Group #31829670 — subgroups

Subgroups — C₇₁×#8798 #31829670

Order 18176.

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

normal subgroups only

This group has 220 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	C₇₁×#1568	C₂	1	2	yes	maximal
2	C₇₁×#1452	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1287	C₂	1	4	yes	maximal
4	#432×C₇₁	C₂²	1	4	yes	—
5	#431×C₇₁	C₂²	1	2	yes	—
6	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
7	#335×C₇₁	—	2	2	no	—
8	#334×C₇₁	—	2	2	no	—
9	#322×C₇₁	C₄	1	4	yes	—
10	C₂²×C₈×C₇₁	Q₈	1	2	yes	—
11	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
12	C₄⋊C₈×C₇₁	—	2	4	no	—
13	C₄⋊C₈×C₇₁	—	2	2	no	—
14	C₂²⋊C₈×C₇₁	—	2	2	no	—
15	C₂²⋊C₈×C₇₁	—	2	4	no	—
16	#97×C₇₁	—	2	4	no	—
17	C₄×C₈×C₇₁	C₂×C₄	1	4	yes	—
18	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
20	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₇₁	#41	1	2	yes	—
23	C₂×C₈×C₇₁	—	2	4	no	—
24	C₂×C₈×C₇₁	—	4	2	no	—
25	C₂×C₈×C₇₁	—	4	4	no	—
26	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
28	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
29	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
30	C₄²×C₇₁	#41	1	2	yes	—
31	C₂³×C₇₁	#125	1	1	yes	characteristic
32	C₂×C₄×C₇₁	—	2	2	no	—
33	C₂×C₄×C₇₁	—	4	2	no	—
34	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₇₁	#117	1	2	yes	—
36	C₂×C₄×C₇₁	#124	1	2	yes	—
37	C₂×C₄×C₇₁	#125	1	1	yes	characteristic
38	C₈×C₇₁	—	2	2	no	—
39	C₈×C₇₁	—	2	2	no	—
40	C₈×C₇₁	—	4	4	no	—
41	C₈×C₇₁	—	8	4	no	—
42	C₂²×C₇₁	—	2	2	no	—
43	C₂²×C₇₁	—	2	1	no	—
44	C₂²×C₇₁	#382	1	1	yes	characteristic
45	C₄×C₇₁	—	2	2	no	—
46	C₄×C₇₁	—	2	2	no	—
47	C₄×C₇₁	—	2	1	no	—
48	C₄×C₇₁	#430	1	2	yes	—
49	#8798	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂×C₇₁	—	4	1	no	—
51	C₂×C₇₁	#1833	1	2	yes	—
52	C₂×C₇₁	#1838	1	1	yes	characteristic
53	#1568	C₂×C₇₁	1	2	yes	—
54	#1452	C₂×C₇₁	1	1	yes	characteristic
55	#1287	C₂×C₇₁	1	4	yes	—
56	C₇₁	#8798	1	1	yes	characteristic, direct product factor, Sylow
57	#432	C₂²×C₇₁	1	4	yes	—
58	#431	C₂²×C₇₁	1	2	yes	—
59	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
60	#335	—	2	2	no	—
61	#334	—	2	2	no	—
62	#322	C₄×C₇₁	1	4	yes	—
63	C₂²×C₈	Q₈×C₇₁	1	2	yes	—
64	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
65	C₄⋊C₈	—	2	4	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₂²⋊C₈	—	2	2	no	—
68	C₂²⋊C₈	—	2	4	no	—
69	#97	—	2	4	no	—
70	C₄×C₈	C₂×C₄×C₇₁	1	4	yes	—
71	C₄×C₈	C₂×C₄×C₇₁	1	2	yes	—
72	C₄×C₈	D₄×C₇₁	1	1	yes	characteristic
73	C₄×C₈	D₄×C₇₁	1	1	yes	characteristic
74	C₂²×C₄	C₂×Q₈×C₇₁	1	1	yes	characteristic
75	C₂²×C₄	#41×C₇₁	1	2	yes	—
76	C₂×C₈	—	2	4	no	—
77	C₂×C₈	—	4	2	no	—
78	C₂×C₈	—	4	4	no	—
79	C₂×C₈	C₄⋊C₄×C₇₁	1	2	yes	—
80	C₂×C₈	C₄⋊C₄×C₇₁	1	2	yes	—
81	C₄²	C₂²×C₄×C₇₁	1	1	yes	characteristic
82	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
83	C₄²	#41×C₇₁	1	2	yes	—
84	C₂³	#125×C₇₁	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
88	C₂×C₄	#117×C₇₁	1	2	yes	—
89	C₂×C₄	#124×C₇₁	1	2	yes	—
90	C₂×C₄	#125×C₇₁	1	1	yes	characteristic
91	C₈	—	2	2	no	—
92	C₈	—	2	2	no	—
93	C₈	—	4	4	no	—
94	C₈	—	8	4	no	—
95	C₂²	—	2	2	no	—
96	C₂²	—	2	1	no	—
97	C₂²	#382×C₇₁	1	1	yes	characteristic
98	C₄	—	2	2	no	—
99	C₄	—	2	2	no	—
100	C₄	—	2	1	no	—
101	C₄	#430×C₇₁	1	2	yes	—
102	C₂	—	4	1	no	—
103	C₂	C₇₁×#1833	1	2	yes	—
104	C₂	C₇₁×#1838	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₁×#1568

