Subgroups — #564×Dic₇₁ #32160174

Order 18176.

1800 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22591526	C₂	1	15	yes	maximal
2	#125×Dic₇₁	C₂	1	15	yes	maximal
3	C₂×#564×C₇₁	C₂	1	1	yes	maximal, characteristic
4	#17985898	C₂²	1	30	yes	—
5	#17985885	C₂²	1	60	yes	—
6	#17985865	C₂²	1	15	yes	—
7	C₄⋊C₄×Dic₇₁	C₂²	1	30	yes	—
8	C₄²×Dic₇₁	C₂²	1	5	yes	—
9	#564×C₇₁	C₄	1	16	yes	complemented
10	C₂×#125×C₇₁	C₂²	1	15	yes	—
11	C₂×C₄⋊Dic₇₁	C₂³	1	30	yes	—
12	C₂×C₄×Dic₇₁	C₂³	1	15	yes	—
13	C₂×C₄×Dic₇₁	C₂³	1	15	yes	—
14	C₇₁⋊#95	C₂³	1	60	yes	—
15	#125×C₇₁	C₂×C₄	1	120	yes	—
16	C₂×C₄⋊C₄×C₇₁	C₂³	1	30	yes	—
17	C₂×C₄²×C₇₁	C₂³	1	5	yes	—
18	C₂²×Dic₇₁	C₂⁴	1	15	yes	—
19	C₂²×Dic₇₁	C₂⁴	1	1	yes	characteristic
20	C₄⋊Dic₇₁	—	4	30	no	—
21	C₄×Dic₇₁	—	2	30	no	—
22	C₄×Dic₇₁	—	4	15	no	—
23	C₂²×C₄×C₇₁	C₂⁴	1	15	yes	—
24	C₄⋊C₄×C₇₁	C₂²×C₄	1	120	yes	—
25	C₄²×C₇₁	C₂²×C₄	1	20	yes	—
26	C₂³×C₇₁	C₂⁵	1	1	yes	characteristic
27	C₂×C₄×C₇₁	—	2	15	no	—
28	C₂×C₄×C₇₁	—	2	15	no	—
29	C₂×C₄×C₇₁	C₂³×C₄	1	30	yes	—
30	C₂×Dic₇₁	—	2	15	no	—
31	C₂×Dic₇₁	—	2	30	no	—
32	C₂×Dic₇₁	2₋¹⁺⁴	1	6	yes	—
33	C₂²×C₇₁	C₂⁴×C₄	1	1	yes	characteristic
34	C₂²×C₇₁	C₂×2₋¹⁺⁴	1	3	yes	—
35	C₂²×C₇₁	#585	1	3	yes	—
36	C₄×C₇₁	—	2	30	no	—
37	Dic₇₁	—	4	15	no	—
38	Dic₇₁	#564	1	4	yes	direct product factor
39	C₄×#564	—	71	1	no	maximal, Sylow
40	C₂×C₇₁	C₄×2₋¹⁺⁴	1	3	yes	—
41	C₂×C₇₁	C₂×#564	1	1	yes	characteristic
42	C₂×C₇₁	#3531	1	3	yes	—
43	C₂×#564	D₇₁	1	1	yes	characteristic
44	#2382	—	71	15	no	—
45	C₄×#125	—	71	15	no	—
46	C₇₁	C₄×#564	1	1	yes	characteristic, complemented, Sylow
47	#564	Dic₇₁	1	16	yes	direct product factor
48	C₂×#125	C₂×D₇₁	1	15	yes	—
49	#389	—	71	30	no	—
50	#387	—	71	60	no	—
51	#382	—	71	15	no	—
52	C₄×C₄⋊C₄	—	71	30	no	—
53	C₄³	—	71	5	no	—
54	#125	C₂×Dic₇₁	1	120	yes	—
55	C₂×C₄⋊C₄	—	71	30	no	—
56	C₂×C₄⋊C₄	C₂²×D₇₁	1	30	yes	—
57	C₂×C₄²	—	71	15	no	—
58	C₂×C₄²	—	71	15	no	—
59	C₂×C₄²	C₂²×D₇₁	1	5	yes	—
60	#95	—	71	60	no	—
61	C₂²×C₄	—	71	1	no	—
62	C₂²×C₄	—	71	15	no	—
63	C₂²×C₄	C₂³×D₇₁	1	15	yes	—
64	C₄⋊C₄	—	284	30	no	—
65	C₄⋊C₄	C₂²×Dic₇₁	1	120	yes	—
66	C₄²	—	284	15	no	—
67	C₄²	—	142	30	no	—
68	C₄²	C₂²×Dic₇₁	1	20	yes	—
69	C₂³	C₂⁴×D₇₁	1	1	yes	characteristic
70	C₂×C₄	—	2	15	no	—
71	C₂×C₄	—	2	15	no	—
72	C₂×C₄	—	71	6	no	—
73	C₂×C₄	—	142	15	no	—
74	C₂×C₄	—	142	30	no	—
75	C₂×C₄	C₂³×Dic₇₁	1	30	yes	—
76	C₂²	C₂⁴×Dic₇₁	1	1	yes	characteristic
77	C₂²	2₋¹⁺⁴×D₇₁	1	3	yes	—
78	C₂²	#17986836	1	3	yes	—
79	C₄	—	2	30	no	—
80	C₄	—	284	15	no	—
81	C₄	—	71	4	no	—
82	C₂	2₋¹⁺⁴×Dic₇₁	1	3	yes	—
83	C₂	#564×D₇₁	1	1	yes	characteristic
84	C₂	#22602488	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₇₁
commutator	C₂²×C₇₁	C₂⁴×C₄
Fitting	C₂×#564×C₇₁	C₂
socle, prime radical, cosocle	C₂³×C₇₁	C₂⁵
hypercenter	C₂×#564	D₇₁
hypocenter	C₇₁	C₄×#564
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×#564

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
9. #564×C₇₁	C₄	semidirect
38. Dic₇₁	#564	direct (N × H)
46. C₇₁	C₄×#564	semidirect
47. #564	Dic₇₁	direct (N × H)

Maximal subgroups

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

subgroup	index
#22591526	2
#125×Dic₇₁	2
C₂×#564×C₇₁	2
C₄×#564	71

Subgroups — #564×Dic71 #32160174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #564×Dic₇₁ #32160174