Subgroups — C₂×#135×C₄₂₁ #41002544

Order 26944.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×C₄₂₁	C₂	1	2	yes	maximal, complemented
2	#135×C₄₂₁	C₂	1	8	yes	maximal, direct product factor
3	C₂×Q₁₆×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×SD₁₆×C₄₂₁	C₂	1	2	yes	maximal, complemented
5	C₂×D₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#41×C₄₂₁	—	2	4	no	—
8	#41×C₄₂₁	C₂²	1	8	yes	complemented
9	C₂×Q₈×C₄₂₁	C₂²	1	2	yes	—
10	C₂×D₄×C₄₂₁	—	2	2	no	—
11	C₂×D₄×C₄₂₁	C₂²	1	2	yes	—
12	C₂²×C₄×C₄₂₁	—	2	2	no	—
13	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
14	Q₁₆×C₄₂₁	C₂²	1	4	yes	complemented
15	SD₁₆×C₄₂₁	C₂²	1	8	yes	complemented
16	D₈×C₄₂₁	C₂²	1	4	yes	complemented
17	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
19	C₂×C₈×C₄₂₁	C₂²	1	4	yes	complemented
20	C₂³×C₄₂₁	—	2	2	no	—
21	C₂³×C₄₂₁	D₄	1	1	yes	characteristic
22	Q₈×C₄₂₁	—	2	2	no	—
23	Q₈×C₄₂₁	C₂³	1	4	yes	—
24	D₄×C₄₂₁	—	2	8	no	—
25	D₄×C₄₂₁	—	2	2	no	—
26	D₄×C₄₂₁	C₂³	1	4	yes	—
27	C₂×C₄×C₄₂₁	—	2	2	no	—
28	C₂×C₄×C₄₂₁	—	2	8	no	—
29	C₂×C₄×C₄₂₁	D₄	1	2	yes	—
30	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
31	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₄₂₁	C₂³	1	2	yes	—
33	C₈×C₄₂₁	C₂³	1	2	yes	—
34	C₈×C₄₂₁	C₂³	1	2	yes	—
35	C₂²×C₄₂₁	—	2	2	no	—
36	C₂²×C₄₂₁	—	2	4	no	—
37	C₂²×C₄₂₁	—	4	4	no	—
38	C₂²×C₄₂₁	C₂×D₄	1	2	yes	—
39	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
40	C₄×C₄₂₁	—	2	4	no	—
41	C₄×C₄₂₁	C₂×D₄	1	2	yes	—
42	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
43	C₄×C₄₂₁	C₂⁴	1	1	yes	characteristic
44	C₂×C₄₂₁	—	2	2	no	—
45	C₂×C₄₂₁	—	4	4	no	—
46	C₂×C₄₂₁	#135	1	2	yes	direct product factor
47	C₂×C₄₂₁	C₂²×D₄	1	1	yes	characteristic
48	C₄₂₁	C₂×#135	1	1	yes	characteristic, direct product factor, Sylow
49	C₂×#135	C₄₂₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂×#41	C₂×C₄₂₁	1	2	yes	complemented
51	#135	C₂×C₄₂₁	1	8	yes	direct product factor
52	C₂×Q₁₆	C₂×C₄₂₁	1	1	yes	characteristic, complemented
53	C₂×SD₁₆	C₂×C₄₂₁	1	2	yes	complemented
54	C₂×D₈	C₂×C₄₂₁	1	1	yes	characteristic, complemented
55	C₂²×C₈	C₂×C₄₂₁	1	1	yes	characteristic, complemented
56	#41	—	2	4	no	—
57	#41	C₂²×C₄₂₁	1	8	yes	complemented
58	C₂×Q₈	C₂²×C₄₂₁	1	2	yes	—
59	C₂×D₄	—	2	2	no	—
60	C₂×D₄	C₂²×C₄₂₁	1	2	yes	—
61	C₂²×C₄	—	2	2	no	—
62	C₂²×C₄	C₂²×C₄₂₁	1	1	yes	characteristic
