Group #31832960 — subgroups

Subgroups — C₇₁×#12088 #31832960

Order 18176.

198 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 90 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×#2176	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#1593	C₂	1	2	yes	maximal
3	C₇₁×#1368	C₂	1	1	yes	maximal, characteristic
4	C₇₁×#1334	C₂	1	2	yes	maximal
5	C₇₁×#1313	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₃₂×C₇₁	C₄	1	1	yes	characteristic
7	#446×C₇₁	—	2	2	no	—
8	C₄×Q₁₆×C₇₁	C₂²	1	2	yes	—
9	#367×C₇₁	C₄	1	1	yes	characteristic
10	#358×C₇₁	C₄	1	2	yes	—
11	#346×C₇₁	C₂²	1	1	yes	characteristic
12	#346×C₇₁	C₂²	1	1	yes	characteristic
13	#334×C₇₁	C₂²	1	2	yes	—
14	Q₈⋊C₈×C₇₁	—	2	2	no	—
15	#322×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₇₁	C₂×C₄	1	2	yes	—
17	C₄×Q₈×C₇₁	—	2	2	no	—
18	Q₃₂×C₇₁	—	2	2	no	—
19	C₂×C₁₆×C₇₁	—	2	1	no	—
20	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
22	#107×C₇₁	C₂×C₄	1	2	yes	—
23	C₄⋊C₈×C₇₁	—	2	2	no	—
24	C₄⋊C₈×C₇₁	—	4	2	no	—
25	#103×C₇₁	—	2	2	no	—
26	#97×C₇₁	—	4	2	no	—
27	C₄×C₈×C₇₁	D₄	1	2	yes	—
28	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₇₁	—	2	2	no	—
30	Q₁₆×C₇₁	—	2	2	no	—
31	Q₁₆×C₇₁	—	2	2	no	—
32	C₂×C₈×C₇₁	—	2	2	no	—
33	C₂×C₈×C₇₁	—	4	2	no	—
34	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
36	C₄⋊C₄×C₇₁	—	2	2	no	—
37	C₄⋊C₄×C₇₁	—	4	2	no	—
38	C₄²×C₇₁	—	4	2	no	—
39	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₇₁	—	2	1	no	—
41	C₁₆×C₇₁	—	4	1	no	—
42	C₁₆×C₇₁	M₁₆	1	2	yes	—
43	Q₈×C₇₁	—	4	2	no	—
44	Q₈×C₇₁	—	4	2	no	—
45	C₂×C₄×C₇₁	—	4	2	no	—
46	C₂×C₄×C₇₁	—	4	2	no	—
47	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
49	C₈×C₇₁	—	2	1	no	—
50	C₈×C₇₁	—	4	2	no	—
51	C₈×C₇₁	—	8	2	no	—
52	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
53	C₈×C₇₁	#131	1	1	yes	characteristic
54	C₂²×C₇₁	#442	1	1	yes	characteristic
55	C₄×C₇₁	—	2	2	no	—
56	C₄×C₇₁	—	8	2	no	—
57	C₄×C₇₁	—	8	2	no	—
58	C₄×C₇₁	#436	1	1	yes	characteristic
59	C₄×C₇₁	#444	1	1	yes	characteristic
60	#12088	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂×C₇₁	#1594	1	1	yes	characteristic
62	C₂×C₇₁	#2180	1	2	yes	—
63	#2176	C₂×C₇₁	1	1	yes	characteristic
64	#1593	C₂×C₇₁	1	2	yes	—
65	#1368	C₂×C₇₁	1	1	yes	characteristic
66	#1334	C₂×C₇₁	1	2	yes	—
67	#1313	C₂×C₇₁	1	1	yes	characteristic
68	C₇₁	#12088	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×Q₃₂	C₄×C₇₁	1	1	yes	characteristic
70	#446	—	2	2	no	—
71	C₄×Q₁₆	C₂²×C₇₁	1	2	yes	—
72	#367	C₄×C₇₁	1	1	yes	characteristic
73	#358	C₄×C₇₁	1	2	yes	—
74	#346	C₂²×C₇₁	1	1	yes	characteristic
75	#346	C₂²×C₇₁	1	1	yes	characteristic
76	#334	C₂²×C₇₁	1	2	yes	—
77	Q₈⋊C₈	—	2	2	no	—
78	#322	C₂²×C₇₁	1	1	yes	characteristic
79	C₂×Q₁₆	C₂×C₄×C₇₁	1	2	yes	—
80	C₄×Q₈	—	2	2	no	—
81	Q₃₂	—	2	2	no	—
82	C₂×C₁₆	—	2	1	no	—
83	C₂×C₁₆	C₂×C₄×C₇₁	1	1	yes	characteristic
84	C₂×C₁₆	C₂×C₄×C₇₁	1	1	yes	characteristic
85	#107	C₂×C₄×C₇₁	1	2	yes	—
86	C₄⋊C₈	—	2	2	no	—
87	C₄⋊C₈	—	4	2	no	—
88	#103	—	2	2	no	—
89	#97	—	4	2	no	—
90	C₄×C₈	D₄×C₇₁	1	2	yes	—
91	C₄×C₈	C₂³×C₇₁	1	1	yes	characteristic
92	C₂×Q₈	—	2	2	no	—
93	Q₁₆	—	2	2	no	—
94	Q₁₆	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	4	2	no	—
97	C₂×C₈	C₂²×C₄×C₇₁	1	1	yes	characteristic
98	C₂×C₈	#41×C₇₁	1	1	yes	characteristic
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄²	—	4	2	no	—
102	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	—	4	1	no	—
105	C₁₆	M₁₆×C₇₁	1	2	yes	—
106	Q₈	—	4	2	no	—
107	Q₈	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	C₄×D₄×C₇₁	1	1	yes	characteristic
111	C₂×C₄	AGL(1,ℤ/8)×C₇₁	1	2	yes	—
112	C₈	—	2	1	no	—
113	C₈	—	4	2	no	—
114	C₈	—	8	2	no	—
115	C₈	C₂×M₁₆×C₇₁	1	1	yes	characteristic
116	C₈	#131×C₇₁	1	1	yes	characteristic
117	C₂²	#442×C₇₁	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	8	2	no	—
120	C₄	—	8	2	no	—
121	C₄	#436×C₇₁	1	1	yes	characteristic
122	C₄	#444×C₇₁	1	1	yes	characteristic
123	C₂	C₇₁×#1594	1	1	yes	characteristic
124	C₂	C₇₁×#2180	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₁×#2176

