Subgroups — #44552616

Order 32128.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₀₀₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#22548661	C₂	1	2	yes	maximal
3	#30632065	C₂	1	2	yes	maximal
4	C₁₆×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₃₂×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₀₀₄	C₂²	1	2	yes	—
7	C₂×#16860800	—	2	2	no	—
8	#22548713	—	2	2	no	—
9	Dic₂₀₀₈	—	2	2	no	—
10	#22548661	—	2	4	no	—
11	C₂×C₂₅₁⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
13	C₂×Q₁₆×C₂₅₁	C₂²	1	2	yes	—
14	Q₃₂×C₂₅₁	—	2	2	no	—
15	C₂×C₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
16	C₂×Dic₅₀₂	—	2	2	no	—
17	Dic₁₀₀₄	—	2	2	no	—
18	Dic₁₀₀₄	—	2	2	no	—
19	#16860800	—	4	4	no	—
20	C₂×C₂₅₁⋊C₈	D₄	1	1	yes	characteristic
21	Dic₂₅₁⋊C₄	—	4	4	no	—
22	C₄×Dic₂₅₁	D₄	1	1	yes	characteristic
23	C₂₅₁⋊C₁₆	D₄	1	2	yes	—
24	C₂×Q₈×C₂₅₁	—	2	2	no	—
25	Q₁₆×C₂₅₁	—	2	2	no	—
26	Q₁₆×C₂₅₁	—	2	2	no	—
27	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
28	C₁₆×C₂₅₁	D₄	1	2	yes	—
29	Dic₅₀₂	—	4	2	no	—
30	Dic₅₀₂	—	4	2	no	—
31	C₂×Dic₂₅₁	—	4	2	no	—
32	C₂×Dic₂₅₁	D₈	1	2	yes	—
33	C₂₅₁⋊C₈	D₈	1	2	yes	—
34	Q₈×C₂₅₁	—	4	2	no	—
35	Q₈×C₂₅₁	—	4	2	no	—
36	C₂×C₄×C₂₅₁	—	4	2	no	—
37	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
38	C₈×C₂₅₁	C₂×D₄	1	1	yes	characteristic
39	C₈×C₂₅₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₂₅₁	C₂×D₈	1	1	yes	characteristic
41	C₄×C₂₅₁	—	8	2	no	—
42	C₄×C₂₅₁	#127	1	1	yes	characteristic
43	C₄×C₂₅₁	C₂×D₈	1	1	yes	characteristic
44	Dic₂₅₁	—	8	2	no	—
45	Dic₂₅₁	Q₃₂	1	4	yes	complemented
46	C₂×C₂₅₁	#493	1	1	yes	characteristic
47	C₂×C₂₅₁	C₂×Q₃₂	1	2	yes	—
48	C₂₅₁	#2247	1	1	yes	characteristic, complemented, Sylow
49	#2247	—	251	1	no	maximal, Sylow
50	C₂×Q₃₂	—	251	2	no	—
51	C₂×Q₃₂	—	251	1	no	—
52	C₂×Q₃₂	D₂₅₁	1	1	yes	characteristic
53	#494	—	251	2	no	—
54	C₄×C₁₆	—	251	1	no	—
55	C₂×Q₁₆	—	502	2	no	—
56	C₂×Q₁₆	—	251	2	no	—
57	C₂×Q₁₆	C₂×D₂₅₁	1	2	yes	—
58	C₄⋊Q₈	—	502	2	no	—
59	Q₃₂	—	2	2	no	—
60	Q₃₂	—	502	4	no	—
61	Q₃₂	—	502	2	no	—
62	C₂×C₁₆	—	251	1	no	—
63	C₂×C₁₆	C₂×D₂₅₁	1	1	yes	characteristic
64	C₄×C₈	—	251	1	no	—
65	C₂×Q₈	—	2	2	no	—
66	C₂×Q₈	—	502	2	no	—
67	Q₁₆	—	2	2	no	—
68	Q₁₆	—	2	2	no	—
69	Q₁₆	—	1004	4	no	—
70	Q₁₆	—	502	2	no	—
71	Q₁₆	—	502	2	no	—
72	C₂×C₈	—	251	1	no	—
73	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
74	C₄⋊C₄	—	1004	4	no	—
75	C₄²	—	251	1	no	—
76	C₁₆	—	251	2	no	—
77	C₁₆	C₂²⋊D₂₅₁	1	2	yes	—
78	Q₈	—	4	2	no	—
79	Q₈	—	4	2	no	—
80	Q₈	—	1004	2	no	—
81	Q₈	—	1004	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	1004	2	no	—
84	C₂×C₄	—	251	2	no	—
85	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
86	C₈	—	251	2	no	—
87	C₈	D₄×D₂₅₁	1	1	yes	characteristic
88	C₈	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
89	C₂²	D₈×D₂₅₁	1	1	yes	characteristic
90	C₄	—	8	2	no	—
91	C₄	—	2008	2	no	—
92	C₄	—	251	4	no	—
93	C₄	#22548711	1	1	yes	characteristic
94	C₄	D₈×D₂₅₁	1	1	yes	characteristic
95	C₂	#30632062	1	1	yes	characteristic
96	C₂	Q₃₂×D₂₅₁	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₂₅₁
commutator, cosocle	C₂×C₈×C₂₅₁	C₂³
Frattini	C₂×C₈	C₂²×D₂₅₁
Fitting	C₂×Q₃₂×C₂₅₁	C₂
socle, prime radical	C₂²×C₂₅₁	C₂×D₈
hypercenter	C₂×Q₃₂	D₂₅₁
hypocenter	C₂₅₁	#2247
LCS	G ▷ C₂×C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
45. Dic₂₅₁	Q₃₂	semidirect
48. C₂₅₁	#2247	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₀₀₈	2
C₂×#22548661	2
#30632065	2
C₁₆×Dic₂₅₁	2
C₂×Q₃₂×C₂₅₁	2
#2247	251