Subgroups — #537×C₄₂₁ #41002509

Order 26944.

294 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	C₂²⋊Q₈×C₄₂₁	C₂	1	4	yes	maximal, complemented
3	#121×C₄₂₁	C₂	1	4	yes	maximal, complemented
4	#121×C₄₂₁	C₂	1	4	yes	maximal, complemented
5	C₂×C₄⋊C₄×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₄₂₁	—	2	8	no	—
7	C₂×Q₈×C₄₂₁	C₂²	1	2	yes	—
8	C₂×D₄×C₄₂₁	C₂²	1	4	yes	—
9	C₂×D₄×C₄₂₁	C₂²	1	2	yes	—
10	C₂×D₄×C₄₂₁	C₂²	1	4	yes	—
11	C₂²×C₄×C₄₂₁	C₂²	1	4	yes	—
12	C₂²×C₄×C₄₂₁	C₂²	1	2	yes	—
13	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₄₂₁	C₂²	1	4	yes	—
15	C₄⋊C₄×C₄₂₁	C₂²	1	4	yes	—
16	C₂²⋊C₄×C₄₂₁	C₂²	1	8	yes	—
17	C₂³×C₄₂₁	C₂³	1	4	yes	—
18	C₂³×C₄₂₁	C₂³	1	1	yes	characteristic
19	Q₈×C₄₂₁	—	2	4	no	—
20	D₄×C₄₂₁	—	2	8	no	—
21	D₄×C₄₂₁	—	2	4	no	—
22	D₄×C₄₂₁	—	4	4	no	—
23	C₂×C₄×C₄₂₁	—	2	8	no	—
24	C₂×C₄×C₄₂₁	—	4	2	no	—
25	C₂×C₄×C₄₂₁	D₄	1	4	yes	complemented
26	C₂×C₄×C₄₂₁	C₂³	1	4	yes	—
27	C₂×C₄×C₄₂₁	C₂³	1	4	yes	—
28	C₂×C₄×C₄₂₁	C₂³	1	2	yes	—
29	C₂²×C₄₂₁	—	2	1	no	—
30	C₂²×C₄₂₁	—	2	1	no	—
31	C₂²×C₄₂₁	—	2	4	no	—
32	C₂²×C₄₂₁	—	2	4	no	—
33	C₂²×C₄₂₁	—	2	4	no	—
34	C₂²×C₄₂₁	C₂×D₄	1	2	yes	—
35	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
36	C₄×C₄₂₁	—	2	4	no	—
37	C₄×C₄₂₁	—	2	4	no	—
38	C₄×C₄₂₁	C₂×D₄	1	4	yes	—
39	C₂×C₄₂₁	—	2	2	no	—
40	C₂×C₄₂₁	—	4	4	no	—
41	C₂×C₄₂₁	C₂²×D₄	1	1	yes	characteristic
42	C₂×C₄₂₁	H(5,𝔽₂)	1	1	yes	characteristic
43	C₂×C₄₂₁	2₋¹⁺⁴	1	1	yes	characteristic
44	C₄₂₁	#537	1	1	yes	characteristic, direct product factor, Sylow
45	#537	C₄₂₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂×#41	C₂×C₄₂₁	1	2	yes	complemented
47	C₂²⋊Q₈	C₂×C₄₂₁	1	4	yes	complemented
48	#121	C₂×C₄₂₁	1	4	yes	complemented
49	#121	C₂×C₄₂₁	1	4	yes	complemented
50	C₂×C₄⋊C₄	C₂×C₄₂₁	1	1	yes	characteristic, complemented
51	#41	—	2	8	no	—
52	C₂×Q₈	C₂²×C₄₂₁	1	2	yes	—
53	C₂×D₄	C₂²×C₄₂₁	1	4	yes	—
54	C₂×D₄	C₂²×C₄₂₁	1	2	yes	—
55	C₂×D₄	C₂²×C₄₂₁	1	4	yes	—
56	C₂²×C₄	C₂²×C₄₂₁	1	4	yes	—
57	C₂²×C₄	C₂²×C₄₂₁	1	2	yes	—
58	C₂²×C₄	C₂²×C₄₂₁	1	1	yes	characteristic
59	C₄⋊C₄	C₂²×C₄₂₁	1	4	yes	—
60	C₄⋊C₄	C₂²×C₄₂₁	1	4	yes	—
61	C₂²⋊C₄	C₂²×C₄₂₁	1	8	yes	—
62	C₂³	C₂³×C₄₂₁	1	4	yes	—
63	C₂³	C₂³×C₄₂₁	1	1	yes	characteristic
64	Q₈	—	2	4	no	—
65	D₄	—	2	8	no	—
66	D₄	—	2	4	no	—
67	D₄	—	4	4	no	—
68	C₂×C₄	—	2	8	no	—
69	C₂×C₄	—	4	2	no	—
70	C₂×C₄	D₄×C₄₂₁	1	4	yes	complemented
71	C₂×C₄	C₂³×C₄₂₁	1	4	yes	—
72	C₂×C₄	C₂³×C₄₂₁	1	4	yes	—
73	C₂×C₄	C₂³×C₄₂₁	1	2	yes	—
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	—	2	4	no	—
77	C₂²	—	2	4	no	—
78	C₂²	—	2	4	no	—
79	C₂²	C₂×D₄×C₄₂₁	1	2	yes	—
80	C₂²	C₂⁴×C₄₂₁	1	1	yes	characteristic
81	C₄	—	2	4	no	—
82	C₄	—	2	4	no	—
83	C₄	C₂×D₄×C₄₂₁	1	4	yes	—
84	C₂	—	2	2	no	—
85	C₂	—	4	4	no	—
86	C₂	C₂²×D₄×C₄₂₁	1	1	yes	characteristic
87	C₂	H(5,𝔽₂)×C₄₂₁	1	1	yes	characteristic
88	C₂	2₋¹⁺⁴×C₄₂₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₂₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₄₂₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×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. C₂²⋊Q₈×C₄₂₁	C₂	semidirect
3. #121×C₄₂₁	C₂	semidirect
4. #121×C₄₂₁	C₂	semidirect
5. C₂×C₄⋊C₄×C₄₂₁	C₂	semidirect
25. C₂×C₄×C₄₂₁	D₄	semidirect
44. C₄₂₁	#537	direct (N × H)
45. #537	C₄₂₁	direct (N × H)
46. C₂×#41	C₂×C₄₂₁	semidirect
47. C₂²⋊Q₈	C₂×C₄₂₁	semidirect
48. #121	C₂×C₄₂₁	semidirect
49. #121	C₂×C₄₂₁	semidirect
50. C₂×C₄⋊C₄	C₂×C₄₂₁	semidirect
70. C₂×C₄	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×#41×C₄₂₁	2
C₂²⋊Q₈×C₄₂₁	2
#121×C₄₂₁	2
C₂×C₄⋊C₄×C₄₂₁	2
#537	421

Subgroups — #537×C421 #41002509

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #537×C₄₂₁ #41002509