Group #46846866 — subgroups

Subgroups — C₂×#31774297 #46846866

Order 34688.

360 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 165 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×#22576151	C₂	1	2	yes	maximal
2	#31774297	C₂	1	8	yes	maximal, direct product factor
3	C₂×#22576093	C₂	1	2	yes	maximal
4	C₂×#22576092	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×Dic₂₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#106×C₂₇₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
8	#22576151	—	2	4	no	—
9	#22576151	C₂²	1	8	yes	—
10	C₂×C₄⋊Dic₂₇₁	—	2	2	no	—
11	C₂×C₄⋊Dic₂₇₁	C₂²	1	2	yes	—
12	C₂×Dic₂₇₁⋊C₄	—	2	2	no	—
13	C₂×C₄×Dic₂₇₁	C₂²	1	1	yes	characteristic
14	#22576093	C₂²	1	8	yes	—
15	#22576092	C₂²	1	4	yes	—
16	C₈×Dic₂₇₁	C₂²	1	4	yes	—
17	C₂²×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₇₁	C₂²	1	2	yes	—
19	#106×C₂₇₁	C₂²	1	4	yes	—
20	C₂²×Dic₂₇₁	—	2	2	no	—
21	C₂²×Dic₂₇₁	D₄	1	2	yes	—
22	C₂×C₂₇₁⋊C₈	Q₈	1	4	yes	—
23	C₂×C₂₇₁⋊C₈	C₂³	1	2	yes	—
24	C₄⋊Dic₂₇₁	—	2	8	no	—
25	C₄⋊Dic₂₇₁	—	2	2	no	—
26	C₄⋊Dic₂₇₁	C₂³	1	4	yes	—
27	Dic₂₇₁⋊C₄	—	2	8	no	—
28	C₄×Dic₂₇₁	D₄	1	2	yes	—
29	C₄×Dic₂₇₁	C₂³	1	2	yes	—
30	C₂²×C₄×C₂₇₁	—	2	2	no	—
31	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₂₇₁	Q₈	1	4	yes	—
33	C₂×C₈×C₂₇₁	C₂³	1	2	yes	—
34	C₄⋊C₄×C₂₇₁	—	2	2	no	—
35	C₄⋊C₄×C₂₇₁	C₂³	1	4	yes	—
36	C₂×Dic₂₇₁	—	2	4	no	—
37	C₂×Dic₂₇₁	—	2	4	no	—
38	C₂×Dic₂₇₁	—	4	4	no	—
39	C₂×Dic₂₇₁	C₂×D₄	1	4	yes	—
40	C₂₇₁⋊C₈	C₂×Q₈	1	4	yes	—
41	C₂³×C₂₇₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₇₁	—	2	4	no	—
43	C₂×C₄×C₂₇₁	—	4	4	no	—
44	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₇₁	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₂₇₁	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
48	C₈×C₂₇₁	C₂×Q₈	1	4	yes	—
49	C₂²×C₂₇₁	C₄⋊Q₈	1	2	yes	—
50	C₂²×C₂₇₁	#135	1	4	yes	—
51	C₂²×C₂₇₁	C₂²×D₄	1	1	yes	characteristic
52	C₄×C₂₇₁	—	4	4	no	—
53	C₄×C₂₇₁	C₄⋊Q₈	1	2	yes	—
54	C₄×C₂₇₁	C₂²×Q₈	1	1	yes	characteristic
55	C₄×C₂₇₁	C₂²×Q₈	1	1	yes	characteristic
56	Dic₂₇₁	—	2	4	no	—
57	Dic₂₇₁	—	4	4	no	—
58	C₂×C₂₇₁	#499	1	4	yes	complemented
59	C₂×C₂₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₂₇₁	C₂×#135	1	2	yes	—
61	C₂₇₁	C₂×#499	1	1	yes	characteristic, complemented, Sylow
62	C₂×#499	—	271	1	no	maximal, Sylow
63	C₂×#125	—	271	2	no	—
64	#499	—	271	8	no	—
65	C₂×#107	—	271	2	no	—
66	C₂×#106	—	271	1	no	—
67	C₂×#106	D₂₇₁	1	1	yes	characteristic
68	C₂×C₄×C₈	—	271	1	no	—
69	C₂²×C₈	—	271	1	no	—
70	C₂²×C₈	C₂×D₂₇₁	1	1	yes	characteristic
71	#125	—	271	8	no	—
72	#125	—	542	4	no	—
73	C₂×C₄⋊C₄	—	271	2	no	—
74	C₂×C₄⋊C₄	—	542	2	no	—
75	C₂×C₄⋊C₄	—	542	2	no	—
76	C₂×C₄⋊C₄	C₂×D₂₇₁	1	2	yes	—
77	C₂×C₄²	—	271	1	no	—
78	#107	—	271	8	no	—
79	#106	—	271	4	no	—
80	#106	C₂×D₂₇₁	1	4	yes	—
81	C₄×C₈	—	271	4	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	271	2	no	—
84	C₂²×C₄	—	542	2	no	—
85	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
86	C₂×C₈	—	271	4	no	—
87	C₂×C₈	—	271	2	no	—
88	C₂×C₈	Dic₅₄₂	1	4	yes	—
89	C₂×C₈	C₂²×D₂₇₁	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	271	4	no	—
92	C₄⋊C₄	—	542	8	no	—
93	C₄⋊C₄	—	542	8	no	—
94	C₄⋊C₄	—	542	2	no	—
95	C₄⋊C₄	C₂²×D₂₇₁	1	4	yes	—
96	C₄²	—	271	2	no	—
97	C₄²	—	271	2	no	—
98	C₂³	D₄×D₂₇₁	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	4	4	no	—
101	C₂×C₄	—	271	4	no	—
102	C₂×C₄	—	542	4	no	—
103	C₂×C₄	—	542	4	no	—
104	C₂×C₄	—	1084	4	no	—
105	C₂×C₄	D₄×D₂₇₁	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₂₇₁	1	2	yes	—
107	C₂×C₄	C₂×Dic₅₄₂	1	2	yes	—
108	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
109	C₈	—	271	4	no	—
110	C₈	C₂×Dic₅₄₂	1	4	yes	—
111	C₂²	#22576159	1	2	yes	—
112	C₂²	#22576181	1	4	yes	—
113	C₂²	C₂×D₄×D₂₇₁	1	1	yes	characteristic
114	C₄	—	4	4	no	—
115	C₄	—	542	4	no	—
116	C₄	—	1084	4	no	—
117	C₄	#22576159	1	2	yes	—
118	C₄	C₂×Q₈×D₂₇₁	1	1	yes	characteristic
119	C₄	C₂²×Dic₅₄₂	1	1	yes	characteristic
120	C₂	#31774297	1	4	yes	direct product factor
121	C₂	C₂×#22576159	1	1	yes	characteristic
122	C₂	C₂×#22576181	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22576151

