Subgroups — #44549967

Order 32128.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632060	C₂	1	1	yes	maximal, characteristic
2	#30632037	C₂	1	2	yes	maximal, complemented
3	#30632035	C₂	1	2	yes	maximal
4	C₈×C₂₅₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#498×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16860799	—	2	2	no	—
7	#22548715	C₂²	1	1	yes	characteristic
8	#22548712	C₂²	1	1	yes	characteristic
9	#22548639	—	2	4	no	—
10	#22548635	—	2	4	no	—
11	C₄×C₂₅₁⋊C₈	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₂₅₁	C₂²	1	2	yes	—
13	#106×C₂₅₁	—	2	2	no	—
14	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₅₀₂	—	2	1	no	—
16	C₂×D₁₀₀₄	—	2	2	no	—
17	C₂×D₁₀₀₄	—	2	1	no	—
18	#16860799	—	4	4	no	—
19	C₂×C₂₅₁⋊C₈	D₄	1	4	yes	—
20	C₄⋊Dic₂₅₁	—	2	2	no	—
21	C₂×Q₈×C₂₅₁	—	2	2	no	—
22	C₂×C₈×C₂₅₁	D₄	1	2	yes	—
23	C₄⋊C₄×C₂₅₁	—	2	2	no	—
24	C₄⋊C₄×C₂₅₁	—	2	2	no	—
25	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
26	C₂²×D₂₅₁	—	4	1	no	—
27	Dic₅₀₂	—	4	2	no	—
28	D₁₀₀₄	—	4	2	no	—
29	D₁₀₀₄	—	4	4	no	—
30	C₂×Dic₂₅₁	—	4	1	no	—
31	C₂₅₁⋊C₈	—	2	4	no	—
32	Q₈×C₂₅₁	—	4	4	no	—
33	C₂×C₄×C₂₅₁	—	4	2	no	—
34	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
35	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
36	C₈×C₂₅₁	SD₁₆	1	4	yes	—
37	C₂²×C₂₅₁	#127	1	1	yes	characteristic
38	C₂×D₂₅₁	—	8	1	no	—
39	C₂×D₂₅₁	—	8	2	no	—
40	C₄×C₂₅₁	—	8	2	no	—
41	C₄×C₂₅₁	C₂×SD₁₆	1	2	yes	—
42	C₄×C₂₅₁	#135	1	4	yes	—
43	Dic₂₅₁	—	8	1	no	—
44	C₂×C₂₅₁	#492	1	1	yes	characteristic
45	C₂×C₂₅₁	#495	1	2	yes	—
46	D₂₅₁	—	16	1	no	—
47	C₂₅₁	#1708	1	1	yes	characteristic, complemented, Sylow
48	#1708	—	251	1	no	maximal, Sylow
49	#498	D₂₅₁	1	1	yes	characteristic, complemented
50	#492	—	251	1	no	—
51	#487	—	251	2	no	—
52	#486	—	251	2	no	—
53	C₈²	—	251	1	no	—
54	C₂×SD₁₆	—	502	2	no	—
55	C₄⋊Q₈	—	251	1	no	—
56	C₄⋊Q₈	C₂×D₂₅₁	1	2	yes	—
57	#127	—	251	1	no	—
58	#106	—	2	2	no	—
59	#103	—	502	4	no	—
60	#102	—	502	4	no	—
61	C₄×C₈	—	251	2	no	—
62	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
63	C₂×Q₈	—	2	2	no	—
64	C₂×Q₈	—	502	1	no	—
65	C₂×D₄	—	502	2	no	—
66	C₂×D₄	—	502	1	no	—
67	SD₁₆	—	1004	4	no	—
68	C₂×C₈	—	251	4	no	—
69	C₂×C₈	C₂²⋊D₂₅₁	1	2	yes	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	502	2	no	—
73	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
74	C₂³	—	1004	1	no	—
75	Q₈	—	4	4	no	—
76	Q₈	—	1004	2	no	—
77	D₄	—	1004	4	no	—
78	D₄	—	1004	2	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	—	1004	1	no	—
81	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
82	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
83	C₈	—	502	4	no	—
84	C₈	#16860797	1	4	yes	—
85	C₂²	—	2008	1	no	—
86	C₂²	—	2008	2	no	—
87	C₂²	#22548711	1	1	yes	characteristic
88	C₄	—	8	2	no	—
89	C₄	—	2008	1	no	—
90	C₄	C₂×#16860797	1	2	yes	—
91	C₄	#22548736	1	4	yes	—
92	C₂	—	4016	1	no	—
93	C₂	#30632061	1	1	yes	characteristic
94	C₂	#30632068	1	2	yes	—

	subgroup	G/H
center	C₂²	#22548711
commutator, cosocle	C₄²×C₂₅₁	C₂³
Frattini	C₄²	C₂²×D₂₅₁
Fitting	#498×C₂₅₁	C₂
socle	C₂²×C₂₅₁	#127
prime radical	#22548712	C₂²
hypercenter	#498	D₂₅₁
hypocenter	C₂₅₁	#1708
LCS	G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

Complements

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

N	H ≅ G/N	type
2. #30632037	C₂	semidirect
4. C₈×C₂₅₁⋊C₈	C₂	semidirect
5. #498×C₂₅₁	C₂	semidirect
47. C₂₅₁	#1708	semidirect
49. #498	D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#30632060	2
#30632037	2
#30632035	2
C₈×C₂₅₁⋊C₈	2
#498×C₂₅₁	2
#1708	251