Group #46836405 — subgroups

Subgroups — #46836405

Order 34688.

272 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 85 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×#16864637	C₂	1	1	yes	maximal, characteristic
2	C₂×#22576095	C₂	1	2	yes	maximal, complemented
3	#31773860	C₂	1	2	yes	maximal
4	#31773662	C₂	1	1	yes	maximal, characteristic, complemented
5	#341×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₂₇₁	C₂²	1	1	yes	characteristic
7	C₂×#16864638	C₂²	1	2	yes	—
8	C₂×#16864637	—	2	4	no	—
9	C₂×#16864637	C₄	1	4	yes	—
10	C₂²×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
11	#22576095	—	2	4	no	—
12	(C₂×D₂₇₁)⋊C₈	—	2	4	no	—
13	C₂×M₁₆×C₂₇₁	C₂²	1	2	yes	—
14	C₂²×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
15	C₂³×D₂₇₁	Q₈	1	1	yes	characteristic
16	C₂²×Dic₂₇₁	Q₈	1	1	yes	characteristic
17	C₂×C₄×D₂₇₁	—	2	2	no	—
18	C₂×C₄×D₂₇₁	—	2	2	no	—
19	C₂×C₄×D₂₇₁	C₂×C₄	1	2	yes	—
20	C₂×C₄×D₂₇₁	D₄	1	2	yes	—
21	#16864638	—	2	2	no	—
22	#16864638	—	2	2	no	—
23	#16864637	—	2	8	no	—
24	C₂×C₂₇₁⋊C₈	—	2	2	no	—
25	C₂×C₂₇₁⋊C₈	C₂×C₄	1	2	yes	—
26	C₂×C₂₇₁⋊C₈	D₄	1	4	yes	—
27	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
28	M₁₆×C₂₇₁	—	2	2	no	—
29	M₁₆×C₂₇₁	—	2	2	no	—
30	C₂×C₈×C₂₇₁	—	2	2	no	—
31	C₂×C₈×C₂₇₁	—	2	2	no	—
32	C₂×C₈×C₂₇₁	C₂×C₄	1	2	yes	—
33	C₂²×D₂₇₁	—	2	2	no	—
34	C₂²×D₂₇₁	—	4	2	no	—
35	C₂²×D₂₇₁	C₄⋊C₄	1	2	yes	—
36	C₂×Dic₂₇₁	—	2	2	no	—
37	C₂×Dic₂₇₁	C₄⋊C₄	1	2	yes	—
38	C₄×D₂₇₁	—	2	2	no	—
39	C₄×D₂₇₁	—	2	2	no	—
40	C₄×D₂₇₁	—	2	4	no	—
41	C₂₇₁⋊C₈	—	4	2	no	—
42	C₂₇₁⋊C₈	C₂²⋊C₄	1	4	yes	—
43	C₂³×C₂₇₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₇₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₇₁	#41	1	2	yes	—
48	C₈×C₂₇₁	—	2	2	no	—
49	C₈×C₂₇₁	—	4	2	no	—
50	C₂²×C₂₇₁	—	2	2	no	—
51	C₂²×C₂₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₂₇₁	C₂²⋊Q₈	1	2	yes	—
53	C₂×D₂₇₁	—	2	2	no	—
54	C₂×D₂₇₁	—	2	4	no	—
55	C₂×D₂₇₁	—	4	4	no	—
56	C₄×C₂₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₄×C₂₇₁	#117	1	1	yes	characteristic