C₂

yes

maximal

C₇₁×#1452

C₂

yes

maximal, characteristic

C₇₁×#1287

C₂

yes

maximal

#432×C₇₁

C₂²

yes

—

#431×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#335×C₇₁

—

#334×C₇₁

—

#322×C₇₁

C₄

yes

—

C₂²×C₈×C₇₁

Q₈

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₂²⋊C₈×C₇₁

—

C₂²⋊C₈×C₇₁

—

#97×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

#41

yes

—

C₂³×C₇₁

#125

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#117

yes

—

C₂×C₄×C₇₁

#124

yes

—

C₂×C₄×C₇₁

#125

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#382

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#430

yes

—

#8798

C₇₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1833

yes

—

C₂×C₇₁

#1838

yes

characteristic

#1568

C₂×C₇₁

yes

—

#1452

C₂×C₇₁

yes

characteristic

#1287

C₂×C₇₁

yes

—

C₇₁

#8798

yes

characteristic, direct product factor, Sylow

#432

C₂²×C₇₁

yes

—

#431

C₂²×C₇₁

yes

—

C₂×C₄×C₈

C₂²×C₇₁

yes

—

—

C₄×C₇₁

yes

—

C₂²×C₈

Q₈×C₇₁

yes

—

C₂×C₄²

C₂³×C₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

C₂×C₄×C₇₁

yes

—

C₄×C₈

C₂×C₄×C₇₁

yes

—

C₄×C₈

D₄×C₇₁

yes

characteristic

C₄×C₈

D₄×C₇₁

yes

characteristic

C₂²×C₄

C₂×Q₈×C₇₁

yes

characteristic

C₂²×C₄

#41×C₇₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₇₁

yes

—

C₂×C₈

C₄⋊C₄×C₇₁

yes

—

C₄²

C₂²×C₄×C₇₁

yes

characteristic

C₄²

C₂×D₄×C₇₁

yes

characteristic

C₄²

#41×C₇₁

yes

—

C₂³

#125×C₇₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄×C₇₁

yes

characteristic

C₂×C₄

#117×C₇₁

yes

—

C₂×C₄

#124×C₇₁

yes

—

C₂×C₄

#125×C₇₁

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

#382×C₇₁

yes

characteristic

C₄

—

C₄

—

100

C₄

—

101

C₄

#430×C₇₁

yes

—

102

C₂

—

103

C₂

C₇₁×#1833

yes

—

104

C₂

C₇₁×#1838

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₁	#382
commutator	C₄²	C₂²×C₄×C₇₁
Frattini, cosocle	C₂×C₄²	C₂³×C₇₁
prime radical	C₂³×C₇₁	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₁

#382

commutator

C₄²

C₂²×C₄×C₇₁

Frattini, cosocle

C₂×C₄²

C₂³×C₇₁

C₂³×C₇₁

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

N	H ≅ G/N	type
49. #8798	C₇₁	direct (N × H)
56. C₇₁	#8798	direct (N × H)

H ≅ G/N

type

49. #8798

C₇₁

direct (N × H)

56. C₇₁

#8798

direct (N × H)

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

subgroup	index
C₇₁×#1568	2
C₇₁×#1452	2
C₇₁×#1287	2
#8798	71

subgroup

index

C₇₁×#1568

C₇₁×#1452

C₇₁×#1287

#8798

Subgroups — C₇₁×#8798 #31829670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C71×#8798 #31829670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#8798 #31829670