C₂

yes

maximal

#31774297

C₂

yes

maximal, direct product factor

C₂×#22576093

C₂

yes

maximal

C₂×#22576092

C₂

yes

maximal, characteristic

C₂×C₈×Dic₂₇₁

C₂

yes

maximal, characteristic

C₂×#106×C₂₇₁

C₂

yes

maximal, characteristic

C₂²×C₂₇₁⋊C₈

C₂²

yes

characteristic

#22576151

—

#22576151

C₂²

yes

—

C₂×C₄⋊Dic₂₇₁

—

C₂×C₄⋊Dic₂₇₁

C₂²

yes

—

C₂×Dic₂₇₁⋊C₄

—

C₂×C₄×Dic₂₇₁

C₂²

yes

characteristic

#22576093

C₂²

yes

—

#22576092

C₂²

yes

—

C₈×Dic₂₇₁

C₂²

yes

—

C₂²×C₈×C₂₇₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₇₁

C₂²

yes

—

#106×C₂₇₁

C₂²

yes

—

C₂²×Dic₂₇₁

—

C₂²×Dic₂₇₁

D₄

yes

—

C₂×C₂₇₁⋊C₈

Q₈

yes

—

C₂×C₂₇₁⋊C₈

C₂³

yes

—

C₄⋊Dic₂₇₁

—

C₄⋊Dic₂₇₁

—

C₄⋊Dic₂₇₁

C₂³

yes

—

Dic₂₇₁⋊C₄

—

C₄×Dic₂₇₁

D₄

yes

—

C₄×Dic₂₇₁

C₂³

yes

—

C₂²×C₄×C₂₇₁

—

C₂²×C₄×C₂₇₁

C₂³

yes

characteristic

C₂×C₈×C₂₇₁

Q₈

yes

—

C₂×C₈×C₂₇₁

C₂³

yes

—

C₄⋊C₄×C₂₇₁

—

C₄⋊C₄×C₂₇₁

C₂³

yes

—

C₂×Dic₂₇₁

—

C₂×Dic₂₇₁

—

C₂×Dic₂₇₁

—

C₂×Dic₂₇₁

C₂×D₄

yes

—

C₂₇₁⋊C₈

C₂×Q₈

yes

—

C₂³×C₂₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₁

C₂×Q₈

yes

—

C₂×C₄×C₂₇₁

C₂×Q₈

yes

—

C₂×C₄×C₂₇₁

C₂⁴

yes

characteristic

C₈×C₂₇₁

C₂×Q₈

yes

—

C₂²×C₂₇₁

C₄⋊Q₈

yes

—

C₂²×C₂₇₁

#135

yes

—

C₂²×C₂₇₁

C₂²×D₄

yes

characteristic

C₄×C₂₇₁

—

C₄×C₂₇₁

C₄⋊Q₈

yes

—

C₄×C₂₇₁

C₂²×Q₈

yes

characteristic

C₄×C₂₇₁

C₂²×Q₈

yes

characteristic

Dic₂₇₁

—

Dic₂₇₁

—

C₂×C₂₇₁

#499

yes

complemented

C₂×C₂₇₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₇₁

C₂×#135

yes

—

C₂₇₁

C₂×#499

yes

characteristic, complemented, Sylow

C₂×#499

—

271

maximal, Sylow

C₂×#125

—

271

—

#499

—

271

—

C₂×#107

—

271

—

C₂×#106

—

271

—

C₂×#106

D₂₇₁

yes

characteristic

C₂×C₄×C₈

—

271

—

C₂²×C₈

—

271

—

C₂²×C₈

C₂×D₂₇₁

yes

characteristic

#125

—

271

—

#125

—

542

—

C₂×C₄⋊C₄

—

271

—

C₂×C₄⋊C₄

—

542

—

C₂×C₄⋊C₄

—

542

—

C₂×C₄⋊C₄

C₂×D₂₇₁

yes

—

C₂×C₄²

—

271

—

#107

—

271

—

#106

—

271

—

#106

C₂×D₂₇₁

yes

—

C₄×C₈

—

271

—

C₂²×C₄

—

C₂²×C₄

—

271

—

C₂²×C₄

—

542

—

C₂²×C₄

C₂²×D₂₇₁

yes

characteristic

C₂×C₈

—

271

—

