Group #46836413 — subgroups

Subgroups — #46836413

Order 34688.

252 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 81 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×#22576168	C₂	1	1	yes	maximal, characteristic
2	C₂×#22576095	C₂	1	2	yes	maximal, complemented
3	#31773879	C₂	1	2	yes	maximal
4	C₂₇₁⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
5	#338×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16864654	C₂²	1	1	yes	characteristic
7	#22576168	—	2	4	no	—
8	C₂×#16864638	C₂²	1	2	yes	—
9	C₂×#16864637	C₄	1	2	yes	—
10	C₂²×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₂₇₁	C₄	1	2	yes	—
12	#22576095	—	2	4	no	—
13	#22576078	—	2	2	no	—
14	#22576076	—	2	2	no	—
15	C₂×M₁₆×C₂₇₁	C₂²	1	2	yes	—
16	C₂×M₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
17	#16864654	—	2	2	no	—
18	#16864654	—	2	2	no	—
19	C₂×Dic₅₄₂	D₄	1	1	yes	characteristic
20	C₂×D₁₀₈₄	D₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₂₇₁	Q₈	1	2	yes	—
22	C₂×C₄×D₂₇₁	C₂×C₄	1	2	yes	—
23	#16864638	—	2	2	no	—
24	#16864638	—	2	2	no	—
25	#16864637	—	2	4	no	—
26	C₂×C₂₇₁⋊C₈	—	2	2	no	—
27	C₂×C₂₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₂₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₂₇₁⋊C₈	D₄	1	4	yes	—
30	C₈×D₂₇₁	—	2	4	no	—
31	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
32	M₁₆×C₂₇₁	—	2	2	no	—
33	M₁₆×C₂₇₁	—	2	2	no	—
34	M₁₆×C₂₇₁	—	2	2	no	—
35	C₂×C₈×C₂₇₁	—	2	2	no	—
36	C₂×C₈×C₂₇₁	C₂×C₄	1	2	yes	—
37	C₂²×D₂₇₁	C₄⋊C₄	1	2	yes	—
38	Dic₅₄₂	—	2	2	no	—
39	D₁₀₈₄	—	2	2	no	—
40	C₂²⋊D₂₇₁	—	2	4	no	—
41	C₂×Dic₂₇₁	C₄⋊C₄	1	2	yes	—
42	C₄×D₂₇₁	—	2	2	no	—
43	C₄×D₂₇₁	—	2	2	no	—
44	C₂₇₁⋊C₈	—	4	2	no	—
45	C₂₇₁⋊C₈	C₂²⋊C₄	1	2	yes	—
46	C₂₇₁⋊C₈	C₂²⋊C₄	1	2	yes	—
47	C₂³×C₂₇₁	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₂₇₁	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₇₁	#41	1	2	yes	—
52	C₈×C₂₇₁	—	2	2	no	—
53	C₈×C₂₇₁	—	4	2	no	—
54	C₂²×C₂₇₁	—	4	1	no	—
55	C₂²×C₂₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₂₇₁	C₂²⋊Q₈	1	2	yes	—
57	C₂×D₂₇₁	—	2	2	no	—
58	C₂×D₂₇₁	—	4	2	no	—
