Subgroups — #29566752

Order 15616.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21443889	C₂	1	1	yes	maximal, characteristic
2	C₈×#11758185	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1370	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11758185	—	2	1	no	—
5	C₄×#11758185	C₄	1	1	yes	characteristic
6	C₄×#11758185	C₄	1	1	yes	characteristic
7	C₆₁×C₄⋊C₁₆	—	2	1	no	—
8	C₈²×C₆₁	C₂²	1	1	yes	characteristic
9	C₂×#11758185	—	2	1	no	—
10	C₂×#11758185	—	4	1	no	—
11	C₂×#11758185	C₈	1	1	yes	characteristic
12	C₂×#11758185	C₈	1	1	yes	characteristic
13	C₂×C₁₆×C₆₁	—	4	1	no	—
14	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
15	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
16	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
17	C₂×C₈×C₆₁	—	2	1	no	—
18	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
19	C₂×C₈×C₆₁	M₁₆	1	1	yes	characteristic
20	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
21	C₂×C₈×C₆₁	Q₁₆	1	1	yes	characteristic
22	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
23	C₁₆×C₆₁	—	8	1	no	—
24	#11758185	—	2	2	no	—
25	#11758185	—	2	1	no	—
26	#11758185	—	8	1	no	—
27	#11758185	C₁₆	1	2	yes	complemented
28	C₂×C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
29	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
30	C₂×C₄×C₆₁	#108	1	1	yes	characteristic
31	C₈×C₆₁	—	2	2	no	—
32	C₈×C₆₁	—	2	1	no	—
33	C₈×C₆₁	—	2	1	no	—
34	C₈×C₆₁	C₂×C₁₆	1	1	yes	characteristic
35	C₈×C₆₁	M₃₂	1	1	yes	characteristic
36	C₈×C₆₁	D₁₆	1	1	yes	characteristic
37	C₈×C₆₁	Q₃₂	1	1	yes	characteristic
38	#7100	—	61	1	no	maximal, Sylow
39	C₂²×C₆₁	#335	1	1	yes	characteristic
40	C₄×C₆₁	—	2	1	no	—
41	C₄×C₆₁	C₄⋊C₁₆	1	1	yes	characteristic
42	C₄×C₆₁	#364	1	1	yes	characteristic
43	C₄×C₆₁	#366	1	1	yes	characteristic
44	C₄×C₆₁	#368	1	1	yes	characteristic
45	#1370	—	61	1	no	—
46	#1370	D₆₁	1	1	yes	characteristic
47	C₈×C₁₆	—	61	1	no	—
48	C₂×C₆₁	#1370	1	1	yes	characteristic
49	C₂×C₆₁	#1371	1	1	yes	characteristic
50	C₂×C₆₁	#1373	1	1	yes	characteristic
51	C₄⋊C₁₆	—	2	1	no	—
52	C₄⋊C₁₆	—	122	1	no	—
53	C₄×C₁₆	—	61	1	no	—
54	C₄×C₁₆	—	61	1	no	—
55	C₈²	C₂×D₆₁	1	1	yes	characteristic
56	C₆₁	#7100	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₁₆	—	4	1	no	—
58	C₂×C₁₆	—	61	1	no	—
59	C₂×C₁₆	—	61	1	no	—
60	C₂×C₁₆	—	122	1	no	—
61	C₂×C₁₆	—	244	1	no	—
62	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
63	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
64	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
65	C₂×C₈	—	2	1	no	—
66	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
67	C₂×C₈	#11758189	1	1	yes	characteristic
68	C₂×C₈	#11758199	1	1	yes	characteristic
69	C₂×C₈	#11758202	1	1	yes	characteristic
70	C₄²	#11758196	1	1	yes	characteristic
71	C₁₆	—	8	1	no	—
72	C₁₆	—	61	2	no	—
73	C₁₆	—	122	2	no	—
74	C₁₆	—	122	1	no	—
75	C₁₆	—	488	1	no	—
76	C₂×C₄	#14519706	1	1	yes	characteristic
77	C₂×C₄	#14519710	1	1	yes	characteristic
78	C₂×C₄	#14519712	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	—	2	1	no	—
81	C₈	—	2	1	no	—
82	C₈	C₁₆×D₆₁	1	1	yes	characteristic
83	C₈	#14519716	1	1	yes	characteristic
84	C₈	#14519718	1	1	yes	characteristic
85	C₈	#14519723	1	1	yes	characteristic
86	C₂²	#16839514	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	#16839580	1	1	yes	characteristic
89	C₄	#16839582	1	1	yes	characteristic
90	C₄	#16839587	1	1	yes	characteristic
91	C₄	#16839591	1	1	yes	characteristic
92	C₂	#21443888	1	1	yes	characteristic
93	C₂	#21443890	1	1	yes	characteristic
94	C₂	#21443894	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#11758199
commutator	C₈×C₆₁	C₂×C₁₆
Frattini	C₈²	C₂×D₆₁
Fitting	C₆₁×#1370	C₂
socle, prime radical	C₂²×C₆₁	#335
cosocle	C₈²×C₆₁	C₂²
hypercenter	#1370	D₆₁
hypocenter	C₆₁	#7100
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈² ◁ #1370

Complements

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

N	H ≅ G/N	type
27. #11758185	C₁₆	semidirect
56. C₆₁	#7100	semidirect

Maximal subgroups

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

subgroup	index
#21443889	2
C₈×#11758185	2
C₆₁×#1370	2
#7100	61