Subgroups — #35325399

Order 21248.

270 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₆₅₆	C₂	1	4	yes	maximal, complemented
2	#24909713	C₂	1	2	yes	maximal, complemented
3	C₄×C₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₃₂₈	—	2	2	no	—
5	C₂×D₁₃₂₈	C₂²	1	4	yes	—
6	#19128078	—	2	2	no	—
7	D₂₆₅₆	—	2	8	no	—
8	C₂×C₃₂×C₈₃	C₂²	1	2	yes	—
9	C₄×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
10	C₂×D₆₆₄	—	2	4	no	—
11	C₂×D₆₆₄	—	4	2	no	—
12	#14581789	—	4	2	no	—
13	D₁₃₂₈	—	2	8	no	—
14	D₁₃₂₈	—	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	complemented
19	C₂×D₃₃₂	—	4	4	no	—
20	C₂×D₃₃₂	—	8	4	no	—
21	D₆₆₄	—	4	4	no	—
22	D₆₆₄	—	4	4	no	—
23	D₆₆₄	—	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	C₂²×D₈₃	—	8	4	no	—
30	C₂×C₄×C₈₃	D₁₆	1	2	yes	—
31	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
32	D₃₃₂	—	8	4	no	—
33	D₃₃₂	—	8	4	no	—
34	D₃₃₂	—	16	8	no	—
35	C₈×C₈₃	D₁₆	1	2	yes	—
36	C₈×C₈₃	#127	1	1	yes	characteristic
37	C₈×C₈₃	C₂×D₈	1	1	yes	characteristic
38	C₂²×C₈₃	C₂×D₁₆	1	1	yes	characteristic
39	C₂×D₈₃	—	16	4	no	—
40	C₂×D₈₃	—	16	8	no	—
41	C₄×C₈₃	D₃₂	1	4	yes	complemented
42	C₄×C₈₃	#493	1	1	yes	characteristic
43	C₄×C₈₃	C₂×D₁₆	1	1	yes	characteristic
44	#13725	—	83	1	no	maximal, Sylow
45	C₂×C₈₃	#2246	1	1	yes	characteristic
46	C₂×C₈₃	C₂×D₃₂	1	2	yes	—
47	D₈₃	—	32	4	no	—
48	C₂×D₃₂	—	83	4	no	—
49	#2246	—	83	2	no	—
50	C₄×C₃₂	D₈₃	1	1	yes	characteristic, complemented
51	C₈₃	#13725	1	1	yes	characteristic, complemented, Sylow
52	C₂×D₁₆	—	83	4	no	—
53	C₂×D₁₆	—	166	2	no	—
54	#493	—	166	2	no	—
55	D₃₂	—	166	8	no	—
56	C₂×C₃₂	C₂×D₈₃	1	2	yes	—
57	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
58	C₂×D₈	—	332	2	no	—
59	C₂×D₈	—	166	4	no	—
60	#127	—	332	2	no	—
61	D₁₆	—	332	4	no	—
62	D₁₆	—	166	8	no	—
63	C₂×C₁₆	D₃₃₂	1	1	yes	characteristic
64	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
65	C₄×C₈	D₃₃₂	1	1	yes	characteristic
66	C₃₂	D₃₃₂	1	4	yes	complemented
67	C₂×D₄	—	332	4	no	—
68	C₂×D₄	—	664	4	no	—
69	D₈	—	332	4	no	—
70	D₈	—	332	4	no	—
71	D₈	—	664	4	no	—
72	C₂×C₈	D₆₆₄	1	1	yes	characteristic
73	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
74	C₄²	D₆₆₄	1	1	yes	characteristic
75	C₁₆	D₆₆₄	1	2	yes	—
76	C₁₆	C₂×D₃₃₂	1	2	yes	—
77	C₂³	—	664	4	no	—
78	D₄	—	1328	8	no	—
79	D₄	—	664	4	no	—
80	D₄	—	664	4	no	—
81	C₂×C₄	D₁₃₂₈	1	2	yes	—
82	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
83	C₈	D₁₃₂₈	1	2	yes	—
84	C₈	#14581789	1	1	yes	characteristic
85	C₈	C₂×D₆₆₄	1	1	yes	characteristic
86	C₂²	—	1328	4	no	—
87	C₂²	—	1328	8	no	—
88	C₂²	C₂×D₁₃₂₈	1	1	yes	characteristic
89	C₄	D₂₆₅₆	1	4	yes	complemented
90	C₄	#19128078	1	1	yes	characteristic
91	C₄	C₂×D₁₃₂₈	1	1	yes	characteristic
92	C₂	—	2656	4	no	—
93	C₂	#24909713	1	1	yes	characteristic
94	C₂	C₂×D₂₆₅₆	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	C₂²×C₈₃	C₂×D₁₆
hypercenter	C₄×C₃₂	D₈₃
hypocenter	C₈₃	#13725
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
1. C₂×D₂₆₅₆	C₂	semidirect
2. #24909713	C₂	semidirect
3. C₄×C₃₂×C₈₃	C₂	semidirect
18. C₃₂×C₈₃	D₄	semidirect
41. C₄×C₈₃	D₃₂	semidirect
50. C₄×C₃₂	D₈₃	semidirect
51. C₈₃	#13725	semidirect
66. C₃₂	D₃₃₂	semidirect
89. C₄	D₂₆₅₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₆₅₆	2
#24909713	2
C₄×C₃₂×C₈₃	2
#13725	83