Group #46836246 — subgroups

Subgroups — #46836246

Order 34688.

232 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 117 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	#31773935	C₂	1	1	yes	maximal, characteristic
2	#31773855	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₂₇₁	C₂	1	1	yes	maximal, characteristic
4	#31773658	C₂	1	2	yes	maximal
5	C₂×#108×C₂₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
7	#22576119	C₂²	1	2	yes	—
8	C₂×C₄×Dic₂₇₁	C₂²	1	1	yes	characteristic
9	#22576094	C₄	1	2	yes	complemented
10	#22576092	C₄	1	2	yes	complemented
11	#22576070	—	2	2	no	—
12	C₈×Dic₂₇₁	—	2	2	no	—
13	C₈×Dic₂₇₁	C₄	1	4	yes	complemented
14	C₂×M₁₆×C₂₇₁	C₂²	1	2	yes	—
15	C₂²×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
16	#108×C₂₇₁	C₄	1	4	yes	complemented
17	C₂²×Dic₂₇₁	—	2	1	no	—
18	C₂×C₂₇₁⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₂₇₁⋊C₈	D₄	1	1	yes	characteristic
20	C₂×C₂₇₁⋊C₈	D₄	1	1	yes	characteristic
21	C₂×C₂₇₁⋊C₈	Q₈	1	2	yes	—
22	C₄⋊Dic₂₇₁	C₂×C₄	1	4	yes	—
23	C₂²⋊Dic₂₇₁	—	2	2	no	—
24	C₄×Dic₂₇₁	—	2	2	no	—
25	C₄×Dic₂₇₁	—	2	1	no	—
26	C₄×Dic₂₇₁	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
28	M₁₆×C₂₇₁	—	2	2	no	—
29	M₁₆×C₂₇₁	C₂×C₄	1	4	yes	—
30	C₂×C₈×C₂₇₁	—	2	2	no	—
31	C₂×C₈×C₂₇₁	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₂₇₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₂₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₂₇₁	C₂×C₄	1	2	yes	—
35	C₂×Dic₂₇₁	—	2	4	no	—
36	C₂×Dic₂₇₁	—	2	2	no	—
37	C₂×Dic₂₇₁	—	2	4	no	—
38	C₂₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
39	C₂₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
40	C₂³×C₂₇₁	#41	1	1	yes	characteristic
41	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₇₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₂₇₁	#41	1	1	yes	characteristic
47	C₈×C₂₇₁	—	2	4	no	—
48	C₈×C₂₇₁	C₄²	1	2	yes	—
49	C₈×C₂₇₁	C₄²	1	2	yes	—
50	C₂²×C₂₇₁	—	2	2	no	—
51	C₂²×C₂₇₁	#117	1	1	yes	characteristic
52	C₂²×C₂₇₁	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₂₇₁	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₂₇₁	C₂×C₄²	1	1	yes	characteristic
55	C₄×C₂₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₂₇₁	C₄×D₄	1	1	yes	characteristic
57	C₄×C₂₇₁	C₄×Q₈	1	1	yes	characteristic
58	Dic₂₇₁	—	2	4	no	—
59	Dic₂₇₁	—	4	4	no	—
60	C₂×C₂₇₁	—	2	1	no	—
61	C₂×C₂₇₁	—	2	1	no	—
62	C₂×C₂₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₂₇₁	#443	1	2	yes	—
64	C₂₇₁	#1772	1	1	yes	characteristic, complemented, Sylow
65	#1772	—	271	1	no	maximal, Sylow
66	C₂×#108	D₂₇₁	1	1	yes	characteristic
67	#427	—	271	1	no	—
68	#405	—	271	2	no	—
69	C₂×C₄×C₈	—	271	1	no	—
70	#339	—	271	2	no	—
71	C₂×M₁₆	C₂×D₂₇₁	1	2	yes	—
72	C₂²×C₈	—	271	1	no	—
73	C₂²×C₈	C₂×D₂₇₁	1	1	yes	characteristic
74	#117	—	271	2	no	—
75	C₂×C₄²	—	271	1	no	—
76	#108	Dic₂₇₁	1	4	yes	complemented
77	#107	—	271	2	no	—
78	#106	—	271	2	no	—
79	#97	—	542	2	no	—
80	C₄×C₈	—	271	4	no	—
81	C₄×C₈	—	542	2	no	—
82	C₂²×C₄	—	542	1	no	—
83	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	C₂×Dic₂₇₁	1	4	yes	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	271	2	no	—
88	C₂×C₈	—	271	1	no	—
89	C₂×C₈	—	271	1	no	—
90	C₂×C₈	—	271	2	no	—
91	C₂×C₈	C₄×D₂₇₁	1	2	yes	—
92	C₂×C₈	C₄×D₂₇₁	1	1	yes	characteristic
93	C₂×C₈	C₄×D₂₇₁	1	1	yes	characteristic
94	C₂×C₈	C₂×Dic₂₇₁	1	2	yes	—
95	C₄⋊C₄	—	271	4	no	—
96	C₂²⋊C₄	—	542	2	no	—
97	C₄²	—	271	2	no	—
98	C₄²	—	542	2	no	—
99	C₄²	—	542	1	no	—
100	C₂³	#16864655	1	1	yes	characteristic
101	C₂×C₄	—	542	4	no	—
102	C₂×C₄	—	542	2	no	—
103	C₂×C₄	—	542	4	no	—
104	C₂×C₄	C₂×C₄×D₂₇₁	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₂₇₁	1	1	yes	characteristic
106	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
107	C₂×C₄	D₄×D₂₇₁	1	1	yes	characteristic
108	C₂×C₄	Q₈×D₂₇₁	1	1	yes	characteristic
109	C₂×C₄	#16864653	1	1	yes	characteristic
110	C₈	—	2	4	no	—
111	C₈	—	271	2	no	—
112	C₈	—	271	2	no	—
113	C₈	C₄×Dic₂₇₁	1	2	yes	—
114	C₈	C₄×Dic₂₇₁	1	2	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	#22576118	1	1	yes	characteristic
117	C₂²	#22576122	1	1	yes	characteristic
118	C₂²	Q₈×Dic₂₇₁	1	1	yes	characteristic
119	C₄	—	542	4	no	—
120	C₄	—	1084	4	no	—
121	C₄	C₂×C₄×Dic₂₇₁	1	1	yes	characteristic
122	C₄	C₄⋊C₄×D₂₇₁	1	1	yes	characteristic
123	C₄	D₄×Dic₂₇₁	1	1	yes	characteristic
124	C₄	#22576125	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	C₄⋊C₄×Dic₂₇₁	1	1	yes	characteristic
128	C₂	#31774013	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31773935

