Group #31887886 — subgroups

Subgroups — #31887886

Order 18176.

382 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 167 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	#22588085	C₂	1	3	yes	maximal
2	C₇₁⋊#1273	C₂	1	3	yes	maximal
3	C₇₁×#1273	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₇₁⋊C₈	C₂²	1	3	yes	—
5	C₄²×Dic₇₁	C₂²	1	1	yes	characteristic
6	#17985753	C₄	1	6	yes	—
7	#17985753	C₄	1	6	yes	—
8	C₂×C₄×C₈×C₇₁	C₂²	1	3	yes	—
9	C₂²×C₇₁⋊C₈	C₂×C₄	1	6	yes	—
10	C₂×C₄×Dic₇₁	C₂×C₄	1	3	yes	—
11	C₂×C₄×Dic₇₁	C₂×C₄	1	3	yes	—
12	C₄×C₇₁⋊C₈	—	2	3	no	—
13	C₄×C₇₁⋊C₈	—	2	3	no	—
14	C₂²×C₈×C₇₁	C₂×C₄	1	6	yes	—
15	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₇₁	—	2	6	no	—
17	C₂²×Dic₇₁	C₄²	1	1	yes	characteristic
18	C₂²×Dic₇₁	C₄²	1	3	yes	—
19	C₂×C₇₁⋊C₈	—	2	6	no	—
20	C₂×C₇₁⋊C₈	—	2	6	no	—
21	C₂×C₇₁⋊C₈	—	2	6	no	—
22	C₄×Dic₇₁	—	2	6	no	—
23	C₄×Dic₇₁	—	2	6	no	—
24	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
25	C₂×C₈×C₇₁	—	2	6	no	—
26	C₂×C₈×C₇₁	—	2	6	no	—
27	C₂×C₈×C₇₁	—	2	6	no	—
28	C₄²×C₇₁	#41	1	3	yes	—
29	C₄²×C₇₁	#41	1	1	yes	characteristic
30	C₂³×C₇₁	C₂×C₄²	1	1	yes	characteristic
31	C₂×C₄×C₇₁	#117	1	3	yes	—
32	C₂×C₄×C₇₁	#117	1	3	yes	—
33	C₂×C₄×C₇₁	#131	1	6	yes	—
34	C₂×C₄×C₇₁	#131	1	6	yes	—
35	C₂×Dic₇₁	—	2	3	no	—
36	C₂×Dic₇₁	—	2	3	no	—
37	C₂×Dic₇₁	—	2	6	no	—
38	C₈×C₇₁	—	4	6	no	—
39	C₇₁⋊C₈	—	4	6	no	—
40	C₂²×C₇₁	#376	1	1	yes	characteristic
41	C₂²×C₇₁	#405	1	3	yes	—
42	C₂²×C₇₁	#405	1	3	yes	—
43	C₄×C₇₁	#433	1	6	yes	—
44	C₄×C₇₁	#433	1	6	yes	—
45	Dic₇₁	—	4	1	no	—
46	Dic₇₁	—	4	3	no	—
47	#8135	—	71	1	no	maximal, Sylow
48	C₂×C₇₁	#1454	1	1	yes	characteristic
49	C₂×C₇₁	#1454	1	3	yes	—
50	C₂×C₇₁	#1746	1	3	yes	—
51	#1744	—	71	3	no	—
52	#1273	—	71	3	no	—
53	#1273	D₇₁	1	1	yes	characteristic
54	C₇₁	#8135	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₄×C₈	—	71	3	no	—
56	C₂×C₄×C₈	C₂×D₇₁	1	3	yes	—
57	C₄³	—	71	1	no	—
58	#336	—	71	6	no	—
59	#336	—	71	6	no	—
60	C₂²×C₈	—	71	6	no	—
61	C₂²×C₈	C₄×D₇₁	1	6	yes	—
62	C₂×C₄²	—	71	3	no	—
63	C₂×C₄²	—	71	3	no	—
64	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
65	C₄×C₈	—	2	6	no	—
66	C₄×C₈	—	142	3	no	—
67	C₄×C₈	—	142	3	no	—
68	C₂²×C₄	—	71	1	no	—
69	C₂²×C₄	—	71	3	no	—
70	C₂²×C₄	C₂×C₄×D₇₁	1	3	yes	—
71	C₂×C₈	—	2	6	no	—
72	C₂×C₈	—	2	6	no	—
73	C₂×C₈	—	2	6	no	—
74	C₂×C₈	—	142	6	no	—
75	C₂×C₈	—	142	6	no	—
76	C₂×C₈	—	142	6	no	—
77	C₄²	—	142	6	no	—
78	C₄²	—	142	6	no	—
79	C₄²	#11765864	1	3	yes	—
80	C₄²	#11765866	1	1	yes	characteristic
81	C₂³	C₄²×D₇₁	1	1	yes	characteristic
82	C₂×C₄	—	142	3	no	—
83	C₂×C₄	—	142	3	no	—
84	C₂×C₄	—	142	6	no	—
85	C₂×C₄	#14556084	1	3	yes	—
86	C₂×C₄	#14556086	1	3	yes	—
87	C₂×C₄	#14556134	1	6	yes	—
88	C₂×C₄	#14556136	1	6	yes	—
89	C₈	—	4	6	no	—
90	C₈	—	284	6	no	—
91	C₂²	#17985844	1	1	yes	characteristic
92	C₂²	#17985953	1	3	yes	—
93	C₂²	#17985956	1	3	yes	—
94	C₄	—	284	1	no	—
95	C₄	—	284	3	no	—
96	C₄	#17986057	1	6	yes	—
97	C₄	#17986059	1	6	yes	—
98	C₂	#22586639	1	1	yes	characteristic
99	C₂	#22586641	1	3	yes	—
100	C₂	#22588090	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22588085

