Subgroups — #29596415

Order 15616.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21443897	C₂	1	6	yes	maximal
2	C₆₁×#1726	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11758185	C₂²	1	4	yes	—
4	C₂×#97×C₆₁	C₄	1	3	yes	—
5	C₂×C₄×C₈×C₆₁	C₂²	1	3	yes	—
6	C₆₁×#376	C₄	1	1	yes	characteristic
7	C₂×#11758185	—	2	12	no	—
8	C₂²×C₈×C₆₁	C₂×C₄	1	3	yes	—
9	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
10	C₂×C₄²×C₆₁	C₂×C₄	1	3	yes	—
11	#97×C₆₁	—	2	6	no	—
12	C₄×C₈×C₆₁	—	2	6	no	—
13	#95×C₆₁	C₈	1	4	yes	—
14	C₂²×C₄×C₆₁	C₂×C₈	1	6	yes	—
15	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
16	C₂×C₈×C₆₁	—	2	3	no	—
17	C₂×C₈×C₆₁	—	2	3	no	—
18	C₂×C₈×C₆₁	M₁₆	1	6	yes	—
19	C₂×C₈×C₆₁	#41	1	6	yes	—
20	C₄²×C₆₁	—	2	3	no	—
21	C₄²×C₆₁	—	2	3	no	—
22	#11758185	—	4	4	no	—
23	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
24	C₂×C₄×C₆₁	—	2	6	no	—
25	C₂×C₄×C₆₁	—	2	12	no	—
26	C₂×C₄×C₆₁	#117	1	3	yes	—
27	C₂×C₄×C₆₁	C₂×M₁₆	1	3	yes	—
28	C₈×C₆₁	—	2	6	no	—
29	C₈×C₆₁	#126	1	4	yes	—
30	#12895	—	61	1	no	maximal, Sylow
31	C₂²×C₆₁	#431	1	3	yes	—
32	C₂²×C₆₁	#504	1	4	yes	—
33	C₄×C₆₁	—	4	6	no	—
34	C₄×C₆₁	#383	1	1	yes	characteristic
35	C₄×C₆₁	#432	1	3	yes	—
36	#1726	D₆₁	1	1	yes	characteristic
37	#1374	—	61	6	no	—
38	C₂×C₆₁	#1839	1	1	yes	characteristic
39	C₂×C₆₁	#2164	1	6	yes	—
40	C₂²×C₁₆	—	61	4	no	—
41	C₂×#97	Dic₆₁	1	3	yes	—
42	C₂×C₄×C₈	C₂×D₆₁	1	3	yes	—
43	#376	Dic₆₁	1	1	yes	characteristic
44	C₆₁	#12895	1	1	yes	characteristic, complemented, Sylow
45	C₂²×C₈	C₂×Dic₆₁	1	3	yes	—
46	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
47	C₂×C₄²	C₂×Dic₆₁	1	3	yes	—
48	C₂×C₁₆	—	122	12	no	—
49	#97	—	2	6	no	—
50	C₄×C₈	—	2	6	no	—
51	#95	#92418	1	4	yes	—
52	C₂²×C₄	C₂×#92418	1	6	yes	—
53	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
54	C₂×C₈	—	2	3	no	—
55	C₂×C₈	—	2	3	no	—
56	C₂×C₈	#11758194	1	6	yes	—
57	C₂×C₈	#11758222	1	6	yes	—
58	C₄²	—	2	3	no	—
59	C₄²	—	2	3	no	—
60	C₁₆	—	244	4	no	—
61	C₂³	C₂²×#92418	1	1	yes	characteristic
62	C₂×C₄	—	2	6	no	—
63	C₂×C₄	—	2	12	no	—
64	C₂×C₄	#14519736	1	3	yes	—
65	C₂×C₄	C₂×#11758194	1	3	yes	—
66	C₈	—	2	6	no	—
67	C₈	#14519771	1	4	yes	—
68	C₂²	#16839815	1	3	yes	—
69	C₂²	#16840179	1	4	yes	—
70	C₄	—	4	6	no	—
71	C₄	#16839630	1	1	yes	characteristic
72	C₄	#16839819	1	3	yes	—
73	C₂	#21445962	1	1	yes	characteristic
74	C₂	#21447626	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₁
commutator, socle, prime radical	C₂³×C₆₁	C₂²×C₈
Fitting	C₆₁×#1726	C₂
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1726	D₆₁
hypocenter	C₆₁	#12895
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
44. C₆₁	#12895	semidirect

Maximal subgroups

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

subgroup	index
#21443897	2
C₆₁×#1726	2
#12895	61