Group #31833384 — subgroups

Subgroups — C₇₁×#12512 #31833384

Order 18176.

268 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 94 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	C₇₁×#2241	C₂	1	2	yes	maximal, complemented
2	C₇₁×#2240	C₂	1	2	yes	maximal
3	C₇₁×#1713	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₁×#1709	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	#500×C₇₁	—	2	1	no	—
7	#500×C₇₁	C₂²	1	2	yes	—
8	#494×C₇₁	—	2	1	no	—
9	#494×C₇₁	C₂²	1	1	yes	characteristic
10	#493×C₇₁	C₂²	1	1	yes	characteristic
11	#486×C₇₁	—	2	1	no	—
12	#486×C₇₁	—	2	1	no	—
13	#358×C₇₁	—	2	4	no	—
14	#357×C₇₁	—	2	4	no	—
15	C₄×C₁₆×C₇₁	C₂²	1	2	yes	—
16	C₈²×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₇₁	—	2	2	no	—
18	C₂×Q₁₆×C₇₁	—	4	2	no	—
19	C₂×D₈×C₇₁	—	2	2	no	—
20	C₄⋊Q₈×C₇₁	—	2	1	no	—
21	C₄⋊Q₈×C₇₁	—	2	2	no	—
22	#127×C₇₁	—	2	1	no	—
23	C₂×C₁₆×C₇₁	D₄	1	4	yes	—
24	#107×C₇₁	—	2	2	no	—
25	#107×C₇₁	—	2	2	no	—
26	#107×C₇₁	—	4	2	no	—
27	#102×C₇₁	—	4	2	no	—
28	#102×C₇₁	—	4	2	no	—
29	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
30	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
31	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
32	C₂×Q₈×C₇₁	—	4	1	no	—
33	C₂×Q₈×C₇₁	—	4	2	no	—
34	C₂×D₄×C₇₁	—	4	1	no	—
35	C₂×D₄×C₇₁	—	4	1	no	—
36	C₂×D₄×C₇₁	—	4	1	no	—
37	Q₁₆×C₇₁	—	4	4	no	—
38	Q₁₆×C₇₁	—	8	4	no	—
39	D₈×C₇₁	—	4	4	no	—
40	C₂×C₈×C₇₁	D₈	1	2	yes	—
41	C₂×C₈×C₇₁	D₈	1	2	yes	—
42	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
43	C₄⋊C₄×C₇₁	—	4	2	no	—
44	C₄⋊C₄×C₇₁	—	4	1	no	—
45	C₄⋊C₄×C₇₁	—	4	1	no	—
46	C₄⋊C₄×C₇₁	—	4	2	no	—
47	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₁₆×C₇₁	—	2	4	no	—
49	C₂³×C₇₁	—	8	1	no	—
50	Q₈×C₇₁	—	8	2	no	—
51	Q₈×C₇₁	—	8	4	no	—
52	D₄×C₇₁	—	8	2	no	—
53	D₄×C₇₁	—	8	2	no	—
54	D₄×C₇₁	—	8	2	no	—
55	C₂×C₄×C₇₁	—	8	1	no	—
56	C₂×C₄×C₇₁	—	8	2	no	—
57	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
60	C₈×C₇₁	—	2	2	no	—
61	C₈×C₇₁	—	2	2	no	—
62	C₈×C₇₁	#135	1	4	yes	—
63	C₂²×C₇₁	—	16	1	no	—
64	C₂²×C₇₁	—	16	1	no	—
65	C₂²×C₇₁	—	16	1	no	—
66	C₂²×C₇₁	#493	1	1	yes	characteristic
67	C₄×C₇₁	—	16	1	no	—
68	C₄×C₇₁	—	16	2	no	—
69	C₄×C₇₁	#495	1	2	yes	—
70	C₄×C₇₁	#509	1	2	yes	—
71	C₄×C₇₁	#510	1	2	yes	—
