Subgroups — C₄×C₄₁×#564 #23810556

Order 10496.

1800 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#564	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2382	C₂	1	15	yes	maximal
3	C₄×#125×C₄₁	C₂	1	15	yes	maximal
4	C₄₁×#564	C₄	1	16	yes	direct product factor
5	C₂×#125×C₄₁	C₂²	1	15	yes	—
6	C₄₁×#389	C₂²	1	30	yes	—
7	C₄₁×#387	C₂²	1	60	yes	—
8	C₄₁×#382	C₂²	1	15	yes	—
9	C₄×C₄⋊C₄×C₄₁	C₂²	1	30	yes	—
10	C₄³×C₄₁	C₂²	1	5	yes	—
11	#125×C₄₁	C₂×C₄	1	120	yes	—
12	C₂×C₄⋊C₄×C₄₁	C₂³	1	30	yes	—
13	C₂×C₄⋊C₄×C₄₁	C₂³	1	30	yes	—
14	C₂×C₄²×C₄₁	C₂³	1	15	yes	—
15	C₂×C₄²×C₄₁	C₂³	1	15	yes	—
16	C₂×C₄²×C₄₁	C₂³	1	5	yes	—
17	#95×C₄₁	C₂³	1	60	yes	—
18	C₂²×C₄×C₄₁	C₂⁴	1	15	yes	—
19	C₂²×C₄×C₄₁	C₂⁴	1	15	yes	—
20	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
21	C₄⋊C₄×C₄₁	—	4	30	no	—
22	C₄⋊C₄×C₄₁	C₂²×C₄	1	120	yes	—
23	C₄²×C₄₁	—	2	30	no	—
24	C₄²×C₄₁	—	4	15	no	—
25	C₄²×C₄₁	C₂²×C₄	1	20	yes	—
26	C₂³×C₄₁	C₂⁵	1	1	yes	characteristic
27	C₂×C₄×C₄₁	—	2	15	no	—
28	C₂×C₄×C₄₁	—	2	15	no	—
29	C₂×C₄×C₄₁	—	2	15	no	—
30	C₂×C₄×C₄₁	—	2	30	no	—
31	C₂×C₄×C₄₁	C₂³×C₄	1	30	yes	—
32	C₂×C₄×C₄₁	2₋¹⁺⁴	1	6	yes	—
33	C₄×#564	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₄₁	C₂⁴×C₄	1	1	yes	characteristic
35	C₂²×C₄₁	C₂×2₋¹⁺⁴	1	3	yes	—
36	C₂²×C₄₁	#585	1	3	yes	—
37	C₄×C₄₁	—	2	30	no	—
38	C₄×C₄₁	—	4	15	no	—
39	C₄×C₄₁	#564	1	4	yes	direct product factor
40	C₂×#564	C₂×C₄₁	1	1	yes	characteristic
41	#2382	C₂×C₄₁	1	15	yes	—
42	C₄×#125	C₂×C₄₁	1	15	yes	—
43	C₂×C₄₁	C₄×2₋¹⁺⁴	1	3	yes	—
44	C₂×C₄₁	C₂×#564	1	1	yes	characteristic
45	C₂×C₄₁	#3531	1	3	yes	—
46	#564	C₄×C₄₁	1	16	yes	direct product factor
47	C₂×#125	C₂²×C₄₁	1	15	yes	—
48	#389	C₂²×C₄₁	1	30	yes	—
49	#387	C₂²×C₄₁	1	60	yes	—
50	#382	C₂²×C₄₁	1	15	yes	—
51	C₄×C₄⋊C₄	C₂²×C₄₁	1	30	yes	—
52	C₄³	C₂²×C₄₁	1	5	yes	—
53	C₄₁	C₄×#564	1	1	yes	characteristic, direct product factor, Sylow
54	#125	C₂×C₄×C₄₁	1	120	yes	—
55	C₂×C₄⋊C₄	C₂³×C₄₁	1	30	yes	—
56	C₂×C₄⋊C₄	C₂³×C₄₁	1	30	yes	—
57	C₂×C₄²	C₂³×C₄₁	1	15	yes	—
58	C₂×C₄²	C₂³×C₄₁	1	15	yes	—
59	C₂×C₄²	C₂³×C₄₁	1	5	yes	—
60	#95	C₂³×C₄₁	1	60	yes	—
61	C₂²×C₄	C₂⁴×C₄₁	1	15	yes	—
62	C₂²×C₄	C₂⁴×C₄₁	1	15	yes	—
63	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
64	C₄⋊C₄	—	4	30	no	—
65	C₄⋊C₄	C₂²×C₄×C₄₁	1	120	yes	—
66	C₄²	—	2	30	no	—
67	C₄²	—	4	15	no	—
68	C₄²	C₂²×C₄×C₄₁	1	20	yes	—
69	C₂³	C₂⁵×C₄₁	1	1	yes	characteristic
70	C₂×C₄	—	2	15	no	—
71	C₂×C₄	—	2	15	no	—
72	C₂×C₄	—	2	15	no	—
73	C₂×C₄	—	2	30	no	—
74	C₂×C₄	C₂³×C₄×C₄₁	1	30	yes	—
75	C₂×C₄	2₋¹⁺⁴×C₄₁	1	6	yes	—
76	C₂²	C₂⁴×C₄×C₄₁	1	1	yes	characteristic
77	C₂²	C₂×2₋¹⁺⁴×C₄₁	1	3	yes	—
78	C₂²	C₄₁×#585	1	3	yes	—
79	C₄	—	2	30	no	—
80	C₄	—	4	15	no	—
81	C₄	C₄₁×#564	1	4	yes	direct product factor
82	C₂	C₄×2₋¹⁺⁴×C₄₁	1	3	yes	—
83	C₂	C₂×C₄₁×#564	1	1	yes	characteristic
84	C₂	C₄₁×#3531	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄×C₄₁	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₄₁
Frattini, cosocle	C₂³	C₂⁵×C₄₁
socle, prime radical	C₂³×C₄₁	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄₁×#564	C₄	direct (N × H)
33. C₄×#564	C₄₁	direct (N × H)
39. C₄×C₄₁	#564	direct (N × H)
46. #564	C₄×C₄₁	direct (N × H)
53. C₄₁	C₄×#564	direct (N × H)
81. C₄	C₄₁×#564	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#564	2
C₄₁×#2382	2
C₄×#125×C₄₁	2
C₄×#564	41

Subgroups — C4×C41×#564 #23810556

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₄₁×#564 #23810556