Group #46846740 — subgroups

Subgroups — #46846740

Order 34688.

484 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 141 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×#22576157	C₂	1	1	yes	maximal, characteristic
2	#31774263	C₂	1	4	yes	maximal, complemented
3	#31774245	C₂	1	4	yes	maximal, complemented
4	#31773901	C₂	1	4	yes	maximal, complemented
5	C₄×#16864638	C₂	1	1	yes	maximal, characteristic, complemented
6	#533×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₁₀₈₄	—	2	2	no	—
8	C₂²×D₁₀₈₄	C₂²	1	2	yes	—
9	C₂×#16864638	C₂²	1	2	yes	—
10	#22576157	—	2	2	no	—
11	#22576157	C₂²	1	2	yes	—
12	#22576157	C₂²	1	2	yes	—
13	#22576080	C₂²	1	16	yes	complemented
14	#22576071	C₂²	1	2	yes	complemented
15	C₄×C₂₇₁⋊C₈	C₂²	1	2	yes	complemented
16	C₄⋊Q₈×C₂₇₁	C₂²	1	2	yes	complemented
17	#125×C₂₇₁	C₂²	1	2	yes	complemented
18	C₂²⋊Q₈×C₂₇₁	—	2	2	no	—
19	C₄×Q₈×C₂₇₁	—	2	2	no	—
20	#117×C₂₇₁	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₇₁	C₂²	1	1	yes	characteristic
22	C₂³×D₂₇₁	—	2	2	no	—
23	C₂×D₁₀₈₄	—	2	2	no	—
24	C₂×D₁₀₈₄	—	2	4	no	—
25	C₂×D₁₀₈₄	—	2	4	no	—
26	C₂×D₁₀₈₄	—	2	8	no	—
27	C₂×D₁₀₈₄	—	4	8	no	—
28	C₂×D₁₀₈₄	C₂³	1	4	yes	—
29	#16864638	—	2	4	no	—
30	C₂×C₂₇₁⋊C₈	C₂³	1	4	yes	—
31	C₂×Q₈×C₂₇₁	—	2	2	no	—
32	C₂²×C₄×C₂₇₁	D₄	1	2	yes	—
33	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₂₇₁	—	2	4	no	—
35	C₄⋊C₄×C₂₇₁	—	2	2	no	—
36	C₄⋊C₄×C₂₇₁	C₂³	1	4	yes	—
37	C₂²⋊C₄×C₂₇₁	—	2	2	no	—
38	C₄²×C₂₇₁	—	2	2	no	—
39	C₄²×C₂₇₁	D₄	1	2	yes	—
40	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
41	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
42	C₂²×D₂₇₁	—	2	4	no	—
43	C₂²×D₂₇₁	—	4	8	no	—
44	C₂²×D₂₇₁	—	4	4	no	—
45	D₁₀₈₄	—	2	8	no	—
46	D₁₀₈₄	—	4	8	no	—
47	D₁₀₈₄	—	4	4	no	—
48	D₁₀₈₄	—	4	8	no	—
49	C₂₇₁⋊C₈	—	2	4	no	—
50	C₂³×C₂₇₁	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₂₇₁	—	4	4	no	—
52	C₂×C₄×C₂₇₁	—	2	4	no	—
53	C₂×C₄×C₂₇₁	—	2	4	no	—
54	C₂×C₄×C₂₇₁	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₂₇₁	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₂₇₁	#41	1	4	yes	—
58	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
59	C₂²×C₂₇₁	—	2	2	no	—
60	C₂²×C₂₇₁	#124	1	2	yes	—
61	C₂²×C₂₇₁	C₂²×D₄	1	1	yes	characteristic
62	C₂×D₂₇₁	—	4	4	no	—
63	C₂×D₂₇₁	—	4	8	no	—
64	C₂×D₂₇₁	—	8	8	no	—
65	C₄×C₂₇₁	—	2	2	no	—
66	C₄×C₂₇₁	—	4	4	no	—
67	C₄×C₂₇₁	#124	1	2	yes	—
68	C₄×C₂₇₁	AGL(1,ℤ/8)	1	4	yes	—
69	C₄×C₂₇₁	C₂×#41	1	2	yes	—
