Group #46836406 — subgroups

Subgroups — #46836406

Order 34688.

272 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×#22576096	C₂	1	2	yes	maximal, complemented
2	#31773861	C₂	1	2	yes	maximal
3	#31773662	C₂	1	2	yes	maximal, complemented
4	C₂²×M₁₆×C₂₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#16864638	C₂²	1	4	yes	—
6	#22576096	—	2	4	no	—
7	C₂²⋊(C₂₇₁⋊C₈)	—	2	4	no	—
8	C₂³×C₄×C₂₇₁	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₂₇₁	—	2	4	no	—
10	C₂×M₁₆×C₂₇₁	C₄	1	4	yes	—
11	C₂²×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
13	#16864638	—	2	4	no	—
14	#16864638	—	2	4	no	—
15	C₂×C₂₇₁⋊C₈	—	2	4	no	—
16	C₂⁴×C₂₇₁	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₂₇₁	—	2	2	no	—
18	C₂²×C₄×C₂₇₁	—	2	2	no	—
19	C₂²×C₄×C₂₇₁	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₂₇₁	D₄	1	2	yes	—
21	C₂²×C₄×C₂₇₁	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
23	M₁₆×C₂₇₁	—	2	8	no	—
24	C₂×C₈×C₂₇₁	—	2	2	no	—
25	C₂×C₈×C₂₇₁	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₂₇₁	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₂₇₁	D₄	1	4	yes	—
28	C₂₇₁⋊C₈	—	4	4	no	—
29	C₂³×C₂₇₁	—	2	2	no	—
30	C₂³×C₂₇₁	—	4	2	no	—
31	C₂³×C₂₇₁	C₄⋊C₄	1	2	yes	—
32	C₂³×C₂₇₁	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₂₇₁	—	2	2	no	—
34	C₂×C₄×C₂₇₁	—	2	2	no	—
35	C₂×C₄×C₂₇₁	—	2	2	no	—
36	C₂×C₄×C₂₇₁	—	2	4	no	—
37	C₂×C₄×C₂₇₁	C₄⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₇₁	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₇₁	#41	1	2	yes	—
42	C₈×C₂₇₁	—	2	2	no	—
43	C₈×C₂₇₁	C₂²⋊C₄	1	4	yes	—
44	C₂²×C₂₇₁	—	2	2	no	—
45	C₂²×C₂₇₁	—	2	2	no	—
46	C₂²×C₂₇₁	—	2	4	no	—
47	C₂²×C₂₇₁	—	4	4	no	—
48	C₂²×C₂₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₂₇₁	C₂²⋊Q₈	1	2	yes	—
50	C₄×C₂₇₁	—	2	2	no	—
51	C₄×C₂₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₄×C₂₇₁	#117	1	1	yes	characteristic
53	C₄×C₂₇₁	#121	1	2	yes	—
54	C₂×C₂₇₁	—	2	2	no	—
55	C₂×C₂₇₁	—	4	2	no	—
56	C₂×C₂₇₁	#380	1	1	yes	characteristic
57	C₂×C₂₇₁	#430	1	2	yes	—
58	C₂₇₁	#1811	1	1	yes	characteristic, complemented, Sylow
59	#1811	—	271	1	no	maximal, Sylow
60	C₂²×M₁₆	D₂₇₁	1	1	yes	characteristic
61	C₂×#108	—	271	2	no	—
62	#407	—	271	2	no	—
63	#341	—	271	2	no	—
64	C₂³×C₄	C₂×D₂₇₁	1	1	yes	characteristic
65	C₂×M₁₆	—	2	4	no	—
66	C₂×M₁₆	—	271	4	no	—
67	C₂×M₁₆	Dic₂₇₁	1	4	yes	—
68	C₂²×C₈	C₂×D₂₇₁	1	1	yes	characteristic
69	C₂²×C₈	C₂×D₂₇₁	1	1	yes	characteristic
70	#108	—	542	4	no	—
71	C₂²⋊C₈	—	542	4	no	—
72	C₂⁴	Dic₅₄₂	1	1	yes	characteristic
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	C₂×Dic₂₇₁	1	2	yes	—
76	C₂²×C₄	D₁₀₈₄	1	2	yes	—
77	C₂²×C₄	Dic₅₄₂	1	1	yes	characteristic
78	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
79	M₁₆	—	2	8	no	—
80	M₁₆	—	542	4	no	—
81	M₁₆	—	542	4	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	542	4	no	—
84	C₂×C₈	C₂×Dic₂₇₁	1	2	yes	—
85	C₂×C₈	C₂×Dic₂₇₁	1	2	yes	—
86	C₂×C₈	C₂²⋊D₂₇₁	1	4	yes	—
87	C₂³	—	2	2	no	—
88	C₂³	—	4	2	no	—
89	C₂³	C₄⋊Dic₂₇₁	1	2	yes	—
90	C₂³	C₂×Dic₅₄₂	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	C₄⋊Dic₂₇₁	1	2	yes	—
96	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
97	C₂×C₄	C₂×C₂²⋊D₂₇₁	1	2	yes	—
98	C₂×C₄	C₂×D₁₀₈₄	1	1	yes	characteristic
99	C₂×C₄	#16864654	1	2	yes	—
100	C₈	—	2	2	no	—
101	C₈	—	1084	4	no	—
102	C₈	C₂²⋊Dic₂₇₁	1	4	yes	—
103	C₂²	—	2	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	—	2	4	no	—
106	C₂²	—	4	4	no	—
107	C₂²	C₂×C₄⋊Dic₂₇₁	1	1	yes	characteristic
108	C₂²	#22576139	1	2	yes	—
109	C₄	—	2	2	no	—
110	C₄	C₂×C₂²⋊Dic₂₇₁	1	1	yes	characteristic
111	C₄	#22576119	1	1	yes	characteristic
112	C₄	#22576134	1	2	yes	—
113	C₂	—	2	2	no	—
114	C₂	—	4	2	no	—
115	C₂	#31773757	1	1	yes	characteristic
116	C₂	#31773946	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22576096

