Subgroups — #35214607

Order 20864.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊(C₁₆₃⋊C₈)	C₂	1	2	yes	maximal
2	#23781420	C₂	1	4	yes	maximal, complemented
3	C₂×#117×C₁₆₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆₃⋊C₈	C₂²	1	4	yes	—
5	C₂²⋊(C₁₆₃⋊C₈)	—	2	4	no	—
6	C₂³×C₄×C₁₆₃	C₂²	1	1	yes	characteristic
7	#117×C₁₆₃	—	2	4	no	—
8	C₂×C₄⋊C₄×C₁₆₃	C₄	1	2	yes	—
9	C₂×C₂²⋊C₄×C₁₆₃	C₄	1	2	yes	—
10	C₂×C₄²×C₁₆₃	C₂²	1	2	yes	—
11	C₂×C₁₆₃⋊C₈	—	2	4	no	—
12	C₂×C₁₆₃⋊C₈	—	2	8	no	—
13	C₂⁴×C₁₆₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₁₆₃	—	2	4	no	—
15	C₂²×C₄×C₁₆₃	C₂×C₄	1	4	yes	—
16	C₂²×C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₆₃	D₄	1	4	yes	—
18	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₁₆₃	—	2	4	no	—
20	C₂²⋊C₄×C₁₆₃	—	2	4	no	—
21	C₄²×C₁₆₃	—	2	4	no	—
22	C₁₆₃⋊C₈	—	4	4	no	—
23	C₂³×C₁₆₃	—	2	2	no	—
24	C₂³×C₁₆₃	—	2	4	no	—
25	C₂³×C₁₆₃	C₂²⋊C₄	1	2	yes	—
26	C₂³×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₆₃	—	2	2	no	—
28	C₂×C₄×C₁₆₃	—	2	8	no	—
29	C₂×C₄×C₁₆₃	—	2	4	no	—
30	C₂×C₄×C₁₆₃	—	2	8	no	—
31	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
33	C₂×C₄×C₁₆₃	#41	1	4	yes	—
34	C₂²×C₁₆₃	—	2	2	no	—
35	C₂²×C₁₆₃	—	2	4	no	—
36	C₂²×C₁₆₃	—	4	4	no	—
37	C₂²×C₁₆₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₁₆₃	#117	1	2	yes	—
39	C₂²×C₁₆₃	#131	1	4	yes	—
40	C₄×C₁₆₃	—	2	2	no	—
41	C₄×C₁₆₃	—	4	4	no	—
42	C₄×C₁₆₃	#123	1	4	yes	—
43	C₂×C₁₆₃	—	4	2	no	—
44	C₂×C₁₆₃	#381	1	1	yes	characteristic
45	C₂×C₁₆₃	#408	1	2	yes	—
46	C₂×C₁₆₃	#433	1	4	yes	—
47	C₁₆₃	#1818	1	1	yes	characteristic, complemented, Sylow
48	#1818	—	163	1	no	maximal, Sylow
49	C₂×#117	D₁₆₃	1	1	yes	characteristic
50	C₂×C₂²⋊C₈	—	163	2	no	—
51	#336	—	163	4	no	—
52	C₂³×C₄	C₂×D₁₆₃	1	1	yes	characteristic
53	C₂²×C₈	—	163	4	no	—
54	#117	—	2	4	no	—
55	C₂×C₄⋊C₄	Dic₁₆₃	1	2	yes	—
56	C₂×C₂²⋊C₄	Dic₁₆₃	1	2	yes	—
57	C₂×C₄²	C₂×D₁₆₃	1	2	yes	—
58	C₂²⋊C₈	—	326	4	no	—
59	C₂⁴	C₂×Dic₁₆₃	1	1	yes	characteristic
60	C₂²×C₄	—	2	4	no	—
61	C₂²×C₄	C₂×Dic₁₆₃	1	4	yes	—
62	C₂²×C₄	C₂×Dic₁₆₃	1	1	yes	characteristic
63	C₂²×C₄	C₂²⋊D₁₆₃	1	4	yes	—
64	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
65	C₂×C₈	—	326	4	no	—
66	C₂×C₈	—	326	8	no	—
67	C₄⋊C₄	—	2	4	no	—
68	C₂²⋊C₄	—	2	4	no	—
69	C₄²	—	2	4	no	—
70	C₂³	—	2	2	no	—
71	C₂³	—	2	4	no	—
72	C₂³	C₂²⋊Dic₁₆₃	1	2	yes	—
73	C₂³	C₂²×Dic₁₆₃	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	8	no	—
76	C₂×C₄	—	2	4	no	—
77	C₂×C₄	—	2	8	no	—
78	C₂×C₄	C₂²⋊Dic₁₆₃	1	2	yes	—
79	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	2	yes	—
80	C₂×C₄	#14579699	1	4	yes	—
81	C₈	—	652	4	no	—
82	C₂²	—	2	4	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	4	4	no	—
85	C₂²	C₂×C₂²⋊Dic₁₆₃	1	1	yes	characteristic
86	C₂²	#19104694	1	2	yes	—
87	C₂²	#19104742	1	4	yes	—
88	C₄	—	2	2	no	—
89	C₄	—	4	4	no	—
90	C₄	#19104718	1	4	yes	—
91	C₂	—	4	2	no	—
92	C₂	#23781528	1	1	yes	characteristic
93	C₂	#23781632	1	2	yes	—
94	C₂	#23781726	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₆₃
commutator, socle	C₂³×C₁₆₃	C₂²×C₄
Fitting	C₂×#117×C₁₆₃	C₂
prime radical	C₂⁴×C₁₆₃	C₂×C₄
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	C₂×#117	D₁₆₃
hypocenter	C₁₆₃	#1818
LCS	G ▷ C₂³×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #23781420	C₂	semidirect
47. C₁₆₃	#1818	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊(C₁₆₃⋊C₈)	2
#23781420	2
C₂×#117×C₁₆₃	2
#1818	163