63	Q₁₆	C₂²×C₄₂₁	1	4	yes	complemented
64	SD₁₆	C₂²×C₄₂₁	1	8	yes	complemented
65	D₈	C₂²×C₄₂₁	1	4	yes	complemented
66	C₂×C₈	C₂²×C₄₂₁	1	1	yes	characteristic
67	C₂×C₈	C₂²×C₄₂₁	1	1	yes	characteristic
68	C₂×C₈	C₂²×C₄₂₁	1	4	yes	complemented
69	C₂³	—	2	2	no	—
70	C₂³	D₄×C₄₂₁	1	1	yes	characteristic
71	Q₈	—	2	2	no	—
72	Q₈	C₂³×C₄₂₁	1	4	yes	—
73	D₄	—	2	8	no	—
74	D₄	—	2	2	no	—
75	D₄	C₂³×C₄₂₁	1	4	yes	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	8	no	—
78	C₂×C₄	D₄×C₄₂₁	1	1	yes	characteristic
79	C₂×C₄	D₄×C₄₂₁	1	2	yes	—
80	C₂×C₄	C₂³×C₄₂₁	1	1	yes	characteristic
81	C₂×C₄	C₂³×C₄₂₁	1	2	yes	—
82	C₈	C₂³×C₄₂₁	1	2	yes	—
83	C₈	C₂³×C₄₂₁	1	2	yes	—
84	C₂²	—	2	2	no	—
85	C₂²	—	2	4	no	—
86	C₂²	—	4	4	no	—
87	C₂²	C₂×D₄×C₄₂₁	1	2	yes	—
88	C₂²	C₂×D₄×C₄₂₁	1	1	yes	characteristic
89	C₄	—	2	4	no	—
90	C₄	C₂×D₄×C₄₂₁	1	2	yes	—
91	C₄	C₂×D₄×C₄₂₁	1	1	yes	characteristic
92	C₄	C₂⁴×C₄₂₁	1	1	yes	characteristic
93	C₂	—	2	2	no	—
94	C₂	—	4	4	no	—
95	C₂	#135×C₄₂₁	1	2	yes	direct product factor
96	C₂	C₂²×D₄×C₄₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₄₂₁	D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₄₂₁
socle	C₂²×C₄₂₁	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₂₁ ◁ C₂²×C₄×C₄₂₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#41×C₄₂₁	C₂	semidirect
2. #135×C₄₂₁	C₂	direct (N × H)
3. C₂×Q₁₆×C₄₂₁	C₂	semidirect
4. C₂×SD₁₆×C₄₂₁	C₂	semidirect
5. C₂×D₈×C₄₂₁	C₂	semidirect
6. C₂²×C₈×C₄₂₁	C₂	semidirect
8. #41×C₄₂₁	C₂²	semidirect
14. Q₁₆×C₄₂₁	C₂²	semidirect
15. SD₁₆×C₄₂₁	C₂²	semidirect
16. D₈×C₄₂₁	C₂²	semidirect
19. C₂×C₈×C₄₂₁	C₂²	semidirect
46. C₂×C₄₂₁	#135	direct (N × H)
48. C₄₂₁	C₂×#135	direct (N × H)
49. C₂×#135	C₄₂₁	direct (N × H)
50. C₂×#41	C₂×C₄₂₁	semidirect
51. #135	C₂×C₄₂₁	direct (N × H)
52. C₂×Q₁₆	C₂×C₄₂₁	semidirect
53. C₂×SD₁₆	C₂×C₄₂₁	semidirect
54. C₂×D₈	C₂×C₄₂₁	semidirect
55. C₂²×C₈	C₂×C₄₂₁	semidirect
57. #41	C₂²×C₄₂₁	semidirect
63. Q₁₆	C₂²×C₄₂₁	semidirect
64. SD₁₆	C₂²×C₄₂₁	semidirect
65. D₈	C₂²×C₄₂₁	semidirect
68. C₂×C₈	C₂²×C₄₂₁	semidirect
95. C₂	#135×C₄₂₁	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×#41×C₄₂₁	2
#135×C₄₂₁	2
C₂×Q₁₆×C₄₂₁	2
C₂×SD₁₆×C₄₂₁	2
C₂×D₈×C₄₂₁	2
C₂²×C₈×C₄₂₁	2
C₂×#135	421

Subgroups — C2×#135×C421 #41002544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#135×C₄₂₁ #41002544