Subgroups — #44552622

Order 32128.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548658	C₂	1	4	yes	maximal, complemented
2	#30632064	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₅₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#494×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₂₀₀₈	—	2	1	no	—
6	C₂×D₂₀₀₈	C₂²	1	2	yes	—
7	#22548712	—	2	1	no	—
8	#22548658	—	2	8	no	—
9	C₂×C₂₅₁⋊C₁₆	C₂²	1	2	yes	—
10	C₂×Q₁₆×C₂₅₁	—	2	1	no	—
11	C₂×Q₁₆×C₂₅₁	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₂₅₁	—	2	1	no	—
13	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
14	C₂×D₁₀₀₄	—	2	2	no	—
15	C₂×D₁₀₀₄	—	4	2	no	—
16	D₂₀₀₈	—	2	4	no	—
17	D₂₀₀₈	—	4	2	no	—
18	C₂₅₁⋊C₁₆	D₄	1	4	yes	complemented
19	C₂×Q₈×C₂₅₁	—	2	2	no	—
20	Q₁₆×C₂₅₁	—	2	4	no	—
21	Q₁₆×C₂₅₁	—	4	2	no	—
22	C₂×C₈×C₂₅₁	D₄	1	1	yes	characteristic
23	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₂₅₁	—	4	2	no	—
25	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
26	C₂²×D₂₅₁	—	4	2	no	—
27	D₁₀₀₄	—	4	2	no	—
28	D₁₀₀₄	—	4	2	no	—
29	D₁₀₀₄	—	8	4	no	—
30	Q₈×C₂₅₁	—	4	2	no	—
31	Q₈×C₂₅₁	—	4	2	no	—
32	C₂×C₄×C₂₅₁	—	4	2	no	—
33	C₂×C₄×C₂₅₁	D₈	1	2	yes	—
34	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
35	C₈×C₂₅₁	D₈	1	2	yes	—
36	C₈×C₂₅₁	C₂×D₄	1	2	yes	—
37	C₂²×C₂₅₁	C₂×D₈	1	1	yes	characteristic
38	C₂×D₂₅₁	—	8	2	no	—
39	C₂×D₂₅₁	—	8	4	no	—
40	C₄×C₂₅₁	—	8	2	no	—
41	C₄×C₂₅₁	SD₃₂	1	4	yes	complemented
42	C₄×C₂₅₁	#127	1	1	yes	characteristic
43	C₄×C₂₅₁	C₂×D₈	1	1	yes	characteristic
44	C₂×C₂₅₁	#493	1	1	yes	characteristic
45	C₂×C₂₅₁	C₂×SD₃₂	1	2	yes	—
46	D₂₅₁	—	16	2	no	—
47	C₂₅₁	#2248	1	1	yes	characteristic, complemented, Sylow
48	#2248	—	251	1	no	maximal, Sylow
49	C₂×SD₃₂	—	251	4	no	—
50	#494	D₂₅₁	1	1	yes	characteristic, complemented
51	#493	—	251	1	no	—
52	C₄×C₁₆	—	251	1	no	—
53	C₂×Q₁₆	—	2	1	no	—
54	C₂×Q₁₆	C₂×D₂₅₁	1	2	yes	—
55	C₂×D₈	—	502	1	no	—
56	C₂×D₈	—	251	2	no	—
57	C₄⋊Q₈	—	2	1	no	—
58	#127	—	502	1	no	—
59	SD₃₂	—	502	8	no	—
60	C₂×C₁₆	—	251	2	no	—
61	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
62	C₂×Q₈	—	2	2	no	—
63	C₂×D₄	—	1004	2	no	—
64	C₂×D₄	—	502	2	no	—
65	Q₁₆	—	2	4	no	—
66	Q₁₆	—	4	2	no	—
67	D₈	—	1004	2	no	—
68	D₈	—	502	4	no	—
69	C₂×C₈	C₂²⋊D₂₅₁	1	1	yes	characteristic
70	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
71	C₄⋊C₄	—	4	2	no	—
72	C₄²	C₂²⋊D₂₅₁	1	1	yes	characteristic
73	C₁₆	—	251	4	no	—
74	C₂³	—	1004	2	no	—
75	Q₈	—	4	2	no	—
76	Q₈	—	4	2	no	—
77	D₄	—	2008	4	no	—
78	D₄	—	1004	2	no	—
79	D₄	—	1004	2	no	—
80	C₂×C₄	—	4	2	no	—
81	C₂×C₄	#16860795	1	2	yes	—
82	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
83	C₈	#16860795	1	2	yes	—
84	C₈	D₄×D₂₅₁	1	2	yes	—
85	C₂²	—	2008	2	no	—
86	C₂²	—	2008	4	no	—
87	C₂²	C₂×#16860795	1	1	yes	characteristic
88	C₄	—	8	2	no	—
89	C₄	#22548658	1	4	yes	complemented
90	C₄	#22548711	1	1	yes	characteristic
91	C₄	C₂×#16860795	1	1	yes	characteristic
92	C₂	—	4016	2	no	—
93	C₂	#30632063	1	1	yes	characteristic
94	C₂	C₂×#22548658	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#16860795
commutator, cosocle	C₂×C₈×C₂₅₁	C₂³
Frattini	C₂×C₈	C₂²×D₂₅₁
Fitting	#494×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂×D₈
prime radical	#30632064	C₂
hypercenter	#494	D₂₅₁
hypocenter	C₂₅₁	#2248
LCS	G ▷ C₂×C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
1. C₂×#22548658	C₂	semidirect
3. C₄×C₂₅₁⋊C₁₆	C₂	semidirect
4. #494×C₂₅₁	C₂	semidirect
18. C₂₅₁⋊C₁₆	D₄	semidirect
41. C₄×C₂₅₁	SD₃₂	semidirect
47. C₂₅₁	#2248	semidirect
50. #494	D₂₅₁	semidirect
89. C₄	#22548658	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22548658	2
#30632064	2
C₄×C₂₅₁⋊C₁₆	2
#494×C₂₅₁	2
#2248	251