Subgroups — #29601033

Order 15616.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21448094	C₂	1	2	yes	maximal
2	#21444019	C₂	1	4	yes	maximal
3	C₄×C₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
4	#16840167	—	2	2	no	—
5	#16839588	—	2	2	no	—
6	#16839588	C₂²	1	4	yes	—
7	C₂×C₃₂×C₆₁	C₂²	1	2	yes	—
8	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
9	#14519777	—	4	2	no	—
10	#14519711	—	2	4	no	—
11	#14519711	—	4	2	no	—
12	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
13	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
14	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
15	C₃₂×C₆₁	Q₈	1	4	yes	complemented
16	C₂×Dic₁₂₂	—	8	2	no	—
17	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
18	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
19	C₄²×C₆₁	D₈	1	1	yes	characteristic
20	C₄⋊Dic₆₁	—	4	4	no	—
21	C₄⋊Dic₆₁	—	8	2	no	—
22	C₁₆×C₆₁	Q₁₆	1	2	yes	—
23	C₁₆×C₆₁	C₂×Q₈	1	2	yes	—
24	C₂×C₄×C₆₁	D₁₆	1	1	yes	characteristic
25	C₂×C₄×C₆₁	D₁₆	1	1	yes	characteristic
26	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
27	C₂×Dic₆₁	—	8	4	no	—
28	Dic₁₂₂	—	16	4	no	—
29	C₈×C₆₁	Q₃₂	1	2	yes	—
30	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
31	C₈×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
32	#13731	—	61	1	no	maximal, Sylow
33	C₂²×C₆₁	C₂×D₁₆	1	1	yes	characteristic
34	C₄×C₆₁	D₃₂	1	2	yes	—
35	C₄×C₆₁	Q₆₄	1	2	yes	—
36	C₄×C₆₁	#500	1	1	yes	characteristic
37	C₄×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
38	Dic₆₁	—	16	4	no	—
39	#2252	—	61	2	no	—
40	#1423	—	61	4	no	—
41	C₄×C₃₂	D₆₁	1	1	yes	characteristic
42	C₂×C₆₁	#2252	1	1	yes	characteristic
43	C₂×C₆₁	C₂×D₃₂	1	1	yes	characteristic
44	C₂×C₆₁	C₂×Q₆₄	1	1	yes	characteristic
45	#500	—	122	2	no	—
46	C₂×C₃₂	C₂×D₆₁	1	2	yes	—
47	#366	—	61	4	no	—
48	#366	—	122	2	no	—
49	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
50	C₆₁	#13731	1	1	yes	characteristic, complemented, Sylow
51	C₄⋊Q₈	—	244	2	no	—
52	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
53	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
54	#107	—	122	4	no	—
55	#107	—	244	2	no	—
56	C₄×C₈	D₂₄₄	1	1	yes	characteristic
57	C₃₂	Dic₁₂₂	1	4	yes	complemented
58	C₂×Q₈	—	488	2	no	—
59	C₂×C₈	D₄₈₈	1	1	yes	characteristic
60	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
61	C₄⋊C₄	—	488	2	no	—
62	C₄⋊C₄	—	244	4	no	—
63	C₄²	D₄₈₈	1	1	yes	characteristic
64	C₁₆	Dic₂₄₄	1	2	yes	—
65	C₁₆	C₂×Dic₁₂₂	1	2	yes	—
66	Q₈	—	976	4	no	—
67	C₂×C₄	—	488	4	no	—
68	C₂×C₄	D₉₇₆	1	1	yes	characteristic
69	C₂×C₄	D₉₇₆	1	1	yes	characteristic
70	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
71	C₈	Dic₄₈₈	1	2	yes	—
72	C₈	#14519777	1	1	yes	characteristic
73	C₈	C₂×Dic₂₄₄	1	1	yes	characteristic
74	C₂²	C₂×D₉₇₆	1	1	yes	characteristic
75	C₄	—	976	4	no	—
76	C₄	D₁₉₅₂	1	2	yes	—
77	C₄	Dic₉₇₆	1	2	yes	—
78	C₄	#16840167	1	1	yes	characteristic
79	C₄	C₂×Dic₄₈₈	1	1	yes	characteristic
80	C₂	#21448094	1	1	yes	characteristic
81	C₂	C₂×D₁₉₅₂	1	1	yes	characteristic
82	C₂	C₂×Dic₉₇₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₉₇₆
commutator, cosocle	C₂×C₁₆×C₆₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₆₁
Fitting	C₄×C₃₂×C₆₁	C₂
socle, prime radical	C₂²×C₆₁	C₂×D₁₆
hypercenter	C₄×C₃₂	D₆₁
hypocenter	C₆₁	#13731
LCS	G ▷ C₂×C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
15. C₃₂×C₆₁	Q₈	semidirect
50. C₆₁	#13731	semidirect
57. C₃₂	Dic₁₂₂	semidirect

Maximal subgroups

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

subgroup	index
#21448094	2
#21444019	2
C₄×C₃₂×C₆₁	2
#13731	61