Group #31829697 — subgroups

Subgroups — C₇₁×#8825 #31829697

Order 18176.

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

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 198 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×#1848	C₂	1	2	yes	maximal
2	C₇₁×#1566	C₂	1	4	yes	maximal
3	C₂×#322×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#108×C₇₁	—	2	4	no	—
5	#423×C₇₁	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
8	#334×C₇₁	—	2	8	no	—
9	#322×C₇₁	C₄	1	4	yes	—
10	C₂×M₁₆×C₇₁	—	2	4	no	—
11	C₂²×C₈×C₇₁	D₄	1	2	yes	—
12	C₂²×C₈×C₇₁	Q₈	1	4	yes	—
13	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
14	#108×C₇₁	—	4	8	no	—
15	C₄⋊C₈×C₇₁	—	2	8	no	—
16	C₄×C₈×C₇₁	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₇₁	D₄	1	4	yes	—
19	C₄×C₈×C₇₁	Q₈	1	2	yes	—
20	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
21	C₂²×C₄×C₇₁	C₂×Q₈	1	2	yes	—
22	M₁₆×C₇₁	—	4	8	no	—
23	C₂×C₈×C₇₁	—	4	4	no	—
24	C₂×C₈×C₇₁	—	4	4	no	—
25	C₂×C₈×C₇₁	C₄⋊C₄	1	8	yes	—
26	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
27	C₂×C₈×C₇₁	SD₁₆	1	8	yes	—
28	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
29	C₄²×C₇₁	C₂×D₄	1	2	yes	—
30	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
31	C₂³×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
32	C₂×C₄×C₇₁	—	2	4	no	—
33	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
36	C₂×C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
37	C₂×C₄×C₇₁	C₂×SD₁₆	1	4	yes	—
38	C₈×C₇₁	—	2	8	no	—
39	C₈×C₇₁	—	8	4	no	—
40	C₈×C₇₁	#106	1	8	yes	—
41	C₂²×C₇₁	—	2	2	no	—
42	C₂²×C₇₁	#384	1	1	yes	characteristic
43	C₂²×C₇₁	#498	1	2	yes	—
44	C₄×C₇₁	—	2	2	no	—
45	C₄×C₇₁	C₂×#106	1	2	yes	—
46	C₄×C₇₁	#430	1	4	yes	—
47	C₄×C₇₁	#492	1	2	yes	—
48	#8825	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂×C₇₁	—	2	2	no	—
50	C₂×C₇₁	#1844	1	1	yes	characteristic
51	C₂×C₇₁	#1849	1	2	yes	—
52	#1848	C₂×C₇₁	1	2	yes	—
53	#1566	C₂×C₇₁	1	4	yes	—
54	C₂×#322	C₂×C₇₁	1	1	yes	characteristic
55	C₇₁	#8825	1	1	yes	characteristic, direct product factor, Sylow
56	C₂×#108	—	2	4	no	—
57	#423	C₂²×C₇₁	1	4	yes	—
58	C₂×C₄×C₈	C₂²×C₇₁	1	2	yes	—
59	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
60	#334	—	2	8	no	—
61	#322	C₄×C₇₁	1	4	yes	—
62	C₂×M₁₆	—	2	4	no	—
63	C₂²×C₈	D₄×C₇₁	1	2	yes	—
64	C₂²×C₈	Q₈×C₇₁	1	4	yes	—
65	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
66	#108	—	4	8	no	—
67	C₄⋊C₈	—	2	8	no	—
68	C₄×C₈	C₂×C₄×C₇₁	1	4	yes	—
69	C₄×C₈	C₂×C₄×C₇₁	1	2	yes	—
70	C₄×C₈	D₄×C₇₁	1	4	yes	—
71	C₄×C₈	Q₈×C₇₁	1	2	yes	—
72	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
73	C₂²×C₄	C₂×Q₈×C₇₁	1	2	yes	—
74	M₁₆	—	4	8	no	—
75	C₂×C₈	—	4	4	no	—
76	C₂×C₈	—	4	4	no	—
77	C₂×C₈	C₄⋊C₄×C₇₁	1	8	yes	—
78	C₂×C₈	C₄⋊C₄×C₇₁	1	4	yes	—
79	C₂×C₈	SD₁₆×C₇₁	1	8	yes	—
80	C₄²	C₂²×C₄×C₇₁	1	1	yes	characteristic
81	C₄²	C₂×D₄×C₇₁	1	2	yes	—
82	C₄²	C₂×Q₈×C₇₁	1	1	yes	characteristic
83	C₂³	C₄⋊Q₈×C₇₁	1	1	yes	characteristic
84	C₂×C₄	—	2	4	no	—
85	C₂×C₄	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
86	C₂×C₄	C₂×C₄⋊C₄×C₇₁	1	2	yes	—
87	C₂×C₄	#127×C₇₁	1	1	yes	characteristic
88	C₂×C₄	C₄⋊Q₈×C₇₁	1	2	yes	—
89	C₂×C₄	C₂×SD₁₆×C₇₁	1	4	yes	—
90	C₈	—	2	8	no	—
91	C₈	—	8	4	no	—
92	C₈	#106×C₇₁	1	8	yes	—
93	C₂²	—	2	2	no	—
94	C₂²	#384×C₇₁	1	1	yes	characteristic
95	C₂²	#498×C₇₁	1	2	yes	—
96	C₄	—	2	2	no	—
97	C₄	C₂×#106×C₇₁	1	2	yes	—
98	C₄	#430×C₇₁	1	4	yes	—
99	C₄	#492×C₇₁	1	2	yes	—
100	C₂	—	2	2	no	—
101	C₂	C₇₁×#1844	1	1	yes	characteristic
102	C₂	C₇₁×#1849	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₁×#1848