58	C₄×C₂₇₁	#121	1	2	yes	—
59	Dic₂₇₁	—	2	2	no	—
60	C₂×C₂₇₁	—	2	2	no	—
61	C₂×C₂₇₁	#380	1	1	yes	characteristic
62	C₂×C₂₇₁	#430	1	2	yes	—
63	D₂₇₁	—	4	2	no	—
64	C₂₇₁	#1811	1	1	yes	characteristic, complemented, Sylow
65	#1811	—	271	1	no	maximal, Sylow
66	C₂²×M₁₆	—	271	1	no	—
67	C₂×#108	—	271	2	no	—
68	#407	—	271	2	no	—
69	#341	—	271	1	no	—
70	#341	D₂₇₁	1	1	yes	characteristic, complemented
71	C₂³×C₄	—	271	1	no	—
72	C₂×M₁₆	—	271	4	no	—
73	C₂×M₁₆	—	271	2	no	—
74	C₂×M₁₆	—	542	4	no	—
75	C₂×M₁₆	C₂×D₂₇₁	1	2	yes	—
76	C₂²×C₈	—	271	1	no	—
77	C₂²×C₈	C₂×D₂₇₁	1	1	yes	characteristic
78	#108	—	542	4	no	—
79	C₂²⋊C₈	—	542	4	no	—
80	C₂⁴	—	271	1	no	—
81	C₂²×C₄	—	271	1	no	—
82	C₂²×C₄	—	271	2	no	—
83	C₂²×C₄	—	271	2	no	—
84	C₂²×C₄	—	542	2	no	—
85	C₂²×C₄	—	542	2	no	—
86	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
87	M₁₆	—	2	2	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	542	8	no	—
90	M₁₆	—	542	2	no	—
91	M₁₆	—	542	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	271	2	no	—
95	C₂×C₈	—	271	4	no	—
96	C₂×C₈	—	542	2	no	—
97	C₂×C₈	C₄×D₂₇₁	1	2	yes	—
98	C₂³	—	271	2	no	—
99	C₂³	—	542	2	no	—
100	C₂³	—	1084	2	no	—
101	C₂³	Q₈×D₂₇₁	1	1	yes	characteristic
102	C₂×C₄	—	271	2	no	—
103	C₂×C₄	—	542	2	no	—
104	C₂×C₄	—	542	4	no	—
105	C₂×C₄	—	542	2	no	—
106	C₂×C₄	—	542	2	no	—
107	C₂×C₄	C₂×C₄×D₂₇₁	1	1	yes	characteristic
108	C₂×C₄	D₄×D₂₇₁	1	2	yes	—
109	C₂×C₄	D₄×D₂₇₁	1	1	yes	characteristic
110	C₂×C₄	#16864654	1	2	yes	—
111	C₈	—	2	2	no	—
112	C₈	—	4	2	no	—
113	C₈	—	271	4	no	—
114	C₈	—	1084	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	—	542	2	no	—
117	C₂²	—	542	4	no	—
118	C₂²	—	1084	4	no	—
119	C₂²	C₄⋊C₄×D₂₇₁	1	1	yes	characteristic
120	C₂²	#22576138	1	2	yes	—
121	C₄	—	542	2	no	—
122	C₄	C₂²⋊C₄×D₂₇₁	1	1	yes	characteristic
123	C₄	#22576117	1	1	yes	characteristic
124	C₄	#22576133	1	2	yes	—
125	C₂	—	2	2	no	—
126	C₂	—	1084	2	no	—
127	C₂	#31773756	1	1	yes	characteristic
128	C₂	#31773944	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#16864637

