Subgroups — #29567692

Order 15616.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21443891	C₂	1	2	yes	maximal
2	C₄×C₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
3	#16839515	—	2	2	no	—
4	C₂×C₃₂×C₆₁	C₄	1	1	yes	characteristic
5	C₂×C₃₂×C₆₁	C₄	1	1	yes	characteristic
6	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
7	C₄⋊#92418	—	4	2	no	—
8	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
9	C₂×C₁₆×C₆₁	Q₈	1	1	yes	characteristic
10	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
11	C₃₂×C₆₁	—	2	1	no	—
12	C₃₂×C₆₁	C₈	1	2	yes	complemented
13	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
14	C₂×C₈×C₆₁	Q₁₆	1	1	yes	characteristic
15	C₄²×C₆₁	D₈	1	1	yes	characteristic
16	C₂×#92418	—	8	2	no	—
17	C₁₆×C₆₁	—	2	1	no	—
18	C₁₆×C₆₁	C₂×C₈	1	1	yes	characteristic
19	C₁₆×C₆₁	M₁₆	1	1	yes	characteristic
20	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
21	C₂×C₄×C₆₁	D₁₆	1	1	yes	characteristic
22	C₂×C₄×C₆₁	Q₃₂	1	1	yes	characteristic
23	C₈×C₆₁	—	2	1	no	—
24	C₈×C₆₁	C₄⋊C₈	1	1	yes	characteristic
25	C₈×C₆₁	#108	1	1	yes	characteristic
26	#92418	—	16	2	no	—
27	#7461	—	61	1	no	maximal, Sylow
28	C₂²×C₆₁	#366	1	1	yes	characteristic
29	C₄×C₆₁	—	2	1	no	—
30	C₄×C₆₁	#335	1	1	yes	characteristic
31	C₄×C₆₁	#368	1	1	yes	characteristic
32	C₄×C₆₁	SD₆₄	1	2	yes	—
33	C₄×C₃₂	D₆₁	1	1	yes	characteristic
34	#1371	—	61	2	no	—
35	C₂×C₆₁	#1371	1	1	yes	characteristic
36	C₂×C₆₁	#1424	1	1	yes	characteristic
37	C₂×C₆₁	#1425	1	1	yes	characteristic
38	C₂×C₃₂	Dic₆₁	1	1	yes	characteristic
39	C₂×C₃₂	Dic₆₁	1	1	yes	characteristic
40	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
41	#335	—	122	2	no	—
42	C₆₁	#7461	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₁₆	Dic₁₂₂	1	1	yes	characteristic
44	C₂×C₁₆	C₂×Dic₆₁	1	1	yes	characteristic
45	C₄⋊C₈	—	244	2	no	—
46	C₄×C₈	D₂₄₄	1	1	yes	characteristic
47	C₃₂	—	2	1	no	—
48	C₃₂	#92418	1	2	yes	complemented
49	C₂×C₈	—	488	2	no	—
50	C₂×C₈	Dic₂₄₄	1	1	yes	characteristic
51	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
52	C₄²	D₄₈₈	1	1	yes	characteristic
53	C₁₆	—	2	1	no	—
54	C₁₆	C₂×#92418	1	1	yes	characteristic
55	C₁₆	#11758194	1	1	yes	characteristic
56	C₂×C₄	#14519711	1	1	yes	characteristic
57	C₂×C₄	D₉₇₆	1	1	yes	characteristic
58	C₂×C₄	Dic₄₈₈	1	1	yes	characteristic
59	C₈	—	2	1	no	—
60	C₈	—	976	2	no	—
61	C₈	C₄⋊#92418	1	1	yes	characteristic
62	C₈	#14519713	1	1	yes	characteristic
63	C₂²	#16839588	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	#16839515	1	1	yes	characteristic
66	C₄	#16839592	1	1	yes	characteristic
67	C₄	#16839601	1	2	yes	—
68	C₂	#21443891	1	1	yes	characteristic
69	C₂	#21444021	1	1	yes	characteristic
70	C₂	#21444023	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₉₇₆
commutator	C₁₆×C₆₁	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₆₁
Fitting	C₄×C₃₂×C₆₁	C₂
socle, prime radical	C₂²×C₆₁	#366
cosocle	C₄×C₁₆×C₆₁	C₂²
hypercenter	C₄×C₃₂	D₆₁
hypocenter	C₆₁	#7461
LCS	G ▷ C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
12. C₃₂×C₆₁	C₈	semidirect
42. C₆₁	#7461	semidirect
48. C₃₂	#92418	semidirect

Maximal subgroups

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

subgroup	index
#21443891	2
C₄×C₃₂×C₆₁	2
#7461	61