Group #46837972 — subgroups

Subgroups — C₂×C₂²⋊(C₂₇₁⋊C₁₆) #46837972

Order 34688.

272 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₂²×C₂₇₁⋊C₁₆	C₂	1	2	yes	maximal, complemented
2	C₂²⋊(C₂₇₁⋊C₁₆)	C₂	1	4	yes	maximal, direct product factor
3	C₂³×C₈×C₂₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₇₁⋊C₁₆	—	2	4	no	—
5	C₂×C₂₇₁⋊C₁₆	C₂²	1	4	yes	complemented
6	C₂³×C₄×C₂₇₁	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₂₇₁	—	2	4	no	—
8	C₂²×C₈×C₂₇₁	C₄	1	1	yes	characteristic
9	C₂²×C₈×C₂₇₁	C₄	1	2	yes	—
10	C₂²×C₈×C₂₇₁	C₂²	1	2	yes	—
11	C₂²×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
12	C₂₇₁⋊C₁₆	—	2	4	no	—
13	C₂⁴×C₂₇₁	C₈	1	1	yes	characteristic
14	C₂²×C₄×C₂₇₁	—	2	2	no	—
15	C₂²×C₄×C₂₇₁	—	2	2	no	—
16	C₂²×C₄×C₂₇₁	C₈	1	1	yes	characteristic
17	C₂²×C₄×C₂₇₁	C₈	1	2	yes	—
18	C₂²×C₄×C₂₇₁	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₂₇₁	—	2	2	no	—
21	C₂×C₈×C₂₇₁	—	2	4	no	—
22	C₂×C₈×C₂₇₁	—	2	4	no	—
23	C₂×C₈×C₂₇₁	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₂₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₂₇₁	D₄	1	4	yes	—
26	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
27	C₂³×C₂₇₁	—	2	2	no	—
28	C₂³×C₂₇₁	—	2	2	no	—
29	C₂³×C₂₇₁	C₁₆	1	4	yes	complemented
30	C₂³×C₂₇₁	C₂×C₈	1	2	yes	—
31	C₂³×C₂₇₁	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₂₇₁	—	2	2	no	—
33	C₂×C₄×C₂₇₁	—	2	4	no	—
34	C₂×C₄×C₂₇₁	—	2	2	no	—
35	C₂×C₄×C₂₇₁	—	2	2	no	—
36	C₂×C₄×C₂₇₁	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₂₇₁	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₂₇₁	C₂×C₈	1	1	yes	characteristic
39	C₂×C₄×C₂₇₁	M₁₆	1	2	yes	—
40	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
41	C₈×C₂₇₁	—	2	2	no	—
42	C₈×C₂₇₁	C₂²⋊C₄	1	2	yes	—
43	C₈×C₂₇₁	C₂×D₄	1	2	yes	—
44	C₂²×C₂₇₁	—	2	8	no	—
45	C₂²×C₂₇₁	—	2	2	no	—
46	C₂²×C₂₇₁	—	2	2	no	—
47	C₂²×C₂₇₁	C₂²⋊C₈	1	2	yes	—
48	C₂²×C₂₇₁	C₂×C₁₆	1	2	yes	—
49	C₂²×C₂₇₁	C₂×C₁₆	1	4	yes	complemented
50	C₂²×C₂₇₁	M₃₂	1	2	yes	—
51	C₂²×C₂₇₁	C₂²×C₈	1	1	yes	characteristic
52	C₄×C₂₇₁	—	2	2	no	—
53	C₄×C₂₇₁	C₂²⋊C₈	1	2	yes	—
54	C₄×C₂₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₄×C₂₇₁	C₂×M₁₆	1	1	yes	characteristic
56	C₂×C₂₇₁	—	2	4	no	—
57	C₂×C₂₇₁	C₂²⋊C₁₆	1	4	yes	complemented
