Group #46836040 — subgroups

Subgroups — #46836040

Order 34688.

176 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 61 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	#31774300	C₂	1	1	yes	maximal, characteristic
2	#31774293	C₂	1	1	yes	maximal, characteristic
3	#31774276	C₂	1	1	yes	maximal, characteristic
4	#31774248	C₂	1	1	yes	maximal, characteristic
5	#31774247	C₂	1	1	yes	maximal, characteristic
6	#31773619	C₂	1	1	yes	maximal, characteristic
7	#500×C₂₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#16864644	—	2	2	no	—
9	#22576160	C₂²	1	1	yes	characteristic
10	#22576160	C₂²	1	1	yes	characteristic
11	#22576093	—	2	2	no	—
12	#22576091	—	2	2	no	—
13	#22576085	—	2	2	no	—
14	#22576084	—	2	2	no	—
15	C₄×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₂₇₁	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₂₇₁	C₂²	1	1	yes	characteristic
19	#107×C₂₇₁	—	2	2	no	—
20	C₄×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₅₄₂	—	2	1	no	—
22	C₂×Dic₅₄₂	—	2	1	no	—
23	#16864644	—	4	4	no	—
24	C₂×C₂₇₁⋊C₈	D₄	1	2	yes	—
25	C₂×C₂₇₁⋊C₈	D₄	1	2	yes	—
26	C₄⋊Dic₂₇₁	—	2	1	no	—
27	C₄⋊Dic₂₇₁	—	2	1	no	—
28	C₄⋊Dic₂₇₁	—	2	1	no	—
29	C₄⋊Dic₂₇₁	—	2	1	no	—
30	C₂×Q₈×C₂₇₁	—	2	1	no	—
31	C₂×Q₈×C₂₇₁	—	2	1	no	—
32	C₂×C₈×C₂₇₁	D₄	1	2	yes	—
33	C₄⋊C₄×C₂₇₁	—	2	1	no	—
34	C₄⋊C₄×C₂₇₁	—	2	1	no	—
35	C₄⋊C₄×C₂₇₁	—	2	1	no	—
36	C₄⋊C₄×C₂₇₁	—	2	1	no	—
37	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
38	Dic₅₄₂	—	4	2	no	—
39	Dic₅₄₂	—	4	2	no	—
40	C₂×Dic₂₇₁	—	4	1	no	—
41	C₂×Dic₂₇₁	—	4	1	no	—
42	C₂₇₁⋊C₈	—	2	2	no	—
43	C₂₇₁⋊C₈	Q₁₆	1	4	yes	complemented
44	Q₈×C₂₇₁	—	4	2	no	—
45	Q₈×C₂₇₁	—	4	2	no	—
46	C₂×C₄×C₂₇₁	—	4	1	no	—
47	C₂×C₄×C₂₇₁	—	4	1	no	—
48	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
51	C₈×C₂₇₁	—	2	2	no	—
52	C₂²×C₂₇₁	#127	1	1	yes	characteristic
53	C₄×C₂₇₁	—	8	1	no	—
54	C₄×C₂₇₁	—	8	1	no	—
55	C₄×C₂₇₁	C₂×Q₁₆	1	2	yes	—
56	C₄×C₂₇₁	AGL(1,ℤ/8)	1	2	yes	—
57	C₄×C₂₇₁	#137	1	2	yes	—
58	Dic₂₇₁	—	8	1	no	—
59	Dic₂₇₁	—	8	1	no	—
60	C₂×C₂₇₁	#494	1	1	yes	characteristic
61	C₂×C₂₇₁	#496	1	1	yes	characteristic
62	C₂×C₂₇₁	#497	1	1	yes	characteristic
63	C₂₇₁	#1723	1	1	yes	characteristic, complemented, Sylow
64	#1723	—	271	1	no	maximal, Sylow
65	#500	—	271	1	no	—
66	#500	D₂₇₁	1	1	yes	characteristic
67	#498	—	271	1	no	—
68	#494	—	271	1	no	—
69	#487	—	271	1	no	—
70	#487	—	271	1	no	—
71	#322	—	271	1	no	—
72	C₂×Q₁₆	—	542	2	no	—
73	C₄⋊Q₈	—	271	1	no	—
74	C₄⋊Q₈	—	271	1	no	—
75	C₄⋊Q₈	C₂×D₂₇₁	1	1	yes	characteristic
76	C₄⋊Q₈	C₂×D₂₇₁	1	1	yes	characteristic
77	#107	—	2	2	no	—
78	#107	—	542	2	no	—
79	#106	—	542	2	no	—
80	#103	—	542	2	no	—
81	#103	—	542	2	no	—
82	C₄×C₈	—	271	1	no	—
83	C₄×C₈	—	271	1	no	—
84	C₄×C₈	C₂×D₂₇₁	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	542	1	no	—
88	C₂×Q₈	—	542	1	no	—
89	Q₁₆	—	1084	4	no	—
90	C₂×C₈	—	271	2	no	—
91	C₂×C₈	—	271	2	no	—
92	C₂×C₈	C₂²⋊D₂₇₁	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	542	1	no	—
98	C₄⋊C₄	—	542	1	no	—
99	C₄⋊C₄	—	542	1	no	—
100	C₄⋊C₄	—	542	1	no	—
101	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	1084	2	no	—
105	Q₈	—	1084	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	1084	1	no	—
109	C₂×C₄	—	1084	1	no	—
110	C₂×C₄	D₄×D₂₇₁	1	1	yes	characteristic
111	C₂×C₄	D₄×D₂₇₁	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₂₇₁	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	271	4	no	—
115	C₈	—	542	2	no	—
116	C₂²	#22576156	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	2168	1	no	—
120	C₄	—	2168	1	no	—
121	C₄	Q₁₆×D₂₇₁	1	2	yes	—
122	C₄	#22576184	1	2	yes	—
123	C₄	#22576191	1	2	yes	—
124	C₂	#31774275	1	1	yes	characteristic
125	C₂	#31774285	1	1	yes	characteristic
126	C₂	#31774289	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31774300

