Group #31834525 — subgroups

Subgroups — C₇₁×#13653 #31834525

Order 18176.

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

normal subgroups only

This group has 188 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	C₂×Q₆₄×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₃₂×C₇₁	C₂	1	2	yes	maximal
3	C₇₁×#1424	C₂	1	1	yes	maximal, characteristic
4	C₇₁×#1416	C₂	1	2	yes	maximal
5	C₇₁×#1397	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₃₂×C₇₁	C₂²	1	2	yes	—
7	C₄×Q₁₆×C₇₁	—	2	2	no	—
8	Q₆₄×C₇₁	C₄	1	4	yes	complemented
9	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
10	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
11	#366×C₇₁	C₂²	1	2	yes	—
12	#358×C₇₁	—	2	2	no	—
13	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×Q₁₆×C₇₁	—	2	2	no	—
15	C₄×Q₈×C₇₁	—	4	2	no	—
16	Q₃₂×C₇₁	—	2	2	no	—
17	Q₃₂×C₇₁	C₂×C₄	1	4	yes	—
18	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
19	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
20	#107×C₇₁	—	2	2	no	—
21	#103×C₇₁	—	4	2	no	—
22	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
23	C₃₂×C₇₁	—	2	1	no	—
24	C₃₂×C₇₁	C₂×C₄	1	2	yes	—
25	C₂×Q₈×C₇₁	—	4	2	no	—
26	Q₁₆×C₇₁	—	2	4	no	—
27	Q₁₆×C₇₁	—	4	2	no	—
28	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
29	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
30	C₄⋊C₄×C₇₁	—	4	2	no	—
31	C₄⋊C₄×C₇₁	—	8	2	no	—
32	C₄²×C₇₁	—	8	2	no	—
33	C₄²×C₇₁	D₈	1	1	yes	characteristic
34	C₁₆×C₇₁	—	2	1	no	—
35	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
36	C₁₆×C₇₁	#41	1	1	yes	characteristic
37	Q₈×C₇₁	—	4	4	no	—
38	Q₈×C₇₁	—	8	2	no	—
39	C₂×C₄×C₇₁	—	8	2	no	—
40	C₂×C₄×C₇₁	—	8	2	no	—
41	C₂×C₄×C₇₁	D₁₆	1	2	yes	—
42	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
43	C₈×C₇₁	—	2	1	no	—
44	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
45	C₈×C₇₁	#135	1	1	yes	characteristic
46	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
47	C₄×C₇₁	—	2	2	no	—
48	C₄×C₇₁	—	8	4	no	—
49	C₄×C₇₁	—	16	2	no	—
50	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
51	C₄×C₇₁	#508	1	1	yes	characteristic
52	#13653	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂×C₇₁	C₄×D₁₆	1	1	yes	characteristic
54	C₂×C₇₁	#2264	1	2	yes	—
55	C₂×Q₆₄	C₂×C₇₁	1	1	yes	characteristic
56	C₄×Q₃₂	C₂×C₇₁	1	2	yes	—
57	#1424	C₂×C₇₁	1	1	yes	characteristic
58	#1416	C₂×C₇₁	1	2	yes	—
59	#1397	C₂×C₇₁	1	1	yes	characteristic
60	C₇₁	#13653	1	1	yes	characteristic, direct product factor, Sylow
61	C₂×Q₃₂	C₂²×C₇₁	1	2	yes	—
62	C₄×Q₁₆	—	2	2	no	—
63	Q₆₄	C₄×C₇₁	1	4	yes	complemented
64	C₂×C₃₂	C₂²×C₇₁	1	1	yes	characteristic
65	C₂×C₃₂	C₂²×C₇₁	1	1	yes	characteristic
66	#366	C₂²×C₇₁	1	2	yes	—
67	#358	—	2	2	no	—
68	C₄×C₁₆	C₂²×C₇₁	1	1	yes	characteristic
69	C₂×Q₁₆	—	2	2	no	—
70	C₄×Q₈	—	4	2	no	—
71	Q₃₂	—	2	2	no	—
72	Q₃₂	C₂×C₄×C₇₁	1	4	yes	—
73	C₂×C₁₆	D₄×C₇₁	1	1	yes	characteristic
74	C₂×C₁₆	C₂³×C₇₁	1	1	yes	characteristic
75	#107	—	2	2	no	—
76	#103	—	4	2	no	—
77	C₄×C₈	D₄×C₇₁	1	1	yes	characteristic
78	C₃₂	—	2	1	no	—
79	C₃₂	C₂×C₄×C₇₁	1	2	yes	—
80	C₂×Q₈	—	4	2	no	—
81	Q₁₆	—	2	4	no	—
82	Q₁₆	—	4	2	no	—
83	C₂×C₈	D₈×C₇₁	1	1	yes	characteristic
84	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
85	C₄⋊C₄	—	4	2	no	—
86	C₄⋊C₄	—	8	2	no	—
87	C₄²	—	8	2	no	—
88	C₄²	D₈×C₇₁	1	1	yes	characteristic
89	C₁₆	—	2	1	no	—
90	C₁₆	C₂²×C₄×C₇₁	1	1	yes	characteristic
91	C₁₆	#41×C₇₁	1	1	yes	characteristic
92	Q₈	—	4	4	no	—
93	Q₈	—	8	2	no	—
94	C₂×C₄	—	8	2	no	—
95	C₂×C₄	—	8	2	no	—
96	C₂×C₄	D₁₆×C₇₁	1	2	yes	—
97	C₂×C₄	C₂×D₈×C₇₁	1	1	yes	characteristic
98	C₈	—	2	1	no	—
99	C₈	C₄×D₄×C₇₁	1	1	yes	characteristic
100	C₈	#135×C₇₁	1	1	yes	characteristic
101	C₂²	C₂×D₁₆×C₇₁	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	8	4	no	—
104	C₄	—	16	2	no	—
105	C₄	C₄×D₈×C₇₁	1	1	yes	characteristic
106	C₄	#508×C₇₁	1	1	yes	characteristic
107	C₂	C₄×D₁₆×C₇₁	1	1	yes	characteristic
108	C₂	C₇₁×#2264	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₆₄×C₇₁

