Group #46836408 — subgroups

Subgroups — #46836408

Order 34688.

260 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 89 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×#22576167	C₂	1	1	yes	maximal, characteristic
2	#31773933	C₂	1	2	yes	maximal, complemented
3	#31773871	C₂	1	1	yes	maximal, characteristic
4	#31773653	C₂	1	2	yes	maximal, complemented
5	#410×C₂₇₁	C₂	1	1	yes	maximal, characteristic
6	#22576167	—	2	2	no	—
7	#22576167	—	2	2	no	—
8	C₂×#16864638	C₄	1	2	yes	complemented
9	C₂×#16864638	C₂²	1	1	yes	characteristic
10	C₂²×C₂₇₁⋊C₈	C₄	1	2	yes	complemented
11	C₂²×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
12	#22576119	C₂²	1	2	yes	—
13	#22576093	—	2	2	no	—
14	#22576091	—	2	2	no	—
15	#22576075	—	2	2	no	—
16	C₂×#41×C₂₇₁	C₂²	1	1	yes	characteristic
17	#117×C₂₇₁	C₂²	1	2	yes	—
18	#99×C₂₇₁	—	2	2	no	—
19	#16864638	—	2	4	no	—
20	#16864638	—	2	2	no	—
21	C₂×C₂₇₁⋊C₈	—	2	2	no	—
22	C₂×C₂₇₁⋊C₈	—	2	2	no	—
23	C₂×C₂₇₁⋊C₈	C₂×C₄	1	2	yes	—
24	C₂×C₂₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₂₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₂₇₁⋊C₈	D₄	1	2	yes	—
27	C₂×C₂₇₁⋊C₈	D₄	1	2	yes	—
28	C₄⋊Dic₂₇₁	—	2	2	no	—
29	C₂²⋊Dic₂₇₁	—	2	2	no	—
30	C₄×Dic₂₇₁	—	2	2	no	—
31	#41×C₂₇₁	—	2	2	no	—
32	#41×C₂₇₁	—	2	2	no	—
33	C₂×Q₈×C₂₇₁	Q₈	1	1	yes	characteristic
34	C₂×D₄×C₂₇₁	D₄	1	2	yes	—
35	C₂×D₄×C₂₇₁	Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₂₇₁	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₂₇₁	—	2	2	no	—
39	C₂²⋊C₄×C₂₇₁	—	2	2	no	—
40	C₄²×C₂₇₁	—	2	2	no	—
41	C₂×Dic₂₇₁	—	4	2	no	—
42	C₂₇₁⋊C₈	—	2	2	no	—
43	C₂₇₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
44	C₂₇₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
45	C₂₇₁⋊C₈	C₂²⋊C₄	1	2	yes	—
46	C₂³×C₂₇₁	C₄⋊C₄	1	2	yes	—
47	C₂³×C₂₇₁	C₂×D₄	1	1	yes	characteristic
48	Q₈×C₂₇₁	—	2	2	no	—
49	D₄×C₂₇₁	—	2	4	no	—
50	D₄×C₂₇₁	—	2	2	no	—
51	C₂×C₄×C₂₇₁	—	2	2	no	—
52	C₂×C₄×C₂₇₁	—	2	2	no	—
53	C₂×C₄×C₂₇₁	—	4	2	no	—
54	C₂×C₄×C₂₇₁	C₄⋊C₄	1	2	yes	—
55	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₂₇₁	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₂₇₁	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₂₇₁	#41	1	2	yes	—
59	C₂²×C₂₇₁	—	2	2	no	—
60	C₂²×C₂₇₁	—	4	1	no	—
61	C₂²×C₂₇₁	—	4	2	no	—
62	C₂²×C₂₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₂₇₁	#121	1	2	yes	—