70	C₂×C₂₇₁	—	2	2	no	—
71	C₂×C₂₇₁	C₂×#124	1	1	yes	characteristic
72	C₂×C₂₇₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
73	D₂₇₁	—	8	4	no	—
74	C₂₇₁	#3138	1	1	yes	characteristic, complemented, Sylow
75	#3138	—	271	1	no	maximal, Sylow
76	#533	D₂₇₁	1	1	yes	characteristic, complemented
77	C₂×#127	—	271	1	no	—
78	#490	—	271	4	no	—
79	#486	—	271	4	no	—
80	#418	—	271	4	no	—
81	C₄×M₁₆	—	271	1	no	—
82	C₂²×D₄	—	271	2	no	—
83	C₂²×D₄	—	542	2	no	—
84	C₂×M₁₆	—	271	2	no	—
85	C₄⋊Q₈	C₂×D₂₇₁	1	2	yes	complemented
86	#127	—	271	2	no	—
87	#127	—	271	2	no	—
88	#127	—	542	2	no	—
89	#125	C₂×D₂₇₁	1	2	yes	complemented
90	C₂²⋊Q₈	—	2	2	no	—
91	C₄×Q₈	—	2	2	no	—
92	#117	C₂×D₂₇₁	1	2	yes	—
93	C₂×C₄²	C₂×D₂₇₁	1	1	yes	characteristic
94	#102	—	271	16	no	—
95	#97	—	271	2	no	—
96	C₄×C₈	—	271	2	no	—
97	C₂⁴	—	542	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×D₄	—	271	4	no	—
100	C₂×D₄	—	542	4	no	—
101	C₂×D₄	—	542	4	no	—
102	C₂×D₄	—	542	2	no	—
103	C₂×D₄	—	542	8	no	—
104	C₂×D₄	—	1084	8	no	—
105	C₂²×C₄	C₂²⋊D₂₇₁	1	2	yes	—
106	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
107	M₁₆	—	542	4	no	—
108	C₂×C₈	—	271	4	no	—
109	C₄⋊C₄	—	2	4	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	C₂²×D₂₇₁	1	4	yes	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	C₂²⋊D₂₇₁	1	2	yes	—
115	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
116	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
117	C₂³	—	542	4	no	—
118	C₂³	—	1084	8	no	—
119	C₂³	—	1084	4	no	—
120	C₂³	C₂×C₂²⋊D₂₇₁	1	1	yes	characteristic
121	Q₈	—	4	4	no	—
122	D₄	—	542	8	no	—
123	D₄	—	1084	8	no	—
124	D₄	—	1084	4	no	—
125	D₄	—	1084	8	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	C₂×C₂²⋊D₂₇₁	1	2	yes	—
129	C₂×C₄	C₂×C₂²⋊D₂₇₁	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₂₇₁	1	2	yes	—
131	C₂×C₄	#16864655	1	4	yes	—
132	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
133	C₈	—	542	4	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	1084	4	no	—
136	C₂²	—	1084	8	no	—
137	C₂²	—	2168	8	no	—
138	C₂²	#22576149	1	2	yes	—
139	C₂²	C₂²×C₂²⋊D₂₇₁	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	4	4	no	—
142	C₄	#22576149	1	2	yes	—
143	C₄	#22576185	1	4	yes	—
144	C₄	C₂×#16864655	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	2168	4	no	—
147	C₂	C₂×#22576149	1	1	yes	characteristic
148	C₂	C₂×#22576185	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22576157

