Group #31906507 — subgroups

Subgroups — #31906507

Order 18176.

228 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 123 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	#22590143	C₂	1	2	yes	maximal
2	#22586261	C₂	1	4	yes	maximal, complemented
3	C₇₁×#1450	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₇₁⋊C₁₆)	C₄	1	4	yes	—
5	C₂²⋊(C₇₁⋊C₁₆)	C₄	1	4	yes	—
6	#17985775	C₂²	1	4	yes	—
7	#431×C₇₁	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
9	#322×C₇₁	C₂²	1	2	yes	—
10	C₈²×C₇₁	C₄	1	2	yes	—
11	C₂×C₇₁⋊C₁₆	C₂×C₄	1	8	yes	—
12	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
14	C₄⋊C₈×C₇₁	—	2	2	no	—
15	C₂²⋊C₈×C₇₁	—	2	2	no	—
16	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
17	C₄×C₈×C₇₁	C₂×C₄	1	4	yes	—
18	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
20	C₇₁⋊C₁₆	—	2	8	no	—
21	C₂²×C₄×C₇₁	C₄²	1	1	yes	characteristic
22	C₂²×C₄×C₇₁	M₁₆	1	2	yes	—
23	C₂×C₈×C₇₁	—	2	4	no	—
24	C₂×C₈×C₇₁	—	2	4	no	—
25	C₂×C₈×C₇₁	C₄²	1	2	yes	—
26	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
27	C₄²×C₇₁	C₄²	1	1	yes	characteristic
28	C₄²×C₇₁	M₁₆	1	2	yes	—
29	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
30	C₂³×C₇₁	#97	1	1	yes	characteristic
31	C₂×C₄×C₇₁	—	2	2	no	—
32	C₂×C₄×C₇₁	—	4	2	no	—
33	C₂×C₄×C₇₁	#97	1	2	yes	—
34	C₂×C₄×C₇₁	#97	1	1	yes	characteristic
35	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
36	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
37	C₈×C₇₁	—	2	2	no	—
38	C₈×C₇₁	—	4	4	no	—
39	C₈×C₇₁	#131	1	4	yes	—
40	C₂²×C₇₁	—	2	2	no	—
41	C₂²×C₇₁	—	2	1	no	—
42	C₂²×C₇₁	C₂×#97	1	1	yes	characteristic
43	C₄×C₇₁	—	2	2	no	—
44	C₄×C₇₁	—	2	2	no	—
45	C₄×C₇₁	—	2	1	no	—
46	C₄×C₇₁	#405	1	2	yes	—
47	#11416	—	71	1	no	maximal, Sylow
48	C₂×C₇₁	—	4	1	no	—
49	C₂×C₇₁	#1452	1	1	yes	characteristic
50	C₂×C₇₁	#2110	1	2	yes	—
51	#2163	—	71	2	no	—
52	#1450	D₇₁	1	1	yes	characteristic
53	#1313	—	71	4	no	—
54	C₇₁	#11416	1	1	yes	characteristic, complemented, Sylow
55	#431	Dic₇₁	1	2	yes	—
56	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
57	C₄⋊C₁₆	—	71	4	no	—
58	C₂²⋊C₁₆	—	71	4	no	—
59	#346	—	71	4	no	—
60	#322	C₂×D₇₁	1	2	yes	—
61	C₈²	Dic₇₁	1	2	yes	—
62	C₂²×C₈	C₄×D₇₁	1	2	yes	—
63	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
64	C₂×C₁₆	—	71	8	no	—
65	C₄⋊C₈	—	2	2	no	—
66	C₂²⋊C₈	—	2	2	no	—
67	C₄×C₈	C₄×D₇₁	1	2	yes	—
68	C₄×C₈	C₂×Dic₇₁	1	4	yes	—
69	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
70	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
71	C₂²×C₄	C₄×Dic₇₁	1	1	yes	characteristic
72	C₂²×C₄	#11765849	1	2	yes	—
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	4	no	—
75	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
76	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
77	C₄²	C₄×Dic₇₁	1	1	yes	characteristic
78	C₄²	#11765849	1	2	yes	—
79	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
80	C₁₆	—	142	8	no	—
81	C₂³	#14556039	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	#14556039	1	2	yes	—
85	C₂×C₄	#14556039	1	1	yes	characteristic
86	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
87	C₂×C₄	C₂×#11765849	1	2	yes	—
88	C₈	—	2	2	no	—
89	C₈	—	4	4	no	—
90	C₈	#14556134	1	4	yes	—
91	C₂²	—	2	1	no	—
92	C₂²	—	2	2	no	—
93	C₂²	C₂×#14556039	1	1	yes	characteristic
94	C₄	—	2	1	no	—
95	C₄	—	2	2	no	—
96	C₄	—	2	2	no	—
97	C₄	#17985954	1	2	yes	—
98	C₂	—	4	1	no	—
99	C₂	#22586632	1	1	yes	characteristic
100	C₂	#22589945	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22590143