C₂

yes

maximal, characteristic

#31773855

C₂

yes

maximal

C₂×C₈×Dic₂₇₁

C₂

yes

maximal, characteristic

#31773658

C₂

yes

maximal

C₂×#108×C₂₇₁

C₂

yes

maximal, characteristic

C₂²×C₂₇₁⋊C₈

C₂²

yes

characteristic

#22576119

C₂²

yes

—

C₂×C₄×Dic₂₇₁

C₂²

yes

characteristic

#22576094

C₄

yes

complemented

#22576092

C₄

yes

complemented

#22576070

—

C₈×Dic₂₇₁

—

C₈×Dic₂₇₁

C₄

yes

complemented

C₂×M₁₆×C₂₇₁

C₂²

yes

—

C₂²×C₈×C₂₇₁

C₂²

yes

characteristic

#108×C₂₇₁

C₄

yes

complemented

C₂²×Dic₂₇₁

—

C₂×C₂₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₇₁⋊C₈

D₄

yes

characteristic

C₂×C₂₇₁⋊C₈

D₄

yes

characteristic

C₂×C₂₇₁⋊C₈

Q₈

yes

—

C₄⋊Dic₂₇₁

C₂×C₄

yes

—

C₂²⋊Dic₂₇₁

—

C₄×Dic₂₇₁

—

C₄×Dic₂₇₁

—

C₄×Dic₂₇₁

C₂×C₄

yes

—

C₂²×C₄×C₂₇₁

C₂³

yes

characteristic

M₁₆×C₂₇₁

—

M₁₆×C₂₇₁

C₂×C₄

yes

—

C₂×C₈×C₂₇₁

—

C₂×C₈×C₂₇₁

C₂×C₄

yes

—

C₂×C₈×C₂₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₇₁

C₂×C₄

yes

—

C₂×Dic₂₇₁

—

C₂×Dic₂₇₁

—

C₂×Dic₂₇₁

—

C₂₇₁⋊C₈

C₄⋊C₄

yes

—

C₂₇₁⋊C₈

C₄⋊C₄

yes

—

C₂³×C₂₇₁

#41

yes

characteristic

C₂×C₄×C₂₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₇₁

#41

yes

characteristic

C₈×C₂₇₁

—

C₈×C₂₇₁

C₄²

yes

—

C₈×C₂₇₁

C₄²

yes

—

C₂²×C₂₇₁

—

C₂²×C₂₇₁

#117

yes

characteristic

C₂²×C₂₇₁

C₄×D₄

yes

characteristic

C₂²×C₂₇₁

C₄×Q₈

yes

characteristic

C₄×C₂₇₁

C₂×C₄²

yes

characteristic

C₄×C₂₇₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₇₁

C₄×D₄

yes

characteristic

C₄×C₂₇₁

C₄×Q₈

yes

characteristic

Dic₂₇₁

—

Dic₂₇₁

—

C₂×C₂₇₁

—

C₂×C₂₇₁

—

C₂×C₂₇₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₇₁

#443

yes

—

C₂₇₁

#1772

yes

characteristic, complemented, Sylow

#1772

—

271

maximal, Sylow

C₂×#108

D₂₇₁

yes

characteristic

#427

—

271

—

#405

—

271

—

C₂×C₄×C₈

—

271

—

#339

—

271

—

C₂×M₁₆

C₂×D₂₇₁

yes

—

C₂²×C₈

—

271

—

C₂²×C₈

C₂×D₂₇₁

yes

characteristic

#117

—

271

—

C₂×C₄²

—

271

—

#108

Dic₂₇₁

yes

complemented

#107

—

271

—

#106

—

271

—

#97

—

542

—

C₄×C₈

—

271

—

C₄×C₈

—

542

—

C₂²×C₄

—

542

—

C₂²×C₄

C₂²×D₂₇₁

yes

characteristic

M₁₆

—

M₁₆

C₂×Dic₂₇₁

yes

—

C₂×C₈

—

C₂×C₈

—

271

—

C₂×C₈

—

271

—

C₂×C₈

—

271

—

C₂×C₈

—

271

—

C₂×C₈

C₄×D₂₇₁

yes

—

C₂×C₈

C₄×D₂₇₁