C₂

yes

maximal, characteristic

C₂×#22576095

C₂

yes

maximal, complemented

#31773860

C₂

yes

maximal

#31773662

C₂

yes

maximal, characteristic, complemented

#341×C₂₇₁

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₂₇₁

C₂²

yes

characteristic

C₂×#16864638

C₂²

yes

—

C₂×#16864637

—

C₂×#16864637

C₄

yes

—

C₂²×C₂₇₁⋊C₈

C₂²

yes

characteristic

#22576095

—

(C₂×D₂₇₁)⋊C₈

—

C₂×M₁₆×C₂₇₁

C₂²

yes

—

C₂²×C₈×C₂₇₁

C₂²

yes

characteristic

C₂³×D₂₇₁

Q₈

yes

characteristic

C₂²×Dic₂₇₁

Q₈

yes

characteristic

C₂×C₄×D₂₇₁

—

C₂×C₄×D₂₇₁

—

C₂×C₄×D₂₇₁

C₂×C₄

yes

—

C₂×C₄×D₂₇₁

D₄

yes

—

#16864638

—

#16864638

—

#16864637

—

C₂×C₂₇₁⋊C₈

—

C₂×C₂₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₇₁⋊C₈

D₄

yes

—

C₂²×C₄×C₂₇₁

C₂³

yes

characteristic

M₁₆×C₂₇₁

—

M₁₆×C₂₇₁

—

C₂×C₈×C₂₇₁

—

C₂×C₈×C₂₇₁

—

C₂×C₈×C₂₇₁

C₂×C₄

yes

—

C₂²×D₂₇₁

—

C₂²×D₂₇₁

—

C₂²×D₂₇₁

C₄⋊C₄

yes

—

C₂×Dic₂₇₁

—

C₂×Dic₂₇₁

C₄⋊C₄

yes

—

C₄×D₂₇₁

—

C₄×D₂₇₁

—

C₄×D₂₇₁

—

C₂₇₁⋊C₈

—

C₂₇₁⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₂₇₁

C₂×Q₈

yes

characteristic

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×C₂₇₁

—

C₂²×C₂₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₇₁

C₂²⋊Q₈

yes

—

C₂×D₂₇₁

—

C₂×D₂₇₁

—

C₂×D₂₇₁

—

C₄×C₂₇₁

C₂×C₂²⋊C₄

yes

C₄×C₂₇₁

yes

C₄×C₂₇₁

yes

—

Dic₂₇₁

—

C₂×C₂₇₁

—

C₂×C₂₇₁

#380

yes

characteristic

C₂×C₂₇₁

#430

yes

—

D₂₇₁

—

C₂₇₁

#1811

yes

characteristic, complemented, Sylow

#1811

—

271

maximal, Sylow

C₂²×M₁₆

—

271

—

C₂×#108

—

271

—

#407

—

271

—

#341

—

271

—

#341

D₂₇₁

yes

characteristic, complemented

C₂³×C₄

—

271

—

C₂×M₁₆

—

271

—

C₂×M₁₆

—

271

—

C₂×M₁₆

—

542

—

C₂×M₁₆

C₂×D₂₇₁

yes

—

C₂²×C₈

—

271

—

C₂²×C₈

C₂×D₂₇₁

yes

characteristic

#108

—

542

—

C₂²⋊C₈

—

542

—

C₂⁴

—

271

—

C₂²×C₄

—

271

—

C₂²×C₄

—

271

—

C₂²×C₄

—

271

—

C₂²×C₄

—

542

—

C₂²×C₄

—

542

—

C₂²×C₄

C₂²×D₂₇₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

542

—

M₁₆

—

542

—

M₁₆

—

542

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

271

—

C₂×C₈

—

271

—

C₂×C₈

—

542

—

C₂×C₈

C₄×D₂₇₁

yes

—

C₂³

—

271

—

C₂³

—

542

—

100

C₂³

—

1084

—

101

C₂³

Q₈×D₂₇₁

yes

characteristic

102

C₂×C₄

—

271

—

103

C₂×C₄

—

542

—

104

C₂×C₄

—

542

—

105

C₂×C₄

—

542

—

106

C₂×C₄

—

542

—

107

C₂×C₄

C₂×C₄×D₂₇₁

yes

characteristic

108

C₂×C₄

D₄×D₂₇₁

yes

—

109

C₂×C₄

D₄×D₂₇₁

yes

characteristic

110

C₂×C₄

#16864654

yes

—

111

C₈

—

112

C₈

—

113

C₈

—

271

—

114

C₈

—

1084

—

115

C₂²

—

116

C₂²

—

542

—

117

C₂²

—

542

—

118

C₂²

—

1084

—

119

C₂²

C₄⋊C₄×D₂₇₁

yes

characteristic

120

C₂²

#22576138

yes

—

121

C₄

—

542

—

122

C₄

C₂²⋊C₄×D₂₇₁

yes

characteristic

123

C₄

#22576117

yes

characteristic

124

C₄

#22576133

yes

—

125

C₂

—

126

C₂

—

1084

—

127

C₂

#31773756

yes

characteristic

128

C₂

#31773944

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₇₁
commutator	C₂×C₄×C₂₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₇₁
Fitting	#341×C₂₇₁	C₂
socle	C₂²×C₂₇₁	C₂×C₄⋊C₄
prime radical	C₂³×D₂₇₁	Q₈
cosocle	C₂²×C₄×C₂₇₁	C₂³
hypercenter	#341	D₂₇₁
hypocenter	C₂₇₁	#1811
LCS	G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₂₇₁

commutator

C₂×C₄×C₂₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₇₁

Fitting

#341×C₂₇₁

C₂

socle

C₂²×C₂₇₁

C₂×C₄⋊C₄

prime radical

C₂³×D₂₇₁

Q₈

cosocle

C₂²×C₄×C₂₇₁

C₂³

D₂₇₁

C₂₇₁

G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
2. C₂×#22576095	C₂	semidirect
4. #31773662	C₂	semidirect
5. #341×C₂₇₁	C₂	semidirect
64. C₂₇₁	#1811	semidirect
70. #341	D₂₇₁	semidirect

H ≅ G/N

type

2. C₂×#22576095

C₂

semidirect

4. #31773662

C₂

semidirect

5. #341×C₂₇₁

C₂

semidirect

64. C₂₇₁

#1811

semidirect

70. #341

D₂₇₁

semidirect

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

subgroup	index
C₂²×#16864637	2
C₂×#22576095	2
#31773860	2
#31773662	2
#341×C₂₇₁	2
#1811	271

subgroup

index

#341×C₂₇₁

271

Subgroups — #46836405

Conjugacy classes of subgroups

Complements

Maximal subgroups