Group #31891126 — subgroups

Subgroups — #31891126

Order 18176.

322 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 147 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	#22588457	C₂	1	2	yes	maximal
2	#22586175	C₂	1	4	yes	maximal
3	C₇₁×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	4	yes	—
5	#17985754	—	2	4	no	—
6	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
7	C₄³×C₇₁	C₂²	1	1	yes	characteristic
8	#336×C₇₁	C₄	1	4	yes	—
9	C₂²×C₇₁⋊C₈	—	2	4	no	—
10	C₄⋊(C₇₁⋊C₈)	—	2	8	no	—
11	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
12	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₇₁	D₄	1	2	yes	—
14	C₂×C₄²×C₇₁	Q₈	1	2	yes	—
15	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₇₁	—	2	4	no	—
17	C₂×C₇₁⋊C₈	—	4	4	no	—
18	C₂×C₇₁⋊C₈	—	4	8	no	—
19	C₂²×C₄×C₇₁	C₄⋊C₄	1	4	yes	—
20	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
22	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
23	C₂×C₈×C₇₁	—	2	4	no	—
24	C₂×C₈×C₇₁	—	2	8	no	—
25	C₄²×C₇₁	—	2	4	no	—
26	C₄²×C₇₁	—	2	4	no	—
27	C₄²×C₇₁	—	4	2	no	—
28	C₄²×C₇₁	#41	1	4	yes	—
29	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₇₁	—	2	2	no	—
31	C₂×C₄×C₇₁	—	2	4	no	—
32	C₂×C₄×C₇₁	—	2	4	no	—
33	C₂×C₄×C₇₁	—	2	4	no	—
34	C₂×C₄×C₇₁	#117	1	2	yes	—
35	C₂×C₄×C₇₁	#124	1	2	yes	—
36	C₂×C₄×C₇₁	#125	1	2	yes	—
37	C₂×C₄×C₇₁	#131	1	4	yes	—
38	C₂×C₄×C₇₁	#135	1	4	yes	—
39	C₈×C₇₁	—	4	4	no	—
40	C₇₁⋊C₈	—	8	4	no	—
41	C₂²×C₇₁	#382	1	1	yes	characteristic
42	C₂²×C₇₁	#424	1	2	yes	—
43	C₂²×C₇₁	#427	1	2	yes	—
44	C₂²×C₇₁	#430	1	2	yes	—
45	C₄×C₇₁	—	2	2	no	—
46	C₄×C₇₁	—	4	4	no	—
47	C₄×C₇₁	#433	1	4	yes	—
48	C₄×C₇₁	#488	1	4	yes	—
49	#8788	—	71	1	no	maximal, Sylow
50	C₂×C₇₁	#1572	1	4	yes	—
51	C₂×C₇₁	#1832	1	1	yes	characteristic
52	C₂×C₇₁	#1835	1	1	yes	characteristic
53	C₂×C₇₁	#1838	1	1	yes	characteristic
54	#1831	—	71	2	no	—
55	#1744	D₇₁	1	1	yes	characteristic
56	#1277	—	71	4	no	—
57	C₇₁	#8788	1	1	yes	characteristic, complemented, Sylow
58	C₂×C₄⋊C₈	—	71	4	no	—
59	C₂×C₄×C₈	C₂×D₇₁	1	2	yes	—
60	C₄³	C₂×D₇₁	1	1	yes	characteristic
61	#336	—	142	4	no	—
62	#336	Dic₇₁	1	4	yes	—
63	C₂²×C₈	—	142	4	no	—
64	C₂²×C₈	C₂×Dic₇₁	1	4	yes	—
65	C₂×C₄²	Dic₁₄₂	1	2	yes	—
66	C₂×C₄²	D₂₈₄	1	2	yes	—
67	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
68	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
69	C₄⋊C₈	—	142	8	no	—
70	C₄×C₈	—	2	4	no	—
71	C₂²×C₄	C₄⋊Dic₇₁	1	4	yes	—
72	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
73	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
74	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
75	C₂×C₈	—	2	4	no	—
76	C₂×C₈	—	2	8	no	—
77	C₂×C₈	—	284	4	no	—
78	C₂×C₈	—	284	8	no	—
79	C₄²	—	2	4	no	—
80	C₄²	—	2	4	no	—
81	C₄²	—	4	2	no	—
82	C₄²	#11765865	1	4	yes	—
83	C₂³	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	#14556087	1	2	yes	—
89	C₂×C₄	#14556116	1	2	yes	—
90	C₂×C₄	#14556120	1	2	yes	—
91	C₂×C₄	#14556135	1	4	yes	—
92	C₂×C₄	#14556150	1	4	yes	—
93	C₈	—	4	4	no	—
94	C₈	—	568	4	no	—
95	C₂²	#17985865	1	1	yes	characteristic
96	C₂²	#17986026	1	2	yes	—
97	C₂²	#17986037	1	2	yes	—
98	C₂²	#17986048	1	2	yes	—
99	C₄	—	2	2	no	—
100	C₄	—	4	4	no	—
101	C₄	#17986060	1	4	yes	—
102	C₄	#17986354	1	4	yes	—
103	C₂	#22587120	1	4	yes	—
104	C₂	#22588460	1	1	yes	characteristic
105	C₂	#22588471	1	1	yes	characteristic
106	C₂	#22588481	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22588457

