Subgroups — #2150×C₂₅₁ #44546737

Order 32128.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×C₂₅₁	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₁₆×C₂₅₁	C₂	1	4	yes	maximal, complemented
4	C₂²×C₈×C₂₅₁	C₄	1	1	yes	characteristic
5	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₂₅₁	C₄	1	1	yes	characteristic
7	M₃₂×C₂₅₁	—	2	4	no	—
8	C₂×C₁₆×C₂₅₁	C₂²	1	4	yes	—
9	C₄×C₈×C₂₅₁	C₄	1	2	yes	—
10	C₄×C₈×C₂₅₁	C₂²	1	2	yes	—
11	C₂²×C₄×C₂₅₁	C₈	1	2	yes	—
12	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₂₅₁	—	2	2	no	—
14	C₂×C₈×C₂₅₁	C₂×C₄	1	2	yes	—
15	C₂×C₈×C₂₅₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₂₅₁	D₄	1	2	yes	—
17	C₂×C₈×C₂₅₁	Q₈	1	2	yes	—
18	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
19	C₄²×C₂₅₁	C₈	1	1	yes	characteristic
20	C₄²×C₂₅₁	C₈	1	1	yes	characteristic
21	C₄²×C₂₅₁	C₂×C₄	1	2	yes	—
22	C₁₆×C₂₅₁	—	2	4	no	—
23	C₂³×C₂₅₁	C₂×C₈	1	1	yes	characteristic
24	C₂×C₄×C₂₅₁	—	2	4	no	—
25	C₂×C₄×C₂₅₁	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₂₅₁	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₂₅₁	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₂₅₁	C₂×C₈	1	2	yes	—
29	C₂×C₄×C₂₅₁	M₁₆	1	2	yes	—
30	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
31	C₈×C₂₅₁	—	2	2	no	—
32	C₈×C₂₅₁	C₄⋊C₄	1	2	yes	—
33	C₈×C₂₅₁	C₂×D₄	1	1	yes	characteristic
34	C₈×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₂₅₁	—	2	2	no	—
36	C₂²×C₂₅₁	C₄⋊C₈	1	2	yes	—
37	C₂²×C₂₅₁	C₂²×C₈	1	1	yes	characteristic
38	C₄×C₂₅₁	—	2	2	no	—
39	C₄×C₂₅₁	C₄⋊C₈	1	2	yes	—
40	C₄×C₂₅₁	M₃₂	1	4	yes	complemented
41	C₄×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₄×C₂₅₁	C₂×M₁₆	1	1	yes	characteristic
43	C₂×C₂₅₁	—	2	2	no	—
44	C₂×C₂₅₁	C₂×C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₂₅₁	C₂×M₃₂	1	2	yes	—
46	C₂₅₁	#2150	1	1	yes	characteristic, direct product factor, Sylow
47	#2150	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂×M₃₂	C₂×C₂₅₁	1	2	yes	—
49	C₂×C₄×C₈	C₂×C₂₅₁	1	1	yes	characteristic
50	C₄⋊C₁₆	C₂×C₂₅₁	1	4	yes	complemented
51	C₂²×C₈	C₄×C₂₅₁	1	1	yes	characteristic
52	C₂²×C₈	C₂²×C₂₅₁	1	1	yes	characteristic
53	C₂×C₄²	C₄×C₂₅₁	1	1	yes	characteristic
54	M₃₂	—	2	4	no	—
55	C₂×C₁₆	C₂²×C₂₅₁	1	4	yes	—
56	C₄×C₈	C₄×C₂₅₁	1	2	yes	—
57	C₄×C₈	C₂²×C₂₅₁	1	2	yes	—
58	C₂²×C₄	C₈×C₂₅₁	1	2	yes	—
59	C₂²×C₄	C₂×C₄×C₂₅₁	1	1	yes	characteristic
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	C₂×C₄×C₂₅₁	1	2	yes	—
62	C₂×C₈	C₂×C₄×C₂₅₁	1	1	yes	characteristic
63	C₂×C₈	D₄×C₂₅₁	1	2	yes	—
64	C₂×C₈	Q₈×C₂₅₁	1	2	yes	—
65	C₂×C₈	C₂³×C₂₅₁	1	1	yes	characteristic
66	C₄²	C₈×C₂₅₁	1	1	yes	characteristic
67	C₄²	C₈×C₂₅₁	1	1	yes	characteristic
68	C₄²	C₂×C₄×C₂₅₁	1	2	yes	—
69	C₁₆	—	2	4	no	—
70	C₂³	C₂×C₈×C₂₅₁	1	1	yes	characteristic
71	C₂×C₄	—	2	4	no	—
72	C₂×C₄	C₄⋊C₄×C₂₅₁	1	2	yes	—
73	C₂×C₄	C₂×C₈×C₂₅₁	1	2	yes	—
74	C₂×C₄	C₂×C₈×C₂₅₁	1	1	yes	characteristic
75	C₂×C₄	C₂×C₈×C₂₅₁	1	2	yes	—
76	C₂×C₄	M₁₆×C₂₅₁	1	2	yes	—
77	C₂×C₄	C₂²×C₄×C₂₅₁	1	1	yes	characteristic
78	C₈	—	2	2	no	—
79	C₈	C₄⋊C₄×C₂₅₁	1	2	yes	—
80	C₈	C₂×D₄×C₂₅₁	1	1	yes	characteristic
81	C₈	C₂×Q₈×C₂₅₁	1	1	yes	characteristic
82	C₂²	—	2	2	no	—
83	C₂²	C₄⋊C₈×C₂₅₁	1	2	yes	—
84	C₂²	C₂²×C₈×C₂₅₁	1	1	yes	characteristic
85	C₄	—	2	2	no	—
86	C₄	C₄⋊C₈×C₂₅₁	1	2	yes	—
87	C₄	M₃₂×C₂₅₁	1	4	yes	complemented
88	C₄	C₂×C₄⋊C₄×C₂₅₁	1	1	yes	characteristic
89	C₄	C₂×M₁₆×C₂₅₁	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	C₂×C₄⋊C₈×C₂₅₁	1	1	yes	characteristic
92	C₂	C₂×M₃₂×C₂₅₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₂₅₁	C₂³
commutator	C₂²	C₂²×C₈×C₂₅₁
Frattini, cosocle	C₂×C₈	C₂³×C₂₅₁
socle	C₂²×C₂₅₁	C₂²×C₈
prime radical	C₂³×C₂₅₁	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄⋊C₁₆×C₂₅₁	C₂	semidirect
40. C₄×C₂₅₁	M₃₂	semidirect
46. C₂₅₁	#2150	direct (N × H)
47. #2150	C₂₅₁	direct (N × H)
50. C₄⋊C₁₆	C₂×C₂₅₁	semidirect
87. C₄	M₃₂×C₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×C₂₅₁	2
C₂×C₄×C₈×C₂₅₁	2
C₄⋊C₁₆×C₂₅₁	2
#2150	251

Subgroups — #2150×C251 #44546737

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2150×C₂₅₁ #44546737