C₂

yes

maximal, characteristic

C₄×Q₃₂×C₇₁

C₂

yes

maximal

C₇₁×#1424

C₂

yes

maximal, characteristic

C₇₁×#1416

C₂

yes

maximal

C₇₁×#1397

C₂

yes

maximal, characteristic

C₂×Q₃₂×C₇₁

C₂²

yes

—

C₄×Q₁₆×C₇₁

—

Q₆₄×C₇₁

C₄

yes

complemented

C₂×C₃₂×C₇₁

C₂²

yes

characteristic

C₂×C₃₂×C₇₁

C₂²

yes

characteristic

#366×C₇₁

C₂²

yes

—

#358×C₇₁

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×Q₁₆×C₇₁

—

C₄×Q₈×C₇₁

—

Q₃₂×C₇₁

—

Q₃₂×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

#107×C₇₁

—

#103×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

characteristic

C₃₂×C₇₁

—

C₃₂×C₇₁

C₂×C₄

yes

—

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

D₈

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

D₈

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

C₂²×C₄

yes

characteristic

C₁₆×C₇₁

#41

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

D₁₆

yes

—

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

C₄×D₄

yes

characteristic

C₈×C₇₁

#135

yes

characteristic

C₂²×C₇₁

C₂×D₁₆

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₄×D₈

yes

C₄×C₇₁

yes

C₇₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₁

C₄×D₁₆

yes

characteristic

C₂×C₇₁

#2264

yes

—

C₂×Q₆₄

C₂×C₇₁

yes

characteristic

C₄×Q₃₂

C₂×C₇₁

yes

—

#1424

C₂×C₇₁

yes

characteristic

#1416

C₂×C₇₁

yes

—

#1397

C₂×C₇₁

yes

characteristic

C₇₁

#13653

yes

characteristic, direct product factor, Sylow

C₂×Q₃₂

C₂²×C₇₁

yes

—

C₄×Q₁₆

—

Q₆₄

C₄×C₇₁

yes

complemented

C₂×C₃₂

C₂²×C₇₁

yes

characteristic

C₂×C₃₂

C₂²×C₇₁

yes

characteristic

#366

C₂²×C₇₁

yes

—

#358

—

C₄×C₁₆

C₂²×C₇₁

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

C₂×C₄×C₇₁

yes

—

C₂×C₁₆

D₄×C₇₁

yes

characteristic

C₂×C₁₆

C₂³×C₇₁

yes

characteristic

#107

—

#103

—

C₄×C₈

D₄×C₇₁

yes

characteristic

C₃₂

—

C₃₂

C₂×C₄×C₇₁

yes

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

D₈×C₇₁

yes

characteristic

C₂×C₈

C₂×D₄×C₇₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

D₈×C₇₁

yes

characteristic

C₁₆

—

C₁₆

C₂²×C₄×C₇₁

yes

characteristic

C₁₆

#41×C₇₁

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₁₆×C₇₁

yes

—

C₂×C₄

C₂×D₈×C₇₁

yes

characteristic

C₈

—

C₈

C₄×D₄×C₇₁

yes

characteristic

100

C₈

#135×C₇₁

yes

characteristic

101

C₂²

C₂×D₁₆×C₇₁

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₄×D₈×C₇₁

yes

characteristic

106

C₄

#508×C₇₁

yes

characteristic

107

C₂

C₄×D₁₆×C₇₁

yes

characteristic

108

C₂

C₇₁×#2264

yes

—

	subgroup	G/H
center, socle, prime radical	C₂²×C₇₁	C₂×D₁₆
commutator	C₁₆	C₂²×C₄×C₇₁
Frattini, cosocle	C₂×C₁₆	C₂³×C₇₁
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₄²×C₇₁ ◁ C₄×C₈×C₇₁ ◁ C₄×C₁₆×C₇₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₇₁

C₂×D₁₆

commutator

C₁₆

C₂²×C₄×C₇₁

Frattini, cosocle

C₂×C₁₆

C₂³×C₇₁

LCS

G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₁ ◁ C₄²×C₇₁ ◁ C₄×C₈×C₇₁ ◁ C₄×C₁₆×C₇₁ ◁ G

N	H ≅ G/N	type
8. Q₆₄×C₇₁	C₄	semidirect
52. #13653	C₇₁	direct (N × H)
60. C₇₁	#13653	direct (N × H)
63. Q₆₄	C₄×C₇₁	semidirect

H ≅ G/N

type

8. Q₆₄×C₇₁

C₄

semidirect

52. #13653

C₇₁

direct (N × H)

60. C₇₁

#13653

direct (N × H)

63. Q₆₄

C₄×C₇₁

semidirect

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

subgroup	index
C₂×Q₆₄×C₇₁	2
C₄×Q₃₂×C₇₁	2
C₇₁×#1424	2
C₇₁×#1416	2
C₇₁×#1397	2
#13653	71

subgroup

index

C₂×Q₆₄×C₇₁

C₄×Q₃₂×C₇₁

C₇₁×#1424

C₇₁×#1416

C₇₁×#1397

#13653

Subgroups — C₇₁×#13653 #31834525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C71×#13653 #31834525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#13653 #31834525