Subgroups — #29594727

Order 15616.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21448071	C₂	1	4	yes	maximal
2	#21445437	C₂	1	1	yes	maximal, characteristic
3	C₈×#11758185	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1713	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519723	—	2	8	no	—
6	#16840167	—	2	2	no	—
7	#16840142	C₂²	1	2	yes	—
8	C₄×#11758185	C₂²	1	2	yes	—
9	C₆₁×#500	—	2	2	no	—
10	C₆₁×#494	C₂²	1	2	yes	—
11	C₈²×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₄₄	—	2	4	no	—
13	#14519777	—	2	2	no	—
14	#14519723	—	4	16	no	—
15	C₂×#11758185	D₄	1	4	yes	—
16	#14519711	—	4	4	no	—
17	C₂×Q₁₆×C₆₁	—	2	4	no	—
18	C₄⋊Q₈×C₆₁	—	2	2	no	—
19	#107×C₆₁	—	4	4	no	—
20	C₄×C₈×C₆₁	D₄	1	2	yes	—
21	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
22	C₂×Q₈×C₆₁	—	4	2	no	—
23	C₂×Dic₁₂₂	—	4	2	no	—
24	Q₁₆×C₆₁	—	4	8	no	—
25	C₂×C₈×C₆₁	D₈	1	4	yes	—
26	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
27	C₄⋊C₄×C₆₁	—	4	4	no	—
28	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
29	C₄⋊Dic₆₁	—	4	4	no	—
30	Dic₂₄₄	—	4	8	no	—
31	#11758185	D₈	1	8	yes	—
32	Q₈×C₆₁	—	8	4	no	—
33	C₂×C₄×C₆₁	—	8	2	no	—
34	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
35	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
36	C₂×Dic₆₁	—	8	2	no	—
37	Dic₁₂₂	—	8	4	no	—
38	C₈×C₆₁	Q₃₂	1	8	yes	complemented
39	C₈×C₆₁	C₂×D₈	1	4	yes	—
40	#12516	—	61	1	no	maximal, Sylow
41	C₂²×C₆₁	#493	1	1	yes	characteristic
42	C₄×C₆₁	—	16	2	no	—
43	C₄×C₆₁	#493	1	2	yes	—
44	C₄×C₆₁	C₂×Q₃₂	1	4	yes	—
45	Dic₆₁	—	16	2	no	—
46	#2247	—	61	4	no	—
47	#1713	—	61	1	no	—
48	#1713	D₆₁	1	1	yes	characteristic
49	C₈×C₁₆	—	61	1	no	—
50	C₂×C₆₁	#1712	1	1	yes	characteristic
51	C₂×C₆₁	#2247	1	2	yes	—
52	C₂×Q₃₂	—	122	8	no	—
53	#500	—	2	2	no	—
54	#500	—	122	2	no	—
55	#494	—	61	2	no	—
56	#494	C₂×D₆₁	1	2	yes	—
57	C₄×C₁₆	—	61	2	no	—
58	C₈²	C₂×D₆₁	1	1	yes	characteristic
59	C₆₁	#12516	1	1	yes	characteristic, complemented, Sylow
60	C₂×Q₁₆	—	2	4	no	—
61	C₂×Q₁₆	—	122	4	no	—
62	C₄⋊Q₈	—	2	2	no	—
63	C₄⋊Q₈	—	122	2	no	—
64	Q₃₂	—	244	16	no	—
65	C₂×C₁₆	—	61	4	no	—
66	#107	—	4	4	no	—
67	#107	—	244	4	no	—
68	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
69	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
70	C₂×Q₈	—	4	2	no	—
71	C₂×Q₈	—	244	2	no	—
72	Q₁₆	—	4	8	no	—
73	Q₁₆	—	244	8	no	—
74	C₂×C₈	#11758199	1	4	yes	—
75	C₂×C₈	D₄×D₆₁	1	2	yes	—
76	C₄⋊C₄	—	4	4	no	—
77	C₄⋊C₄	—	244	4	no	—
78	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
79	C₁₆	—	61	8	no	—
80	Q₈	—	8	4	no	—
81	Q₈	—	488	4	no	—
82	C₂×C₄	—	8	2	no	—
83	C₂×C₄	—	488	2	no	—
84	C₂×C₄	#14519773	1	1	yes	characteristic
85	C₂×C₄	C₂×#11758199	1	2	yes	—
86	C₈	#14519723	1	8	yes	complemented
87	C₈	D₈×D₆₁	1	4	yes	—
88	C₂²	#16840138	1	1	yes	characteristic
89	C₄	—	16	2	no	—
90	C₄	—	976	2	no	—
91	C₄	#16840137	1	2	yes	—
92	C₄	C₂×#14519723	1	4	yes	—
93	C₂	#21445433	1	1	yes	characteristic
94	C₂	#21448071	1	2	yes	—

	subgroup	G/H
center	C₂²	#16840138
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1713	C₂
socle, prime radical	C₂²×C₆₁	#493
hypercenter	#1713	D₆₁
hypocenter	C₆₁	#12516
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

Complements

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

N	H ≅ G/N	type
38. C₈×C₆₁	Q₃₂	semidirect
59. C₆₁	#12516	semidirect
86. C₈	#14519723	semidirect

Maximal subgroups

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

subgroup	index
#21448071	2
#21445437	2
C₈×#11758185	2
C₆₁×#1713	2
#12516	61