C₂

yes

maximal, complemented

#31773861

C₂

yes

maximal

#31773662

C₂

yes

maximal, complemented

C₂²×M₁₆×C₂₇₁

C₂

yes

maximal, characteristic

C₂×#16864638

C₂²

yes

—

#22576096

—

C₂²⋊(C₂₇₁⋊C₈)

—

C₂³×C₄×C₂₇₁

C₂²

yes

characteristic

C₂×M₁₆×C₂₇₁

—

C₂×M₁₆×C₂₇₁

C₄

yes

—

C₂²×C₈×C₂₇₁

C₂²

yes

characteristic

C₂²×C₈×C₂₇₁

C₂²

yes

characteristic

#16864638

—

#16864638

—

C₂×C₂₇₁⋊C₈

—

C₂⁴×C₂₇₁

Q₈

yes

characteristic

C₂²×C₄×C₂₇₁

—

C₂²×C₄×C₂₇₁

—

C₂²×C₄×C₂₇₁

C₂×C₄

yes

—

C₂²×C₄×C₂₇₁

D₄

yes

—

C₂²×C₄×C₂₇₁

Q₈

yes

characteristic

C₂²×C₄×C₂₇₁

C₂³

yes

characteristic

M₁₆×C₂₇₁

—

C₂×C₈×C₂₇₁

—

C₂×C₈×C₂₇₁

C₂×C₄

yes

—

C₂×C₈×C₂₇₁

C₂×C₄

yes

—

C₂×C₈×C₂₇₁

D₄

yes

—

C₂₇₁⋊C₈

—

C₂³×C₂₇₁

—

C₂³×C₂₇₁

—

C₂³×C₂₇₁

C₄⋊C₄

yes

—

C₂³×C₂₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₁

C₂×D₄

yes

—

C₂×C₄×C₂₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₁

#41

yes

—

C₈×C₂₇₁

—

C₈×C₂₇₁

C₂²⋊C₄

yes

—

C₂²×C₂₇₁

—

C₂²×C₂₇₁

—

C₂²×C₂₇₁

—

C₂²×C₂₇₁

—

C₂²×C₂₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₇₁

C₂²⋊Q₈

yes

—

C₄×C₂₇₁

—

C₄×C₂₇₁

C₂×C₂²⋊C₄

yes

C₄×C₂₇₁

yes

C₄×C₂₇₁

yes

—

C₂×C₂₇₁

—

C₂×C₂₇₁

—

C₂×C₂₇₁

#380

yes

characteristic

C₂×C₂₇₁

#430

yes

—

C₂₇₁

#1811

yes

characteristic, complemented, Sylow

#1811

—

271

maximal, Sylow

C₂²×M₁₆

D₂₇₁

yes

characteristic

C₂×#108

—

271

—

#407

—

271

—

#341

—

271

—

C₂³×C₄

C₂×D₂₇₁

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

271

—

C₂×M₁₆

Dic₂₇₁

yes

—

C₂²×C₈

C₂×D₂₇₁

yes

characteristic

C₂²×C₈

C₂×D₂₇₁

yes

characteristic

#108

—

542

—

C₂²⋊C₈

—

542

—

C₂⁴

Dic₅₄₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₂₇₁

yes

—

C₂²×C₄

D₁₀₈₄

yes

—

C₂²×C₄

Dic₅₄₂

yes

characteristic

C₂²×C₄

C₂²×D₂₇₁

yes

characteristic

M₁₆

—

M₁₆

—

542

—

M₁₆

—

542

—

C₂×C₈

—

