Group #31912157 — subgroups

Subgroups — #31912157

Order 18176.

290 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 79 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	#22590565	C₂	1	4	yes	maximal, complemented
2	#22587955	C₂	1	1	yes	maximal, characteristic
3	#22586257	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₁×#1710	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986376	—	2	2	no	—
6	#17986376	C₂²	1	2	yes	—
7	#17985799	—	2	8	no	—
8	C₄×C₇₁⋊C₁₆	C₂²	1	2	yes	—
9	#500×C₇₁	C₂²	1	2	yes	—
10	#487×C₇₁	—	2	2	no	—
11	C₈²×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×D₅₆₈	—	2	4	no	—
13	C₂×D₅₆₈	—	4	4	no	—
14	#14556125	—	2	2	no	—
15	C₂×C₇₁⋊C₁₆	D₄	1	4	yes	—
16	C₄⋊Q₈×C₇₁	—	2	2	no	—
17	#107×C₇₁	—	2	4	no	—
18	#103×C₇₁	—	4	4	no	—
19	C₄×C₈×C₇₁	D₄	1	2	yes	—
20	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
21	C₂×D₂₈₄	—	4	2	no	—
22	C₂×D₂₈₄	—	4	4	no	—
23	D₅₆₈	—	4	8	no	—
24	D₅₆₈	—	8	8	no	—
25	C₇₁⋊C₁₆	—	2	4	no	—
26	C₂×Q₈×C₇₁	—	4	2	no	—
27	C₂×C₈×C₇₁	D₈	1	4	yes	—
28	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
29	C₄⋊C₄×C₇₁	—	4	2	no	—
30	C₄⋊C₄×C₇₁	—	4	2	no	—
31	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
32	C₂²×D₇₁	—	8	2	no	—
33	Q₈×C₇₁	—	8	4	no	—
34	C₂×C₄×C₇₁	—	8	2	no	—
35	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
36	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
37	D₂₈₄	—	8	4	no	—
38	D₂₈₄	—	8	8	no	—
39	C₈×C₇₁	—	2	4	no	—
40	C₈×C₇₁	#135	1	4	yes	—
41	C₂²×C₇₁	#493	1	1	yes	characteristic
42	C₂×D₇₁	—	16	2	no	—
43	C₂×D₇₁	—	16	4	no	—
44	C₄×C₇₁	—	16	2	no	—
45	C₄×C₇₁	#495	1	2	yes	—
46	C₄×C₇₁	#509	1	4	yes	—
47	#12513	—	71	1	no	maximal, Sylow
48	C₂×C₇₁	#1709	1	1	yes	characteristic
49	C₂×C₇₁	#2250	1	2	yes	—
50	D₇₁	—	32	2	no	—
51	#2240	—	71	4	no	—
52	#1712	—	71	1	no	—
53	#1710	D₇₁	1	1	yes	characteristic, complemented
54	#1311	—	71	1	no	—
55	C₇₁	#12513	1	1	yes	characteristic, complemented, Sylow
56	#500	C₂×D₇₁	1	2	yes	—
57	#493	—	71	2	no	—
58	#493	—	142	2	no	—
59	#487	—	2	2	no	—
60	#357	—	142	8	no	—
61	C₄×C₁₆	—	71	2	no	—
62	C₈²	C₂×D₇₁	1	1	yes	characteristic
63	C₂×D₈	—	284	4	no	—
64	C₂×D₈	—	142	4	no	—
65	C₄⋊Q₈	—	2	2	no	—
66	#127	—	142	2	no	—
67	C₂×C₁₆	—	71	4	no	—
68	#107	—	2	4	no	—
69	#103	—	4	4	no	—
70	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
71	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
72	C₂×Q₈	—	4	2	no	—
73	C₂×D₄	—	284	2	no	—
74	C₂×D₄	—	284	4	no	—
75	D₈	—	284	8	no	—
76	D₈	—	568	8	no	—
77	C₂×C₈	#11765850	1	4	yes	—
78	C₂×C₈	D₄×D₇₁	1	2	yes	—
79	C₄⋊C₄	—	4	2	no	—
80	C₄⋊C₄	—	4	2	no	—
81	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
82	C₁₆	—	142	4	no	—
83	C₂³	—	568	2	no	—
84	Q₈	—	8	4	no	—
85	D₄	—	568	4	no	—
86	D₄	—	568	8	no	—
87	C₂×C₄	—	8	2	no	—
88	C₂×C₄	#14556124	1	1	yes	characteristic
89	C₂×C₄	C₂×#11765850	1	2	yes	—
90	C₈	—	2	4	no	—
91	C₈	#14556151	1	4	yes	—
92	C₂²	—	1136	2	no	—
93	C₂²	—	1136	4	no	—
94	C₂²	#17986375	1	1	yes	characteristic
95	C₄	—	16	2	no	—
96	C₄	#17986384	1	2	yes	—
97	C₄	#17986434	1	4	yes	—
98	C₂	—	2272	2	no	—
99	C₂	#22587947	1	1	yes	characteristic
100	C₂	#22590607	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22590565