C₂

yes

maximal, characteristic

#31774293

C₂

yes

maximal, characteristic

#31774276

C₂

yes

maximal, characteristic

#31774248

C₂

yes

maximal, characteristic

#31774247

C₂

yes

maximal, characteristic

#31773619

C₂

yes

maximal, characteristic

#500×C₂₇₁

C₂

yes

maximal, characteristic

C₂×#16864644

—

#22576160

C₂²

yes

characteristic

#22576160

C₂²

yes

—

—

—

—

C₄×C₂₇₁⋊C₈

C₂²

yes

characteristic

C₄×C₂₇₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₂₇₁

C₂²

yes

characteristic

C₄⋊Q₈×C₂₇₁

C₂²

yes

characteristic

#107×C₂₇₁

—

C₄×C₈×C₂₇₁

C₂²

yes

characteristic

C₂×Dic₅₄₂

—

C₂×Dic₅₄₂

—

#16864644

—

C₂×C₂₇₁⋊C₈

D₄

yes

—

C₂×C₂₇₁⋊C₈

D₄

yes

—

C₄⋊Dic₂₇₁

—

C₄⋊Dic₂₇₁

—

C₄⋊Dic₂₇₁

—

C₄⋊Dic₂₇₁

—

C₂×Q₈×C₂₇₁

—

C₂×Q₈×C₂₇₁

—

C₂×C₈×C₂₇₁

D₄

yes

—

C₄⋊C₄×C₂₇₁

—

C₄⋊C₄×C₂₇₁

—

C₄⋊C₄×C₂₇₁

—

C₄⋊C₄×C₂₇₁

—

C₄²×C₂₇₁

C₂³

yes

characteristic

Dic₅₄₂

—

Dic₅₄₂

—

C₂×Dic₂₇₁

—

C₂×Dic₂₇₁

—

C₂₇₁⋊C₈

—

C₂₇₁⋊C₈

Q₁₆

yes

complemented

Q₈×C₂₇₁

—

Q₈×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₁

C₂×D₄

yes

characteristic

C₈×C₂₇₁

—

C₂²×C₂₇₁

#127

yes

characteristic

C₄×C₂₇₁

—

C₄×C₂₇₁

—

C₄×C₂₇₁

C₂×Q₁₆

yes

—

C₄×C₂₇₁

AGL(1,ℤ/8)

