Group #31891263 — subgroups

Subgroups — #31891263

Order 18176.

338 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 187 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	#22588512	C₂	1	2	yes	maximal
2	#22587096	C₂	1	4	yes	maximal
3	C₂×#322×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556064	—	2	4	no	—
5	#17986023	C₂²	1	4	yes	—
6	#17985750	—	2	8	no	—
7	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
9	#322×C₇₁	C₄	1	4	yes	—
10	C₂×#11765849	—	2	4	no	—
11	#14556064	—	4	8	no	—
12	C₄⋊(C₇₁⋊C₈)	—	2	8	no	—
13	C₂²×C₈×C₇₁	D₄	1	2	yes	—
14	C₂²×C₈×C₇₁	Q₈	1	4	yes	—
15	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
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	#11765849	—	4	8	no	—
21	C₂×C₇₁⋊C₈	—	4	4	no	—
22	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
23	C₂²×C₄×C₇₁	C₂×Q₈	1	2	yes	—
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	#106	1	8	yes	—
40	C₇₁⋊C₈	—	8	4	no	—
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	—	71	1	no	maximal, Sylow
49	C₂×C₇₁	—	2	2	no	—
50	C₂×C₇₁	#1844	1	1	yes	characteristic
51	C₂×C₇₁	#1849	1	2	yes	—
52	#1848	—	71	2	no	—
53	#1566	—	71	4	no	—
54	C₂×#322	D₇₁	1	1	yes	characteristic
55	C₇₁	#8825	1	1	yes	characteristic, complemented, Sylow
56	C₂×#108	—	142	4	no	—
57	#423	—	71	4	no	—
58	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂×D₇₁	1	2	yes	—
60	#334	—	142	8	no	—
61	#322	Dic₇₁	1	4	yes	—
62	C₂×M₁₆	—	142	4	no	—
63	C₂²×C₈	Dic₁₄₂	1	4	yes	—
64	C₂²×C₈	D₂₈₄	1	2	yes	—
65	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
66	#108	—	284	8	no	—
67	C₄⋊C₈	—	142	8	no	—
68	C₄×C₈	Dic₁₄₂	1	2	yes	—
69	C₄×C₈	D₂₈₄	1	4	yes	—
70	C₄×C₈	C₂×Dic₇₁	1	4	yes	—
71	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
72	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
73	C₂²×C₄	C₂×Dic₁₄₂	1	2	yes	—
74	M₁₆	—	284	8	no	—
75	C₂×C₈	—	4	4	no	—
76	C₂×C₈	—	284	4	no	—
77	C₂×C₈	C₄⋊Dic₇₁	1	8	yes	—
78	C₂×C₈	C₄⋊Dic₇₁	1	4	yes	—
79	C₂×C₈	#11765854	1	8	yes	—
80	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
81	C₄²	C₂×D₂₈₄	1	2	yes	—
82	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
83	C₂³	#14556128	1	1	yes	characteristic
84	C₂×C₄	—	2	4	no	—
85	C₂×C₄	C₂×C₄⋊Dic₇₁	1	2	yes	—
86	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
87	C₂×C₄	#14556125	1	1	yes	characteristic
88	C₂×C₄	#14556128	1	2	yes	—
89	C₂×C₄	C₂×#11765854	1	4	yes	—
90	C₈	—	2	8	no	—
91	C₈	—	568	4	no	—
92	C₈	#14556060	1	8	yes	—
93	C₂²	—	2	2	no	—
94	C₂²	#17985869	1	1	yes	characteristic
95	C₂²	#17986397	1	2	yes	—
96	C₄	—	2	2	no	—
97	C₄	C₂×#14556060	1	2	yes	—
98	C₄	#17986048	1	4	yes	—
99	C₄	#17986372	1	2	yes	—
100	C₂	—	2	2	no	—
101	C₂	#22588497	1	1	yes	characteristic
102	C₂	#22588515	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22588512

C₂

yes

maximal

#22587096

C₂

yes

maximal

C₂×#322×C₇₁

C₂

yes

maximal, characteristic

C₂×#14556064

—

#17986023

C₂²

yes

—

#17985750

—

C₂×C₄×C₈×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#322×C₇₁

C₄

yes

—

C₂×#11765849

—

#14556064

—

C₄⋊(C₇₁⋊C₈)

—

C₂²×C₈×C₇₁

D₄

yes

—

C₂²×C₈×C₇₁

Q₈

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

Q₈

yes

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×Q₈

yes

—

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

#106

yes

—

C₇₁⋊C₈

—

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

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1844

yes

characteristic

C₂×C₇₁

#1849

yes

—

#1848

—

#1566

—

C₂×#322

D₇₁

yes

characteristic

C₇₁

#8825

yes

characteristic, complemented, Sylow

C₂×#108

—

142

—

#423

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×C₄×C₈

C₂×D₇₁

yes

—

#334

—

142

—

#322

Dic₇₁

yes

—

C₂×M₁₆

—

142

—

C₂²×C₈

Dic₁₄₂

yes

—

C₂²×C₈

D₂₈₄

yes

—

C₂×C₄²

C₂²×D₇₁

yes

characteristic

#108

—

284

—

C₄⋊C₈

—

142

—

C₄×C₈

Dic₁₄₂

yes

—

C₄×C₈

D₂₈₄

yes

—

C₄×C₈

C₂×Dic₇₁

yes

—

C₄×C₈

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂×D₂₈₄

yes

characteristic

C₂²×C₄

C₂×Dic₁₄₂

yes

—

M₁₆

—

284

—

C₂×C₈

—

C₂×C₈

—

284

—

C₂×C₈

C₄⋊Dic₇₁

yes

—

C₂×C₈

C₄⋊Dic₇₁

yes

—

C₂×C₈

#11765854

yes

—

C₄²

C₂²×Dic₇₁

yes

characteristic

C₄²

C₂×D₂₈₄

yes

—

C₄²

C₂×Dic₁₄₂

yes

characteristic

C₂³

#14556128

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×C₄⋊Dic₇₁

yes

—

C₂×C₄

C₂×C₄⋊Dic₇₁

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

C₂×#11765854

yes

—

C₈

—

C₈

—

568

—

C₈

#14556060

yes

—

C₂²

—

C₂²

#17985869

yes

characteristic

C₂²

#17986397

yes

—

C₄

—

C₄

C₂×#14556060

yes

—

C₄

#17986048

yes

—

C₄

#17986372

yes

—

100

C₂

—

101

C₂

#22588497

yes

characteristic

102

C₂

#22588515

yes

—

	subgroup	G/H
center	C₂×C₄	#14556125
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×#322×C₇₁	C₂
socle	C₂²×C₇₁	#384
prime radical	C₂³×C₇₁	C₄⋊Q₈
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×#322	D₇₁
hypocenter	C₇₁	#8825
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#14556125

commutator

C₄²×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₂×#322×C₇₁

C₂

socle

C₂²×C₇₁

#384

prime radical

C₂³×C₇₁

C₄⋊Q₈

cosocle

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
55. C₇₁	#8825	semidirect

H ≅ G/N

type

55. C₇₁

#8825

semidirect

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

subgroup	index
#22588512	2
#22587096	2
C₂×#322×C₇₁	2
#8825	71

subgroup

index

#22588512

#22587096

C₂×#322×C₇₁

#8825

Subgroups — #31891263

Conjugacy classes of subgroups

Complements

Maximal subgroups