C₂

yes

maximal

C₇₁⋊#1273

C₂

yes

maximal

C₇₁×#1273

C₂

yes

maximal, characteristic

C₂×C₄×C₇₁⋊C₈

C₂²

yes

—

C₄²×Dic₇₁

C₂²

yes

characteristic

#17985753

C₄

yes

—

#17985753

C₄

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

—

C₂²×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₇₁

C₂×C₄

yes

—

C₂×C₄×Dic₇₁

C₂×C₄

yes

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

—

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄×C₈×C₇₁

—

C₂²×Dic₇₁

C₄²

yes

characteristic

C₂²×Dic₇₁

C₄²

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄²×C₇₁

#41

yes

—

C₄²×C₇₁

#41

yes

characteristic

C₂³×C₇₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₁

#117

yes

—

C₂×C₄×C₇₁

#117

yes

—

C₂×C₄×C₇₁

#131

yes

—

C₂×C₄×C₇₁

#131

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

#376

yes

characteristic

C₂²×C₇₁

#405

yes

—

C₂²×C₇₁

#405

yes

—

C₄×C₇₁

#433

yes

—

C₄×C₇₁

#433

yes

—

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#1746

yes

—

#1744

—

#1273

—

#1273

D₇₁

yes

characteristic

C₇₁

#8135

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₁

yes

—

C₄³

—

#336

—

#336

—

C₂²×C₈

—

C₂²×C₈

C₄×D₇₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₇₁

yes

characteristic

C₄×C₈

—

C₄×C₈

—

142

—

C₄×C₈

—

142

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×D₇₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₄²

—

142

—

C₄²

—

142

—

C₄²

#11765864

yes

—

C₄²

#11765866

yes

characteristic

C₂³

C₄²×D₇₁

yes

characteristic

C₂×C₄

—

142

—

C₂×C₄

—

142

—

C₂×C₄

—

142

—

C₂×C₄

#14556084

yes

—

C₂×C₄

#14556086

yes

—

C₂×C₄

#14556134

yes

—

C₂×C₄

#14556136

yes

—

C₈

—

C₈

—

284

—

C₂²

#17985844

yes

characteristic

C₂²

#17985953

yes

—

C₂²

#17985956

yes

—

C₄

—

284

—

C₄

—

284

—

C₄

#17986057

yes

—

C₄

#17986059

yes

—

C₂

#22586639

yes

characteristic

C₂

#22586641

yes

—

100

C₂

#22588090

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₁
commutator, socle, prime radical	C₂³×C₇₁	C₂×C₄²
Fitting	C₇₁×#1273	C₂
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1273	D₇₁
hypocenter	C₇₁	#8135
LCS	G ▷ C₂³×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₇₁

commutator, socle, prime radical

C₂³×C₇₁

C₂×C₄²

Fitting

C₇₁×#1273

C₂

cosocle

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂³×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
54. C₇₁	#8135	semidirect

H ≅ G/N

type

54. C₇₁

#8135

semidirect

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

subgroup	index
#22588085	2
C₇₁⋊#1273	2
C₇₁×#1273	2
#8135	71

subgroup

index

#22588085

C₇₁⋊#1273

C₇₁×#1273

#8135

Subgroups — #31887886

Conjugacy classes of subgroups

Complements

Maximal subgroups