Subgroups — #24905508

Order 10624.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128078	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₈₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	#335×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₆₆₄	—	2	1	no	—
5	C₂×D₆₆₄	C₄	1	2	yes	—
6	#14581789	—	2	1	no	—
7	C₂×C₈₃⋊C₁₆	—	2	1	no	—
8	C₄⋊C₈×C₈₃	—	2	1	no	—
9	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
10	C₂×D₃₃₂	—	2	2	no	—
11	C₂×D₃₃₂	—	4	1	no	—
12	C₂×D₃₃₂	—	4	1	no	—
13	D₆₆₄	—	4	2	no	—
14	D₆₆₄	—	4	1	no	—
15	D₆₆₄	—	4	1	no	—
16	C₈₃⋊C₁₆	—	4	1	no	—
17	C₂×C₈×C₈₃	—	4	1	no	—
18	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
20	C₄²×C₈₃	D₄	1	1	yes	characteristic
21	C₂²×D₈₃	—	4	2	no	—
22	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₈₃	D₈	1	1	yes	characteristic
24	C₂×C₄×C₈₃	SD₁₆	1	1	yes	characteristic
25	D₃₃₂	—	4	2	no	—
26	D₃₃₂	—	4	2	no	—
27	D₃₃₂	—	8	2	no	—
28	D₃₃₂	—	8	1	no	—
29	D₃₃₂	—	8	1	no	—
30	C₈×C₈₃	—	2	1	no	—
31	C₈×C₈₃	—	8	1	no	—
32	C₈×C₈₃	D₈	1	1	yes	characteristic
33	C₈×C₈₃	SD₁₆	1	1	yes	characteristic
34	C₂²×C₈₃	#102	1	1	yes	characteristic
35	C₂×D₈₃	—	8	2	no	—
36	C₂×D₈₃	—	8	2	no	—
37	C₂×D₈₃	—	8	2	no	—
38	C₄×C₈₃	—	2	1	no	—
39	C₄×C₈₃	#100	1	1	yes	characteristic
40	C₄×C₈₃	#102	1	1	yes	characteristic
41	C₄×C₈₃	D₁₆	1	1	yes	characteristic
42	C₄×C₈₃	SD₃₂	1	1	yes	characteristic
43	C₂×C₈₃	#331	1	1	yes	characteristic
44	C₂×C₈₃	#357	1	1	yes	characteristic
45	C₂×C₈₃	#361	1	1	yes	characteristic
46	D₈₃	—	16	2	no	—
47	#1361	—	83	1	no	maximal, Sylow
48	C₈₃	#1361	1	1	yes	characteristic, complemented, Sylow
49	#493	—	83	1	no	—
50	C₄⋊C₁₆	—	83	1	no	—
51	#335	D₈₃	1	1	yes	characteristic, complemented
52	C₂×D₈	—	83	2	no	—
53	C₂×D₈	—	166	1	no	—
54	#127	—	166	1	no	—
55	C₂×C₁₆	—	166	1	no	—
56	C₄⋊C₈	—	2	1	no	—
57	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
58	C₂×D₄	—	332	1	no	—
59	C₂×D₄	—	332	1	no	—
60	C₂×D₄	—	166	2	no	—
61	D₈	—	332	2	no	—
62	D₈	—	332	1	no	—
63	D₈	—	332	1	no	—
64	C₂×C₈	—	4	1	no	—
65	C₂×C₈	C₄×D₈₃	1	1	yes	characteristic
66	C₂×C₈	D₃₃₂	1	1	yes	characteristic
67	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
68	C₁₆	—	332	1	no	—
69	C₂³	—	332	2	no	—
70	D₄	—	332	2	no	—
71	D₄	—	332	2	no	—
72	D₄	—	664	2	no	—
73	D₄	—	664	1	no	—
74	D₄	—	664	1	no	—
75	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
76	C₂×C₄	#13050575	1	1	yes	characteristic
77	C₂×C₄	#13050577	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	8	1	no	—
80	C₈	D₆₆₄	1	1	yes	characteristic
81	C₈	#13050579	1	1	yes	characteristic
82	C₂²	—	664	2	no	—
83	C₂²	—	664	2	no	—
84	C₂²	—	664	2	no	—
85	C₂²	#14581712	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	#14581708	1	1	yes	characteristic
88	C₄	#14581714	1	1	yes	characteristic
89	C₄	#14581733	1	1	yes	characteristic
90	C₄	#14581735	1	1	yes	characteristic
91	C₂	—	1328	2	no	—
92	C₂	#19127444	1	1	yes	characteristic
93	C₂	#19127501	1	1	yes	characteristic
94	C₂	#19127513	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14581712
commutator	C₂×C₈×C₈₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₈₃
Fitting	#335×C₈₃	C₂
socle	C₂²×C₈₃	#102
prime radical	#19128078	C₂
cosocle	C₄×C₈×C₈₃	C₂²
hypercenter	#335	D₈₃
hypocenter	C₈₃	#1361
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₈₃⋊C₁₆)	C₂	semidirect
3. #335×C₈₃	C₂	semidirect
48. C₈₃	#1361	semidirect
51. #335	D₈₃	semidirect