Subgroups — #35325402

Order 21248.

230 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₃₂₈	C₂	1	4	yes	maximal
2	#24909716	C₂	1	2	yes	maximal
3	C₄×C₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₆₆₄	—	2	2	no	—
5	C₂×Dic₆₆₄	C₂²	1	4	yes	—
6	#19128081	—	2	2	no	—
7	Dic₁₃₂₈	—	2	8	no	—
8	C₂×C₃₂×C₈₃	C₂²	1	2	yes	—
9	C₄×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
10	C₂×Dic₃₃₂	—	2	4	no	—
11	C₂×Dic₃₃₂	—	4	2	no	—
12	#14581792	—	4	2	no	—
13	Dic₆₆₄	—	2	8	no	—
14	Dic₆₆₄	—	4	4	no	—
15	C₂×C₁₆×C₈₃	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
17	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
18	C₃₂×C₈₃	D₄	1	4	yes	—
19	C₂×Dic₁₆₆	—	4	4	no	—
20	Dic₃₃₂	—	4	4	no	—
21	Dic₃₃₂	—	4	4	no	—
22	Dic₃₃₂	—	8	4	no	—
23	C₄⋊Dic₈₃	—	8	4	no	—
24	C₂×C₈×C₈₃	D₈	1	1	yes	characteristic
25	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
26	C₄²×C₈₃	D₈	1	1	yes	characteristic
27	C₁₆×C₈₃	D₈	1	2	yes	—
28	C₁₆×C₈₃	C₂×D₄	1	2	yes	—
29	C₂×C₄×C₈₃	D₁₆	1	2	yes	—
30	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
31	C₂×Dic₈₃	—	8	4	no	—
32	Dic₁₆₆	—	8	4	no	—
33	Dic₁₆₆	—	8	4	no	—
34	C₈×C₈₃	D₁₆	1	2	yes	—
35	C₈×C₈₃	#127	1	1	yes	characteristic
36	C₈×C₈₃	C₂×D₈	1	1	yes	characteristic
37	C₂²×C₈₃	C₂×D₁₆	1	1	yes	characteristic
38	C₄×C₈₃	Q₆₄	1	4	yes	complemented
39	C₄×C₈₃	#493	1	1	yes	characteristic
40	C₄×C₈₃	C₂×D₁₆	1	1	yes	characteristic
41	Dic₈₃	—	16	4	no	—
42	#13726	—	83	1	no	maximal, Sylow
43	C₂×C₈₃	#2246	1	1	yes	characteristic
44	C₂×C₈₃	C₂×Q₆₄	1	2	yes	—
45	C₂×Q₆₄	—	83	4	no	—
46	#2247	—	83	2	no	—
47	C₄×C₃₂	D₈₃	1	1	yes	characteristic
48	C₈₃	#13726	1	1	yes	characteristic, complemented, Sylow
49	C₂×Q₃₂	—	83	4	no	—
50	C₂×Q₃₂	—	166	2	no	—
51	#494	—	166	2	no	—
52	Q₆₄	—	166	8	no	—
53	C₂×C₃₂	C₂×D₈₃	1	2	yes	—
54	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
55	C₂×Q₁₆	—	332	2	no	—
56	C₂×Q₁₆	—	166	4	no	—
57	C₄⋊Q₈	—	332	2	no	—
58	Q₃₂	—	332	4	no	—
59	Q₃₂	—	166	8	no	—
60	C₂×C₁₆	D₃₃₂	1	1	yes	characteristic
61	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
62	C₄×C₈	D₃₃₂	1	1	yes	characteristic
63	C₃₂	D₃₃₂	1	4	yes	—
64	C₂×Q₈	—	332	4	no	—
65	Q₁₆	—	332	4	no	—
66	Q₁₆	—	332	4	no	—
67	Q₁₆	—	664	4	no	—
68	C₂×C₈	D₆₆₄	1	1	yes	characteristic
69	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
70	C₄⋊C₄	—	664	4	no	—
71	C₄²	D₆₆₄	1	1	yes	characteristic
72	C₁₆	D₆₆₄	1	2	yes	—
73	C₁₆	C₂×D₃₃₂	1	2	yes	—
74	Q₈	—	664	4	no	—
75	Q₈	—	664	4	no	—
76	C₂×C₄	—	664	4	no	—
77	C₂×C₄	D₁₃₂₈	1	2	yes	—
78	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
79	C₈	D₁₃₂₈	1	2	yes	—
80	C₈	#14581789	1	1	yes	characteristic
81	C₈	C₂×D₆₆₄	1	1	yes	characteristic
82	C₂²	C₂×D₁₃₂₈	1	1	yes	characteristic
83	C₄	—	1328	4	no	—
84	C₄	Dic₁₃₂₈	1	4	yes	complemented
85	C₄	#19128078	1	1	yes	characteristic
86	C₄	C₂×D₁₃₂₈	1	1	yes	characteristic
87	C₂	#24909713	1	1	yes	characteristic
88	C₂	C₂×Dic₁₃₂₈	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₃₂₈
commutator, cosocle	C₂×C₁₆×C₈₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₃
Fitting	C₄×C₃₂×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	C₂×D₁₆
hypercenter	C₄×C₃₂	D₈₃
hypocenter	C₈₃	#13726
LCS	G ▷ C₂×C₁₆×C₈₃ ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
38. C₄×C₈₃	Q₆₄	semidirect
48. C₈₃	#13726	semidirect
84. C₄	Dic₁₃₂₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₃₂₈	2
#24909716	2
C₄×C₃₂×C₈₃	2
#13726	83