Group #31884547 — subgroups

Subgroups — #31884547

Order 18176.

148 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 63 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	#22590156	C₂	1	2	yes	maximal
2	C₇₁×#2151	C₂	1	1	yes	maximal, characteristic
3	#17985820	—	2	2	no	—
4	#17985781	—	2	2	no	—
5	#17985777	C₄	1	4	yes	—
6	C₂×M₃₂×C₇₁	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₇₁	C₄	1	1	yes	characteristic
8	#405×C₇₁	C₂²	1	1	yes	characteristic
9	C₄⋊C₁₆×C₇₁	—	2	2	no	—
10	#14556068	—	2	4	no	—
11	C₂×M₁₆×C₇₁	D₄	1	1	yes	characteristic
12	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
13	#117×C₇₁	Q₈	1	1	yes	characteristic
14	M₃₂×C₇₁	—	4	1	no	—
15	C₂×C₁₆×C₇₁	—	2	1	no	—
16	C₂×C₁₆×C₇₁	—	2	1	no	—
17	C₂×C₁₆×C₇₁	—	4	1	no	—
18	#97×C₇₁	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₇₁	D₄	1	2	yes	—
20	C₇₁⋊C₁₆	—	4	4	no	—
21	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
22	M₁₆×C₇₁	—	4	1	no	—
23	C₂×C₈×C₇₁	—	2	1	no	—
24	C₂×C₈×C₇₁	—	2	1	no	—
25	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
26	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
27	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
28	C₄⋊C₄×C₇₁	—	2	1	no	—
29	C₂²⋊C₄×C₇₁	—	2	1	no	—
30	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
31	C₁₆×C₇₁	—	4	1	no	—
32	C₁₆×C₇₁	—	4	1	no	—
33	C₂³×C₇₁	#108	1	1	yes	characteristic
34	C₂×C₄×C₇₁	—	2	1	no	—
35	C₂×C₄×C₇₁	—	2	1	no	—
36	C₂×C₄×C₇₁	—	2	2	no	—
37	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
38	C₂×C₄×C₇₁	#108	1	1	yes	characteristic
39	C₈×C₇₁	—	2	2	no	—
40	C₈×C₇₁	—	2	1	no	—
41	C₈×C₇₁	#100	1	1	yes	characteristic
42	C₈×C₇₁	#101	1	1	yes	characteristic
43	C₂²×C₇₁	—	2	1	no	—
44	C₂²×C₇₁	—	4	1	no	—
45	C₂²×C₇₁	#341	1	1	yes	characteristic
46	C₄×C₇₁	—	2	1	no	—
47	C₄×C₇₁	—	4	2	no	—
48	C₄×C₇₁	H(3,ℤ/4)	1	1	yes	characteristic
49	C₄×C₇₁	#343	1	1	yes	characteristic
50	#7239	—	71	1	no	maximal, Sylow
51	C₂×C₇₁	—	2	1	no	—
52	C₂×C₇₁	—	4	1	no	—
53	C₂×C₇₁	#1290	1	1	yes	characteristic
54	#2166	—	71	2	no	—
55	#2151	D₇₁	1	1	yes	characteristic
56	C₇₁	#7239	1	1	yes	characteristic, complemented, Sylow
57	C₂×M₃₂	Dic₇₁	1	1	yes	characteristic
58	C₂²×C₁₆	Dic₇₁	1	1	yes	characteristic
59	#405	C₂×D₇₁	1	1	yes	characteristic
60	#364	—	142	2	no	—
61	C₄⋊C₁₆	—	2	2	no	—
62	#349	—	142	2	no	—
63	#347	—	71	4	no	—
64	C₂×M₁₆	D₂₈₄	1	1	yes	characteristic
65	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
66	#117	Dic₁₄₂	1	1	yes	characteristic
67	M₃₂	—	4	1	no	—
68	M₃₂	—	142	4	no	—
69	C₂×C₁₆	—	2	1	no	—
70	C₂×C₁₆	—	2	1	no	—
71	C₂×C₁₆	—	4	1	no	—
72	#97	C₄×D₇₁	1	2	yes	—
73	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
74	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
75	M₁₆	—	4	1	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
79	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
80	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
81	C₄⋊C₄	—	2	1	no	—
82	C₂²⋊C₄	—	2	1	no	—
83	C₄²	Dic₇₁⋊C₄	1	2	yes	—
84	C₁₆	—	4	1	no	—
85	C₁₆	—	4	1	no	—
86	C₁₆	—	284	4	no	—
87	C₂³	#14556064	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
92	C₂×C₄	#14556064	1	1	yes	characteristic
93	C₈	—	2	2	no	—
94	C₈	—	2	1	no	—
95	C₈	#14556045	1	1	yes	characteristic
96	C₈	#14556047	1	1	yes	characteristic
97	C₂²	—	2	1	no	—
98	C₂²	—	4	1	no	—
99	C₂²	#17985764	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	4	2	no	—
102	C₄	#17985756	1	1	yes	characteristic
103	C₄	#17985768	1	1	yes	characteristic
104	C₂	—	2	1	no	—
105	C₂	—	4	1	no	—
106	C₂	#22586210	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590156

