Subgroups — #44548359

Order 32128.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30631469	C₂	1	1	yes	maximal, characteristic
2	C₂₅₁⋊#338	C₂	1	1	yes	maximal, characteristic
3	#470×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#16860794	—	2	1	no	—
5	#22548655	—	2	1	no	—
6	C₂×Q₁₆×C₂₅₁	C₄	1	1	yes	characteristic
7	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
8	C₂²⋊Q₈×C₂₅₁	—	2	1	no	—
9	#106×C₂₅₁	C₄	1	1	yes	characteristic
10	#103×C₂₅₁	—	2	1	no	—
11	#16860794	—	4	1	no	—
12	#16860794	—	4	1	no	—
13	C₂×C₂₅₁⋊C₈	—	2	2	no	—
14	C₂₅₁⋊C₁₆	—	4	1	no	—
15	C₂×Q₈×C₂₅₁	—	2	1	no	—
16	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
17	Q₁₆×C₂₅₁	—	4	1	no	—
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	—	2	1	no	—
22	C₄⋊C₄×C₂₅₁	—	4	1	no	—
23	C₂²⋊C₄×C₂₅₁	—	4	1	no	—
24	C₂₅₁⋊C₈	—	4	2	no	—
25	C₂³×C₂₅₁	SD₁₆	1	1	yes	characteristic
26	Q₈×C₂₅₁	—	4	1	no	—
27	Q₈×C₂₅₁	—	4	1	no	—
28	C₂×C₄×C₂₅₁	—	2	1	no	—
29	C₂×C₄×C₂₅₁	—	2	1	no	—
30	C₂×C₄×C₂₅₁	—	4	1	no	—
31	C₂×C₄×C₂₅₁	—	4	1	no	—
32	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₅₁	D₈	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	1	no	—
41	C₄×C₂₅₁	—	8	1	no	—
42	C₄×C₂₅₁	#99	1	1	yes	characteristic
43	C₄×C₂₅₁	#104	1	1	yes	characteristic
44	C₂×C₂₅₁	—	2	1	no	—
45	C₂×C₂₅₁	—	4	1	no	—
46	C₂×C₂₅₁	#327	1	1	yes	characteristic
47	C₂₅₁	#1342	1	1	yes	characteristic, complemented, Sylow
48	#1342	—	251	1	no	maximal, Sylow
49	#470	D₂₅₁	1	1	yes	characteristic
50	#349	—	251	1	no	—
51	#338	—	251	1	no	—
52	C₂×Q₁₆	Dic₂₅₁	1	1	yes	characteristic
53	C₂×M₁₆	—	502	1	no	—
54	C₂×M₁₆	C₂×D₂₅₁	1	1	yes	characteristic
55	C₂²⋊Q₈	—	2	1	no	—
56	M₃₂	—	502	1	no	—
57	#106	Dic₂₅₁	1	1	yes	characteristic
58	#103	—	2	1	no	—
59	C₂×Q₈	—	2	1	no	—
60	C₂²×C₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
61	Q₁₆	—	4	1	no	—
62	M₁₆	—	4	1	no	—
63	M₁₆	—	1004	1	no	—
64	M₁₆	—	1004	1	no	—
65	C₂×C₈	—	502	2	no	—
66	C₂×C₈	C₂×Dic₂₅₁	1	1	yes	characteristic
67	C₂×C₈	C₂²⋊D₂₅₁	1	1	yes	characteristic
68	C₄⋊C₄	—	2	1	no	—
69	C₄⋊C₄	—	4	1	no	—
70	C₂²⋊C₄	—	4	1	no	—
71	C₁₆	—	1004	1	no	—
72	C₂³	#16860798	1	1	yes	characteristic
73	Q₈	—	4	1	no	—
74	Q₈	—	4	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	—	4	1	no	—
79	C₂×C₄	C₂²⋊Dic₂₅₁	1	1	yes	characteristic
80	C₂×C₄	#16860795	1	1	yes	characteristic
81	C₈	—	2	1	no	—
82	C₈	—	2	1	no	—
83	C₈	—	1004	2	no	—
84	C₂²	—	2	1	no	—
85	C₂²	—	4	1	no	—
86	C₂²	#22548636	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	8	1	no	—
89	C₄	—	8	1	no	—
90	C₄	#22548630	1	1	yes	characteristic
91	C₄	#22548642	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	4	1	no	—
94	C₂	#30631421	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#30631421
commutator	C₂×C₈×C₂₅₁	C₂×C₄
Frattini	C₂×M₁₆	C₂×D₂₅₁
Fitting	#470×C₂₅₁	C₂
socle	C₂×C₂₅₁	#327
prime radical	C₂³×C₂₅₁	SD₁₆
cosocle	C₂×M₁₆×C₂₅₁	C₂²
hypercenter	#470	D₂₅₁
hypocenter	C₂₅₁	#1342
LCS	G ▷ C₂×C₈×C₂₅₁ ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #470

Complements

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

N	H ≅ G/N	type
47. C₂₅₁	#1342	semidirect

Maximal subgroups

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

subgroup	index
#30631469	2
C₂₅₁⋊#338	2
#470×C₂₅₁	2
#1342	251