58	C₂×C₂₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₂₇₁	C₂²×C₁₆	1	1	yes	characteristic
60	C₂×C₂₇₁	C₂×M₃₂	1	1	yes	characteristic
61	C₂₇₁	C₂×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
62	C₂×C₂²⋊C₁₆	—	271	1	no	maximal, Sylow
63	C₂³×C₈	D₂₇₁	1	1	yes	characteristic
64	C₂²×C₁₆	—	271	2	no	—
65	C₂²⋊C₁₆	—	271	4	no	—
66	C₂³×C₄	Dic₂₇₁	1	1	yes	characteristic
67	C₂²×C₈	—	2	4	no	—
68	C₂²×C₈	Dic₂₇₁	1	1	yes	characteristic
69	C₂²×C₈	Dic₂₇₁	1	2	yes	—
70	C₂²×C₈	C₂×D₂₇₁	1	2	yes	—
71	C₂²×C₈	C₂×D₂₇₁	1	1	yes	characteristic
72	C₂×C₁₆	—	271	4	no	—
73	C₂×C₁₆	—	542	4	no	—
74	C₂⁴	C₂₇₁⋊C₈	1	1	yes	characteristic
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	—	2	2	no	—
77	C₂²×C₄	C₂₇₁⋊C₈	1	1	yes	characteristic
78	C₂²×C₄	C₂₇₁⋊C₈	1	2	yes	—
79	C₂²×C₄	C₂×Dic₂₇₁	1	2	yes	—
80	C₂²×C₄	C₂×Dic₂₇₁	1	1	yes	characteristic
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	—	2	4	no	—
84	C₂×C₈	C₂×Dic₂₇₁	1	2	yes	—
85	C₂×C₈	C₂×Dic₂₇₁	1	1	yes	characteristic
86	C₂×C₈	C₂²⋊D₂₇₁	1	4	yes	—
87	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
88	C₁₆	—	542	4	no	—
89	C₂³	—	2	2	no	—
90	C₂³	—	2	2	no	—
91	C₂³	C₂₇₁⋊C₁₆	1	4	yes	complemented
92	C₂³	C₂×C₂₇₁⋊C₈	1	2	yes	—
93	C₂³	C₂×C₂₇₁⋊C₈	1	1	yes	characteristic
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	C₂²⋊Dic₂₇₁	1	2	yes	—
99	C₂×C₄	C₂×C₂₇₁⋊C₈	1	2	yes	—
100	C₂×C₄	C₂×C₂₇₁⋊C₈	1	1	yes	characteristic
101	C₂×C₄	#16864638	1	2	yes	—
102	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
103	C₈	—	2	2	no	—
104	C₈	C₂²⋊Dic₂₇₁	1	2	yes	—
105	C₈	C₂×C₂²⋊D₂₇₁	1	2	yes	—
106	C₂²	—	2	2	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	2	8	no	—
109	C₂²	C₂²⋊(C₂₇₁⋊C₈)	1	2	yes	—
110	C₂²	C₂×C₂₇₁⋊C₁₆	1	2	yes	—
111	C₂²	C₂×C₂₇₁⋊C₁₆	1	4	yes	complemented
112	C₂²	#22576100	1	2	yes	—
113	C₂²	C₂²×C₂₇₁⋊C₈	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	C₂²⋊(C₂₇₁⋊C₈)	1	2	yes	—
116	C₄	C₂×C₂²⋊Dic₂₇₁	1	1	yes	characteristic
117	C₄	C₂×#16864638	1	1	yes	characteristic
118	C₂	—	2	4	no	—
119	C₂	C₂²⋊(C₂₇₁⋊C₁₆)	1	4	yes	direct product factor
120	C₂	C₂×C₂²⋊(C₂₇₁⋊C₈)	1	1	yes	characteristic
121	C₂	C₂²×C₂₇₁⋊C₁₆	1	1	yes	characteristic
122	C₂	C₂×#22576100	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₂₇₁⋊C₁₆

