Subgroups — #24909781

Order 10624.

104 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	#19127541	C₂	1	2	yes	maximal, complemented
3	#19127538	C₂	1	2	yes	maximal, complemented
4	#19127535	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#14581814	—	2	1	no	—
7	Dic₆₆₄	C₂²	1	1	yes	characteristic
8	#14581737	—	2	1	no	—
9	D₁₃₂₈	C₂²	1	1	yes	characteristic
10	C₈₃⋊C₃₂	C₂²	1	2	yes	—
11	C₂×Q₁₆×C₈₃	—	2	1	no	—
12	Q₃₂×C₈₃	—	2	1	no	—
13	Q₃₂×C₈₃	C₂²	1	2	yes	—
14	C₂×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
15	#13050590	—	4	1	no	—
16	Dic₃₃₂	—	2	1	no	—
17	#13050579	—	4	1	no	—
18	D₆₆₄	—	2	1	no	—
19	C₂×Q₈×C₈₃	—	4	1	no	—
20	Q₁₆×C₈₃	—	2	2	no	—
21	Q₁₆×C₈₃	—	4	1	no	—
22	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
23	C₁₆×C₈₃	D₄	1	1	yes	characteristic
24	C₁₆×C₈₃	C₂³	1	1	yes	characteristic
25	Q₈×C₈₃	—	4	2	no	—
26	Q₈×C₈₃	—	8	1	no	—
27	C₂×C₄×C₈₃	—	8	1	no	—
28	C₂×C₄×C₈₃	D₈	1	1	yes	characteristic
29	C₂²⋊D₈₃	—	8	1	no	—
30	D₃₃₂	—	4	1	no	—
31	C₄×D₈₃	—	8	1	no	—
32	Dic₁₆₆	—	4	1	no	—
33	C₈×C₈₃	D₈	1	1	yes	characteristic
34	C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₈₃	D₁₆	1	1	yes	characteristic
36	C₂×D₈₃	—	8	1	no	—
37	C₄×C₈₃	—	8	2	no	—
38	C₄×C₈₃	D₁₆	1	1	yes	characteristic
39	C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
40	Dic₈₃	—	8	1	no	—
41	C₂×C₈₃	—	2	1	no	—
42	C₂×C₈₃	C₂×D₁₆	1	1	yes	characteristic
43	D₈₃	—	16	1	no	—
44	#2264	—	83	1	no	maximal, Sylow
45	C₈₃	#2264	1	1	yes	characteristic, complemented, Sylow
46	#508	—	83	1	no	—
47	C₂×Q₃₂	D₈₃	1	1	yes	characteristic, complemented
48	Q₆₄	—	83	2	no	—
49	SD₆₄	—	83	2	no	—
50	M₆₄	—	83	1	no	—
51	#135	—	166	1	no	—
52	C₂×Q₁₆	—	2	1	no	—
53	Q₃₂	—	2	1	no	—
54	Q₃₂	—	83	1	no	—
55	Q₃₂	C₂×D₈₃	1	2	yes	—
56	SD₃₂	—	166	1	no	—
57	D₁₆	—	83	1	no	—
58	C₂×C₁₆	C₂×D₈₃	1	1	yes	characteristic
59	C₃₂	—	83	2	no	—
60	#41	—	332	1	no	—
61	C₂×Q₈	—	4	1	no	—
62	Q₁₆	—	2	2	no	—
63	Q₁₆	—	4	1	no	—
64	Q₁₆	—	166	1	no	—
65	SD₁₆	—	332	1	no	—
66	D₈	—	166	1	no	—
67	C₂×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
68	C₁₆	C₂²⋊D₈₃	1	1	yes	characteristic
69	C₁₆	C₂²×D₈₃	1	1	yes	characteristic
70	Q₈	—	4	2	no	—
71	Q₈	—	8	1	no	—
72	Q₈	—	332	1	no	—
73	D₄	—	332	1	no	—
74	D₄	—	664	1	no	—
75	C₂×C₄	—	8	1	no	—
76	C₂×C₄	—	664	1	no	—
77	C₂×C₄	#13050575	1	1	yes	characteristic
78	C₈	#13050575	1	1	yes	characteristic
79	C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
80	C₂²	—	664	1	no	—
81	C₂²	#14581733	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	664	1	no	—
84	C₄	#14581733	1	1	yes	characteristic
85	C₄	C₂×#13050575	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	1328	1	no	—
88	C₂	C₂×#14581733	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#14581733
commutator, cosocle	C₁₆×C₈₃	C₂³
Frattini	C₁₆	C₂²×D₈₃
Fitting	C₂×Q₃₂×C₈₃	C₂
socle	C₂×C₈₃	C₂×D₁₆
prime radical	#19128133	C₂
hypercenter	C₂×Q₃₂	D₈₃
hypocenter	C₈₃	#2264
LCS	G ▷ C₁₆×C₈₃ ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #19127541	C₂	semidirect
3. #19127538	C₂	semidirect
4. #19127535	C₂	semidirect
5. C₂×Q₃₂×C₈₃	C₂	semidirect
45. C₈₃	#2264	semidirect
47. C₂×Q₃₂	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
#19127541	2
#19127538	2
#19127535	2
C₂×Q₃₂×C₈₃	2
#2264	83