Subgroups — #469×C₄₂₁ #41002441

Order 26944.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×M₁₆×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#121×C₄₂₁	C₂	1	2	yes	maximal, complemented
4	#106×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#102×C₄₂₁	C₂	1	2	yes	maximal, complemented
6	C₂×D₄×C₄₂₁	—	2	2	no	—
7	C₂×D₄×C₄₂₁	C₂²	1	2	yes	—
8	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
9	D₈×C₄₂₁	—	2	2	no	—
10	M₁₆×C₄₂₁	—	2	2	no	—
11	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₄₂₁	C₂²	1	2	yes	—
14	C₂²⋊C₄×C₄₂₁	—	2	2	no	—
15	C₂³×C₄₂₁	—	2	2	no	—
16	C₂³×C₄₂₁	D₄	1	1	yes	characteristic
17	D₄×C₄₂₁	—	2	2	no	—
18	D₄×C₄₂₁	—	2	2	no	—
19	D₄×C₄₂₁	—	4	4	no	—
20	C₂×C₄×C₄₂₁	—	2	1	no	—
21	C₂×C₄×C₄₂₁	—	2	1	no	—
22	C₂×C₄×C₄₂₁	—	2	2	no	—
23	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
25	C₈×C₄₂₁	—	2	1	no	—
26	C₈×C₄₂₁	D₄	1	2	yes	complemented
27	C₂²×C₄₂₁	—	2	1	no	—
28	C₂²×C₄₂₁	—	2	2	no	—
29	C₂²×C₄₂₁	—	4	2	no	—
30	C₂²×C₄₂₁	—	4	2	no	—
31	C₂²×C₄₂₁	—	4	2	no	—
32	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
33	C₄×C₄₂₁	—	2	1	no	—
34	C₄×C₄₂₁	—	4	2	no	—
35	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
36	C₄×C₄₂₁	#41	1	1	yes	characteristic
37	C₂×C₄₂₁	—	4	1	no	—
38	C₂×C₄₂₁	—	8	2	no	—
39	C₂×C₄₂₁	#121	1	1	yes	characteristic
40	C₂×C₄₂₁	AGL(1,ℤ/8)	1	2	yes	—
41	C₄₂₁	#469	1	1	yes	characteristic, direct product factor, Sylow
42	#469	C₄₂₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂×D₈	C₂×C₄₂₁	1	1	yes	characteristic, complemented
44	C₂×M₁₆	C₂×C₄₂₁	1	1	yes	characteristic, complemented
45	#121	C₂×C₄₂₁	1	2	yes	complemented
46	#106	C₂×C₄₂₁	1	1	yes	characteristic, complemented
47	#102	C₂×C₄₂₁	1	2	yes	complemented
48	C₂×D₄	—	2	2	no	—
49	C₂×D₄	C₂²×C₄₂₁	1	2	yes	—
50	C₂²×C₄	C₂²×C₄₂₁	1	1	yes	characteristic
51	D₈	—	2	2	no	—
52	M₁₆	—	2	2	no	—
53	C₂×C₈	C₂²×C₄₂₁	1	1	yes	characteristic
54	C₂×C₈	C₂²×C₄₂₁	1	1	yes	characteristic
55	C₄⋊C₄	C₂²×C₄₂₁	1	2	yes	—
56	C₂²⋊C₄	—	2	2	no	—
57	C₂³	—	2	2	no	—
58	C₂³	D₄×C₄₂₁	1	1	yes	characteristic
59	D₄	—	2	2	no	—
60	D₄	—	2	2	no	—
61	D₄	—	4	4	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	D₄×C₄₂₁	1	1	yes	characteristic
66	C₂×C₄	C₂³×C₄₂₁	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	D₄×C₄₂₁	1	2	yes	complemented
69	C₂²	—	2	1	no	—
70	C₂²	—	2	2	no	—
71	C₂²	—	4	2	no	—
72	C₂²	—	4	2	no	—
73	C₂²	—	4	2	no	—
74	C₂²	C₂×D₄×C₄₂₁	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	2	no	—
77	C₄	C₂×D₄×C₄₂₁	1	1	yes	characteristic
78	C₄	#41×C₄₂₁	1	1	yes	characteristic
79	C₂	—	4	1	no	—
80	C₂	—	8	2	no	—
81	C₂	#121×C₄₂₁	1	1	yes	characteristic
82	C₂	AGL(1,ℤ/8)×C₄₂₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₂₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₄₂₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₂₁ ◁ C₂²×C₄×C₄₂₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×D₈×C₄₂₁	C₂	semidirect
2. C₂×M₁₆×C₄₂₁	C₂	semidirect
3. #121×C₄₂₁	C₂	semidirect
4. #106×C₄₂₁	C₂	semidirect
5. #102×C₄₂₁	C₂	semidirect
26. C₈×C₄₂₁	D₄	semidirect
41. C₄₂₁	#469	direct (N × H)
42. #469	C₄₂₁	direct (N × H)
43. C₂×D₈	C₂×C₄₂₁	semidirect
44. C₂×M₁₆	C₂×C₄₂₁	semidirect
45. #121	C₂×C₄₂₁	semidirect
46. #106	C₂×C₄₂₁	semidirect
47. #102	C₂×C₄₂₁	semidirect
68. C₈	D₄×C₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₈×C₄₂₁	2
C₂×M₁₆×C₄₂₁	2
#121×C₄₂₁	2
#106×C₄₂₁	2
#102×C₄₂₁	2
#469	421

Subgroups — #469×C421 #41002441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #469×C₄₂₁ #41002441