Subgroups — #44550448

Order 32128.

272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30631651	C₂	1	2	yes	maximal
2	#30631456	C₂	1	4	yes	maximal, complemented
3	C₂×#117×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#16860794	C₂²	1	4	yes	—
5	C₂²⋊(C₂₅₁⋊C₈)	—	2	4	no	—
6	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
7	#117×C₂₅₁	—	2	4	no	—
8	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
9	C₂×C₂²⋊C₄×C₂₅₁	C₄	1	2	yes	—
10	C₂×C₄²×C₂₅₁	C₄	1	2	yes	—
11	#16860794	—	2	8	no	—
12	C₂×C₂₅₁⋊C₈	—	2	4	no	—
13	C₂⁴×C₂₅₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₂₅₁	—	2	4	no	—
15	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₂₅₁	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₂₅₁	D₄	1	4	yes	—
18	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₂₅₁	—	2	4	no	—
20	C₂²⋊C₄×C₂₅₁	—	2	4	no	—
21	C₄²×C₂₅₁	—	2	4	no	—
22	C₂₅₁⋊C₈	—	4	4	no	—
23	C₂³×C₂₅₁	—	2	2	no	—
24	C₂³×C₂₅₁	—	4	2	no	—
25	C₂³×C₂₅₁	C₂²⋊C₄	1	2	yes	—
26	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₂₅₁	—	2	2	no	—
28	C₂×C₄×C₂₅₁	—	2	8	no	—
29	C₂×C₄×C₂₅₁	—	2	4	no	—
30	C₂×C₄×C₂₅₁	—	4	4	no	—
31	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
33	C₂×C₄×C₂₅₁	#41	1	4	yes	—
34	C₂²×C₂₅₁	—	2	2	no	—
35	C₂²×C₂₅₁	—	2	2	no	—
36	C₂²×C₂₅₁	—	2	4	no	—
37	C₂²×C₂₅₁	—	4	4	no	—
38	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₂₅₁	#117	1	2	yes	—
40	C₄×C₂₅₁	—	2	2	no	—
41	C₄×C₂₅₁	—	4	4	no	—
42	C₄×C₂₅₁	#123	1	4	yes	—
43	C₂×C₂₅₁	—	2	2	no	—
44	C₂×C₂₅₁	—	4	2	no	—
45	C₂×C₂₅₁	#381	1	1	yes	characteristic
46	C₂×C₂₅₁	#413	1	2	yes	—
47	C₂₅₁	#1822	1	1	yes	characteristic, complemented, Sylow
48	#1822	—	251	1	no	maximal, Sylow
49	C₂×#117	D₂₅₁	1	1	yes	characteristic
50	#407	—	251	2	no	—
51	#343	—	251	4	no	—
52	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
53	C₂×M₁₆	—	251	4	no	—
54	#117	—	2	4	no	—
55	C₂×C₄⋊C₄	C₂×D₂₅₁	1	2	yes	—
56	C₂×C₂²⋊C₄	Dic₂₅₁	1	2	yes	—
57	C₂×C₄²	Dic₂₅₁	1	2	yes	—
58	C₂²⋊C₈	—	502	4	no	—
59	C₂⁴	C₂×Dic₂₅₁	1	1	yes	characteristic
60	C₂²×C₄	—	2	4	no	—
61	C₂²×C₄	C₂×Dic₂₅₁	1	1	yes	characteristic
62	C₂²×C₄	C₂×Dic₂₅₁	1	4	yes	—
63	C₂²×C₄	C₂²⋊D₂₅₁	1	4	yes	—
64	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
65	M₁₆	—	502	8	no	—
66	C₂×C₈	—	502	4	no	—
67	C₄⋊C₄	—	2	4	no	—
68	C₂²⋊C₄	—	2	4	no	—
69	C₄²	—	2	4	no	—
70	C₂³	—	2	2	no	—
71	C₂³	—	4	2	no	—
72	C₂³	C₂²⋊Dic₂₅₁	1	2	yes	—
73	C₂³	C₂²×Dic₂₅₁	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	8	no	—
76	C₂×C₄	—	2	4	no	—
77	C₂×C₄	—	4	4	no	—
78	C₂×C₄	C₂²⋊Dic₂₅₁	1	2	yes	—
79	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
80	C₂×C₄	#16860810	1	4	yes	—
81	C₈	—	1004	4	no	—
82	C₂²	—	2	2	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	2	4	no	—
85	C₂²	—	4	4	no	—
86	C₂²	C₂×C₂²⋊Dic₂₅₁	1	1	yes	characteristic
87	C₂²	#22548674	1	2	yes	—
88	C₄	—	2	2	no	—
89	C₄	—	4	4	no	—
90	C₄	#22548698	1	4	yes	—
91	C₂	—	2	2	no	—
92	C₂	—	4	2	no	—
93	C₂	#30631550	1	1	yes	characteristic
94	C₂	#30631671	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₅₁
commutator	C₂³×C₂₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	C₂×#117×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₂₅₁	C₂×C₄
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	C₂×#117	D₂₅₁
hypocenter	C₂₅₁	#1822
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #30631456	C₂	semidirect
47. C₂₅₁	#1822	semidirect

Maximal subgroups

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

subgroup	index
#30631651	2
#30631456	2
C₂×#117×C₂₅₁	2
#1822	251