Subgroups — C₂×#498×C₂₅₁ #44547744

Order 32128.

424 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈×C₂₅₁	C₂	1	2	yes	maximal
2	#498×C₂₅₁	C₂	1	8	yes	maximal, direct product factor
3	C₂×#106×C₂₅₁	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₂₅₁	—	2	2	no	—
6	C₂²×C₈×C₂₅₁	C₂²	1	2	yes	—
7	C₄⋊Q₈×C₂₅₁	—	2	4	no	—
8	C₄⋊Q₈×C₂₅₁	C₂²	1	8	yes	—
9	C₂×C₄⋊C₄×C₂₅₁	—	2	2	no	—
10	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	4	yes	—
11	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
12	#106×C₂₅₁	C₂²	1	16	yes	—
13	C₄×C₈×C₂₅₁	C₂²	1	4	yes	—
14	C₂×Q₈×C₂₅₁	—	2	8	no	—
15	C₂×Q₈×C₂₅₁	—	4	8	no	—
16	C₂²×C₄×C₂₅₁	—	2	4	no	—
17	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
18	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
19	C₂×C₈×C₂₅₁	Q₈	1	8	yes	complemented
20	C₂×C₈×C₂₅₁	C₂³	1	4	yes	—
21	C₄⋊C₄×C₂₅₁	—	2	8	no	—
22	C₄⋊C₄×C₂₅₁	—	2	4	no	—
23	C₄⋊C₄×C₂₅₁	C₂³	1	8	yes	—
24	C₄²×C₂₅₁	D₄	1	2	yes	—
25	C₄²×C₂₅₁	C₂³	1	2	yes	—
26	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
27	Q₈×C₂₅₁	—	4	16	no	—
28	C₂×C₄×C₂₅₁	—	2	8	no	—
29	C₂×C₄×C₂₅₁	—	4	8	no	—
30	C₂×C₄×C₂₅₁	SD₁₆	1	8	yes	—
31	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₂₅₁	C₂×D₄	1	4	yes	—
33	C₂×C₄×C₂₅₁	C₂×Q₈	1	4	yes	—
34	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
35	C₈×C₂₅₁	C₂×Q₈	1	8	yes	complemented
36	C₂²×C₂₅₁	C₄⋊Q₈	1	2	yes	—
37	C₂²×C₂₅₁	C₂×SD₁₆	1	4	yes	—
38	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
39	C₄×C₂₅₁	—	4	8	no	—
40	C₄×C₂₅₁	C₄⋊Q₈	1	2	yes	—
41	C₄×C₂₅₁	C₂×SD₁₆	1	8	yes	—
42	C₄×C₂₅₁	C₂²×Q₈	1	2	yes	—
43	C₂×C₂₅₁	#498	1	4	yes	direct product factor
44	C₂×C₂₅₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
45	C₂×C₂₅₁	C₂²×SD₁₆	1	2	yes	—
46	C₂₅₁	C₂×#498	1	1	yes	characteristic, direct product factor, Sylow
47	C₂×#498	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂×C₄⋊Q₈	C₂×C₂₅₁	1	2	yes	—
49	#498	C₂×C₂₅₁	1	8	yes	direct product factor
50	C₂×#106	C₂×C₂₅₁	1	4	yes	—
51	C₂×C₄×C₈	C₂×C₂₅₁	1	1	yes	characteristic
52	C₂²×Q₈	—	2	2	no	—
53	C₂²×C₈	C₂²×C₂₅₁	1	2	yes	—
54	C₄⋊Q₈	—	2	4	no	—
55	C₄⋊Q₈	C₂²×C₂₅₁	1	8	yes	—
56	C₂×C₄⋊C₄	—	2	2	no	—
57	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	4	yes	—
58	C₂×C₄²	C₂²×C₂₅₁	1	1	yes	characteristic
59	#106	C₂²×C₂₅₁	1	16	yes	—
60	C₄×C₈	C₂²×C₂₅₁	1	4	yes	—
61	C₂×Q₈	—	2	8	no	—
62	C₂×Q₈	—	4	8	no	—
63	C₂²×C₄	—	2	4	no	—
64	C₂²×C₄	D₄×C₂₅₁	1	2	yes	—
65	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
66	C₂×C₈	Q₈×C₂₅₁	1	8	yes	complemented
67	C₂×C₈	C₂³×C₂₅₁	1	4	yes	—
68	C₄⋊C₄	—	2	8	no	—
69	C₄⋊C₄	—	2	4	no	—
70	C₄⋊C₄	C₂³×C₂₅₁	1	8	yes	—
71	C₄²	D₄×C₂₅₁	1	2	yes	—
72	C₄²	C₂³×C₂₅₁	1	2	yes	—
73	C₂³	C₂×D₄×C₂₅₁	1	1	yes	characteristic
74	Q₈	—	4	16	no	—
75	C₂×C₄	—	2	8	no	—
76	C₂×C₄	—	4	8	no	—
77	C₂×C₄	SD₁₆×C₂₅₁	1	8	yes	—
78	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
79	C₂×C₄	C₂×D₄×C₂₅₁	1	4	yes	—
80	C₂×C₄	C₂×Q₈×C₂₅₁	1	4	yes	—
81	C₂×C₄	C₂⁴×C₂₅₁	1	1	yes	characteristic
82	C₈	C₂×Q₈×C₂₅₁	1	8	yes	complemented
83	C₂²	C₄⋊Q₈×C₂₅₁	1	2	yes	—
84	C₂²	C₂×SD₁₆×C₂₅₁	1	4	yes	—
85	C₂²	C₂²×D₄×C₂₅₁	1	1	yes	characteristic
86	C₄	—	4	8	no	—
87	C₄	C₄⋊Q₈×C₂₅₁	1	2	yes	—
88	C₄	C₂×SD₁₆×C₂₅₁	1	8	yes	—
89	C₄	C₂²×Q₈×C₂₅₁	1	2	yes	—
90	C₂	#498×C₂₅₁	1	4	yes	direct product factor
91	C₂	C₂×C₄⋊Q₈×C₂₅₁	1	1	yes	characteristic
92	C₂	C₂²×SD₁₆×C₂₅₁	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₅₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×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
2. #498×C₂₅₁	C₂	direct (N × H)
19. C₂×C₈×C₂₅₁	Q₈	semidirect
35. C₈×C₂₅₁	C₂×Q₈	semidirect
43. C₂×C₂₅₁	#498	direct (N × H)
46. C₂₅₁	C₂×#498	direct (N × H)
47. C₂×#498	C₂₅₁	direct (N × H)
49. #498	C₂×C₂₅₁	direct (N × H)
66. C₂×C₈	Q₈×C₂₅₁	semidirect
82. C₈	C₂×Q₈×C₂₅₁	semidirect
90. C₂	#498×C₂₅₁	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×C₄⋊Q₈×C₂₅₁	2
#498×C₂₅₁	2
C₂×#106×C₂₅₁	2
C₂×C₄×C₈×C₂₅₁	2
C₂×#498	251

Subgroups — C2×#498×C251 #44547744

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#498×C₂₅₁ #44547744