Subgroups — #24909348

Order 10624.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128133	C₂	1	1	yes	maximal, characteristic
2	#19127817	C₂	1	2	yes	maximal, complemented
3	#19127507	C₂	1	2	yes	maximal, complemented
4	#19127476	C₂	1	1	yes	maximal, characteristic, complemented
5	#368×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#14581814	C₂²	1	2	yes	—
7	#14581800	—	2	2	no	—
8	Dic₆₆₄	C₄	1	1	yes	characteristic, complemented
9	#14581737	C₄	1	2	yes	complemented
10	D₁₃₂₈	C₄	1	1	yes	characteristic, complemented
11	C₂×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
12	#14581718	—	2	2	no	—
13	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
15	#108×C₈₃	C₂²	1	2	yes	—
16	#13050590	—	2	2	no	—
17	Dic₃₃₂	C₂×C₄	1	2	yes	—
18	#13050579	—	2	2	no	—
19	D₆₆₄	C₂×C₄	1	2	yes	—
20	#13050573	—	4	2	no	—
21	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
22	C₈×D₈₃	—	4	2	no	—
23	C₄×Dic₈₃	D₄	1	1	yes	characteristic
24	C₈₃⋊C₁₆	—	2	1	no	—
25	M₁₆×C₈₃	—	2	2	no	—
26	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
27	C₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
30	C₂²⋊D₈₃	—	4	2	no	—
31	C₂×Dic₈₃	—	2	1	no	—
32	D₃₃₂	—	2	2	no	—
33	C₄×D₈₃	—	4	2	no	—
34	Dic₁₆₆	—	2	2	no	—
35	C₈×C₈₃	—	4	2	no	—
36	C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
37	C₈×C₈₃	#41	1	1	yes	characteristic
38	C₈₃⋊C₈	D₈	1	2	yes	—
39	C₂²×C₈₃	#135	1	1	yes	characteristic
40	C₂×D₈₃	—	4	2	no	—
41	C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
42	C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
43	Dic₈₃	—	4	1	no	—
44	Dic₈₃	—	4	2	no	—
45	C₂×C₈₃	—	2	1	no	—
46	C₂×C₈₃	C₄×D₈	1	1	yes	characteristic
47	D₈₃	—	8	2	no	—
48	#2179	—	83	1	no	maximal, Sylow
49	C₈₃	#2179	1	1	yes	characteristic, complemented, Sylow
50	#508	—	83	1	no	—
51	#443	—	83	2	no	—
52	#368	D₈₃	1	1	yes	characteristic, complemented
53	#359	—	83	2	no	—
54	#346	—	83	1	no	—
55	#135	—	83	2	no	—
56	#131	—	166	2	no	—
57	Q₃₂	—	83	1	no	—
58	SD₃₂	—	83	2	no	—
59	D₁₆	—	83	1	no	—
60	C₂×C₁₆	—	83	1	no	—
61	C₂×C₁₆	C₂×D₈₃	1	1	yes	characteristic
62	#108	C₂×D₈₃	1	2	yes	—
63	#104	—	166	2	no	—
64	C₄×C₈	—	83	1	no	—
65	#41	—	166	2	no	—
66	Q₁₆	—	83	2	no	—
67	SD₁₆	—	166	2	no	—
68	D₈	—	83	2	no	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	332	2	no	—
71	C₂×C₈	—	332	2	no	—
72	C₂×C₈	—	83	1	no	—
73	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
74	C₄²	—	83	1	no	—
75	C₁₆	—	166	1	no	—
76	C₁₆	C₄×D₈₃	1	1	yes	characteristic
77	C₁₆	C₄×D₈₃	1	1	yes	characteristic
78	Q₈	—	166	2	no	—
79	D₄	—	332	2	no	—
80	D₄	—	166	2	no	—
81	C₂×C₄	—	332	2	no	—
82	C₂×C₄	—	166	1	no	—
83	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
84	C₈	—	4	2	no	—
85	C₈	—	83	2	no	—
86	C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
87	C₈	#13050591	1	1	yes	characteristic
88	C₂²	—	332	2	no	—
89	C₂²	#14581815	1	1	yes	characteristic
90	C₄	—	332	1	no	—
91	C₄	—	332	2	no	—
92	C₄	#14581754	1	1	yes	characteristic
93	C₄	D₈×D₈₃	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	664	2	no	—
96	C₂	#19127783	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×D₈₃
commutator	C₈×C₈₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₈₃
Fitting	#368×C₈₃	C₂
socle	C₂×C₈₃	C₄×D₈
prime radical	#19128133	C₂
cosocle	C₂×C₈×C₈₃	C₂³
hypercenter	#368	D₈₃
hypocenter	C₈₃	#2179
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

Complements

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

N	H ≅ G/N	type
2. #19127817	C₂	semidirect
3. #19127507	C₂	semidirect
4. #19127476	C₂	semidirect
5. #368×C₈₃	C₂	semidirect
8. Dic₆₆₄	C₄	semidirect
9. #14581737	C₄	semidirect
10. D₁₃₂₈	C₄	semidirect
49. C₈₃	#2179	semidirect
52. #368	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128133	2
#19127817	2
#19127507	2
#19127476	2
#368×C₈₃	2
#2179	83