Group #46836412 — subgroups

Subgroups — #46836412

Order 34688.

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

normal subgroups only

This group has 252 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	C₂×#22576095	C₂	1	2	yes	maximal, complemented
3	#31773881	C₂	1	1	yes	maximal, characteristic
4	C₂₇₁⋊#338	C₂	1	2	yes	maximal, complemented
5	#413×C₂₇₁	C₂	1	1	yes	maximal, characteristic
6	#22576167	—	2	4	no	—
7	C₂×#16864638	C₄	1	2	yes	—
8	C₂×#16864638	C₂²	1	2	yes	—
9	C₂×#16864638	C₂²	1	1	yes	characteristic
10	C₂²×C₂₇₁⋊C₈	C₄	1	2	yes	—
11	C₂²×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
12	#22576095	—	2	4	no	—
13	#22576079	—	2	1	no	—
14	#22576077	—	2	1	no	—
15	C₂×#41×C₂₇₁	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₂₇₁	C₂²	1	2	yes	—
17	#101×C₂₇₁	—	2	1	no	—
18	#100×C₂₇₁	—	2	1	no	—
19	#16864638	—	2	4	no	—
20	#16864638	—	2	2	no	—
21	#16864638	—	2	2	no	—
22	#16864638	—	2	2	no	—
23	C₂×C₂₇₁⋊C₈	—	2	4	no	—
24	C₂×C₂₇₁⋊C₈	—	2	2	no	—
25	C₂×C₂₇₁⋊C₈	C₂×C₄	1	2	yes	—
26	C₂×C₂₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₂₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₂₇₁⋊C₈	D₄	1	4	yes	—
29	#41×C₂₇₁	—	2	2	no	—
30	#41×C₂₇₁	—	2	2	no	—
31	C₂×Q₈×C₂₇₁	D₄	1	1	yes	characteristic
32	C₂×D₄×C₂₇₁	D₄	1	1	yes	characteristic
33	C₂×D₄×C₂₇₁	Q₈	1	2	yes	—
34	C₂²×C₄×C₂₇₁	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
36	M₁₆×C₂₇₁	—	2	2	no	—
37	M₁₆×C₂₇₁	—	2	2	no	—
38	C₂×C₈×C₂₇₁	—	2	2	no	—
39	C₂₇₁⋊C₈	—	2	2	no	—
40	C₂₇₁⋊C₈	—	4	2	no	—
41	C₂₇₁⋊C₈	C₂²⋊C₄	1	2	yes	—
42	C₂₇₁⋊C₈	C₂²⋊C₄	1	2	yes	—
43	C₂³×C₂₇₁	C₄⋊C₄	1	2	yes	—
44	C₂³×C₂₇₁	C₂×Q₈	1	1	yes	characteristic
45	Q₈×C₂₇₁	—	2	2	no	—
46	D₄×C₂₇₁	—	2	4	no	—
47	D₄×C₂₇₁	—	2	2	no	—
48	C₂×C₄×C₂₇₁	—	2	2	no	—
49	C₂×C₄×C₂₇₁	—	2	2	no	—
50	C₂×C₄×C₂₇₁	C₄⋊C₄	1	2	yes	—
51	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₂₇₁	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₇₁	#41	1	2	yes	—
55	C₈×C₂₇₁	—	4	2	no	—
56	C₂²×C₂₇₁	—	2	2	no	—
57	C₂²×C₂₇₁	—	4	1	no	—
58	C₂²×C₂₇₁	—	4	2	no	—
59	C₂²×C₂₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₂₇₁	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₂₇₁	—	2	2	no	—