C₂

yes

maximal

#22586261

C₂

yes

maximal, complemented

C₇₁×#1450

C₂

yes

maximal, characteristic

C₄⋊(C₇₁⋊C₁₆)

C₄

yes

—

C₂²⋊(C₇₁⋊C₁₆)

C₄

yes

—

#17985775

C₂²

yes

—

#431×C₇₁

C₄

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#322×C₇₁

C₂²

yes

—

C₈²×C₇₁

C₄

yes

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₁

—

C₂²⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₄²

yes

characteristic

C₂²×C₄×C₇₁

M₁₆

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄²

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

—

C₄²×C₇₁

C₄²

yes

characteristic

C₄²×C₇₁

M₁₆

yes

—

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₂³×C₇₁

#97

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#97

yes

—

C₂×C₄×C₇₁

#97

yes

characteristic

C₂×C₄×C₇₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₁

C₂×M₁₆

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#131

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×#97

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#405

yes

—

#11416

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1452

yes

characteristic

C₂×C₇₁

#2110

yes

—

#2163

—

#1450

D₇₁

yes

characteristic

#1313

—

C₇₁

#11416

yes

characteristic, complemented, Sylow

#431

Dic₇₁

yes

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#322

C₂×D₇₁

yes

—

C₈²

Dic₇₁

yes

—

C₂²×C₈

C₄×D₇₁

yes

—

C₂×C₄²

C₂×Dic₇₁

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₄×D₇₁

yes

—

C₄×C₈

C₂×Dic₇₁

yes

—

C₄×C₈

C₂×Dic₇₁

yes

characteristic

C₄×C₈

C₂²×D₇₁

yes

characteristic

C₂²×C₄

C₄×Dic₇₁

yes

characteristic

C₂²×C₄

#11765849

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×Dic₇₁

yes

—

C₂×C₈

C₂×C₄×D₇₁

yes

—

C₄²

C₄×Dic₇₁

yes

characteristic

C₄²

#11765849

yes

—

C₄²

C₂²×Dic₇₁

yes

characteristic

C₁₆

—

142

—

C₂³

#14556039

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#14556039

yes

—

C₂×C₄

#14556039

yes

characteristic

C₂×C₄

C₂×C₄×Dic₇₁

yes

characteristic

C₂×C₄

C₂×#11765849

yes

—

C₈

—

C₈

—

C₈

#14556134

yes

—

C₂²

—

C₂²

—

C₂²

C₂×#14556039

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#17985954

yes

—

C₂

—

C₂

#22586632

yes

characteristic

100

C₂

#22589945

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₇₁
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1450	C₂
socle	C₂²×C₇₁	C₂×#97
prime radical	C₂³×C₇₁	#97
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#1450	D₇₁
hypocenter	C₇₁	#11416
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ #1450

subgroup

G/H

center

C₂×C₄

C₂×C₄×Dic₇₁

commutator

C₂×C₄×C₇₁

C₂×C₄²

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1450

C₂

socle

C₂²×C₇₁

C₂×#97

prime radical

C₂³×C₇₁

#97

cosocle

C₄×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₄ ◁ #1450

N	H ≅ G/N	type
2. #22586261	C₂	semidirect
54. C₇₁	#11416	semidirect

H ≅ G/N

type

2. #22586261

C₂

semidirect

54. C₇₁

#11416

semidirect

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

subgroup	index
#22590143	2
#22586261	2
C₇₁×#1450	2
#11416	71

subgroup

index

#22590143

#22586261

C₇₁×#1450

#11416

Subgroups — #31906507

Conjugacy classes of subgroups

Complements

Maximal subgroups