Subgroups — #24907635

Order 10624.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14581728	C₂	1	4	yes	maximal
2	#19127725	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13050574	C₂²	1	4	yes	—
5	#14581728	—	2	8	no	—
6	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
7	C₂²×C₈×C₈₃	C₂²	1	2	yes	—
8	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₈₃	C₄	1	2	yes	—
10	C₄×C₈×C₈₃	C₄	1	2	yes	—
11	#13050574	—	2	8	no	—
12	C₂×C₈₃⋊C₈	—	2	4	no	—
13	C₂²×C₄×C₈₃	Q₈	1	2	yes	—
14	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
15	C₂×C₈×C₈₃	C₂×C₄	1	4	yes	—
16	C₂×C₈×C₈₃	D₄	1	4	yes	—
17	C₂×C₈×C₈₃	Q₈	1	4	yes	—
18	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₈₃	D₄	1	2	yes	—
21	C₂³×C₈₃	C₂×Q₈	1	1	yes	characteristic
22	C₂×C₄×C₈₃	—	2	4	no	—
23	C₂×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
24	C₂×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
25	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
27	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
29	C₈×C₈₃	C₄⋊C₄	1	8	yes	—
30	C₈₃⋊C₈	—	4	4	no	—
31	C₂²×C₈₃	—	2	2	no	—
32	C₂²×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₈₃	C₄⋊Q₈	1	2	yes	—
34	C₄×C₈₃	—	2	2	no	—
35	C₄×C₈₃	#108	1	4	yes	complemented
36	C₄×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
37	C₄×C₈₃	#127	1	1	yes	characteristic
38	C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₈₃	—	2	2	no	—
40	C₂×C₈₃	#384	1	1	yes	characteristic
41	C₂×C₈₃	C₂×#108	1	2	yes	—
42	#1848	—	83	1	no	maximal, Sylow
43	C₈₃	#1848	1	1	yes	characteristic, complemented, Sylow
44	C₂×#108	—	83	4	no	—
45	#423	—	83	2	no	—
46	C₂×C₄×C₈	D₈₃	1	1	yes	characteristic
47	C₂×M₁₆	—	83	4	no	—
48	C₂²×C₈	C₂×D₈₃	1	2	yes	—
49	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
50	#108	—	166	8	no	—
51	C₄⋊C₈	—	166	4	no	—
52	C₄×C₈	Dic₈₃	1	2	yes	—
53	C₄×C₈	Dic₈₃	1	2	yes	—
54	C₂²×C₄	Dic₁₆₆	1	2	yes	—
55	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
56	M₁₆	—	166	8	no	—
57	C₂×C₈	—	166	4	no	—
58	C₂×C₈	Dic₁₆₆	1	4	yes	—
59	C₂×C₈	D₃₃₂	1	4	yes	—
60	C₂×C₈	C₂×Dic₈₃	1	4	yes	—
61	C₄²	D₃₃₂	1	2	yes	—
62	C₄²	C₂×Dic₈₃	1	1	yes	characteristic
63	C₄²	C₂×Dic₈₃	1	1	yes	characteristic
64	C₂³	C₂×Dic₁₆₆	1	1	yes	characteristic
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	C₄⋊Dic₈₃	1	2	yes	—
67	C₂×C₄	C₄⋊Dic₈₃	1	2	yes	—
68	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
69	C₂×C₄	C₂×D₃₃₂	1	2	yes	—
70	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
71	C₂×C₄	C₂×Dic₁₆₆	1	2	yes	—
72	C₈	—	332	4	no	—
73	C₈	C₄⋊Dic₈₃	1	8	yes	—
74	C₂²	—	2	2	no	—
75	C₂²	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
76	C₂²	#14581792	1	2	yes	—
77	C₄	—	2	2	no	—
78	C₄	#14581728	1	4	yes	complemented
79	C₄	C₂×C₄⋊Dic₈₃	1	2	yes	—
80	C₄	#14581789	1	1	yes	characteristic
81	C₄	#14581792	1	1	yes	characteristic
82	C₂	—	2	2	no	—
83	C₂	#19127571	1	1	yes	characteristic
84	C₂	C₂×#14581728	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₃₃₂
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	C₂×C₄×C₈×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₄⋊C₄
prime radical	C₂³×C₈₃	C₂×Q₈
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	C₂×C₄×C₈	D₈₃
hypocenter	C₈₃	#1848
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
35. C₄×C₈₃	#108	semidirect
43. C₈₃	#1848	semidirect
78. C₄	#14581728	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14581728	2
#19127725	2
C₂×C₄×C₈×C₈₃	2
#1848	83