Subgroups — C₂³×#16860800 #44565087

Order 32128.

1464 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₅₀₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×#16860800	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₂₅₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂³×Q₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₀₂	—	2	7	no	—
6	C₂²×Dic₅₀₂	C₂²	1	14	yes	—
7	C₂³×Dic₂₅₁	—	2	1	no	—
8	C₂×#16860800	C₂²	1	112	yes	direct product factor
9	C₂²×C₂₅₁⋊C₈	C₂²	1	14	yes	—
10	C₂²×Q₈×C₂₅₁	—	2	7	no	—
11	C₂²×Q₈×C₂₅₁	C₂²	1	14	yes	—
12	C₂³×C₄×C₂₅₁	—	2	1	no	—
13	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
14	C₂×Dic₅₀₂	—	2	42	no	—
15	C₂×Dic₅₀₂	C₂³	1	28	yes	—
16	C₂²×Dic₂₅₁	—	2	14	no	—
17	#16860800	C₂³	1	64	yes	direct product factor
18	C₂×C₂₅₁⋊C₈	C₂³	1	28	yes	—
19	C₂⁴×C₂₅₁	D₄	1	1	yes	characteristic
20	C₂×Q₈×C₂₅₁	—	2	42	no	—
21	C₂×Q₈×C₂₅₁	C₂³	1	28	yes	—
22	C₂²×C₄×C₂₅₁	—	2	14	no	—
23	C₂²×C₄×C₂₅₁	D₄	1	7	yes	—
24	C₂²×C₄×C₂₅₁	C₂³	1	7	yes	—
25	Dic₅₀₂	—	2	28	no	—
26	Dic₅₀₂	C₂⁴	1	8	yes	—
27	C₂×Dic₂₅₁	—	2	28	no	—
28	C₂₅₁⋊C₈	C₂⁴	1	8	yes	—
29	C₂³×C₂₅₁	Q₁₆	1	8	yes	complemented
30	C₂³×C₂₅₁	C₂×D₄	1	7	yes	—
31	Q₈×C₂₅₁	—	2	28	no	—
32	Q₈×C₂₅₁	C₂⁴	1	8	yes	—
33	C₂×C₄×C₂₅₁	—	2	28	no	—
34	C₂×C₄×C₂₅₁	C₂×D₄	1	21	yes	—
35	C₂×C₄×C₂₅₁	C₂⁴	1	7	yes	—
36	C₂²×C₂₅₁	C₂×Q₁₆	1	28	yes	complemented
37	C₂²×C₂₅₁	C₂²×D₄	1	7	yes	—
38	C₄×C₂₅₁	—	2	8	no	—
39	C₄×C₂₅₁	C₂²×D₄	1	7	yes	—
40	C₄×C₂₅₁	C₂⁵	1	1	yes	characteristic
41	Dic₂₅₁	—	2	8	no	—
42	C₂×C₂₅₁	C₂²×Q₁₆	1	14	yes	complemented
43	C₂×C₂₅₁	C₂³×D₄	1	1	yes	characteristic
44	C₂₅₁	C₂³×Q₁₆	1	1	yes	characteristic, complemented, Sylow
45	C₂³×Q₁₆	—	251	1	no	maximal, Sylow
46	C₂³×Q₈	—	251	1	no	—
47	C₂³×Q₈	D₂₅₁	1	1	yes	characteristic
48	C₂²×Q₁₆	—	251	28	no	—
49	C₂³×C₈	—	251	1	no	—
50	C₂²×Q₈	—	2	7	no	—
51	C₂²×Q₈	—	502	7	no	—
52	C₂²×Q₈	—	251	14	no	—
53	C₂²×Q₈	C₂×D₂₅₁	1	14	yes	—
54	C₂³×C₄	—	2	1	no	—
55	C₂³×C₄	—	502	1	no	—
56	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
57	C₂×Q₁₆	—	251	112	no	—
58	C₂²×C₈	—	251	14	no	—
59	C₂⁴	C₂²⋊D₂₅₁	1	1	yes	characteristic
60	C₂×Q₈	—	2	42	no	—
61	C₂×Q₈	—	502	42	no	—
62	C₂×Q₈	—	251	28	no	—
63	C₂×Q₈	C₂²×D₂₅₁	1	28	yes	—
64	C₂²×C₄	—	2	14	no	—
65	C₂²×C₄	—	502	14	no	—
66	C₂²×C₄	C₂²⋊D₂₅₁	1	7	yes	—
67	C₂²×C₄	C₂²×D₂₅₁	1	7	yes	—
68	Q₁₆	—	251	64	no	—
69	C₂×C₈	—	251	28	no	—
70	C₂³	#16860800	1	8	yes	direct product factor
71	C₂³	C₂×C₂²⋊D₂₅₁	1	7	yes	—
72	Q₈	—	2	28	no	—
73	Q₈	—	502	28	no	—
74	Q₈	—	251	8	no	—
75	Q₈	C₂³×D₂₅₁	1	8	yes	—
76	C₂×C₄	—	2	28	no	—
77	C₂×C₄	—	502	28	no	—
78	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	21	yes	—
79	C₂×C₄	C₂³×D₂₅₁	1	7	yes	—
80	C₈	—	251	8	no	—
81	C₂²	C₂×#16860800	1	28	yes	direct product factor
82	C₂²	C₂²×C₂²⋊D₂₅₁	1	7	yes	—
83	C₄	—	2	8	no	—
84	C₄	—	502	8	no	—
85	C₄	C₂²×C₂²⋊D₂₅₁	1	7	yes	—
86	C₄	C₂⁴×D₂₅₁	1	1	yes	characteristic
87	C₂	C₂²×#16860800	1	14	yes	direct product factor
88	C₂	C₂³×C₂²⋊D₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²⋊D₂₅₁
commutator, cosocle	C₄×C₂₅₁	C₂⁵
Frattini	C₄	C₂⁴×D₂₅₁
Fitting	C₂³×Q₈×C₂₅₁	C₂
socle, prime radical	C₂⁴×C₂₅₁	D₄
hypercenter	C₂³×Q₈	D₂₅₁
hypocenter	C₂₅₁	C₂³×Q₁₆
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂²×#16860800	C₂	direct (N × H)
8. C₂×#16860800	C₂²	direct (N × H)
17. #16860800	C₂³	direct (N × H)
29. C₂³×C₂₅₁	Q₁₆	semidirect
36. C₂²×C₂₅₁	C₂×Q₁₆	semidirect
42. C₂×C₂₅₁	C₂²×Q₁₆	semidirect
44. C₂₅₁	C₂³×Q₁₆	semidirect
70. C₂³	#16860800	direct (N × H)
81. C₂²	C₂×#16860800	direct (N × H)
87. C₂	C₂²×#16860800	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₅₀₂	2
C₂²×#16860800	2
C₂³×C₂₅₁⋊C₈	2
C₂³×Q₈×C₂₅₁	2
C₂³×Q₁₆	251

Subgroups — C23×#16860800 #44565087

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#16860800 #44565087