Group #31828113 — subgroups

Subgroups — C₇₁×#7241 #31828113

Order 18176.

206 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 110 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×C₄⋊C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#1567	C₂	1	2	yes	maximal
3	C₂²×C₁₆×C₇₁	C₄	1	2	yes	—
4	#423×C₇₁	—	2	2	no	—
5	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
6	C₄⋊C₁₆×C₇₁	—	2	2	no	—
7	#335×C₇₁	—	2	2	no	—
8	#335×C₇₁	C₄	1	4	yes	—
9	C₂×M₁₆×C₇₁	—	4	2	no	—
10	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
12	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
13	C₂×C₁₆×C₇₁	—	2	2	no	—
14	C₂×C₁₆×C₇₁	—	4	2	no	—
15	C₄⋊C₈×C₇₁	—	2	4	no	—
16	C₄⋊C₈×C₇₁	—	4	2	no	—
17	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₇₁	D₄	1	2	yes	—
19	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
20	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
21	C₂²×C₄×C₇₁	Q₁₆	1	1	yes	characteristic
22	M₁₆×C₇₁	—	8	2	no	—
23	M₁₆×C₇₁	—	8	2	no	—
24	C₂×C₈×C₇₁	—	2	1	no	—
25	C₂×C₈×C₇₁	—	2	1	no	—
26	C₂×C₈×C₇₁	—	4	1	no	—
27	C₂×C₈×C₇₁	—	4	4	no	—
28	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
29	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
31	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
32	C₄²×C₇₁	SD₁₆	1	2	yes	—
33	C₁₆×C₇₁	—	4	2	no	—
34	C₂³×C₇₁	#107	1	1	yes	characteristic
35	C₂×C₄×C₇₁	—	2	1	no	—
36	C₂×C₄×C₇₁	—	2	1	no	—
37	C₂×C₄×C₇₁	—	4	1	no	—
38	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
39	C₂×C₄×C₇₁	#102	1	2	yes	—
40	C₂×C₄×C₇₁	#103	1	2	yes	—
41	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
42	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
43	C₂×C₄×C₇₁	SD₃₂	1	2	yes	—
44	C₂×C₄×C₇₁	Q₃₂	1	1	yes	characteristic
45	C₈×C₇₁	—	2	2	no	—
46	C₈×C₇₁	—	2	1	no	—
47	C₈×C₇₁	—	8	4	no	—
48	C₈×C₇₁	#100	1	1	yes	characteristic
49	C₈×C₇₁	#101	1	1	yes	characteristic
50	C₂²×C₇₁	—	2	1	no	—
51	C₂²×C₇₁	—	2	1	no	—
52	C₂²×C₇₁	#340	1	1	yes	characteristic
53	C₂²×C₇₁	#366	1	1	yes	characteristic
54	C₂²×C₇₁	#367	1	1	yes	characteristic
55	C₄×C₇₁	—	2	1	no	—
56	C₄×C₇₁	—	2	2	no	—
57	C₄×C₇₁	#338	1	1	yes	characteristic
58	C₄×C₇₁	#343	1	1	yes	characteristic
59	C₄×C₇₁	#357	1	2	yes	—
60	C₄×C₇₁	#358	1	2	yes	—
61	#7241	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂×C₇₁	—	2	1	no	—
63	C₂×C₇₁	—	2	1	no	—
64	C₂×C₇₁	#1292	1	1	yes	characteristic
65	C₂×C₇₁	#1380	1	1	yes	characteristic
66	C₂×C₇₁	#1385	1	1	yes	characteristic
67	C₂×C₄⋊C₁₆	C₂×C₇₁	1	1	yes	characteristic
68	#1567	C₂×C₇₁	1	2	yes	—
69	C₇₁	#7241	1	1	yes	characteristic, direct product factor, Sylow
70	C₂²×C₁₆	C₄×C₇₁	1	2	yes	—
71	#423	—	2	2	no	—
72	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
73	C₄⋊C₁₆	—	2	2	no	—
74	#335	—	2	2	no	—
75	#335	C₄×C₇₁	1	4	yes	—
76	C₂×M₁₆	—	4	2	no	—
77	C₂²×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
78	C₂²×C₈	Q₈×C₇₁	1	1	yes	characteristic
79	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
80	C₂×C₁₆	—	2	2	no	—
81	C₂×C₁₆	—	4	2	no	—
82	C₄⋊C₈	—	2	4	no	—
83	C₄⋊C₈	—	4	2	no	—
84	C₄×C₈	C₂×C₄×C₇₁	1	2	yes	—
85	C₄×C₈	D₄×C₇₁	1	2	yes	—
86	C₂²×C₄	C₄⋊C₄×C₇₁	1	1	yes	characteristic
87	C₂²×C₄	D₈×C₇₁	1	1	yes	characteristic
88	C₂²×C₄	Q₁₆×C₇₁	1	1	yes	characteristic
89	M₁₆	—	8	2	no	—
90	M₁₆	—	8	2	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	4	1	no	—
94	C₂×C₈	—	4	4	no	—
95	C₂×C₈	C₄²×C₇₁	1	1	yes	characteristic
96	C₂×C₈	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
97	C₂×C₈	C₄⋊C₄×C₇₁	1	2	yes	—
98	C₄²	C₂²⋊C₄×C₇₁	1	2	yes	—
99	C₄²	SD₁₆×C₇₁	1	2	yes	—
100	C₁₆	—	4	2	no	—
101	C₂³	#107×C₇₁	1	1	yes	characteristic
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	#95×C₇₁	1	1	yes	characteristic
106	C₂×C₄	#102×C₇₁	1	2	yes	—
107	C₂×C₄	#103×C₇₁	1	2	yes	—
108	C₂×C₄	#106×C₇₁	1	1	yes	characteristic
109	C₂×C₄	D₁₆×C₇₁	1	1	yes	characteristic
110	C₂×C₄	SD₃₂×C₇₁	1	2	yes	—
111	C₂×C₄	Q₃₂×C₇₁	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	2	1	no	—
114	C₈	—	8	4	no	—
115	C₈	#100×C₇₁	1	1	yes	characteristic
116	C₈	#101×C₇₁	1	1	yes	characteristic
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	#340×C₇₁	1	1	yes	characteristic
120	C₂²	#366×C₇₁	1	1	yes	characteristic
121	C₂²	#367×C₇₁	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	2	2	no	—
124	C₄	#338×C₇₁	1	1	yes	characteristic
125	C₄	#343×C₇₁	1	1	yes	characteristic
126	C₄	#357×C₇₁	1	2	yes	—
127	C₄	#358×C₇₁	1	2	yes	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	C₇₁×#1292	1	1	yes	characteristic
131	C₂	C₇₁×#1380	1	1	yes	characteristic
132	C₂	C₇₁×#1385	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊C₁₆×C₇₁