64	C₄×C₂₇₁	—	2	2	no	—
65	C₄×C₂₇₁	—	8	2	no	—
66	C₄×C₂₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₂₇₁	#117	1	1	yes	characteristic
68	C₄×C₂₇₁	C₂²⋊Q₈	1	2	yes	—
69	Dic₂₇₁	—	8	2	no	—
70	C₂×C₂₇₁	—	2	2	no	—
71	C₂×C₂₇₁	—	2	1	no	—
72	C₂×C₂₇₁	—	4	2	no	—
73	C₂×C₂₇₁	#380	1	1	yes	characteristic
74	C₂₇₁	#1812	1	1	yes	characteristic, complemented, Sylow
75	#1812	—	271	1	no	maximal, Sylow
76	C₂×#131	—	271	1	no	—
77	#427	—	271	2	no	—
78	#410	—	271	1	no	—
79	#410	D₂₇₁	1	1	yes	characteristic
80	H(3,ℤ/4)	—	271	2	no	—
81	C₂×#41	C₂×D₂₇₁	1	1	yes	characteristic
82	#131	—	542	2	no	—
83	#131	—	542	2	no	—
84	C₂×M₁₆	—	271	2	no	—
85	C₂×M₁₆	—	271	1	no	—
86	C₂²×C₈	—	271	1	no	—
87	C₂²×C₈	—	271	2	no	—
88	#117	—	271	2	no	—
89	#117	C₂×D₂₇₁	1	2	yes	—
90	#107	—	542	2	no	—
91	#106	—	542	2	no	—
92	#99	—	2	2	no	—
93	#99	—	542	2	no	—
94	#41	—	2	2	no	—
95	#41	—	2	2	no	—
96	C₂×Q₈	Dic₅₄₂	1	1	yes	characteristic
97	C₂×D₄	C₂²⋊D₂₇₁	1	2	yes	—
98	C₂×D₄	Dic₅₄₂	1	1	yes	characteristic
99	C₂²×C₄	C₄×D₂₇₁	1	2	yes	—
100	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
101	M₁₆	—	542	4	no	—
102	M₁₆	—	542	2	no	—
103	C₂×C₈	—	271	1	no	—
104	C₂×C₈	—	271	1	no	—
105	C₂×C₈	—	271	2	no	—
106	C₂×C₈	—	271	2	no	—
107	C₂×C₈	—	271	2	no	—
108	C₂×C₈	—	542	2	no	—
109	C₂×C₈	—	542	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	542	2	no	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₂²⋊C₄	—	542	2	no	—
114	C₄²	—	2	2	no	—
115	C₄²	—	542	2	no	—
116	C₂³	Dic₂₇₁⋊C₄	1	2	yes	—
117	C₂³	C₂×C₂²⋊D₂₇₁	1	1	yes	characteristic
118	Q₈	—	2	2	no	—
119	D₄	—	2	4	no	—
120	D₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	1084	2	no	—
125	C₂×C₄	Dic₂₇₁⋊C₄	1	2	yes	—
126	C₂×C₄	C₂×C₄×D₂₇₁	1	1	yes	characteristic
127	C₂×C₄	D₄×D₂₇₁	1	2	yes	—
128	C₂×C₄	C₂×Dic₅₄₂	1	1	yes	characteristic
129	C₂×C₄	#16864653	1	2	yes	—
130	C₈	—	271	1	no	—
131	C₈	—	271	1	no	—
132	C₈	—	271	2	no	—
133	C₈	—	542	2	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	2	no	—
137	C₂²	C₂×Dic₂₇₁⋊C₄	1	1	yes	characteristic
138	C₂²	#22576132	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	8	2	no	—
141	C₄	—	2168	2	no	—
142	C₄	C₂²⋊C₄×D₂₇₁	1	1	yes	characteristic
143	C₄	#22576116	1	1	yes	characteristic
144	C₄	#22576137	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	4	2	no	—
148	C₂	#31773755	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22576167

