×C₈

C₂×Dic₄₇

yes

—

#95

—

188

—

C₂²×C₄

—

188

—

C₂²×C₄

D₃₇₆

yes

—

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

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

C₄⋊C₄

—

188

—

C₄⋊C₄

—

188

—

C₄²

Dic₁₈₈

yes

—

C₄²

C₄⋊Dic₄₇

yes

—

C₁₆

—

C₂³

C₂×D₃₇₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

376

—

C₂×C₄

—

376

—

C₂×C₄

#13093958

yes

—

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

characteristic

C₂×C₄

#13094012

yes

—

C₂×C₄

C₂×#10625325

yes

—

100

C₂×C₄

C₂×Dic₁₈₈

yes

characteristic

101

C₈

#13093956

yes

—

102

C₈

#13093983

yes

—

103

C₈

#13094016

yes

—

104

C₂²

C₂×#13093958

yes

characteristic

105

C₂²

#15689470

yes

—

106

C₂²

#15689555

yes

—

107

C₄

—

108

C₄

—

752

—

109

C₄

#15688989

yes

characteristic

110

C₄

C₂×#13093956

yes

characteristic

111

C₄

#15689474

yes

—

112

C₂

#20243516

yes

characteristic

113

C₂

#20245172

yes

—

114

C₂

#20245620

yes

—

	subgroup	G/H
center	C₂³	C₂×D₃₇₆
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄×C₁₆×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₄₇
hypocenter	C₄₇	#12889
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂³

C₂×D₃₇₆

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₂×C₄×C₁₆×C₄₇

C₂

socle, prime radical

C₂³×C₄₇

C₂×D₈

cosocle

C₂²×C₈×C₄₇

C₂³

hypercenter

C₂×C₄×C₁₆

D₄₇

hypocenter

C₄₇

#12889

LCS

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
9. C₄×C₁₆×C₄₇	C₄	semidirect
66. C₄×C₁₆	Dic₄₇	semidirect
67. C₄₇	#12889	semidirect

H ≅ G/N

type

9. C₄×C₁₆×C₄₇

C₄

semidirect

66. C₄×C₁₆

Dic₄₇

semidirect

67. C₄₇

#12889

semidirect

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

subgroup	index
#20243552	2
#20241480	2
C₂×C₄×C₁₆×C₄₇	2
#12889	47

subgroup

index

#20243552

#20241480

C₂×C₄×C₁₆×C₄₇

#12889

Subgroups — #26132661

Conjugacy classes of subgroups

Complements

Maximal subgroups