59	C₄×C₂₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₂₇₁	#117	1	1	yes	characteristic
61	C₄×C₂₇₁	#121	1	2	yes	—
62	Dic₂₇₁	—	2	2	no	—
63	C₂×C₂₇₁	—	2	1	no	—
64	C₂×C₂₇₁	—	2	2	no	—
65	C₂×C₂₇₁	#380	1	1	yes	characteristic
66	D₂₇₁	—	4	2	no	—
67	C₂₇₁	#1813	1	1	yes	characteristic, complemented, Sylow
68	#1813	—	271	1	no	maximal, Sylow
69	C₂×#131	—	271	1	no	—
70	C₂×#108	—	271	2	no	—
71	#413	—	271	2	no	—
72	#338	—	271	1	no	—
73	#338	D₂₇₁	1	1	yes	characteristic, complemented
74	C₂×#41	—	271	1	no	—
75	#131	—	542	4	no	—
76	C₂×M₁₆	—	271	2	no	—
77	C₂×M₁₆	—	271	2	no	—
78	C₂×M₁₆	C₂×D₂₇₁	1	2	yes	—
79	C₂×M₁₆	C₂×D₂₇₁	1	1	yes	characteristic
80	C₂²×C₈	—	271	1	no	—
81	C₂²×C₈	—	271	2	no	—
82	#108	—	542	4	no	—
83	#101	—	542	2	no	—
84	#100	—	542	2	no	—
85	#41	—	542	2	no	—
86	#41	—	542	2	no	—
87	C₂×Q₈	—	271	1	no	—
88	C₂×D₄	—	271	2	no	—
89	C₂×D₄	—	271	1	no	—
90	C₂²×C₄	—	271	2	no	—
91	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	542	4	no	—
96	M₁₆	—	542	2	no	—
97	M₁₆	—	542	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	271	1	no	—
100	C₂×C₈	—	271	1	no	—
101	C₂×C₈	—	271	4	no	—
102	C₂×C₈	—	542	4	no	—
103	C₂×C₈	—	542	2	no	—
104	C₂×C₈	C₄×D₂₇₁	1	2	yes	—
105	C₂³	—	271	2	no	—
106	C₂³	Q₈×D₂₇₁	1	1	yes	characteristic
107	Q₈	—	542	2	no	—
108	D₄	—	542	4	no	—
109	D₄	—	542	2	no	—
110	C₂×C₄	—	271	2	no	—
111	C₂×C₄	—	542	2	no	—
112	C₂×C₄	—	542	2	no	—
113	C₂×C₄	C₂×C₄×D₂₇₁	1	1	yes	characteristic
114	C₂×C₄	D₄×D₂₇₁	1	1	yes	characteristic
115	C₂×C₄	D₄×D₂₇₁	1	2	yes	—
116	C₂×C₄	#16864654	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	271	2	no	—
120	C₈	—	271	2	no	—
121	C₈	—	1084	2	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	542	2	no	—
124	C₂²	—	1084	2	no	—
125	C₂²	C₄⋊C₄×D₂₇₁	1	1	yes	characteristic
126	C₂²	#22576138	1	2	yes	—
127	C₄	—	542	2	no	—
128	C₄	C₂²⋊C₄×D₂₇₁	1	1	yes	characteristic
129	C₄	#22576117	1	1	yes	characteristic
130	C₄	#22576133	1	2	yes	—
131	C₂	—	2	1	no	—
132	C₂	—	2	2	no	—
133	C₂	—	1084	2	no	—
134	C₂	#31773756	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22576168

