Subgroups — #29596435

Order 15616.

278 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21447585	C₂	1	3	yes	maximal
2	#21447578	C₂	1	3	yes	maximal
3	C₆₁×#1747	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519717	C₂²	1	3	yes	—
5	C₂²×#11758185	C₂²	1	1	yes	characteristic
6	#16839583	—	2	6	no	—
7	C₄⋊#11758185	—	2	6	no	—
8	C₆₁×#405	C₂²	1	3	yes	—
9	C₆₁×#338	C₄	1	1	yes	characteristic
10	C₆₁×H(3,ℤ/4)	C₄	1	3	yes	—
11	#14519717	—	2	6	no	—
12	C₂×#11758185	—	2	3	no	—
13	C₂×#11758185	—	2	3	no	—
14	C₂×M₁₆×C₆₁	C₂×C₄	1	3	yes	—
15	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
16	#117×C₆₁	C₂×C₄	1	3	yes	—
17	#97×C₆₁	Q₈	1	6	yes	—
18	C₄×C₈×C₆₁	D₄	1	6	yes	—
19	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
20	M₁₆×C₆₁	—	2	6	no	—
21	C₂×C₈×C₆₁	C₄⋊C₄	1	6	yes	—
22	C₂×C₈×C₆₁	C₂×D₄	1	3	yes	—
23	C₂×C₈×C₆₁	C₂×Q₈	1	3	yes	—
24	C₄⋊C₄×C₆₁	—	2	3	no	—
25	C₂²⋊C₄×C₆₁	—	2	3	no	—
26	C₄²×C₆₁	C₄⋊C₄	1	6	yes	—
27	#11758185	—	4	1	no	—
28	#11758185	—	4	3	no	—
29	C₂³×C₆₁	#131	1	1	yes	characteristic
30	C₂×C₄×C₆₁	—	2	6	no	—
31	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
32	C₂×C₄×C₆₁	#131	1	3	yes	—
33	C₈×C₆₁	—	2	6	no	—
34	C₈×C₆₁	#127	1	1	yes	characteristic
35	C₈×C₆₁	C₄⋊Q₈	1	3	yes	—
36	#12902	—	61	1	no	maximal, Sylow
37	C₂²×C₆₁	—	4	1	no	—
38	C₂²×C₆₁	#424	1	3	yes	—
39	C₄×C₆₁	—	4	6	no	—
40	C₄×C₆₁	#384	1	1	yes	characteristic
41	C₄×C₆₁	#424	1	3	yes	—
42	#2153	—	61	3	no	—
43	#2151	—	61	3	no	—
44	#1747	D₆₁	1	1	yes	characteristic
45	C₂×C₆₁	—	2	3	no	—
46	C₂×C₆₁	#1843	1	1	yes	characteristic
47	C₂×M₃₂	—	61	3	no	—
48	C₂²×C₁₆	—	61	1	no	—
49	#405	C₂×D₆₁	1	3	yes	—
50	#364	—	122	6	no	—
51	C₄⋊C₁₆	—	122	6	no	—
52	#338	Dic₆₁	1	1	yes	characteristic
53	H(3,ℤ/4)	Dic₆₁	1	3	yes	—
54	C₆₁	#12902	1	1	yes	characteristic, complemented, Sylow
55	C₂×M₁₆	C₂×Dic₆₁	1	3	yes	—
56	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
57	#117	C₂×Dic₆₁	1	3	yes	—
58	M₃₂	—	122	6	no	—
59	C₂×C₁₆	—	122	3	no	—
60	C₂×C₁₆	—	122	3	no	—
61	#97	Dic₁₂₂	1	6	yes	—
62	C₄×C₈	D₂₄₄	1	6	yes	—
63	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
64	M₁₆	—	2	6	no	—
65	C₂×C₈	C₄⋊Dic₆₁	1	6	yes	—
66	C₂×C₈	C₂×Dic₁₂₂	1	3	yes	—
67	C₂×C₈	C₂×D₂₄₄	1	3	yes	—
68	C₄⋊C₄	—	2	3	no	—
69	C₂²⋊C₄	—	2	3	no	—
70	C₄²	C₄⋊Dic₆₁	1	6	yes	—
71	C₁₆	—	244	1	no	—
72	C₁₆	—	244	3	no	—
73	C₂³	#14519784	1	1	yes	characteristic
74	C₂×C₄	—	2	6	no	—
75	C₂×C₄	C₂×C₄⋊Dic₆₁	1	3	yes	—
76	C₂×C₄	#14519784	1	3	yes	—
77	C₈	—	2	6	no	—
78	C₈	#14519774	1	1	yes	characteristic
79	C₈	#14519777	1	3	yes	—
80	C₂²	—	4	1	no	—
81	C₂²	#16839789	1	3	yes	—
82	C₄	—	4	6	no	—
83	C₄	#16839632	1	1	yes	characteristic
84	C₄	#16839789	1	3	yes	—
85	C₂	—	2	3	no	—
86	C₂	#21445973	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14519774
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1747	C₂
socle	C₂×C₆₁	#1843
prime radical	C₂³×C₆₁	#131
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1747	D₆₁
hypocenter	C₆₁	#12902
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1747

Complements

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

N	H ≅ G/N	type
54. C₆₁	#12902	semidirect

Maximal subgroups

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

subgroup	index
#21447585	2
#21447578	2
C₆₁×#1747	2
#12902	61