C₂

yes

maximal, complemented

C₂²⋊(C₂₇₁⋊C₁₆)

C₂

yes

maximal, direct product factor

C₂³×C₈×C₂₇₁

C₂

yes

maximal, characteristic

C₂×C₂₇₁⋊C₁₆

—

C₂×C₂₇₁⋊C₁₆

C₂²

yes

complemented

C₂³×C₄×C₂₇₁

C₄

yes

characteristic

C₂²×C₈×C₂₇₁

—

C₂²×C₈×C₂₇₁

C₄

yes

characteristic

C₂²×C₈×C₂₇₁

C₄

yes

—

C₂²×C₈×C₂₇₁

C₂²

yes

—

C₂²×C₈×C₂₇₁

C₂²

yes

characteristic

C₂₇₁⋊C₁₆

—

C₂⁴×C₂₇₁

C₈

yes

characteristic

C₂²×C₄×C₂₇₁

—

C₂²×C₄×C₂₇₁

—

C₂²×C₄×C₂₇₁

C₈

yes

characteristic

C₂²×C₄×C₂₇₁

C₈

yes

—

C₂²×C₄×C₂₇₁

C₂×C₄

yes

—

C₂²×C₄×C₂₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₇₁

—

C₂×C₈×C₂₇₁

—

C₂×C₈×C₂₇₁

—

C₂×C₈×C₂₇₁

C₂×C₄

yes

—

C₂×C₈×C₂₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₇₁

D₄

yes

—

C₂×C₈×C₂₇₁

C₂³

yes

characteristic

C₂³×C₂₇₁

—

C₂³×C₂₇₁

—

C₂³×C₂₇₁

C₁₆

yes

complemented

C₂³×C₂₇₁

C₂×C₈

yes

—

C₂³×C₂₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₇₁

C₂×C₈

yes

—

C₂×C₄×C₂₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₂₇₁

M₁₆

yes

—

C₂×C₄×C₂₇₁

C₂²×C₄

yes

characteristic

C₈×C₂₇₁

—

C₈×C₂₇₁

C₂²⋊C₄

yes

—

C₈×C₂₇₁

C₂×D₄

yes

—

C₂²×C₂₇₁

—

C₂²×C₂₇₁

—

C₂²×C₂₇₁

—

C₂²×C₂₇₁

C₂²⋊C₈

yes

—

C₂²×C₂₇₁

C₂×C₁₆

yes

—

C₂²×C₂₇₁

C₂×C₁₆

yes

complemented

C₂²×C₂₇₁

M₃₂

yes

—

C₂²×C₂₇₁

C₂²×C₈

yes

characteristic

C₄×C₂₇₁

—

C₄×C₂₇₁

C₂²⋊C₈

yes

—

C₄×C₂₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₇₁

C₂×M₁₆

yes

characteristic

C₂×C₂₇₁

—

C₂×C₂₇₁

C₂²⋊C₁₆

yes

complemented

C₂×C₂₇₁

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₂₇₁

C₂²×C₁₆

yes

characteristic

C₂×C₂₇₁

C₂×M₃₂

yes

characteristic

C₂₇₁

C₂×C₂²⋊C₁₆

yes

characteristic, complemented, Sylow

C₂×C₂²⋊C₁₆

—

271

maximal, Sylow

C₂³×C₈

D₂₇₁

yes

characteristic

C₂²×C₁₆

—

271

—

C₂²⋊C₁₆

—

271

—

C₂³×C₄

Dic₂₇₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

Dic₂₇₁

yes

characteristic

C₂²×C₈

Dic₂₇₁

yes

—

C₂²×C₈

C₂×D₂₇₁

yes

—

C₂²×C₈

C₂×D₂₇₁

yes

characteristic

C₂×C₁₆

—

271

—

C₂×C₁₆

—

542

—

C₂⁴

C₂₇₁⋊C₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂₇₁⋊C₈

yes

characteristic

C₂²×C₄

C₂₇₁⋊C₈

yes

—

C₂²×C₄

C₂×Dic₂₇₁

yes

—

C₂²×C₄

C₂×Dic₂₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₂₇₁

yes

—

C₂×C₈

C₂×Dic₂₇₁

yes

characteristic

C₂×C₈

C₂²⋊D₂₇₁

yes

—

C₂×C₈

C₂²×D₂₇₁

yes

characteristic

C₁₆

—

542

—

C₂³

—

C₂³

—

C₂³

C₂₇₁⋊C₁₆

yes

complemented

C₂³

C₂×C₂₇₁⋊C₈

yes

—

C₂³

C₂×C₂₇₁⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₂₇₁

yes

—

C₂×C₄

C₂×C₂₇₁⋊C₈

yes

—

100

C₂×C₄

C₂×C₂₇₁⋊C₈

yes

characteristic

101

C₂×C₄

#16864638

yes

—

102

C₂×C₄

C₂²×Dic₂₇₁

yes

characteristic

103

C₈

—

104

C₈

C₂²⋊Dic₂₇₁

yes

—

105

C₈

C₂×C₂²⋊D₂₇₁

yes

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂²⋊(C₂₇₁⋊C₈)