C₂

yes

maximal, characteristic

C₇₁×#1593

C₂

yes

maximal

C₇₁×#1368

C₂

yes

maximal, characteristic

C₇₁×#1334

C₂

yes

maximal

C₇₁×#1313

C₂

yes

maximal, characteristic

C₂×Q₃₂×C₇₁

C₄

yes

characteristic

#446×C₇₁

—

C₄×Q₁₆×C₇₁

C₂²

yes

—

#367×C₇₁

C₄

yes

characteristic

#358×C₇₁

C₄

yes

—

#346×C₇₁

C₂²

yes

characteristic

#346×C₇₁

C₂²

yes

characteristic

#334×C₇₁

C₂²

yes

—

Q₈⋊C₈×C₇₁

—

#322×C₇₁

C₂²

yes

characteristic

C₂×Q₁₆×C₇₁

C₂×C₄

yes

—

C₄×Q₈×C₇₁

—

Q₃₂×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

#107×C₇₁

C₂×C₄

yes

—

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

#103×C₇₁

—

#97×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

#41

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₁₆×C₇₁

M₁₆

yes

—

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄×D₄

yes

characteristic

C₂×C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₂×M₁₆

yes

C₈×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₇₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₁

#1594

yes

characteristic

C₂×C₇₁

#2180

yes

—

#2176

C₂×C₇₁

yes

characteristic

#1593

C₂×C₇₁

yes

—

#1368

C₂×C₇₁

yes

characteristic

#1334

C₂×C₇₁

yes

—

#1313

C₂×C₇₁

yes

characteristic

C₇₁

#12088

yes

characteristic, direct product factor, Sylow

C₂×Q₃₂

C₄×C₇₁

yes

characteristic

#446

—

C₄×Q₁₆

C₂²×C₇₁

yes

—

#367

C₄×C₇₁

yes

characteristic

#358

C₄×C₇₁

yes

—

#346

C₂²×C₇₁

yes

characteristic

#346

C₂²×C₇₁

yes

characteristic

#334

C₂²×C₇₁

yes

—

Q₈⋊C₈

—

#322

C₂²×C₇₁

yes

characteristic

C₂×Q₁₆

C₂×C₄×C₇₁

yes

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₇₁

yes

characteristic

C₂×C₁₆

C₂×C₄×C₇₁

yes

characteristic

#107

C₂×C₄×C₇₁

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#97

—

C₄×C₈

D₄×C₇₁

yes

—

C₄×C₈

C₂³×C₇₁

yes

characteristic

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄×C₇₁

yes

characteristic

C₂×C₈

#41×C₇₁

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

C₂×D₄×C₇₁

yes

characteristic

103

C₁₆

—

104

C₁₆

—

105

C₁₆

M₁₆×C₇₁

yes

—

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₄×D₄×C₇₁

yes

characteristic

111

C₂×C₄

AGL(1,ℤ/8)×C₇₁

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

C₂×M₁₆×C₇₁

yes

characteristic

116

C₈

#131×C₇₁

yes

characteristic

117

C₂²

#442×C₇₁

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#436×C₇₁

yes

characteristic

122

C₄

#444×C₇₁

yes

characteristic

123

C₂

C₇₁×#1594

yes

characteristic

124

C₂

C₇₁×#2180

yes

—

	subgroup	G/H
center, socle, prime radical	C₂²×C₇₁	#442
commutator	C₂×C₈	C₂²×C₄×C₇₁
Frattini, cosocle	C₄×C₈	C₂³×C₇₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₄²×C₇₁ ◁ #322×C₇₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₇₁

#442

commutator

C₂×C₈

C₂²×C₄×C₇₁

Frattini, cosocle

C₄×C₈

C₂³×C₇₁

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₁ ◁ C₄²×C₇₁ ◁ #322×C₇₁ ◁ G

N	H ≅ G/N	type
60. #12088	C₇₁	direct (N × H)
68. C₇₁	#12088	direct (N × H)

H ≅ G/N

type

60. #12088

C₇₁

direct (N × H)

68. C₇₁

#12088

direct (N × H)

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

subgroup	index
C₇₁×#2176	2
C₇₁×#1593	2
C₇₁×#1368	2
C₇₁×#1334	2
C₇₁×#1313	2
#12088	71

subgroup

index

C₇₁×#2176

C₇₁×#1593

C₇₁×#1368

C₇₁×#1334

C₇₁×#1313

#12088

Subgroups — C₇₁×#12088 #31832960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C71×#12088 #31832960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#12088 #31832960