Subgroups — #1844×C₂₅₁ #44546431

Order 32128.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#106×C₂₅₁	C₂	1	4	yes	maximal
2	C₂×C₄×C₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic
3	#384×C₂₅₁	C₂	1	2	yes	maximal
4	C₂²×C₈×C₂₅₁	C₂²	1	2	yes	—
5	C₂×C₄⋊C₄×C₂₅₁	—	2	4	no	—
6	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	4	yes	—
7	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
8	#106×C₂₅₁	—	2	8	no	—
9	C₄×C₈×C₂₅₁	C₄	1	4	yes	complemented
10	C₂²×C₄×C₂₅₁	—	2	4	no	—
11	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
12	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
13	C₂×C₈×C₂₅₁	C₂×C₄	1	4	yes	—
14	C₂×C₈×C₂₅₁	D₄	1	4	yes	—
15	C₂×C₈×C₂₅₁	Q₈	1	4	yes	—
16	C₄⋊C₄×C₂₅₁	—	2	8	no	—
17	C₄⋊C₄×C₂₅₁	—	4	8	no	—
18	C₄²×C₂₅₁	C₂×C₄	1	2	yes	—
19	C₄²×C₂₅₁	Q₈	1	2	yes	—
20	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₅₁	—	4	4	no	—
22	C₂×C₄×C₂₅₁	—	4	8	no	—
23	C₂×C₄×C₂₅₁	C₄⋊C₄	1	4	yes	—
24	C₂×C₄×C₂₅₁	SD₁₆	1	8	yes	—
25	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
27	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
28	C₂×C₄×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
29	C₈×C₂₅₁	C₄⋊C₄	1	8	yes	complemented
30	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₂₅₁	#127	1	1	yes	characteristic
32	C₂²×C₂₅₁	C₄⋊Q₈	1	1	yes	characteristic
33	C₂²×C₂₅₁	C₂×SD₁₆	1	4	yes	—
34	C₄×C₂₅₁	—	8	4	no	—
35	C₄×C₂₅₁	#106	1	8	yes	complemented
36	C₄×C₂₅₁	C₂×C₄⋊C₄	1	2	yes	—
37	C₄×C₂₅₁	C₄⋊Q₈	1	2	yes	—
38	C₂×C₂₅₁	#384	1	1	yes	characteristic
39	C₂×C₂₅₁	C₂×#106	1	2	yes	—
40	C₂×C₂₅₁	#492	1	2	yes	—
41	C₂×C₂₅₁	#498	1	2	yes	—
42	C₂₅₁	#1844	1	1	yes	characteristic, direct product factor, Sylow
43	#1844	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×#106	C₂×C₂₅₁	1	4	yes	—
45	C₂×C₄×C₈	C₂×C₂₅₁	1	1	yes	characteristic
46	#384	C₂×C₂₅₁	1	2	yes	—
47	C₂²×C₈	C₂²×C₂₅₁	1	2	yes	—
48	C₂×C₄⋊C₄	—	2	4	no	—
49	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	4	yes	—
50	C₂×C₄²	C₂²×C₂₅₁	1	1	yes	characteristic
51	#106	—	2	8	no	—
52	C₄×C₈	C₄×C₂₅₁	1	4	yes	complemented
53	C₂²×C₄	—	2	4	no	—
54	C₂²×C₄	D₄×C₂₅₁	1	2	yes	—
55	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
56	C₂×C₈	C₂×C₄×C₂₅₁	1	4	yes	—
57	C₂×C₈	D₄×C₂₅₁	1	4	yes	—
58	C₂×C₈	Q₈×C₂₅₁	1	4	yes	—
59	C₄⋊C₄	—	2	8	no	—
60	C₄⋊C₄	—	4	8	no	—
61	C₄²	C₂×C₄×C₂₅₁	1	2	yes	—
62	C₄²	Q₈×C₂₅₁	1	2	yes	—
63	C₂³	C₂×D₄×C₂₅₁	1	1	yes	characteristic
64	C₂×C₄	—	4	4	no	—
65	C₂×C₄	—	4	8	no	—
66	C₂×C₄	C₄⋊C₄×C₂₅₁	1	4	yes	—
67	C₂×C₄	SD₁₆×C₂₅₁	1	8	yes	—
68	C₂×C₄	C₂²×C₄×C₂₅₁	1	1	yes	characteristic
69	C₂×C₄	C₂×D₄×C₂₅₁	1	2	yes	—
70	C₂×C₄	C₂×Q₈×C₂₅₁	1	2	yes	—
71	C₂×C₄	C₂×Q₈×C₂₅₁	1	1	yes	characteristic
72	C₈	C₄⋊C₄×C₂₅₁	1	8	yes	complemented
73	C₂²	C₂×C₄⋊C₄×C₂₅₁	1	1	yes	characteristic
74	C₂²	#127×C₂₅₁	1	1	yes	characteristic
75	C₂²	C₄⋊Q₈×C₂₅₁	1	1	yes	characteristic
76	C₂²	C₂×SD₁₆×C₂₅₁	1	4	yes	—
77	C₄	—	8	4	no	—
78	C₄	#106×C₂₅₁	1	8	yes	complemented
79	C₄	C₂×C₄⋊C₄×C₂₅₁	1	2	yes	—
80	C₄	C₄⋊Q₈×C₂₅₁	1	2	yes	—
81	C₂	#384×C₂₅₁	1	1	yes	characteristic
82	C₂	C₂×#106×C₂₅₁	1	2	yes	—
83	C₂	#492×C₂₅₁	1	2	yes	—
84	C₂	#498×C₂₅₁	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₅₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₅₁
Frattini, cosocle	C₂²×C₄	C₂³×C₂₅₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₅₁ ◁ C₂×C₄²×C₂₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₄×C₈×C₂₅₁	C₄	semidirect
29. C₈×C₂₅₁	C₄⋊C₄	semidirect
35. C₄×C₂₅₁	#106	semidirect
42. C₂₅₁	#1844	direct (N × H)
43. #1844	C₂₅₁	direct (N × H)
52. C₄×C₈	C₄×C₂₅₁	semidirect
72. C₈	C₄⋊C₄×C₂₅₁	semidirect
78. C₄	#106×C₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#106×C₂₅₁	2
C₂×C₄×C₈×C₂₅₁	2
#384×C₂₅₁	2
#1844	251

Subgroups — #1844×C251 #44546431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1844×C₂₅₁ #44546431