yes

—

C₄×C₂₇₁

#137

yes

—

Dic₂₇₁

—

Dic₂₇₁

—

C₂×C₂₇₁

yes

C₂×C₂₇₁

yes

C₂×C₂₇₁

yes

C₂₇₁

yes

characteristic, complemented, Sylow

#1723

—

271

maximal, Sylow

#500

—

271

—

#500

D₂₇₁

yes

characteristic

#498

—

271

—

#494

—

271

—

#487

—

271

—

#487

—

271

—

#322

—

271

—

C₂×Q₁₆

—

542

—

C₄⋊Q₈

—

271

—

C₄⋊Q₈

—

271

—

C₄⋊Q₈

C₂×D₂₇₁

yes

characteristic

C₄⋊Q₈

C₂×D₂₇₁

yes

characteristic

#107

—

#107

—

542

—

#106

—

542

—

#103

—

542

—

#103

—

542

—

C₄×C₈

—

271

—

C₄×C₈

—

271

—

C₄×C₈

C₂×D₂₇₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

542

—

C₂×Q₈

—

542

—

Q₁₆

—

1084

—

C₂×C₈

—

271

—

C₂×C₈

—

271

—

C₂×C₈

C₂²⋊D₂₇₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

542

—

C₄⋊C₄

—

542

—

C₄⋊C₄

—

542

—

100

C₄⋊C₄

—

542

—

101

C₄²

C₂²×D₂₇₁

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

1084

—

105

Q₈

—

1084

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

1084

—

109

C₂×C₄

—

1084

—

110

C₂×C₄

D₄×D₂₇₁

yes

characteristic

111

C₂×C₄

D₄×D₂₇₁

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₂₇₁

yes

characteristic

113

C₈

—

114

C₈

—

271

—

115

C₈

—

542

—

116

C₂²

#22576156

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

2168

—

120

C₄

—

2168

—

121

C₄

Q₁₆×D₂₇₁

yes

—

122

C₄

#22576184

yes

—

123

C₄

#22576191

yes

—

124

C₂

#31774275

yes

characteristic

125

C₂

#31774285

yes

characteristic

126

C₂

#31774289

yes

characteristic

	subgroup	G/H
center	C₂²	#22576156
commutator, cosocle	C₄²×C₂₇₁	C₂³
Frattini	C₄²	C₂²×D₂₇₁
Fitting	#500×C₂₇₁	C₂
socle, prime radical	C₂²×C₂₇₁	#127
hypercenter	#500	D₂₇₁
hypocenter	C₂₇₁	#1723
LCS	G ▷ C₄²×C₂₇₁ ▷ C₂²×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#22576156

commutator, cosocle

C₄²×C₂₇₁

C₂³

Frattini

C₄²

C₂²×D₂₇₁

Fitting

#500×C₂₇₁

C₂

C₂²×C₂₇₁

D₂₇₁

C₂₇₁

G ▷ C₄²×C₂₇₁ ▷ C₂²×C₂₇₁ ▷ C₂₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
43. C₂₇₁⋊C₈	Q₁₆	semidirect
63. C₂₇₁	#1723	semidirect

H ≅ G/N

type

43. C₂₇₁⋊C₈

Q₁₆

semidirect

63. C₂₇₁

#1723

semidirect

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

subgroup	index
#31774300	2
#31774293	2
#31774276	2
#31774248	2
#31774247	2
#31773619	2
#500×C₂₇₁	2
#1723	271

subgroup

index

#500×C₂₇₁

271

Subgroups — #46836040

Conjugacy classes of subgroups

Complements

Maximal subgroups