C₂

yes

maximal, characteristic

C₇₁×#1567

C₂

yes

maximal

C₂²×C₁₆×C₇₁

C₄

yes

—

#423×C₇₁

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₁

—

#335×C₇₁

—

#335×C₇₁

C₄

yes

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

Q₈

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₂²×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₁

D₈

yes

characteristic

C₂²×C₄×C₇₁

Q₁₆

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄²

yes

characteristic

C₂×C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₄²×C₇₁

C₂²⋊C₄

yes

—

C₄²×C₇₁

SD₁₆

yes

—

C₁₆×C₇₁

—

C₂³×C₇₁

#107

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#95

yes

characteristic

C₂×C₄×C₇₁

#102

yes

—

C₂×C₄×C₇₁

#103

yes

—

C₂×C₄×C₇₁

#106

yes

characteristic

C₂×C₄×C₇₁

D₁₆

yes

characteristic

C₂×C₄×C₇₁

SD₃₂

yes

—

C₂×C₄×C₇₁

Q₃₂

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₄×C₇₁

#358

yes

—

#7241

C₇₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₄⋊C₁₆

C₂×C₇₁

yes

characteristic

#1567

C₂×C₇₁

yes

—

C₇₁

#7241

yes

characteristic, direct product factor, Sylow

C₂²×C₁₆

C₄×C₇₁

yes

—

#423

—

C₂×C₄×C₈

C₂²×C₇₁

yes

characteristic

C₄⋊C₁₆

—

#335

—

#335

C₄×C₇₁

yes

—

C₂×M₁₆

—

C₂²×C₈

C₂×C₄×C₇₁

yes

characteristic

C₂²×C₈

Q₈×C₇₁

yes

characteristic

C₂×C₄²

D₄×C₇₁

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₇₁

yes

—

C₄×C₈

D₄×C₇₁

yes

—

C₂²×C₄

C₄⋊C₄×C₇₁

yes

characteristic

C₂²×C₄

D₈×C₇₁

yes

characteristic

C₂²×C₄

Q₁₆×C₇₁

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²×C₇₁

yes

characteristic

C₂×C₈

C₂²⋊C₄×C₇₁

yes

characteristic

C₂×C₈

C₄⋊C₄×C₇₁

yes

—

C₄²

C₂²⋊C₄×C₇₁

yes

—

C₄²

SD₁₆×C₇₁

yes

—

100

C₁₆

—

101

C₂³

#107×C₇₁

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

#95×C₇₁

yes

characteristic

106

C₂×C₄

#102×C₇₁

yes

—

107

C₂×C₄

#103×C₇₁

yes

—

108

C₂×C₄

#106×C₇₁

yes

characteristic

109

C₂×C₄

D₁₆×C₇₁

yes

characteristic

110

C₂×C₄

SD₃₂×C₇₁

yes

—

111

C₂×C₄

Q₃₂×C₇₁

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

#100×C₇₁

yes

characteristic

116

C₈

#101×C₇₁

yes

characteristic

117

C₂²

—

118

C₂²

—

119

C₂²

#340×C₇₁

yes

characteristic

120

C₂²

#366×C₇₁

yes

characteristic

121

C₂²

#367×C₇₁

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

#338×C₇₁

yes

characteristic

125

C₄

#343×C₇₁

yes

characteristic

126

C₄

#357×C₇₁

yes

—

127

C₄

#358×C₇₁

yes

—

128

C₂

—

129

C₂

—

130

C₂

C₇₁×#1292

yes

characteristic

131

C₂

C₇₁×#1380

yes

characteristic

132

C₂

C₇₁×#1385

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₁	#340
commutator	C₂×C₈	C₄²×C₇₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₇₁
prime radical	C₂³×C₇₁	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ C₂×C₄×C₈×C₇₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₁

#340

commutator

C₂×C₈

C₄²×C₇₁

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₇₁

prime radical

C₂³×C₇₁

#107

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ C₂×C₄×C₈×C₇₁ ◁ G

N	H ≅ G/N	type
61. #7241	C₇₁	direct (N × H)
69. C₇₁	#7241	direct (N × H)

H ≅ G/N

type

61. #7241

C₇₁

direct (N × H)

69. C₇₁

#7241

direct (N × H)

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

subgroup	index
C₂×C₄⋊C₁₆×C₇₁	2
C₇₁×#1567	2
#7241	71

subgroup

index

C₂×C₄⋊C₁₆×C₇₁

C₇₁×#1567

#7241

Subgroups — C₇₁×#7241 #31828113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C71×#7241 #31828113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#7241 #31828113