Subgroups — #29596413

Order 15616.

342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21443897	C₂	1	6	yes	maximal
2	C₄²×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11758185	C₂²	1	4	yes	—
4	C₂×C₄×C₈×C₆₁	C₄	1	3	yes	—
5	C₂×C₄×C₈×C₆₁	C₂²	1	3	yes	—
6	C₄³×C₆₁	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	C₈	1	3	yes	—
11	C₂×C₄²×C₆₁	C₈	1	1	yes	characteristic
12	C₂×C₄²×C₆₁	C₂×C₄	1	3	yes	—
13	C₄×C₈×C₆₁	—	2	6	no	—
14	C₄×C₈×C₆₁	—	2	6	no	—
15	C₂²×C₄×C₆₁	C₂×C₈	1	3	yes	—
16	C₂²×C₄×C₆₁	C₂×C₈	1	3	yes	—
17	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₈×C₆₁	—	2	3	no	—
19	C₂×C₈×C₆₁	—	2	3	no	—
20	C₂×C₈×C₆₁	M₁₆	1	6	yes	—
21	C₂×C₈×C₆₁	#41	1	6	yes	—
22	C₄²×C₆₁	—	2	6	no	—
23	C₄²×C₆₁	—	2	3	no	—
24	C₄²×C₆₁	—	2	3	no	—
25	C₄²×C₆₁	C₁₆	1	4	yes	complemented
26	#11758185	—	4	4	no	—
27	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
28	C₂×C₄×C₆₁	—	2	3	no	—
29	C₂×C₄×C₆₁	—	2	3	no	—
30	C₂×C₄×C₆₁	—	2	3	no	—
31	C₂×C₄×C₆₁	—	2	3	no	—
32	C₂×C₄×C₆₁	C₂×C₁₆	1	6	yes	—
33	C₂×C₄×C₆₁	M₃₂	1	6	yes	—
34	C₂×C₄×C₆₁	#117	1	3	yes	—
35	C₂×C₄×C₆₁	C₂×M₁₆	1	3	yes	—
36	C₈×C₆₁	—	2	6	no	—
37	C₈×C₆₁	#126	1	4	yes	—
38	#12894	—	61	1	no	maximal, Sylow
39	C₂²×C₆₁	#431	1	3	yes	—
40	C₂²×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
41	C₂²×C₆₁	C₂×M₃₂	1	3	yes	—
42	C₄×C₆₁	—	2	6	no	—
43	C₄×C₆₁	—	2	6	no	—
44	C₄×C₆₁	#383	1	1	yes	characteristic
45	C₄×C₆₁	#432	1	3	yes	—
46	C₄²×C₈	D₆₁	1	1	yes	characteristic
47	#1374	—	61	6	no	—
48	C₂×C₆₁	#1839	1	1	yes	characteristic
49	C₂×C₆₁	#2162	1	3	yes	—
50	C₂×C₆₁	#2163	1	3	yes	—
51	C₂²×C₁₆	—	61	4	no	—
52	C₂×C₄×C₈	Dic₆₁	1	3	yes	—
53	C₂×C₄×C₈	C₂×D₆₁	1	3	yes	—
54	C₄³	Dic₆₁	1	1	yes	characteristic
55	C₆₁	#12894	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₈	C₂×Dic₆₁	1	3	yes	—
57	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
58	C₂×C₄²	#92418	1	3	yes	—
59	C₂×C₄²	#92418	1	1	yes	characteristic
60	C₂×C₄²	C₂×Dic₆₁	1	3	yes	—
61	C₂×C₁₆	—	122	12	no	—
62	C₄×C₈	—	2	6	no	—
63	C₄×C₈	—	2	6	no	—
64	C₂²×C₄	C₂×#92418	1	3	yes	—
65	C₂²×C₄	C₂×#92418	1	3	yes	—
66	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
67	C₂×C₈	—	2	3	no	—
68	C₂×C₈	—	2	3	no	—
69	C₂×C₈	#11758194	1	6	yes	—
70	C₂×C₈	#11758222	1	6	yes	—
71	C₄²	—	2	6	no	—
72	C₄²	—	2	3	no	—
73	C₄²	—	2	3	no	—
74	C₄²	#11758185	1	4	yes	complemented
75	C₁₆	—	244	4	no	—
76	C₂³	C₂²×#92418	1	1	yes	characteristic
77	C₂×C₄	—	2	3	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	C₂×#11758185	1	6	yes	—
82	C₂×C₄	#14519717	1	6	yes	—
83	C₂×C₄	#14519736	1	3	yes	—
84	C₂×C₄	C₂×#11758194	1	3	yes	—
85	C₈	—	2	6	no	—
86	C₈	#14519771	1	4	yes	—
87	C₂²	#16839815	1	3	yes	—
88	C₂²	C₂²×#11758185	1	1	yes	characteristic
89	C₂²	C₂×#14519717	1	3	yes	—
90	C₄	—	2	6	no	—
91	C₄	—	2	6	no	—
92	C₄	#16839630	1	1	yes	characteristic
93	C₄	#16839819	1	3	yes	—
94	C₂	#21445962	1	1	yes	characteristic
95	C₂	#21447618	1	3	yes	—
96	C₂	#21447622	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₁
commutator	C₂²×C₆₁	C₂²×C₁₆
Fitting	C₄²×C₈×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₂²×C₈
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₄²×C₈	D₆₁
hypocenter	C₆₁	#12894
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
25. C₄²×C₆₁	C₁₆	semidirect
55. C₆₁	#12894	semidirect
74. C₄²	#11758185	semidirect

Maximal subgroups

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

subgroup	index
#21443897	2
C₄²×C₈×C₆₁	2
#12894	61