C₂

yes

maximal, complemented

#22587955

C₂

yes

maximal, characteristic

#22586257

C₂

yes

maximal, characteristic, complemented

C₇₁×#1710

C₂

yes

maximal, characteristic, complemented

#17986376

—

#17986376

C₂²

yes

—

#17985799

—

C₄×C₇₁⋊C₁₆

C₂²

yes

—

#500×C₇₁

C₂²

yes

—

#487×C₇₁

—

C₈²×C₇₁

C₂²

yes

characteristic

C₂×D₅₆₈

—

C₂×D₅₆₈

—

#14556125

—

C₂×C₇₁⋊C₁₆

D₄

yes

—

C₄⋊Q₈×C₇₁

—

#107×C₇₁

—

#103×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×D₂₈₄

—

C₂×D₂₈₄

—

D₅₆₈

—

D₅₆₈

—

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₂²×D₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

—

D₂₈₄

—

D₂₈₄

—

C₈×C₇₁

—

C₈×C₇₁

#135

yes

—

C₂²×C₇₁

#493

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#495

yes

—

C₄×C₇₁

#509

yes

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

D₇₁

—

#2240

—

#1712

—

#1710

D₇₁

yes

characteristic, complemented

#1311

—

C₇₁

#12513

yes

characteristic, complemented, Sylow

#500

C₂×D₇₁

yes

—

#493

—

#493

—

142

—

#487

—

#357

—

142

—

C₄×C₁₆

—

C₈²

C₂×D₇₁

yes

characteristic

C₂×D₈

—

284

—

C₂×D₈

—

142

—

C₄⋊Q₈

—

#127

—

142

—

C₂×C₁₆

—

#107

—

#103

—

C₄×C₈

C₂²⋊D₇₁

yes

—

C₄×C₈

C₂²×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

284

—

C₂×D₄

—

284

—

D₈

—

284

—

D₈

—

568

—

C₂×C₈

#11765850

yes

—

C₂×C₈

D₄×D₇₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

C₁₆

—

142

—

C₂³

—

568

—

Q₈

—

D₄

—

568

—

D₄

—

568

—

C₂×C₄

—

C₂×C₄

#14556124

yes

characteristic

C₂×C₄

C₂×#11765850

yes

—

C₈

—

C₈

#14556151

yes

—

C₂²

—

1136

—

C₂²

—

1136

—

C₂²

#17986375

yes

characteristic

C₄

—

C₄

#17986384

yes

—

C₄

#17986434

yes

—

C₂

—

2272

—

C₂

#22587947

yes

characteristic

100

C₂

#22590607

yes

—

	subgroup	G/H
center	C₂²	#17986375
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1710	C₂
socle	C₂²×C₇₁	#493
prime radical	#22587955	C₂
hypercenter	#1710	D₇₁
hypocenter	C₇₁	#12513
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#17986375

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1710

C₂

C₂²×C₇₁

C₂

D₇₁

C₇₁

G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
1. #22590565	C₂	semidirect
3. #22586257	C₂	semidirect
4. C₇₁×#1710	C₂	semidirect
53. #1710	D₇₁	semidirect
55. C₇₁	#12513	semidirect

H ≅ G/N

type

1. #22590565

C₂

semidirect

3. #22586257

C₂

semidirect

4. C₇₁×#1710

C₂

semidirect

53. #1710

D₇₁

semidirect

55. C₇₁

#12513

semidirect

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

subgroup	index
#22590565	2
#22587955	2
#22586257	2
C₇₁×#1710	2
#12513	71

subgroup

index

C₇₁×#1710

Subgroups — #31912157

Conjugacy classes of subgroups

Complements

Maximal subgroups