Subgroups — #2844×C₂₅₁ #44547431

Order 32128.

356 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#400×C₂₅₁	C₂	1	6	yes	maximal
2	#395×C₂₅₁	C₂	1	6	yes	maximal
3	#382×C₂₅₁	C₂	1	3	yes	maximal
4	C₂²×Q₈×C₂₅₁	C₂²	1	2	yes	—
5	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	3	yes	—
6	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	3	yes	—
7	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	6	yes	—
8	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
9	#95×C₂₅₁	C₂²	1	12	yes	—
10	#95×C₂₅₁	C₂²	1	6	yes	—
11	#95×C₂₅₁	C₂²	1	2	yes	—
12	C₂×Q₈×C₂₅₁	—	4	6	no	—
13	C₂²×C₄×C₂₅₁	C₂³	1	6	yes	—
14	C₂²×C₄×C₂₅₁	C₂³	1	6	yes	—
15	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
16	C₄⋊C₄×C₂₅₁	—	4	3	no	—
17	C₄⋊C₄×C₂₅₁	—	4	3	no	—
18	C₄⋊C₄×C₂₅₁	—	4	6	no	—
19	C₄²×C₂₅₁	Q₈	1	4	yes	complemented
20	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
21	Q₈×C₂₅₁	—	4	8	no	—
22	C₂×C₄×C₂₅₁	—	2	3	no	—
23	C₂×C₄×C₂₅₁	—	2	3	no	—
24	C₂×C₄×C₂₅₁	—	2	6	no	—
25	C₂×C₄×C₂₅₁	—	2	6	no	—
26	C₂×C₄×C₂₅₁	—	2	12	no	—
27	C₂×C₄×C₂₅₁	—	2	12	no	—
28	C₂×C₄×C₂₅₁	C₂×Q₈	1	6	yes	—
29	C₂²×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
30	C₂²×C₂₅₁	H(5,𝔽₂)	1	3	yes	—
31	C₂²×C₂₅₁	2₋¹⁺⁴	1	3	yes	—
32	C₄×C₂₅₁	—	2	6	no	—
33	C₄×C₂₅₁	—	4	6	no	—
34	C₄×C₂₅₁	—	4	6	no	—
35	C₂×C₂₅₁	#544	1	3	yes	—
36	C₂×C₂₅₁	H(3,𝔽₄)	1	1	yes	characteristic
37	C₂×C₂₅₁	#563	1	3	yes	—
38	C₂₅₁	#2844	1	1	yes	characteristic, direct product factor, Sylow
39	#2844	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	#400	C₂×C₂₅₁	1	6	yes	—
41	#395	C₂×C₂₅₁	1	6	yes	—
42	#382	C₂×C₂₅₁	1	3	yes	—
43	C₂²×Q₈	C₂²×C₂₅₁	1	2	yes	—
44	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	3	yes	—
45	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	3	yes	—
46	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	6	yes	—
47	C₂×C₄²	C₂²×C₂₅₁	1	1	yes	characteristic
48	#95	C₂²×C₂₅₁	1	12	yes	—
49	#95	C₂²×C₂₅₁	1	6	yes	—
50	#95	C₂²×C₂₅₁	1	2	yes	—
51	C₂×Q₈	—	4	6	no	—
52	C₂²×C₄	C₂³×C₂₅₁	1	6	yes	—
53	C₂²×C₄	C₂³×C₂₅₁	1	6	yes	—
54	C₂²×C₄	C₂³×C₂₅₁	1	3	yes	—
55	C₄⋊C₄	—	4	3	no	—
56	C₄⋊C₄	—	4	3	no	—
57	C₄⋊C₄	—	4	6	no	—
58	C₄²	Q₈×C₂₅₁	1	4	yes	complemented
59	C₂³	C₂⁴×C₂₅₁	1	1	yes	characteristic
60	Q₈	—	4	8	no	—
61	C₂×C₄	—	2	3	no	—
62	C₂×C₄	—	2	3	no	—
63	C₂×C₄	—	2	6	no	—
64	C₂×C₄	—	2	6	no	—
65	C₂×C₄	—	2	12	no	—
66	C₂×C₄	—	2	12	no	—
67	C₂×C₄	C₂×Q₈×C₂₅₁	1	6	yes	—
68	C₂²	C₂²×Q₈×C₂₅₁	1	1	yes	characteristic
69	C₂²	H(5,𝔽₂)×C₂₅₁	1	3	yes	—
70	C₂²	2₋¹⁺⁴×C₂₅₁	1	3	yes	—
71	C₄	—	2	6	no	—
72	C₄	—	4	6	no	—
73	C₄	—	4	6	no	—
74	C₂	#544×C₂₅₁	1	3	yes	—
75	C₂	H(3,𝔽₄)×C₂₅₁	1	1	yes	characteristic
76	C₂	#563×C₂₅₁	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₅₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₅₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
19. C₄²×C₂₅₁	Q₈	semidirect
38. C₂₅₁	#2844	direct (N × H)
39. #2844	C₂₅₁	direct (N × H)
58. C₄²	Q₈×C₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#400×C₂₅₁	2
#395×C₂₅₁	2
#382×C₂₅₁	2
#2844	251

Subgroups — #2844×C251 #44547431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2844×C₂₅₁ #44547431