Subgroups — #44548352

Order 32128.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30631945	C₂	1	1	yes	maximal, characteristic
2	#30631469	C₂	1	1	yes	maximal, characteristic
3	H(3,ℤ/4)×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₀₀₄	C₄	1	1	yes	characteristic, complemented
5	#22548694	—	2	1	no	—
6	#22548655	—	2	1	no	—
7	#22548647	C₄	1	1	yes	characteristic, complemented
8	#22548640	—	2	1	no	—
9	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
10	#117×C₂₅₁	—	2	1	no	—
11	C₂×Dic₅₀₂	—	2	1	no	—
12	Dic₁₀₀₄	—	4	1	no	—
13	C₄⋊Dic₂₅₁	—	2	1	no	—
14	Dic₂₅₁⋊C₄	—	4	1	no	—
15	C₂²⋊Dic₂₅₁	—	4	1	no	—
16	C₂₅₁⋊C₁₆	—	4	1	no	—
17	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
18	M₁₆×C₂₅₁	—	4	1	no	—
19	C₂×C₈×C₂₅₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₂₅₁	D₄	1	1	yes	characteristic
21	C₄⋊C₄×C₂₅₁	—	4	1	no	—
22	C₂²⋊C₄×C₂₅₁	—	4	1	no	—
23	C₄²×C₂₅₁	—	2	2	no	—
24	Dic₅₀₂	—	4	1	no	—
25	Dic₅₀₂	—	4	1	no	—
26	C₂×Dic₂₅₁	—	4	1	no	—
27	C₂×Dic₂₅₁	—	4	1	no	—
28	C₂³×C₂₅₁	D₈	1	1	yes	characteristic
29	C₂×C₄×C₂₅₁	—	2	1	no	—
30	C₂×C₄×C₂₅₁	—	2	1	no	—
31	C₂×C₄×C₂₅₁	—	4	2	no	—
32	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₅₁	SD₁₆	1	1	yes	characteristic
34	C₈×C₂₅₁	—	2	1	no	—
35	C₈×C₂₅₁	—	2	1	no	—
36	C₂²×C₂₅₁	—	2	1	no	—
37	C₂²×C₂₅₁	—	4	1	no	—
38	C₂²×C₂₅₁	#102	1	1	yes	characteristic
39	C₄×C₂₅₁	—	2	1	no	—
40	C₄×C₂₅₁	—	8	2	no	—
41	C₄×C₂₅₁	#99	1	1	yes	characteristic
42	C₄×C₂₅₁	#104	1	1	yes	characteristic
43	Dic₂₅₁	—	8	1	no	—
44	Dic₂₅₁	—	8	1	no	—
45	C₂×C₂₅₁	—	2	1	no	—
46	C₂×C₂₅₁	—	4	1	no	—
47	C₂×C₂₅₁	#327	1	1	yes	characteristic
48	C₂₅₁	#1340	1	1	yes	characteristic, complemented, Sylow
49	#1340	—	251	1	no	maximal, Sylow
50	#470	—	251	1	no	—
51	#349	—	251	1	no	—
52	H(3,ℤ/4)	D₂₅₁	1	1	yes	characteristic
53	C₂×Q₁₆	—	251	1	no	—
54	C₂×M₁₆	C₂×D₂₅₁	1	1	yes	characteristic
55	C₂²⋊Q₈	—	502	1	no	—
56	#117	—	2	1	no	—
57	M₃₂	—	502	1	no	—
58	#106	—	251	1	no	—
59	#103	—	502	1	no	—
60	C₂×Q₈	—	502	1	no	—
61	C₂²×C₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
62	Q₁₆	—	1004	1	no	—
63	M₁₆	—	4	1	no	—
64	C₂×C₈	C₄×D₂₅₁	1	1	yes	characteristic
65	C₂×C₈	D₁₀₀₄	1	1	yes	characteristic
66	C₄⋊C₄	—	4	1	no	—
67	C₄⋊C₄	—	1004	1	no	—
68	C₄⋊C₄	—	502	1	no	—
69	C₂²⋊C₄	—	4	1	no	—
70	C₂²⋊C₄	—	1004	1	no	—
71	C₄²	—	2	2	no	—
72	C₁₆	—	1004	1	no	—
73	C₂³	#16860795	1	1	yes	characteristic
74	Q₈	—	1004	1	no	—
75	Q₈	—	1004	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	4	2	no	—
79	C₂×C₄	—	1004	1	no	—
80	C₂×C₄	—	1004	1	no	—
81	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	1	yes	characteristic
82	C₂×C₄	#16860797	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	—	2	1	no	—
85	C₂²	—	2	1	no	—
86	C₂²	—	4	1	no	—
87	C₂²	#22548635	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	8	2	no	—
90	C₄	—	2008	1	no	—
91	C₄	—	2008	1	no	—
92	C₄	#22548629	1	1	yes	characteristic
93	C₄	#22548643	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	4	1	no	—
96	C₂	#30631420	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#30631420
commutator	C₂×C₈×C₂₅₁	C₂×C₄
Frattini	C₂×M₁₆	C₂×D₂₅₁
Fitting	H(3,ℤ/4)×C₂₅₁	C₂
socle	C₂×C₂₅₁	#327
prime radical	C₂³×C₂₅₁	D₈
cosocle	C₂×M₁₆×C₂₅₁	C₂²
hypercenter	H(3,ℤ/4)	D₂₅₁
hypocenter	C₂₅₁	#1340
LCS	G ▷ C₂×C₈×C₂₅₁ ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
4. C₂×Dic₁₀₀₄	C₄	semidirect
7. #22548647	C₄	semidirect
48. C₂₅₁	#1340	semidirect

Maximal subgroups

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

subgroup	index
#30631945	2
#30631469	2
H(3,ℤ/4)×C₂₅₁	2
#1340	251