62	C₄×C₂₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₂₇₁	#117	1	1	yes	characteristic
64	C₄×C₂₇₁	#121	1	2	yes	—
65	C₂×C₂₇₁	—	2	1	no	—
66	C₂×C₂₇₁	—	2	2	no	—
67	C₂×C₂₇₁	—	4	2	no	—
68	C₂×C₂₇₁	#380	1	1	yes	characteristic
69	C₂₇₁	#1813	1	1	yes	characteristic, complemented, Sylow
70	#1813	—	271	1	no	maximal, Sylow
71	C₂×#131	—	271	1	no	—
72	C₂×#108	—	271	2	no	—
73	#413	—	271	1	no	—
74	#413	D₂₇₁	1	1	yes	characteristic
75	#338	—	271	2	no	—
76	C₂×#41	C₂×D₂₇₁	1	1	yes	characteristic
77	#131	—	542	4	no	—
78	C₂×M₁₆	—	271	2	no	—
79	C₂×M₁₆	—	271	1	no	—
80	C₂×M₁₆	—	271	2	no	—
81	C₂×M₁₆	C₂×D₂₇₁	1	2	yes	—
82	C₂²×C₈	—	271	1	no	—
83	C₂²×C₈	—	271	2	no	—
84	#108	—	542	4	no	—
85	#101	—	2	1	no	—
86	#101	—	542	1	no	—
87	#100	—	2	1	no	—
88	#100	—	542	1	no	—
89	#41	—	2	2	no	—
90	#41	—	2	2	no	—
91	C₂×Q₈	C₂²⋊D₂₇₁	1	1	yes	characteristic
92	C₂×D₄	C₂²⋊D₂₇₁	1	1	yes	characteristic
93	C₂×D₄	Dic₅₄₂	1	2	yes	—
94	C₂²×C₄	C₄×D₂₇₁	1	2	yes	—
95	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
96	M₁₆	—	2	2	no	—
97	M₁₆	—	2	2	no	—
98	M₁₆	—	542	4	no	—
99	M₁₆	—	542	2	no	—
100	M₁₆	—	542	2	no	—
101	M₁₆	—	542	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	271	1	no	—
104	C₂×C₈	—	271	1	no	—
105	C₂×C₈	—	271	4	no	—
106	C₂×C₈	—	271	2	no	—
107	C₂×C₈	—	542	4	no	—
108	C₂×C₈	—	542	2	no	—
109	C₂³	Dic₂₇₁⋊C₄	1	2	yes	—
110	C₂³	C₂×Dic₅₄₂	1	1	yes	characteristic
111	Q₈	—	2	2	no	—
112	D₄	—	2	4	no	—
113	D₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	Dic₂₇₁⋊C₄	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₂₇₁	1	1	yes	characteristic
118	C₂×C₄	D₄×D₂₇₁	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₂₇₁	1	1	yes	characteristic
120	C₂×C₄	#16864653	1	2	yes	—
121	C₈	—	4	2	no	—
122	C₈	—	271	2	no	—
123	C₈	—	271	2	no	—
124	C₈	—	542	2	no	—
125	C₈	—	1084	2	no	—
126	C₂²	—	2	2	no	—
127	C₂²	—	4	1	no	—
128	C₂²	—	4	2	no	—
129	C₂²	C₂×Dic₂₇₁⋊C₄	1	1	yes	characteristic
130	C₂²	#22576137	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	C₂²⋊C₄×D₂₇₁	1	1	yes	characteristic
133	C₄	#22576116	1	1	yes	characteristic
134	C₄	#22576132	1	2	yes	—
135	C₂	—	2	1	no	—
136	C₂	—	2	2	no	—
137	C₂	—	4	2	no	—
138	C₂	#31773755	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22576167

