Group #31901961 — subgroups

Subgroups — #31901961

Order 18176.

350 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 97 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	#22588473	C₂	1	3	yes	maximal
2	#22586221	C₂	1	3	yes	maximal
3	C₇₁×#2845	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556039	C₂²	1	3	yes	—
5	#17985869	C₂²	1	1	yes	characteristic
6	#17985761	—	2	6	no	—
7	#398×C₇₁	—	2	6	no	—
8	#382×C₇₁	C₂²	1	3	yes	—
9	C₂²×C₇₁⋊C₈	—	2	3	no	—
10	C₂×C₄⋊Dic₇₁	—	2	3	no	—
11	#14556039	—	2	3	no	—
12	#14556039	Q₈	1	6	yes	—
13	C₂×C₄⋊C₄×C₇₁	—	2	6	no	—
14	C₂×C₄⋊C₄×C₇₁	D₄	1	6	yes	—
15	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
16	#95×C₇₁	—	2	2	no	—
17	#95×C₇₁	—	2	6	no	—
18	C₂²×Dic₇₁	—	4	1	no	—
19	C₂×C₇₁⋊C₈	—	2	6	no	—
20	C₂×C₇₁⋊C₈	—	4	3	no	—
21	C₂×C₇₁⋊C₈	—	4	3	no	—
22	C₄⋊Dic₇₁	—	4	6	no	—
23	C₂²×C₄×C₇₁	—	2	6	no	—
24	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
25	C₄⋊C₄×C₇₁	—	4	6	no	—
26	C₄⋊C₄×C₇₁	—	8	6	no	—
27	C₄²×C₇₁	C₂×Q₈	1	3	yes	—
28	C₄²×C₇₁	#41	1	1	yes	characteristic
29	C₂³×C₇₁	#120	1	1	yes	characteristic
30	C₂×C₄×C₇₁	—	2	3	no	—
31	C₂×C₄×C₇₁	—	2	3	no	—
32	C₂×C₄×C₇₁	—	4	6	no	—
33	C₂×C₄×C₇₁	—	4	6	no	—
34	C₂×C₄×C₇₁	—	4	6	no	—
35	C₂×C₄×C₇₁	C₂²⋊Q₈	1	3	yes	—
36	C₂×C₄×C₇₁	C₄⋊Q₈	1	3	yes	—
37	C₂×Dic₇₁	—	8	3	no	—
38	C₇₁⋊C₈	—	4	6	no	—
39	C₂²×C₇₁	#395	1	1	yes	characteristic
40	C₂²×C₇₁	#454	1	3	yes	—
41	C₂²×C₇₁	#455	1	3	yes	—
42	C₄×C₇₁	—	2	3	no	—
43	C₄×C₇₁	—	2	3	no	—
44	C₄×C₇₁	—	8	6	no	—
45	Dic₇₁	—	16	1	no	—
46	#10586	—	71	1	no	maximal, Sylow
47	C₂×C₇₁	#1658	1	1	yes	characteristic
48	C₂×C₇₁	#1662	1	3	yes	—
49	C₂×C₇₁	#2061	1	3	yes	—
50	#2845	D₇₁	1	1	yes	characteristic
51	#1836	—	71	3	no	—
52	#1295	—	71	3	no	—
53	C₇₁	#10586	1	1	yes	characteristic, complemented, Sylow
54	C₂×#97	—	71	3	no	—
55	#398	—	2	6	no	—
56	#384	—	71	1	no	—
57	#382	C₂×D₇₁	1	3	yes	—
58	#340	—	142	6	no	—
59	C₂²×C₈	—	142	3	no	—
60	C₂×C₄⋊C₄	—	2	6	no	—
61	C₂×C₄⋊C₄	—	142	3	no	—
62	C₂×C₄⋊C₄	C₂²⋊D₇₁	1	6	yes	—
63	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
64	#97	—	71	6	no	—
65	#97	—	142	3	no	—
66	#95	—	2	2	no	—
67	#95	—	2	6	no	—
68	C₂²×C₄	—	2	6	no	—
69	C₂²×C₄	—	284	1	no	—
70	C₂²×C₄	C₂×C₂²⋊D₇₁	1	3	yes	—
71	C₂×C₈	—	284	3	no	—
72	C₂×C₈	—	284	3	no	—
73	C₂×C₈	—	142	6	no	—
74	C₄⋊C₄	—	4	6	no	—
75	C₄⋊C₄	—	8	6	no	—
76	C₄⋊C₄	—	284	6	no	—
77	C₄²	Q₈×D₇₁	1	3	yes	—
78	C₄²	#11765866	1	1	yes	characteristic
79	C₂³	#14556097	1	1	yes	characteristic
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	3	no	—
82	C₂×C₄	—	4	6	no	—
83	C₂×C₄	—	4	6	no	—
84	C₂×C₄	—	4	6	no	—
85	C₂×C₄	—	568	3	no	—
86	C₂×C₄	#14556104	1	3	yes	—
87	C₂×C₄	#14556126	1	3	yes	—
88	C₈	—	284	6	no	—
89	C₂²	#17985918	1	1	yes	characteristic
90	C₂²	#17986172	1	3	yes	—
91	C₂²	#17986177	1	3	yes	—
92	C₄	—	2	3	no	—
93	C₄	—	2	3	no	—
94	C₄	—	8	6	no	—
95	C₄	—	1136	1	no	—
96	C₂	#22587642	1	1	yes	characteristic
97	C₂	#22587663	1	3	yes	—
98	C₂	#22589687	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22588473