C₂

yes

maximal, characteristic

C₂×#22576095

C₂

yes

maximal, complemented

#31773879

C₂

yes

maximal

C₂₇₁⋊#338

C₂

yes

maximal, characteristic, complemented

#338×C₂₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#16864654

C₂²

yes

characteristic

#22576168

—

C₂×#16864638

C₂²

yes

—

C₂×#16864637

C₄

yes

—

C₂²×C₂₇₁⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₂₇₁

C₄

yes

—

#22576095

—

#22576078

—

#22576076

—

C₂×M₁₆×C₂₇₁

C₂²

yes

—

C₂×M₁₆×C₂₇₁

C₂²

yes

characteristic

#16864654

—

#16864654

—

C₂×Dic₅₄₂

D₄

yes

characteristic

C₂×D₁₀₈₄

D₄

yes

characteristic

C₂×C₂²⋊D₂₇₁

Q₈

yes

—

C₂×C₄×D₂₇₁

C₂×C₄

yes

—

#16864638

—

#16864638

—

#16864637

—

C₂×C₂₇₁⋊C₈

—

C₂×C₂₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₇₁⋊C₈

D₄

yes

—

C₈×D₂₇₁

—

C₂²×C₄×C₂₇₁

C₂³

yes

characteristic

M₁₆×C₂₇₁

—

M₁₆×C₂₇₁

—

M₁₆×C₂₇₁

—

C₂×C₈×C₂₇₁

—

C₂×C₈×C₂₇₁

C₂×C₄

yes

—

C₂²×D₂₇₁

C₄⋊C₄

yes

—

Dic₅₄₂

—

D₁₀₈₄

—

C₂²⋊D₂₇₁

—

C₂×Dic₂₇₁

C₄⋊C₄

yes

—

C₄×D₂₇₁

—

C₄×D₂₇₁

—

C₂₇₁⋊C₈

—

C₂₇₁⋊C₈

C₂²⋊C₄

yes

—

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

—

C₄×C₂₇₁

C₂×C₂²⋊C₄

yes

C₄×C₂₇₁

yes

C₄×C₂₇₁

yes

—

Dic₂₇₁

—

C₂×C₂₇₁

—

C₂×C₂₇₁

—

C₂×C₂₇₁

#380

yes

characteristic

D₂₇₁

—

C₂₇₁

#1813

yes

characteristic, complemented, Sylow

#1813

—

271

maximal, Sylow

C₂×#131

—

271

—

C₂×#108

—

271

—

#413

—

271

—

#338

—

271

—

#338

D₂₇₁

yes

characteristic, complemented

C₂×#41

—

271

—

#131

—

542

—

C₂×M₁₆

—

271

—

C₂×M₁₆

—

271

—

C₂×M₁₆

C₂×D₂₇₁

yes

—

C₂×M₁₆

C₂×D₂₇₁

yes

characteristic

C₂²×C₈

—

271

—

C₂²×C₈

—

271

—

#108

—

542

—

#101

—

542

—

#100

—

542

—

#41

—

542

—

#41

—

542

—

C₂×Q₈

—

271

—

C₂×D₄

—

271

—

C₂×D₄

—

271

—

C₂²×C₄

—

271

—

C₂²×C₄

C₂²×D₂₇₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

542

—

M₁₆

—

542

—

M₁₆

—

542

—

C₂×C₈

—

C₂×C₈

—

271

—

100

C₂×C₈

—

271

—

101

C₂×C₈

—

271

—

102

C₂×C₈

—

542

—

103

C₂×C₈

—

542

—

104

C₂×C₈

C₄×D₂₇₁

yes

—

105

C₂³

—

271

—

106

C₂³

Q₈×D₂₇₁

yes

characteristic

107

Q₈

—

542

—

108

D₄

—

542

—

109

D₄

—

542

—

110

C₂×C₄

—

271

—

111

C₂×C₄

—

542

—

112

C₂×C₄

—

542

—

113

C₂×C₄

C₂×C₄×D₂₇₁

yes

characteristic

114

C₂×C₄

D₄×D₂₇₁

yes

characteristic

115

C₂×C₄

D₄×D₂₇₁

yes

—

116

C₂×C₄

#16864654

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

271

—

120

C₈

—

271

—

121

C₈

—

1084

—

122

C₂²

—

123

C₂²

—

542

—

124

C₂²

—

1084

—

125

C₂²

C₄⋊C₄×D₂₇₁

yes

characteristic

126

C₂²

#22576138

yes

—

127

C₄

—

542

—

128

C₄

C₂²⋊C₄×D₂₇₁

yes

characteristic

129

C₄

#22576117

yes

characteristic

130

C₄

#22576133

yes

—

131

C₂

—

132

C₂

—

133

C₂

—

1084

—

134

C₂

#31773756

yes

characteristic

	subgroup	G/H
center	C₄	#22576117
commutator	C₂×C₄×C₂₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₇₁
Fitting	#338×C₂₇₁	C₂
socle	C₂×C₂₇₁	#380
prime radical	C₂×#16864654	C₂²
cosocle	C₂²×C₄×C₂₇₁	C₂³
hypercenter	#338	D₂₇₁
hypocenter	C₂₇₁	#1813
LCS	G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#22576117

commutator

C₂×C₄×C₂₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₇₁

Fitting

#338×C₂₇₁

C₂

C₂×C₂₇₁

C₂²

C₂²×C₄×C₂₇₁

C₂³

D₂₇₁

C₂₇₁

G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
2. C₂×#22576095	C₂	semidirect
4. C₂₇₁⋊#338	C₂	semidirect
5. #338×C₂₇₁	C₂	semidirect
67. C₂₇₁	#1813	semidirect
73. #338	D₂₇₁	semidirect

H ≅ G/N

type

2. C₂×#22576095

C₂

semidirect

4. C₂₇₁⋊#338

C₂

semidirect

5. #338×C₂₇₁

C₂

semidirect

67. C₂₇₁

#1813

semidirect

73. #338

D₂₇₁

semidirect

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

subgroup	index
C₂×#22576168	2
C₂×#22576095	2
#31773879	2
C₂₇₁⋊#338	2
#338×C₂₇₁	2
#1813	271

subgroup

index

C₂×#22576168

C₂×#22576095

#31773879

C₂₇₁⋊#338

#338×C₂₇₁

#1813

271

Subgroups — #46836413

Conjugacy classes of subgroups

Complements

Maximal subgroups