C₂

yes

maximal, characteristic

C₂×#22576095

C₂

yes

maximal, complemented

#31773881

C₂

yes

maximal, characteristic

C₂₇₁⋊#338

C₂

yes

maximal, complemented

#413×C₂₇₁

C₂

yes

maximal, characteristic

#22576167

—

C₂×#16864638

C₄

yes

—

C₂×#16864638

C₂²

yes

—

C₂×#16864638

C₂²

yes

characteristic

C₂²×C₂₇₁⋊C₈

C₄

yes

—

C₂²×C₂₇₁⋊C₈

C₂²

yes

—

—

—

C₂×#41×C₂₇₁

C₂²

yes

characteristic

C₂×M₁₆×C₂₇₁

C₂²

yes

—

#101×C₂₇₁

—

#100×C₂₇₁

—

—

—

—

—

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

—

#41×C₂₇₁

—

#41×C₂₇₁

—

C₂×Q₈×C₂₇₁

D₄

yes

characteristic

C₂×D₄×C₂₇₁

D₄

yes

characteristic

C₂×D₄×C₂₇₁

Q₈

yes

—

C₂²×C₄×C₂₇₁

C₂×C₄

yes

—

C₂²×C₄×C₂₇₁

C₂³

yes

characteristic

M₁₆×C₂₇₁

—

M₁₆×C₂₇₁

—

C₂×C₈×C₂₇₁

—

C₂₇₁⋊C₈

—

C₂₇₁⋊C₈

—

C₂₇₁⋊C₈

C₂²⋊C₄

yes

—

C₂₇₁⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₂₇₁

C₄⋊C₄

yes

—

C₂³×C₂₇₁

C₂×Q₈

yes

characteristic

Q₈×C₂₇₁

—

D₄×C₂₇₁

—

D₄×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×D₄

yes

characteristic

C₂×C₄×C₂₇₁

#41

yes

—

C₈×C₂₇₁

—

C₂²×C₂₇₁

—

C₂²×C₂₇₁

—

C₂²×C₂₇₁

—

C₂²×C₂₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₇₁

C₂²⋊Q₈

yes

—

C₄×C₂₇₁

—

C₄×C₂₇₁

C₂×C₂²⋊C₄

yes

C₄×C₂₇₁

yes

C₄×C₂₇₁

yes

—

C₂×C₂₇₁

—

C₂×C₂₇₁

—

C₂×C₂₇₁

—

C₂×C₂₇₁

#380

yes

characteristic

C₂₇₁

#1813

yes

characteristic, complemented, Sylow

#1813

—

271

maximal, Sylow

C₂×#131

—

271

—

C₂×#108

—

271

—

#413

—

271

—

#413

D₂₇₁

yes

characteristic

#338

—

271

—

C₂×#41

C₂×D₂₇₁

yes

characteristic

#131

—

542

—

C₂×M₁₆

—

271

—

C₂×M₁₆

—

271

—

C₂×M₁₆

—

271

—

C₂×M₁₆

C₂×D₂₇₁

yes

—

C₂²×C₈

—

271

—

C₂²×C₈

—

271

—

#108

—

542

—

#101

—

#101

—

542

—

#100

—

#100

—

542

—

#41

—

#41

—

C₂×Q₈

C₂²⋊D₂₇₁

yes

characteristic

C₂×D₄

C₂²⋊D₂₇₁

yes

characteristic

C₂×D₄

Dic₅₄₂

yes

—

C₂²×C₄

C₄×D₂₇₁

yes

—

C₂²×C₄

C₂²×D₂₇₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

542

—

M₁₆

—

542

—

100

M₁₆

—

542

—

101

M₁₆

—

542

—

102

C₂×C₈

—

103

C₂×C₈

—

271

—

104

C₂×C₈

—

271

—

105

C₂×C₈

—

271

—

106

C₂×C₈

—

271

—

107

C₂×C₈

—

542

—

108

C₂×C₈

—

542

—

109

C₂³

Dic₂₇₁⋊C₄

yes

—

110

C₂³

C₂×Dic₅₄₂

yes

characteristic

111

Q₈

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

Dic₂₇₁⋊C₄

yes

—

117

C₂×C₄

C₂×C₄×D₂₇₁

yes

characteristic

118

C₂×C₄

D₄×D₂₇₁

yes

—

119

C₂×C₄

C₂×C₂²⋊D₂₇₁

yes

characteristic

120

C₂×C₄

#16864653

yes

—

121

C₈

—

122

C₈

—

271

—

123

C₈

—

271

—

124

C₈

—

542

—

125

C₈

—

1084

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×Dic₂₇₁⋊C₄

yes

characteristic

130

C₂²

#22576137

yes

—

131

C₄

—

132

C₄

C₂²⋊C₄×D₂₇₁

yes

characteristic

133

C₄

#22576116

yes

characteristic

134

C₄

#22576132

yes

—

135

C₂

—

136

C₂

—

137

C₂

—

138

C₂

#31773755

yes

characteristic

	subgroup	G/H
center	C₄	#22576116
commutator	C₂×C₄×C₂₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₇₁
Fitting	#413×C₂₇₁	C₂
socle	C₂×C₂₇₁	#380
prime radical	C₂×#41×C₂₇₁	C₂²
cosocle	C₂²×C₄×C₂₇₁	C₂³
hypercenter	#413	D₂₇₁
hypocenter	C₂₇₁	#1813
LCS	G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

#22576116

commutator

C₂×C₄×C₂₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₇₁

Fitting

#413×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 ◁ #413

N	H ≅ G/N	type
2. C₂×#22576095	C₂	semidirect
4. C₂₇₁⋊#338	C₂	semidirect
69. C₂₇₁	#1813	semidirect

H ≅ G/N

type

2. C₂×#22576095

C₂

semidirect

4. C₂₇₁⋊#338

C₂

semidirect

69. C₂₇₁

#1813

semidirect

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

subgroup	index
C₂×#22576167	2
C₂×#22576095	2
#31773881	2
C₂₇₁⋊#338	2
#413×C₂₇₁	2
#1813	271

subgroup

index

C₂×#22576167

C₂×#22576095

#31773881

C₂₇₁⋊#338

#413×C₂₇₁

#1813

271

Subgroups — #46836412

Conjugacy classes of subgroups

Complements

Maximal subgroups