C₂×C₈

—

542

—

C₂×C₈

C₂×Dic₂₇₁

yes

—

C₂×C₈

C₂×Dic₂₇₁

yes

—

C₂×C₈

C₂²⋊D₂₇₁

yes

—

C₂³

—

C₂³

—

C₂³

C₄⋊Dic₂₇₁

yes

—

C₂³

C₂×Dic₅₄₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₂₇₁

yes

—

C₂×C₄

C₂²×Dic₂₇₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₇₁

yes

—

C₂×C₄

C₂×D₁₀₈₄

yes

characteristic

C₂×C₄

#16864654

yes

—

100

C₈

—

101

C₈

—

1084

—

102

C₈

C₂²⋊Dic₂₇₁

yes

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×C₄⋊Dic₂₇₁

yes

characteristic

108

C₂²

#22576139

yes

—

109

C₄

—

110

C₄

C₂×C₂²⋊Dic₂₇₁

yes

characteristic

111

C₄

#22576119

yes

characteristic

112

C₄

#22576134

yes

—

113

C₂

—

114

C₂

—

115

C₂

#31773757

yes

characteristic

116

C₂

#31773946

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₀₈₄
commutator	C₂×C₄×C₂₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₇₁
Fitting	C₂²×M₁₆×C₂₇₁	C₂
socle	C₂²×C₂₇₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₂₇₁	Q₈
cosocle	C₂²×C₄×C₂₇₁	C₂³
hypercenter	C₂²×M₁₆	D₂₇₁
hypocenter	C₂₇₁	#1811
LCS	G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₁₀₈₄

commutator

C₂×C₄×C₂₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₇₁

Fitting

C₂²×M₁₆×C₂₇₁

C₂

socle

C₂²×C₂₇₁

C₂×C₄⋊C₄

prime radical

C₂⁴×C₂₇₁

Q₈

cosocle

C₂²×C₄×C₂₇₁

C₂³

hypercenter

C₂²×M₁₆

D₂₇₁

hypocenter

C₂₇₁

#1811

LCS

G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

N	H ≅ G/N	type
1. C₂×#22576096	C₂	semidirect
3. #31773662	C₂	semidirect
58. C₂₇₁	#1811	semidirect

H ≅ G/N

type

1. C₂×#22576096

C₂

semidirect

3. #31773662

C₂

semidirect

58. C₂₇₁

#1811

semidirect

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

subgroup	index
C₂×#22576096	2
#31773861	2
#31773662	2
C₂²×M₁₆×C₂₇₁	2
#1811	271

subgroup

index

C₂²×M₁₆×C₂₇₁

271

Subgroups — #46836406

Conjugacy classes of subgroups

Complements

Maximal subgroups