72	#12512	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂×C₇₁	—	32	1	no	—
74	C₂×C₇₁	#1709	1	1	yes	characteristic
75	C₂×C₇₁	#2250	1	1	yes	characteristic
76	C₂×C₇₁	#2251	1	1	yes	characteristic
77	#2241	C₂×C₇₁	1	2	yes	complemented
78	#2240	C₂×C₇₁	1	2	yes	—
79	#1713	C₂×C₇₁	1	1	yes	characteristic, complemented
80	#1709	C₂×C₇₁	1	1	yes	characteristic
81	#1311	C₂×C₇₁	1	1	yes	characteristic, complemented
82	C₇₁	#12512	1	1	yes	characteristic, direct product factor, Sylow
83	#500	—	2	1	no	—
84	#500	C₂²×C₇₁	1	2	yes	—
85	#494	—	2	1	no	—
86	#494	C₂²×C₇₁	1	1	yes	characteristic
87	#493	C₂²×C₇₁	1	1	yes	characteristic
88	#486	—	2	1	no	—
89	#486	—	2	1	no	—
90	#358	—	2	4	no	—
91	#357	—	2	4	no	—
92	C₄×C₁₆	C₂²×C₇₁	1	2	yes	—
93	C₈²	C₂²×C₇₁	1	1	yes	characteristic
94	C₂×Q₁₆	—	2	2	no	—
95	C₂×Q₁₆	—	4	2	no	—
96	C₂×D₈	—	2	2	no	—
97	C₄⋊Q₈	—	2	1	no	—
98	C₄⋊Q₈	—	2	2	no	—
99	#127	—	2	1	no	—
100	C₂×C₁₆	D₄×C₇₁	1	4	yes	—
101	#107	—	2	2	no	—
102	#107	—	2	2	no	—
103	#107	—	4	2	no	—
104	#102	—	4	2	no	—
105	#102	—	4	2	no	—
106	C₄×C₈	D₄×C₇₁	1	1	yes	characteristic
107	C₄×C₈	D₄×C₇₁	1	1	yes	characteristic
108	C₄×C₈	C₂³×C₇₁	1	1	yes	characteristic
109	C₂×Q₈	—	4	1	no	—
110	C₂×Q₈	—	4	2	no	—
111	C₂×D₄	—	4	1	no	—
112	C₂×D₄	—	4	1	no	—
113	C₂×D₄	—	4	1	no	—
114	Q₁₆	—	4	4	no	—
115	Q₁₆	—	8	4	no	—
116	D₈	—	4	4	no	—
117	C₂×C₈	D₈×C₇₁	1	2	yes	—
118	C₂×C₈	D₈×C₇₁	1	2	yes	—
119	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	4	1	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
125	C₁₆	—	2	4	no	—
126	C₂³	—	8	1	no	—
127	Q₈	—	8	2	no	—
128	Q₈	—	8	4	no	—
129	D₄	—	8	2	no	—
130	D₄	—	8	2	no	—
131	D₄	—	8	2	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	—	8	2	no	—
134	C₂×C₄	#127×C₇₁	1	1	yes	characteristic
135	C₂×C₄	C₂×D₈×C₇₁	1	1	yes	characteristic
136	C₂×C₄	C₂×D₈×C₇₁	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	2	2	no	—
139	C₈	#135×C₇₁	1	4	yes	—
140	C₂²	—	16	1	no	—
141	C₂²	—	16	1	no	—
142	C₂²	—	16	1	no	—
143	C₂²	#493×C₇₁	1	1	yes	characteristic
144	C₄	—	16	1	no	—
145	C₄	—	16	2	no	—
146	C₄	#495×C₇₁	1	2	yes	—
147	C₄	#509×C₇₁	1	2	yes	—
148	C₄	#510×C₇₁	1	2	yes	—
149	C₂	—	32	1	no	—
150	C₂	C₇₁×#1709	1	1	yes	characteristic
151	C₂	C₇₁×#2250	1	1	yes	characteristic
152	C₂	C₇₁×#2251	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₁×#2241