C₂

yes

maximal

C₇₁×#2151

C₂

yes

maximal, characteristic

#17985820

—

#17985781

—

#17985777

C₄

yes

—

C₂×M₃₂×C₇₁

C₄

yes

characteristic

C₂²×C₁₆×C₇₁

C₄

yes

characteristic

#405×C₇₁

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₁

—

#14556068

—

C₂×M₁₆×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

#117×C₇₁

Q₈

yes

characteristic

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

#97×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₄⋊C₄

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄²

yes

characteristic

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

C₄⋊C₄

yes

—

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₂³×C₇₁

#108

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#341

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

D₇₁

yes

characteristic

C₇₁

#7239

yes

characteristic, complemented, Sylow

C₂×M₃₂

Dic₇₁

yes

characteristic

C₂²×C₁₆

Dic₇₁

yes

characteristic

#405

C₂×D₇₁

yes

characteristic

#364

—

142

—

C₄⋊C₁₆

—

#349

—

142

—

#347

—

C₂×M₁₆

D₂₈₄

yes

characteristic

C₂²×C₈

C₂×Dic₇₁

yes

characteristic

#117

Dic₁₄₂

yes

characteristic

M₃₂

—

M₃₂

—

142

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

C₄×D₇₁

yes

—

C₄×C₈

C₂²⋊D₇₁

yes

—

C₂²×C₄

C₄⋊Dic₇₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×Dic₇₁

yes

characteristic

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

—

C₂×C₈

C₂²⋊Dic₇₁

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

Dic₇₁⋊C₄

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

284

—

C₂³

#14556064

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₇₁⋊#95

yes

characteristic

C₂×C₄

#14556064

yes

characteristic

C₈

—

C₈

—

C₈

yes

C₈

yes

C₂²

—

C₂²

—

C₂²

#17985764

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

#17985756

yes

characteristic

103

C₄

#17985768

yes

characteristic

104

C₂

—

105

C₂

—

106

C₂

#22586210

yes

characteristic

	subgroup	G/H
center	C₄	#17985756
commutator	C₂×C₈×C₇₁	C₄²
Frattini	#405	C₂×D₇₁
Fitting	C₇₁×#2151	C₂
socle	C₂×C₇₁	#1290
prime radical	C₂³×C₇₁	#108
cosocle	#405×C₇₁	C₂²
hypercenter	#2151	D₇₁
hypocenter	C₇₁	#7239
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2151

subgroup

G/H

center

C₄

#17985756

commutator

C₂×C₈×C₇₁

C₄²

Frattini

#405

C₂×D₇₁

Fitting

C₇₁×#2151

C₂

C₂×C₇₁

C₂³×C₇₁

#405×C₇₁

C₂²

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2151

N	H ≅ G/N	type
56. C₇₁	#7239	semidirect

H ≅ G/N

type

56. C₇₁

#7239

semidirect

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

subgroup	index
#22590156	2
C₇₁×#2151	2
#7239	71

subgroup

index

#22590156

C₇₁×#2151

#7239

Subgroups — #31884547

Conjugacy classes of subgroups

Complements

Maximal subgroups