C₂×C₈

—

271

—

C₂×C₈

Dic₅₄₂

yes

—

C₂×C₈

C₂²×D₂₇₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

271

—

C₄⋊C₄

—

542

—

C₄⋊C₄

—

542

—

C₄⋊C₄

—

542

—

C₄⋊C₄

C₂²×D₂₇₁

yes

—

C₄²

—

271

—

C₄²

—

271

—

C₂³

D₄×D₂₇₁

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

271

—

102

C₂×C₄

—

542

—

103

C₂×C₄

—

542

—

104

C₂×C₄

—

1084

—

105

C₂×C₄

D₄×D₂₇₁

yes

characteristic

106

C₂×C₄

Q₈×D₂₇₁

yes

—

107

C₂×C₄

C₂×Dic₅₄₂

yes

—

108

C₂×C₄

C₂³×D₂₇₁

yes

characteristic

109

C₈

—

271

—

110

C₈

C₂×Dic₅₄₂

yes

—

111

C₂²

#22576159

yes

—

112

C₂²

#22576181

yes

—

113

C₂²

C₂×D₄×D₂₇₁

yes

characteristic

114

C₄

—

115

C₄

—

542

—

116

C₄

—

1084

—

117

C₄

#22576159

yes

—

118

C₄

C₂×Q₈×D₂₇₁

yes

characteristic

119

C₄

C₂²×Dic₅₄₂

yes

characteristic

120

C₂

#31774297

yes

direct product factor

121

C₂

C₂×#22576159

yes

characteristic

122

C₂

C₂×#22576181

yes

—

	subgroup	G/H
center	C₂³	D₄×D₂₇₁
commutator, cosocle	C₂×C₄×C₂₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₇₁
Fitting	C₂×#106×C₂₇₁	C₂
socle, prime radical	C₂³×C₂₇₁	C₂×D₄
hypercenter	C₂×#106	D₂₇₁
hypocenter	C₂₇₁	C₂×#499
LCS	G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₂₇₁

commutator, cosocle

C₂×C₄×C₂₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₇₁

Fitting

C₂×#106×C₂₇₁

C₂

socle, prime radical

C₂³×C₂₇₁

C₂×D₄

D₂₇₁

C₂₇₁

G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
2. #31774297	C₂	direct (N × H)
58. C₂×C₂₇₁	#499	semidirect
61. C₂₇₁	C₂×#499	semidirect
120. C₂	#31774297	direct (N × H)

H ≅ G/N

type

2. #31774297

C₂

direct (N × H)

58. C₂×C₂₇₁

#499

semidirect

61. C₂₇₁

C₂×#499

semidirect

120. C₂

#31774297

direct (N × H)

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

subgroup	index
C₂×#22576151	2
#31774297	2
C₂×#22576093	2
C₂×#22576092	2
C₂×C₈×Dic₂₇₁	2
C₂×#106×C₂₇₁	2
C₂×#499	271

subgroup

index

C₂×C₈×Dic₂₇₁

C₂×#106×C₂₇₁

C₂×#499

271

Subgroups — C₂×#31774297 #46846866

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#31774297 #46846866

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31774297 #46846866