C₂

yes

maximal

C₇₁×#1566

C₂

yes

maximal

C₂×#322×C₇₁

C₂

yes

maximal, characteristic

C₂×#108×C₇₁

—

#423×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#334×C₇₁

—

#322×C₇₁

C₄

yes

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

—

C₂²×C₈×C₇₁

Q₈

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

#108×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

Q₈

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×Q₈

yes

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

—

C₄²×C₇₁

C₂×Q₈

yes

characteristic

C₂³×C₇₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₄⋊Q₈

yes

—

C₂×C₄×C₇₁

C₂×SD₁₆

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#106

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

#384

yes

characteristic

C₂²×C₇₁

#498

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂×#106

yes

—

C₄×C₇₁

#430

yes

—

C₄×C₇₁

#492

yes

—

#8825

C₇₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1844

yes

characteristic

C₂×C₇₁

#1849

yes

—

#1848

C₂×C₇₁

yes

—

#1566

C₂×C₇₁

yes

—

C₂×#322

C₂×C₇₁

yes

characteristic

C₇₁

#8825

yes

characteristic, direct product factor, Sylow

C₂×#108

—

#423

C₂²×C₇₁

yes

—

C₂×C₄×C₈

C₂²×C₇₁

yes

—

C₂×C₄×C₈

C₂²×C₇₁

yes

characteristic

#334

—

#322

C₄×C₇₁

yes

—

C₂×M₁₆

—

C₂²×C₈

D₄×C₇₁

yes

—

C₂²×C₈

Q₈×C₇₁

yes

—

C₂×C₄²

C₂³×C₇₁

yes

characteristic

#108

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₇₁

yes

—

C₄×C₈

C₂×C₄×C₇₁

yes

—

C₄×C₈

D₄×C₇₁

yes

—

C₄×C₈

Q₈×C₇₁

yes

—

C₂²×C₄

C₂×D₄×C₇₁

yes

characteristic

C₂²×C₄

C₂×Q₈×C₇₁

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₇₁

yes

—

C₂×C₈

C₄⋊C₄×C₇₁

yes

—

C₂×C₈

SD₁₆×C₇₁

yes

—

C₄²

C₂²×C₄×C₇₁

yes

characteristic

C₄²

C₂×D₄×C₇₁

yes

—

C₄²

C₂×Q₈×C₇₁

yes

characteristic

C₂³

C₄⋊Q₈×C₇₁

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄×C₇₁

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄×C₇₁

yes

—

C₂×C₄

#127×C₇₁

yes

characteristic

C₂×C₄

C₄⋊Q₈×C₇₁

yes

—

C₂×C₄

C₂×SD₁₆×C₇₁

yes

—

C₈

—

C₈

—

C₈

#106×C₇₁

yes

—

C₂²

—

C₂²

#384×C₇₁

yes

characteristic

C₂²

#498×C₇₁

yes

—

C₄

—

C₄

C₂×#106×C₇₁

yes

—

C₄

#430×C₇₁

yes

—

C₄

#492×C₇₁

yes

—

100

C₂

—

101

C₂

C₇₁×#1844

yes

characteristic

102

C₂

C₇₁×#1849

yes

—

	subgroup	G/H
center	C₂×C₄×C₇₁	#127
commutator	C₄²	C₂²×C₄×C₇₁
Frattini, cosocle	C₂×C₄²	C₂³×C₇₁
socle	C₂²×C₇₁	#384
prime radical	C₂³×C₇₁	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

subgroup

G/H

center

C₂×C₄×C₇₁

#127

commutator

C₄²

C₂²×C₄×C₇₁

Frattini, cosocle

C₂×C₄²

C₂³×C₇₁

socle

C₂²×C₇₁

#384

prime radical

C₂³×C₇₁

C₄⋊Q₈

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

N	H ≅ G/N	type
48. #8825	C₇₁	direct (N × H)
55. C₇₁	#8825	direct (N × H)

H ≅ G/N

type

48. #8825

C₇₁

direct (N × H)

55. C₇₁

#8825

direct (N × H)

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

subgroup	index
C₇₁×#1848	2
C₇₁×#1566	2
C₂×#322×C₇₁	2
#8825	71

subgroup

index

C₇₁×#1848

C₇₁×#1566

C₂×#322×C₇₁

#8825

Subgroups — C₇₁×#8825 #31829697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C71×#8825 #31829697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#8825 #31829697