C₂

yes

maximal, complemented

C₇₁×#2240

C₂

yes

maximal

C₇₁×#1713

C₂

yes

maximal, characteristic, complemented

C₇₁×#1709

C₂

yes

maximal, characteristic

C₇₁×#1311

C₂

yes

maximal, characteristic, complemented

#500×C₇₁

—

#500×C₇₁

C₂²

yes

—

#494×C₇₁

—

#494×C₇₁

C₂²

yes

characteristic

#493×C₇₁

C₂²

yes

characteristic

#486×C₇₁

—

#486×C₇₁

—

#358×C₇₁

—

#357×C₇₁

—

C₄×C₁₆×C₇₁

C₂²

yes

—

C₈²×C₇₁

C₂²

yes

characteristic

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₂×D₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

#127×C₇₁

—

C₂×C₁₆×C₇₁

D₄

yes

—

#107×C₇₁

—

#107×C₇₁

—

#107×C₇₁

—

#102×C₇₁

—

#102×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

D₈×C₇₁

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

—

C₂³×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#135

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#493

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#495

yes

—

C₄×C₇₁

#509

yes

—

C₄×C₇₁

#510

yes

—

#12512

C₇₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

complemented

#2240

C₂×C₇₁

yes

—

#1713

C₂×C₇₁

yes

characteristic, complemented

#1709

C₂×C₇₁

yes

characteristic

#1311

C₂×C₇₁

yes

characteristic, complemented

C₇₁

#12512

yes

characteristic, direct product factor, Sylow

#500

—

#500

C₂²×C₇₁

yes

—

#494

—

#494

C₂²×C₇₁

yes

characteristic

#493

C₂²×C₇₁

yes

—

—

—

—

C₄×C₁₆

C₂²×C₇₁

yes

—

C₈²

C₂²×C₇₁

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

100

C₂×C₁₆

D₄×C₇₁

yes

—

101

#107

—

102

#107

—

103

#107

—

104

#102

—

105

#102

—

106

C₄×C₈

D₄×C₇₁

yes

characteristic

107

C₄×C₈

D₄×C₇₁

yes

characteristic

108

C₄×C₈

C₂³×C₇₁

yes

characteristic

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

Q₁₆

—

115

Q₁₆

—

116

D₈

—

117

C₂×C₈

D₈×C₇₁

yes

—

118

C₂×C₈

D₈×C₇₁

yes

—

119

C₂×C₈

C₂×D₄×C₇₁

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄²

C₂×D₄×C₇₁

yes

characteristic

125

C₁₆

—

126

C₂³

—

127

Q₈

—

128

Q₈

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

#127×C₇₁

yes

characteristic

135

C₂×C₄

C₂×D₈×C₇₁

yes

characteristic

136

C₂×C₄

C₂×D₈×C₇₁

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

#135×C₇₁

yes

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#493×C₇₁

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

#495×C₇₁

yes

—

147

C₄

#509×C₇₁

yes

—

148

C₄

#510×C₇₁

yes

—

149

C₂

—

150

C₂

C₇₁×#1709

yes

characteristic

151

C₂

C₇₁×#2250

yes

characteristic

152

C₂

C₇₁×#2251

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₁	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇₁
prime radical	#493×C₇₁	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₄²×C₇₁ ◁ C₈²×C₇₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₁

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₇₁

prime radical

#493×C₇₁

C₂²

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₁ ◁ C₄²×C₇₁ ◁ C₈²×C₇₁ ◁ G

N	H ≅ G/N	type
1. C₇₁×#2241	C₂	semidirect
3. C₇₁×#1713	C₂	semidirect
5. C₇₁×#1311	C₂	semidirect
72. #12512	C₇₁	direct (N × H)
77. #2241	C₂×C₇₁	semidirect
79. #1713	C₂×C₇₁	semidirect
81. #1311	C₂×C₇₁	semidirect
82. C₇₁	#12512	direct (N × H)

H ≅ G/N

type

1. C₇₁×#2241

C₂

semidirect

3. C₇₁×#1713

C₂

semidirect

5. C₇₁×#1311

C₂

semidirect

72. #12512

C₇₁

direct (N × H)

77. #2241

C₂×C₇₁

semidirect

79. #1713

C₂×C₇₁

semidirect

81. #1311

C₂×C₇₁

semidirect

82. C₇₁

#12512

direct (N × H)

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

subgroup	index
C₇₁×#2241	2
C₇₁×#2240	2
C₇₁×#1713	2
C₇₁×#1709	2
C₇₁×#1311	2
#12512	71

subgroup

index

C₇₁×#2241

C₇₁×#2240

C₇₁×#1713

C₇₁×#1709

C₇₁×#1311

#12512

Subgroups — C₇₁×#12512 #31833384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C71×#12512 #31833384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#12512 #31833384