Subgroups — #410×C₄₂₁ #41002382

Order 26944.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×C₄₂₁	C₂	1	1	yes	maximal, characteristic
2	#117×C₄₂₁	C₂	1	2	yes	maximal, complemented
3	#99×C₄₂₁	C₂	1	4	yes	maximal, complemented
4	#41×C₄₂₁	—	2	2	no	—
5	#41×C₄₂₁	—	2	2	no	—
6	C₂×Q₈×C₄₂₁	C₄	1	1	yes	characteristic, complemented
7	C₂×D₄×C₄₂₁	C₄	1	1	yes	characteristic, complemented
8	C₂×D₄×C₄₂₁	C₂²	1	2	yes	—
9	C₂²×C₄×C₄₂₁	C₄	1	2	yes	complemented
10	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×C₄₂₁	—	2	2	no	—
12	C₂²⋊C₄×C₄₂₁	C₂²	1	4	yes	—
13	C₄²×C₄₂₁	—	2	2	no	—
14	C₂³×C₄₂₁	C₂×C₄	1	2	yes	—
15	C₂³×C₄₂₁	C₂³	1	1	yes	characteristic
16	Q₈×C₄₂₁	—	2	2	no	—
17	D₄×C₄₂₁	—	2	4	no	—
18	D₄×C₄₂₁	—	2	2	no	—
19	C₂×C₄×C₄₂₁	—	2	2	no	—
20	C₂×C₄×C₄₂₁	—	2	2	no	—
21	C₂×C₄×C₄₂₁	—	2	4	no	—
22	C₂×C₄×C₄₂₁	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₄₂₁	D₄	1	2	yes	—
26	C₂×C₄×C₄₂₁	D₄	1	2	yes	—
27	C₂²×C₄₂₁	—	2	2	no	—
28	C₂²×C₄₂₁	—	4	1	no	—
29	C₂²×C₄₂₁	—	4	2	no	—
30	C₂²×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄₂₁	C₂×D₄	1	2	yes	—
32	C₄×C₄₂₁	—	2	2	no	—
33	C₄×C₄₂₁	—	4	4	no	—
34	C₄×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
35	C₄×C₄₂₁	C₂²⋊C₄	1	2	yes	—
36	C₄×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄₂₁	—	2	2	no	—
38	C₂×C₄₂₁	—	2	1	no	—
39	C₂×C₄₂₁	—	4	2	no	—
40	C₂×C₄₂₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₄₂₁	#410	1	1	yes	characteristic, direct product factor, Sylow
42	#410	C₄₂₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂×#41	C₂×C₄₂₁	1	1	yes	characteristic
44	#117	C₂×C₄₂₁	1	2	yes	complemented
45	#99	C₂×C₄₂₁	1	4	yes	complemented
46	#41	—	2	2	no	—
47	#41	—	2	2	no	—
48	C₂×Q₈	C₄×C₄₂₁	1	1	yes	characteristic, complemented
49	C₂×D₄	C₄×C₄₂₁	1	1	yes	characteristic, complemented
50	C₂×D₄	C₂²×C₄₂₁	1	2	yes	—
51	C₂²×C₄	C₄×C₄₂₁	1	2	yes	complemented
52	C₂²×C₄	C₂²×C₄₂₁	1	1	yes	characteristic
53	C₄⋊C₄	—	2	2	no	—
54	C₂²⋊C₄	C₂²×C₄₂₁	1	4	yes	—
55	C₄²	—	2	2	no	—
56	C₂³	C₂×C₄×C₄₂₁	1	2	yes	—
57	C₂³	C₂³×C₄₂₁	1	1	yes	characteristic
58	Q₈	—	2	2	no	—
59	D₄	—	2	4	no	—
60	D₄	—	2	2	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	4	no	—
64	C₂×C₄	C₂×C₄×C₄₂₁	1	2	yes	—
65	C₂×C₄	C₂×C₄×C₄₂₁	1	1	yes	characteristic
66	C₂×C₄	C₂×C₄×C₄₂₁	1	1	yes	characteristic
67	C₂×C₄	D₄×C₄₂₁	1	2	yes	—
68	C₂×C₄	D₄×C₄₂₁	1	2	yes	—
69	C₂²	—	2	2	no	—
70	C₂²	—	4	1	no	—
71	C₂²	—	4	2	no	—
72	C₂²	C₂²×C₄×C₄₂₁	1	1	yes	characteristic
73	C₂²	C₂×D₄×C₄₂₁	1	2	yes	—
74	C₄	—	2	2	no	—
75	C₄	—	4	4	no	—
76	C₄	C₂²⋊C₄×C₄₂₁	1	1	yes	characteristic
77	C₄	C₂²⋊C₄×C₄₂₁	1	1	yes	characteristic
78	C₄	C₂²⋊C₄×C₄₂₁	1	2	yes	—
79	C₂	—	2	2	no	—
80	C₂	—	2	1	no	—
81	C₂	—	4	2	no	—
82	C₂	C₂×C₂²⋊C₄×C₄₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₄₂₁	C₂²⋊C₄
commutator	C₂²	C₂²×C₄×C₄₂₁
Frattini, cosocle	C₂³	C₂³×C₄₂₁
socle	C₂×C₄₂₁	C₂×C₂²⋊C₄
prime radical	C₂×#41×C₄₂₁	C₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₄₂₁ ◁ C₂²×C₄×C₄₂₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #117×C₄₂₁	C₂	semidirect
3. #99×C₄₂₁	C₂	semidirect
6. C₂×Q₈×C₄₂₁	C₄	semidirect
7. C₂×D₄×C₄₂₁	C₄	semidirect
9. C₂²×C₄×C₄₂₁	C₄	semidirect
41. C₄₂₁	#410	direct (N × H)
42. #410	C₄₂₁	direct (N × H)
44. #117	C₂×C₄₂₁	semidirect
45. #99	C₂×C₄₂₁	semidirect
48. C₂×Q₈	C₄×C₄₂₁	semidirect
49. C₂×D₄	C₄×C₄₂₁	semidirect
51. C₂²×C₄	C₄×C₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×C₄₂₁	2
#117×C₄₂₁	2
#99×C₄₂₁	2
#410	421

Subgroups — #410×C421 #41002382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #410×C₄₂₁ #41002382