C₂

yes

maximal, characteristic

#31774263

C₂

yes

maximal, complemented

#31774245

C₂

yes

maximal, complemented

#31773901

C₂

yes

maximal, complemented

C₄×#16864638

C₂

yes

maximal, characteristic, complemented

#533×C₂₇₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₀₈₄

—

C₂²×D₁₀₈₄

C₂²

yes

—

C₂×#16864638

C₂²

yes

—

#22576157

—

#22576157

C₂²

yes

—

#22576157

C₂²

yes

—

#22576080

C₂²

yes

complemented

#22576071

C₂²

yes

complemented

C₄×C₂₇₁⋊C₈

C₂²

yes

complemented

C₄⋊Q₈×C₂₇₁

C₂²

yes

complemented

#125×C₂₇₁

C₂²

yes

complemented

C₂²⋊Q₈×C₂₇₁

—

C₄×Q₈×C₂₇₁

—

#117×C₂₇₁

C₂²

yes

—

C₂×C₄²×C₂₇₁

C₂²

yes

characteristic

C₂³×D₂₇₁

—

C₂×D₁₀₈₄

—

C₂×D₁₀₈₄

—

C₂×D₁₀₈₄

—

C₂×D₁₀₈₄

—

C₂×D₁₀₈₄

—

C₂×D₁₀₈₄

C₂³

yes

—

#16864638

—

C₂×C₂₇₁⋊C₈

C₂³

yes

—

C₂×Q₈×C₂₇₁

—

C₂²×C₄×C₂₇₁

D₄

yes

—

C₂²×C₄×C₂₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₇₁

—

C₄⋊C₄×C₂₇₁

—

C₄⋊C₄×C₂₇₁

C₂³

yes

—

C₂²⋊C₄×C₂₇₁

—

C₄²×C₂₇₁

—

C₄²×C₂₇₁

D₄

yes

—

C₄²×C₂₇₁

C₂³

yes

characteristic

C₄²×C₂₇₁

C₂³

yes

characteristic

C₂²×D₂₇₁

—

C₂²×D₂₇₁

—

C₂²×D₂₇₁

—

D₁₀₈₄

—

D₁₀₈₄

—

D₁₀₈₄

—

D₁₀₈₄

—

C₂₇₁⋊C₈

—

C₂³×C₂₇₁

C₂×D₄

yes

characteristic

Q₈×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

C₂×D₄

yes

—

C₂×C₄×C₂₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₁

C₂×D₄

yes

—

C₂×C₄×C₂₇₁

#41

yes

—

C₂×C₄×C₂₇₁

C₂⁴

yes

characteristic

C₂²×C₂₇₁

—

C₂²×C₂₇₁

#124

yes

—

C₂²×C₂₇₁

C₂²×D₄

yes

characteristic

C₂×D₂₇₁

—

C₂×D₂₇₁

—

C₂×D₂₇₁

—

C₄×C₂₇₁

—

C₄×C₂₇₁

—

C₄×C₂₇₁

#124

yes

—

C₄×C₂₇₁

AGL(1,ℤ/8)

yes

—

C₄×C₂₇₁

C₂×#41

yes

—

C₂×C₂₇₁

—

C₂×C₂₇₁

C₂×#124

yes

characteristic

C₂×C₂₇₁

C₂×AGL(1,ℤ/8)