yes

—

110

C₂²

C₂×C₂₇₁⋊C₁₆

yes

—

111

C₂²

C₂×C₂₇₁⋊C₁₆

yes

complemented

112

C₂²

#22576100

yes

—

113

C₂²

C₂²×C₂₇₁⋊C₈

yes

characteristic

114

C₄

—

115

C₄

C₂²⋊(C₂₇₁⋊C₈)

yes

—

116

C₄

C₂×C₂²⋊Dic₂₇₁

yes

characteristic

117

C₄

C₂×#16864638

yes

characteristic

118

C₂

—

119

C₂

C₂²⋊(C₂₇₁⋊C₁₆)

yes

direct product factor

120

C₂

C₂×C₂²⋊(C₂₇₁⋊C₈)

yes

characteristic

121

C₂

C₂²×C₂₇₁⋊C₁₆

yes

characteristic

122

C₂

C₂×#22576100

yes

characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₂₇₁
commutator	C₂×C₂₇₁	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×D₂₇₁
Fitting	C₂³×C₈×C₂₇₁	C₂
socle	C₂³×C₂₇₁	C₂×C₈
prime radical	C₂⁴×C₂₇₁	C₈
cosocle	C₂×C₈×C₂₇₁	C₂³
hypercenter	C₂³×C₈	D₂₇₁
hypocenter	C₂₇₁	C₂×C₂²⋊C₁₆
LCS	G ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

subgroup

G/H

center

C₂²×C₈

C₂×D₂₇₁

commutator

C₂×C₂₇₁

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×D₂₇₁

Fitting

C₂³×C₈×C₂₇₁

C₂

socle

C₂³×C₂₇₁

C₂×C₈

prime radical

C₂⁴×C₂₇₁

C₈

cosocle

C₂×C₈×C₂₇₁

C₂³

hypercenter

C₂³×C₈

D₂₇₁

hypocenter

C₂₇₁

C₂×C₂²⋊C₁₆

LCS

G ▷ C₂×C₂₇₁ ▷ C₂₇₁

UCS

C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

N	H ≅ G/N	type
1. C₂²×C₂₇₁⋊C₁₆	C₂	semidirect
2. C₂²⋊(C₂₇₁⋊C₁₆)	C₂	direct (N × H)
5. C₂×C₂₇₁⋊C₁₆	C₂²	semidirect
29. C₂³×C₂₇₁	C₁₆	semidirect
49. C₂²×C₂₇₁	C₂×C₁₆	semidirect
57. C₂×C₂₇₁	C₂²⋊C₁₆	semidirect
61. C₂₇₁	C₂×C₂²⋊C₁₆	semidirect
91. C₂³	C₂₇₁⋊C₁₆	semidirect
111. C₂²	C₂×C₂₇₁⋊C₁₆	semidirect
119. C₂	C₂²⋊(C₂₇₁⋊C₁₆)	direct (N × H)

H ≅ G/N

type

1. C₂²×C₂₇₁⋊C₁₆

C₂

semidirect

2. C₂²⋊(C₂₇₁⋊C₁₆)

C₂

direct (N × H)

5. C₂×C₂₇₁⋊C₁₆

C₂²

semidirect

29. C₂³×C₂₇₁

C₁₆

semidirect

49. C₂²×C₂₇₁

C₂×C₁₆

semidirect

57. C₂×C₂₇₁

C₂²⋊C₁₆

semidirect

61. C₂₇₁

C₂×C₂²⋊C₁₆

semidirect

91. C₂³

C₂₇₁⋊C₁₆

semidirect

111. C₂²

C₂×C₂₇₁⋊C₁₆

semidirect

119. C₂

C₂²⋊(C₂₇₁⋊C₁₆)

direct (N × H)

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

subgroup	index
C₂²×C₂₇₁⋊C₁₆	2
C₂²⋊(C₂₇₁⋊C₁₆)	2
C₂³×C₈×C₂₇₁	2
C₂×C₂²⋊C₁₆	271

subgroup

index

C₂²×C₂₇₁⋊C₁₆

C₂²⋊(C₂₇₁⋊C₁₆)

C₂³×C₈×C₂₇₁

C₂×C₂²⋊C₁₆

271

Subgroups — C₂×C₂²⋊(C₂₇₁⋊C₁₆) #46837972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C22⋊(C271⋊C16) #46837972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂²⋊(C₂₇₁⋊C₁₆) #46837972