Subgroups — #44550502

Order 32128.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30631711	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₂₅₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#30631451	C₂	1	4	yes	maximal
4	#423×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#16860794	C₂²	1	2	yes	—
6	C₂²×C₂₅₁⋊C₈	C₂²	1	2	yes	—
7	C₄⋊(C₂₅₁⋊C₈)	—	2	2	no	—
8	C₄×C₂₅₁⋊C₈	C₄	1	2	yes	—
9	C₄×C₂₅₁⋊C₈	C₄	1	2	yes	—
10	C₂×M₁₆×C₂₅₁	C₂²	1	2	yes	—
11	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₂₅₁	—	2	2	no	—
13	#16860794	—	2	4	no	—
14	C₂×C₂₅₁⋊C₈	—	2	2	no	—
15	C₂×C₂₅₁⋊C₈	—	2	2	no	—
16	C₂×C₂₅₁⋊C₈	—	2	2	no	—
17	C₂×C₂₅₁⋊C₈	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₂₅₁	Q₈	1	2	yes	—
19	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
20	M₁₆×C₂₅₁	—	2	4	no	—
21	C₂×C₈×C₂₅₁	—	2	2	no	—
22	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₂₅₁	D₄	1	2	yes	—
25	C₂₅₁⋊C₈	—	2	4	no	—
26	C₂₅₁⋊C₈	—	4	2	no	—
27	C₂³×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄×C₂₅₁	—	2	4	no	—
29	C₂×C₄×C₂₅₁	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₂₅₁	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₂₅₁	#41	1	2	yes	—
34	C₂×C₄×C₂₅₁	#41	1	2	yes	—
35	C₈×C₂₅₁	—	4	2	no	—
36	C₂²×C₂₅₁	—	2	2	no	—
37	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₂₅₁	#125	1	2	yes	—
39	C₄×C₂₅₁	—	2	2	no	—
40	C₄×C₂₅₁	#108	1	4	yes	—
41	C₄×C₂₅₁	#117	1	2	yes	—
42	C₄×C₂₅₁	#124	1	1	yes	characteristic
43	C₄×C₂₅₁	#124	1	1	yes	characteristic
44	C₂×C₂₅₁	—	2	2	no	—
45	C₂×C₂₅₁	#382	1	1	yes	characteristic
46	C₂×C₂₅₁	C₂×#108	1	2	yes	—
47	C₂₅₁	#1837	1	1	yes	characteristic, complemented, Sylow
48	#1837	—	251	1	no	maximal, Sylow
49	#423	—	251	1	no	—
50	#423	D₂₅₁	1	1	yes	characteristic
51	C₂×C₄×C₈	—	251	1	no	—
52	#341	—	251	4	no	—
53	C₂×M₁₆	—	251	2	no	—
54	C₂×M₁₆	C₂×D₂₅₁	1	2	yes	—
55	C₂²×C₈	—	251	2	no	—
56	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
57	C₄⋊C₈	—	2	2	no	—
58	C₄⋊C₈	—	502	2	no	—
59	C₄×C₈	—	251	2	no	—
60	C₄×C₈	—	251	2	no	—
61	C₂²×C₄	Dic₅₀₂	1	2	yes	—
62	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
63	M₁₆	—	2	4	no	—
64	M₁₆	—	502	4	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	502	2	no	—
67	C₂×C₈	—	502	2	no	—
68	C₂×C₈	—	502	2	no	—
69	C₂×C₈	—	251	4	no	—
70	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
71	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
72	C₄²	C₂²⋊D₂₅₁	1	2	yes	—
73	C₂³	C₂×Dic₅₀₂	1	1	yes	characteristic
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	Dic₂₅₁⋊C₄	1	2	yes	—
76	C₂×C₄	Dic₂₅₁⋊C₄	1	2	yes	—
77	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
79	C₂×C₄	#16860809	1	2	yes	—
80	C₂×C₄	#16860811	1	2	yes	—
81	C₈	—	4	2	no	—
82	C₈	—	1004	2	no	—
83	C₈	—	502	4	no	—
84	C₂²	—	2	2	no	—
85	C₂²	C₂×Dic₂₅₁⋊C₄	1	1	yes	characteristic
86	C₂²	#22548706	1	2	yes	—
87	C₄	—	2	2	no	—
88	C₄	#22548650	1	4	yes	—
89	C₄	#22548673	1	2	yes	—
90	C₄	#22548702	1	1	yes	characteristic
91	C₄	#22548704	1	1	yes	characteristic
92	C₂	—	2	2	no	—
93	C₂	#30631552	1	1	yes	characteristic
94	C₂	C₂×#22548650	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₅₁
commutator	C₂×C₄×C₂₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	#423×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂×C₄⋊C₄
prime radical	C₂³×C₂₅₁	C₂×Q₈
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	#423	D₂₅₁
hypocenter	C₂₅₁	#1837
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
47. C₂₅₁	#1837	semidirect

Maximal subgroups

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

subgroup	index
#30631711	2
C₂×C₄×C₂₅₁⋊C₈	2
#30631451	2
#423×C₂₅₁	2
#1837	251