yes

—

D₂₇₁

—

C₂₇₁

#3138

yes

characteristic, complemented, Sylow

#3138

—

271

maximal, Sylow

#533

D₂₇₁

yes

characteristic, complemented

C₂×#127

—

271

—

#490

—

271

—

#486

—

271

—

#418

—

271

—

C₄×M₁₆

—

271

—

C₂²×D₄

—

271

—

C₂²×D₄

—

542

—

C₂×M₁₆

—

271

—

C₄⋊Q₈

C₂×D₂₇₁

yes

complemented

#127

—

271

—

#127

—

271

—

#127

—

542

—

#125

C₂×D₂₇₁

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₂₇₁

yes

—

C₂×C₄²

C₂×D₂₇₁

yes

characteristic

#102

—

271

—

#97

—

271

—

C₄×C₈

—

271

—

C₂⁴

—

542

—

C₂×Q₈

—

C₂×D₄

—

271

—

100

C₂×D₄

—

542

—

101

C₂×D₄

—

542

—

102

C₂×D₄

—

542

—

103

C₂×D₄

—

542

—

104

C₂×D₄

—

1084

—

105

C₂²×C₄

C₂²⋊D₂₇₁

yes

—

106

C₂²×C₄

C₂²×D₂₇₁

yes

characteristic

107

M₁₆

—

542

—

108

C₂×C₈

—

271

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂²×D₂₇₁

yes

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

C₂²⋊D₂₇₁

yes

—

115

C₄²

C₂²×D₂₇₁

yes

characteristic

116

C₄²

C₂²×D₂₇₁

yes

characteristic

117

C₂³

—

542

—

118

C₂³

—

1084

—

119

C₂³

—

1084

—

120

C₂³

C₂×C₂²⋊D₂₇₁

yes

characteristic

121

Q₈

—

122

D₄

—

542

—

123

D₄

—

1084

—

124

D₄

—

1084

—

125

D₄

—

1084

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×C₂²⋊D₂₇₁

yes

—

129

C₂×C₄

C₂×C₂²⋊D₂₇₁

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₂₇₁

yes

—

131

C₂×C₄

#16864655

yes

—

132

C₂×C₄

C₂³×D₂₇₁

yes

characteristic

133

C₈

—

542

—

134

C₂²

—

135

C₂²

—

1084

—

136

C₂²

—

1084

—

137

C₂²

—

2168

—

138

C₂²

#22576149

yes

—

139

C₂²

C₂²×C₂²⋊D₂₇₁

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

#22576149

yes

—

143

C₄

#22576185

yes

—

144

C₄

C₂×#16864655

yes

—

145

C₂

—

146

C₂

—

2168

—

147

C₂

C₂×#22576149

yes

characteristic

148

C₂

C₂×#22576185

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₇₁
commutator, cosocle	C₂×C₄×C₂₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₇₁
Fitting	#533×C₂₇₁	C₂
socle	C₂²×C₂₇₁	C₂²×D₄
prime radical	C₂×#22576157	C₂
hypercenter	#533	D₂₇₁
hypocenter	C₂₇₁	#3138
LCS	G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₂₇₁

commutator, cosocle

C₂×C₄×C₂₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₇₁

Fitting

#533×C₂₇₁

C₂

socle

C₂²×C₂₇₁

C₂²×D₄

C₂

D₂₇₁

C₂₇₁

G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #31774263	C₂	semidirect
3. #31774245	C₂	semidirect
4. #31773901	C₂	semidirect
5. C₄×#16864638	C₂	semidirect
6. #533×C₂₇₁	C₂	semidirect
13. #22576080	C₂²	semidirect
14. #22576071	C₂²	semidirect
15. C₄×C₂₇₁⋊C₈	C₂²	semidirect
16. C₄⋊Q₈×C₂₇₁	C₂²	semidirect
17. #125×C₂₇₁	C₂²	semidirect
74. C₂₇₁	#3138	semidirect
76. #533	D₂₇₁	semidirect
85. C₄⋊Q₈	C₂×D₂₇₁	semidirect
89. #125	C₂×D₂₇₁	semidirect

H ≅ G/N

type

2. #31774263

C₂

semidirect

3. #31774245

C₂

semidirect

4. #31773901

C₂

semidirect

5. C₄×#16864638

C₂

semidirect

6. #533×C₂₇₁

C₂

semidirect

13. #22576080

C₂²

semidirect

14. #22576071

C₂²

semidirect

15. C₄×C₂₇₁⋊C₈

C₂²

semidirect

16. C₄⋊Q₈×C₂₇₁

C₂²

semidirect

17. #125×C₂₇₁

C₂²

semidirect

74. C₂₇₁

#3138

semidirect

76. #533

D₂₇₁

semidirect

85. C₄⋊Q₈

C₂×D₂₇₁

semidirect

89. #125

C₂×D₂₇₁

semidirect

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

subgroup	index
C₂×#22576157	2
#31774263	2
#31774245	2
#31773901	2
C₄×#16864638	2
#533×C₂₇₁	2
#3138	271

subgroup

index

#533×C₂₇₁

271

Subgroups — #46846740

Conjugacy classes of subgroups

Complements

Maximal subgroups