Subgroups — #30607354

Order 15616.

278 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462160	C₂	1	3	yes	maximal
2	#21462156	C₂	1	3	yes	maximal
3	#21445575	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519851	C₂²	1	3	yes	—
5	C₂²×#11758187	C₂²	1	1	yes	characteristic
6	#16840684	—	2	6	no	—
7	#16840683	—	2	6	no	—
8	#16839720	C₂²	1	3	yes	—
9	#16839519	C₄	1	3	yes	—
10	C₆₁×#338	C₄	1	1	yes	characteristic
11	#14519851	—	2	6	no	—
12	C₂×#11758187	—	2	3	no	—
13	C₂×#11758187	—	2	3	no	—
14	C₂²×#92418	C₂³	1	1	yes	characteristic
15	#14519736	C₂×C₄	1	3	yes	—
16	#14519687	Q₈	1	6	yes	—
17	C₈×Dic₆₁	D₄	1	6	yes	—
18	C₂×M₁₆×C₆₁	C₂×C₄	1	3	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	C₂²⋊Dic₆₁	—	2	3	no	—
23	C₄⋊Dic₆₁	—	2	3	no	—
24	C₄×Dic₆₁	C₄⋊C₄	1	6	yes	—
25	C₂×#92418	C₂×D₄	1	3	yes	—
26	C₂×#92418	C₂×Q₈	1	3	yes	—
27	#11758187	—	4	1	no	—
28	#11758187	—	4	3	no	—
29	C₂³×C₆₁	#131	1	1	yes	characteristic
30	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
31	C₂×C₄×C₆₁	#131	1	3	yes	—
32	C₂×Dic₆₁	—	2	6	no	—
33	C₈×C₆₁	—	2	6	no	—
34	#92418	#127	1	1	yes	characteristic
35	#92418	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	#384	1	1	yes	characteristic
40	C₄×C₆₁	#424	1	3	yes	—
41	Dic₆₁	—	4	6	no	—
42	#2153	—	61	3	no	—
43	#2151	—	61	3	no	—
44	#1747	—	61	1	no	—
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	—	61	3	no	—
50	#364	—	122	6	no	—
51	C₄⋊C₁₆	—	122	6	no	—
52	#338	#6923	1	1	yes	characteristic
53	H(3,ℤ/4)	—	61	3	no	—
54	C₆₁	#12902	1	1	yes	characteristic, complemented, Sylow
55	C₂×M₁₆	C₂×#6923	1	3	yes	—
56	C₂²×C₈	—	61	1	no	—
57	#117	—	61	3	no	—
58	M₃₂	—	122	6	no	—
59	C₂×C₁₆	—	122	3	no	—
60	C₂×C₁₆	—	122	3	no	—
61	#97	—	61	6	no	—
62	C₄×C₈	—	61	6	no	—
63	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
64	M₁₆	—	2	6	no	—
65	C₂×C₈	—	61	3	no	—
66	C₂×C₈	—	61	3	no	—
67	C₂×C₈	C₄⋊#6923	1	6	yes	—
68	C₄⋊C₄	—	122	3	no	—
69	C₂²⋊C₄	—	122	3	no	—
70	C₄²	—	61	6	no	—
71	C₁₆	—	244	1	no	—
72	C₁₆	—	244	3	no	—
73	C₂³	#14519865	1	1	yes	characteristic
74	C₂×C₄	—	122	6	no	—
75	C₂×C₄	C₂×C₄⋊#6923	1	3	yes	—
76	C₂×C₄	#14519864	1	3	yes	—
77	C₈	—	2	6	no	—
78	C₈	—	61	1	no	—
79	C₈	—	61	3	no	—
80	C₂²	—	4	1	no	—
81	C₂²	#16840769	1	3	yes	—
82	C₄	—	244	6	no	—
83	C₄	#16840708	1	1	yes	characteristic
84	C₄	#16840768	1	3	yes	—
85	C₂	—	2	3	no	—
86	C₂	#21461753	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16840708
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₆₁×#338	C₄
socle	C₂×C₆₁	#1843
prime radical	C₂³×C₆₁	#131
cosocle	C₂²×#92418	C₂³
hypercenter	#338	#6923
hypocenter	C₆₁	#12902
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

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
#21462160	2
#21462156	2
#21445575	2
#12902	61