Subgroups — #35212694

Order 20864.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782055	C₂	1	1	yes	maximal, characteristic
2	#23781460	C₂	1	1	yes	maximal, characteristic, complemented
3	#355×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14579686	—	2	1	no	—
5	C₂×#14579684	—	2	1	no	—
6	#19104732	C₄	1	1	yes	characteristic, complemented
7	#19104650	—	2	1	no	—
8	#19104646	C₄	1	1	yes	characteristic, complemented
9	#124×C₁₆₃	C₂²	1	1	yes	characteristic
10	#101×C₁₆₃	—	2	1	no	—
11	C₂×D₆₅₂	—	2	1	no	—
12	C₂×D₆₅₂	—	4	1	no	—
13	#14579686	—	4	2	no	—
14	#14579684	—	4	2	no	—
15	C₂×C₁₆₃⋊C₈	—	2	1	no	—
16	C₂²⋊Dic₁₆₃	—	4	1	no	—
17	C₂×Q₈×C₁₆₃	D₄	1	1	yes	characteristic
18	C₂×D₄×C₁₆₃	D₄	1	1	yes	characteristic
19	M₁₆×C₁₆₃	—	4	1	no	—
20	C₂²⋊C₄×C₁₆₃	—	4	1	no	—
21	C₄²×C₁₆₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₁₆₃	—	4	1	no	—
23	D₆₅₂	—	4	1	no	—
24	D₆₅₂	—	4	1	no	—
25	D₆₅₂	—	8	2	no	—
26	C₂×Dic₁₆₃	—	8	1	no	—
27	C₁₆₃⋊C₈	—	2	2	no	—
28	C₂³×C₁₆₃	—	2	1	no	—
29	Q₈×C₁₆₃	—	4	1	no	—
30	D₄×C₁₆₃	—	4	1	no	—
31	C₂×C₄×C₁₆₃	—	2	1	no	—
32	C₂×C₄×C₁₆₃	—	2	1	no	—
33	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₈×C₁₆₃	—	8	1	no	—
35	C₂²×C₁₆₃	—	4	1	no	—
36	C₂²×C₁₆₃	—	8	1	no	—
37	C₂²×C₁₆₃	#99	1	1	yes	characteristic
38	C₂×D₁₆₃	—	8	2	no	—
39	C₂×D₁₆₃	—	8	1	no	—
40	C₄×C₁₆₃	—	2	1	no	—
41	C₄×C₁₆₃	—	4	1	no	—
42	C₄×C₁₆₃	—	4	1	no	—
43	Dic₁₆₃	—	16	1	no	—
44	C₂×C₁₆₃	—	2	1	no	—
45	C₂×C₁₆₃	—	8	1	no	—
46	C₂×C₁₆₃	#353	1	1	yes	characteristic
47	D₁₆₃	—	16	1	no	—
48	C₁₆₃	#1406	1	1	yes	characteristic, complemented, Sylow
49	#1406	—	163	1	no	maximal, Sylow
50	#496	—	163	1	no	—
51	#355	D₁₆₃	1	1	yes	characteristic, complemented
52	#354	—	163	1	no	—
53	C₂×SD₁₆	—	326	1	no	—
54	C₂×D₈	—	326	1	no	—
55	#127	—	163	1	no	—
56	#124	C₂×D₁₆₃	1	1	yes	characteristic
57	#101	—	2	1	no	—
58	#99	—	326	1	no	—
59	#97	—	163	1	no	—
60	C₂×Q₈	C₂²⋊D₁₆₃	1	1	yes	characteristic
61	C₂×D₄	—	326	1	no	—
62	C₂×D₄	—	652	1	no	—
63	C₂×D₄	D₆₅₂	1	1	yes	characteristic
64	SD₁₆	—	652	2	no	—
65	D₈	—	652	2	no	—
66	M₁₆	—	4	1	no	—
67	C₂×C₈	—	326	1	no	—
68	C₂²⋊C₄	—	4	1	no	—
69	C₂²⋊C₄	—	652	1	no	—
70	C₄²	C₄×D₁₆₃	1	1	yes	characteristic
71	C₂³	—	2	1	no	—
72	C₂³	—	652	1	no	—
73	Q₈	—	4	1	no	—
74	D₄	—	4	1	no	—
75	D₄	—	1304	2	no	—
76	D₄	—	652	1	no	—
77	D₄	—	652	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	2	1	no	—
80	C₂×C₄	—	1304	1	no	—
81	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
82	C₈	—	8	1	no	—
83	C₈	—	326	2	no	—
84	C₂²	—	4	1	no	—
85	C₂²	—	8	1	no	—
86	C₂²	—	1304	2	no	—
87	C₂²	—	1304	1	no	—
88	C₂²	#19104649	1	1	yes	characteristic
89	C₄	—	2	1	no	—
90	C₄	—	4	1	no	—
91	C₄	—	4	1	no	—
92	C₄	—	2608	1	no	—
93	C₂	—	2	1	no	—
94	C₂	—	8	1	no	—
95	C₂	—	2608	1	no	—
96	C₂	#23781457	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23781457
commutator	C₄²×C₁₆₃	C₂×C₄
Frattini	#124	C₂×D₁₆₃
Fitting	#355×C₁₆₃	C₂
socle	C₂×C₁₆₃	#353
prime radical	#23782055	C₂
cosocle	#124×C₁₆₃	C₂²
hypercenter	#355	D₁₆₃
hypocenter	C₁₆₃	#1406
LCS	G ▷ C₄²×C₁₆₃ ▷ C₂×C₄×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #355

Complements

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

N	H ≅ G/N	type
2. #23781460	C₂	semidirect
3. #355×C₁₆₃	C₂	semidirect
6. #19104732	C₄	semidirect
8. #19104646	C₄	semidirect
48. C₁₆₃	#1406	semidirect
51. #355	D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
#23782055	2
#23781460	2
#355×C₁₆₃	2
#1406	163