C₂

yes

maximal, characteristic

#31773933

C₂

yes

maximal, complemented

#31773871

C₂

yes

maximal, characteristic

#31773653

C₂

yes

maximal, complemented

#410×C₂₇₁

C₂

yes

maximal, characteristic

#22576167

—

#22576167

—

C₂×#16864638

C₄

yes

complemented

C₂×#16864638

C₂²

yes

characteristic

C₂²×C₂₇₁⋊C₈

C₄

yes

complemented

C₂²×C₂₇₁⋊C₈

C₂²

yes

characteristic

#22576119

C₂²

yes

—

#22576093

—

#22576091

—

#22576075

—

C₂×#41×C₂₇₁

C₂²

yes

characteristic

#117×C₂₇₁

C₂²

yes

—

#99×C₂₇₁

—

#16864638

—

#16864638

—

C₂×C₂₇₁⋊C₈

—

C₂×C₂₇₁⋊C₈

—

C₂×C₂₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₇₁⋊C₈

D₄

yes

—

C₂×C₂₇₁⋊C₈

D₄

yes

—

C₄⋊Dic₂₇₁

—

C₂²⋊Dic₂₇₁

—

C₄×Dic₂₇₁

—

#41×C₂₇₁

—

#41×C₂₇₁

—

C₂×Q₈×C₂₇₁

Q₈

yes

characteristic

C₂×D₄×C₂₇₁

D₄

yes

—

C₂×D₄×C₂₇₁

Q₈

yes

characteristic

C₂²×C₄×C₂₇₁

C₂×C₄

yes

—

C₂²×C₄×C₂₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₇₁

—

C₂²⋊C₄×C₂₇₁

—

C₄²×C₂₇₁

—

C₂×Dic₂₇₁

—

C₂₇₁⋊C₈

—

C₂₇₁⋊C₈

C₂²⋊C₄

yes

characteristic

C₂₇₁⋊C₈

C₂²⋊C₄

yes

characteristic

C₂₇₁⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₂₇₁

C₄⋊C₄

yes

—

C₂³×C₂₇₁

C₂×D₄

yes

characteristic

Q₈×C₂₇₁

—

D₄×C₂₇₁

—

D₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₁

C₂×D₄

yes

—

C₂×C₄×C₂₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₇₁

#41

yes

—

C₂²×C₂₇₁

—

C₂²×C₂₇₁

—

C₂²×C₂₇₁

—

C₂²×C₂₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₇₁

#121

yes

—

C₄×C₂₇₁

—

C₄×C₂₇₁

—

C₄×C₂₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₇₁

#117

yes

characteristic

C₄×C₂₇₁

C₂²⋊Q₈

yes

—

Dic₂₇₁

—

C₂×C₂₇₁

—

C₂×C₂₇₁

—

C₂×C₂₇₁

—

C₂×C₂₇₁

#380

yes

characteristic

C₂₇₁

#1812

yes

characteristic, complemented, Sylow

#1812

—

271

maximal, Sylow

C₂×#131

—

271

—

#427

—

271

—

#410

—

271

—

#410

D₂₇₁

yes

characteristic

H(3,ℤ/4)

—

271

—

C₂×#41

C₂×D₂₇₁

yes

characteristic

#131

—

542

—

#131

—

542

—

C₂×M₁₆

—

271

—

C₂×M₁₆

—

271

—

C₂²×C₈

—

271

—

C₂²×C₈

—

271

—

#117

—

271

—

#117

C₂×D₂₇₁

yes

—

#107

—

542

—

#106

—

542

—

#99

—

#99

—

542

—

#41

—

#41

—

C₂×Q₈

Dic₅₄₂

yes

characteristic

C₂×D₄

C₂²⋊D₂₇₁

yes

—

C₂×D₄

Dic₅₄₂

yes

characteristic

C₂²×C₄

C₄×D₂₇₁

yes

—

100

C₂²×C₄

C₂²×D₂₇₁

yes

characteristic

101

M₁₆

—

542

—

102

M₁₆

—

542

—

103

C₂×C₈

—

271

—

104

C₂×C₈

—

271

—

105

C₂×C₈

—

271

—

106

C₂×C₈

—

271

—

107

C₂×C₈

—

271

—

108

C₂×C₈

—

542

—

109

C₂×C₈

—

542

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

542

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

542

—

114

C₄²

—

115

C₄²

—

542

—

116

C₂³

Dic₂₇₁⋊C₄

yes

—

117

C₂³

C₂×C₂²⋊D₂₇₁

yes

characteristic

118

Q₈

—

119

D₄

—

120

D₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

1084

—

125

C₂×C₄

Dic₂₇₁⋊C₄

yes

—

126

C₂×C₄

C₂×C₄×D₂₇₁

yes

characteristic

127

C₂×C₄

D₄×D₂₇₁

yes

—

128

C₂×C₄

C₂×Dic₅₄₂

yes

characteristic

129

C₂×C₄

#16864653

yes

—

130

C₈

—

271

—

131

C₈

—

271

—

132

C₈

—

271

—

133

C₈

—

542

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂×Dic₂₇₁⋊C₄

yes

characteristic

138

C₂²

#22576132

yes

—

139

C₄

—

140

C₄

—

141

C₄

—

2168

—

142

C₄

C₂²⋊C₄×D₂₇₁

yes

characteristic

143

C₄

#22576116

yes

characteristic

144

C₄

#22576137

yes

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#31773755

yes

characteristic

	subgroup	G/H
center	C₄	#22576116
commutator	C₂×C₄×C₂₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₇₁
Fitting	#410×C₂₇₁	C₂
socle	C₂×C₂₇₁	#380
prime radical	C₂×#41×C₂₇₁	C₂²
cosocle	C₂²×C₄×C₂₇₁	C₂³
hypercenter	#410	D₂₇₁
hypocenter	C₂₇₁	#1812
LCS	G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

subgroup

G/H

center

C₄

#22576116

commutator

C₂×C₄×C₂₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₇₁

Fitting

#410×C₂₇₁

C₂

socle

C₂×C₂₇₁

#380

prime radical

C₂×#41×C₂₇₁

C₂²

cosocle

C₂²×C₄×C₂₇₁

C₂³

D₂₇₁

C₂₇₁

G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

N	H ≅ G/N	type
2. #31773933	C₂	semidirect
4. #31773653	C₂	semidirect
8. C₂×#16864638	C₄	semidirect
10. C₂²×C₂₇₁⋊C₈	C₄	semidirect
74. C₂₇₁	#1812	semidirect

H ≅ G/N

type

2. #31773933

C₂

semidirect

4. #31773653

C₂

semidirect

8. C₂×#16864638

C₄

semidirect

10. C₂²×C₂₇₁⋊C₈

C₄

semidirect

74. C₂₇₁

#1812

semidirect

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

subgroup	index
C₂×#22576167	2
#31773933	2
#31773871	2
#31773653	2
#410×C₂₇₁	2
#1812	271

subgroup

index

#410×C₂₇₁

271

Subgroups — #46836408

Conjugacy classes of subgroups

Complements

Maximal subgroups