C₂

yes

maximal

#22586175

C₂

yes

maximal

C₇₁×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

#17985754

—

C₂×C₄×C₈×C₇₁

C₂²

yes

—

C₄³×C₇₁

C₂²

yes

characteristic

#336×C₇₁

C₄

yes

—

C₂²×C₇₁⋊C₈

—

C₄⋊(C₇₁⋊C₈)

—

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

D₄

yes

—

C₂×C₄²×C₇₁

Q₈

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄×C₈×C₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂²×C₄×C₇₁

C₄⋊C₄

yes

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

#41

yes

—

C₂³×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#117

yes

—

C₂×C₄×C₇₁

#124

yes

—

C₂×C₄×C₇₁

#125

yes

—

C₂×C₄×C₇₁

#131

yes

—

C₂×C₄×C₇₁

#135

yes

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

#382

yes

characteristic

C₂²×C₇₁

#424

yes

—

C₂²×C₇₁

#427

yes

—

C₂²×C₇₁

#430

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#433

yes

—

C₄×C₇₁

#488

yes

—

#8788

—

maximal, Sylow

C₂×C₇₁

#1572

yes

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

D₇₁

yes

characteristic

#1277

—

C₇₁

#8788

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₇₁

yes

—

C₄³

C₂×D₇₁

yes

characteristic

#336

—

142

—

#336

Dic₇₁

yes

—

C₂²×C₈

—

142

—

C₂²×C₈

C₂×Dic₇₁

yes

—

C₂×C₄²

Dic₁₄₂

yes

—

C₂×C₄²

D₂₈₄

yes

—

C₂×C₄²

C₂×Dic₇₁

yes

—

C₂×C₄²

C₂²×D₇₁

yes

characteristic

C₄⋊C₈

—

142

—

C₄×C₈

—

C₂²×C₄

C₄⋊Dic₇₁

yes

—

C₂²×C₄

C₂²×Dic₇₁

yes

characteristic

C₂²×C₄

C₂×D₂₈₄

yes

characteristic

C₂²×C₄

C₂×Dic₁₄₂

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

284

—

C₂×C₈

—

284

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#11765865

yes

—

C₂³

C₂×C₄⋊Dic₇₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#14556087

yes

—

C₂×C₄

#14556116

yes

—

C₂×C₄

#14556120

yes

—

C₂×C₄

#14556135

yes

—

C₂×C₄

#14556150

yes

—

C₈

—

C₈

—

568

—

C₂²

#17985865

yes

characteristic

C₂²

#17986026

yes

—

C₂²

#17986037

yes

—

C₂²

#17986048

yes

—

C₄

—

100

C₄

—

101

C₄

#17986060

yes

—

102

C₄

#17986354

yes

—

103

C₂

#22587120

yes

—

104

C₂

#22588460

yes

characteristic

105

C₂

#22588471

yes

characteristic

106

C₂

#22588481

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₈₄
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1744	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1744	D₇₁
hypocenter	C₇₁	#8788
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₂₈₄

commutator

C₂²×C₄×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1744

C₂

socle, prime radical

C₂³×C₇₁

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
57. C₇₁	#8788	semidirect

H ≅ G/N

type

57. C₇₁

#8788

semidirect

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

subgroup	index
#22588457	2
#22586175	2
C₇₁×#1744	2
#8788	71

subgroup

index

#22588457

#22586175

C₇₁×#1744

#8788

Subgroups — #31891126

Conjugacy classes of subgroups

Complements

Maximal subgroups