C₂

yes

maximal

#22586221

C₂

yes

maximal

C₇₁×#2845

C₂

yes

maximal, characteristic

C₂×#14556039

C₂²

yes

—

#17985869

C₂²

yes

characteristic

#17985761

—

#398×C₇₁

—

#382×C₇₁

C₂²

yes

—

C₂²×C₇₁⋊C₈

—

C₂×C₄⋊Dic₇₁

—

#14556039

—

#14556039

Q₈

yes

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

D₄

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

#95×C₇₁

—

#95×C₇₁

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×Q₈

yes

—

C₄²×C₇₁

yes

C₂³×C₇₁

yes

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₁

C₄⋊Q₈

yes

—

C₂×Dic₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

#395

yes

characteristic

C₂²×C₇₁

#454

yes

—

C₂²×C₇₁

#455

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#2061

yes

—

#2845

D₇₁

yes

characteristic

#1836

—

#1295

—

C₇₁

#10586

yes

characteristic, complemented, Sylow

C₂×#97

—

#398

—

#384

—

#382

C₂×D₇₁

yes

—

#340

—

142

—

C₂²×C₈

—

142

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄⋊C₄

C₂²⋊D₇₁

yes

—

C₂×C₄²

C₂²×D₇₁

yes

characteristic

#97

—

#97

—

142

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

284

—

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

—

C₂×C₈

—

284

—

C₂×C₈

—

284

—

C₂×C₈

—

142

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

284

—

C₄²

Q₈×D₇₁

yes

—

C₄²

yes

C₂³

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

568

—

C₂×C₄

#14556104

yes

—

C₂×C₄

#14556126

yes

—

C₈

—

284

—

C₂²

#17985918

yes

characteristic

C₂²

#17986172

yes

—

C₂²

#17986177

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

1136

—

C₂

#22587642

yes

characteristic

C₂

#22587663

yes

—

C₂

#22589687

yes

—

	subgroup	G/H
center	C₂³	#14556097
commutator, cosocle	C₂×C₄²×C₇₁	C₂³
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#2845	C₂
socle, prime radical	C₂³×C₇₁	#120
hypercenter	#2845	D₇₁
hypocenter	C₇₁	#10586
LCS	G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

subgroup

G/H

center

C₂³

#14556097

commutator, cosocle

C₂×C₄²×C₇₁

C₂³

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#2845

C₂

C₂³×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

N	H ≅ G/N	type
53. C₇₁	#10586	semidirect

H ≅ G/N

type

53. C₇₁

#10586

semidirect

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

subgroup	index
#22588473	2
#22586221	2
C₇₁×#2845	2
#10586	71

subgroup

index

#22588473

#22586221

C₇₁×#2845

#10586

Subgroups — #31901961

Conjugacy classes of subgroups

Complements

Maximal subgroups