yes

characteristic

C₂×C₈

C₄×D₂₇₁

yes

characteristic

C₂×C₈

C₂×Dic₂₇₁

yes

—

C₄⋊C₄

—

271

—

C₂²⋊C₄

—

542

—

C₄²

—

271

—

C₄²

—

542

—

C₄²

—

542

—

100

C₂³

#16864655

yes

characteristic

101

C₂×C₄

—

542

—

102

C₂×C₄

—

542

—

103

C₂×C₄

—

542

—

104

C₂×C₄

C₂×C₄×D₂₇₁

yes

characteristic

105

C₂×C₄

C₂×C₄×D₂₇₁

yes

characteristic

106

C₂×C₄

C₂²×Dic₂₇₁

yes

characteristic

107

C₂×C₄

D₄×D₂₇₁

yes

characteristic

108

C₂×C₄

Q₈×D₂₇₁

yes

characteristic

109

C₂×C₄

#16864653

yes

characteristic

110

C₈

—

111

C₈

—

271

—

112

C₈

—

271

—

113

C₈

C₄×Dic₂₇₁

yes

—

114

C₈

C₄×Dic₂₇₁

yes

—

115

C₂²

—

116

C₂²

#22576118

yes

characteristic

117

C₂²

#22576122

yes

characteristic

118

C₂²

Q₈×Dic₂₇₁

yes

characteristic

119

C₄

—

542

—

120

C₄

—

1084

—

121

C₄

C₂×C₄×Dic₂₇₁

yes

characteristic

122

C₄

C₄⋊C₄×D₂₇₁

yes

characteristic

123

C₄

D₄×Dic₂₇₁

yes

characteristic

124

C₄

#22576125

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

C₄⋊C₄×Dic₂₇₁

yes

characteristic

128

C₂

#31774013

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₇₁
commutator	C₄×C₂₇₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₇₁
Fitting	C₂×#108×C₂₇₁	C₂
socle	C₂²×C₂₇₁	C₄×D₄
prime radical	C₂³×C₂₇₁	#41
cosocle	C₂²×C₄×C₂₇₁	C₂³
hypercenter	C₂×#108	D₂₇₁
hypocenter	C₂₇₁	#1772
LCS	G ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₂₇₁

commutator

C₄×C₂₇₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₇₁

Fitting

C₂×#108×C₂₇₁

C₂

socle

C₂²×C₂₇₁

C₄×D₄

prime radical

C₂³×C₂₇₁

#41

cosocle

C₂²×C₄×C₂₇₁

C₂³

D₂₇₁

C₂₇₁

G ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
9. #22576094	C₄	semidirect
10. #22576092	C₄	semidirect
13. C₈×Dic₂₇₁	C₄	semidirect
16. #108×C₂₇₁	C₄	semidirect
64. C₂₇₁	#1772	semidirect
76. #108	Dic₂₇₁	semidirect

H ≅ G/N

type

9. #22576094

C₄

semidirect

10. #22576092

C₄

semidirect

13. C₈×Dic₂₇₁

C₄

semidirect

16. #108×C₂₇₁

C₄

semidirect

64. C₂₇₁

#1772

semidirect

76. #108

Dic₂₇₁

semidirect

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

subgroup	index
#31773935	2
#31773855	2
C₂×C₈×Dic₂₇₁	2
#31773658	2
C₂×#108×C₂₇₁	2
#1772	271

subgroup

index

C₂×C₈×Dic₂₇₁

C₂×#108×C₂₇₁

271

Subgroups — #46836246

Conjugacy classes of subgroups

Complements

Maximal subgroups