Subgroups — #35222914

Order 20864.

504 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19104735	C₂	1	2	yes	maximal
2	#23781571	C₂	1	8	yes	maximal
3	#23781531	C₂	1	4	yes	maximal
4	C₄³×C₁₆₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₂₆	C₂²	1	4	yes	—
6	#19104735	—	2	8	no	—
7	C₂×C₄⋊Dic₁₆₃	C₂²	1	4	yes	—
8	C₂×C₄⋊Dic₁₆₃	C₂²	1	4	yes	—
9	C₁₆₃⋊#95	C₂²	1	16	yes	—
10	C₂×C₄²×C₁₆₃	C₂²	1	4	yes	—
11	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₁₆₃	C₂²	1	2	yes	—
13	C₂×Dic₃₂₆	—	2	8	no	—
14	C₂×Dic₃₂₆	—	2	16	no	—
15	C₂²×Dic₁₆₃	C₂³	1	8	yes	—
16	C₄⋊Dic₁₆₃	—	2	8	no	—
17	C₄⋊Dic₁₆₃	—	2	8	no	—
18	C₂²×C₄×C₁₆₃	C₂³	1	4	yes	—
19	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₁₆₃	C₂³	1	2	yes	—
21	C₄²×C₁₆₃	—	2	8	no	—
22	C₄²×C₁₆₃	D₄	1	4	yes	—
23	C₄²×C₁₆₃	Q₈	1	8	yes	complemented
24	Dic₃₂₆	—	4	16	no	—
25	C₂×Dic₁₆₃	—	2	8	no	—
26	C₂×Dic₁₆₃	—	2	16	no	—
27	C₂³×C₁₆₃	C₂⁴	1	1	yes	characteristic
28	C₂×C₄×C₁₆₃	—	2	8	no	—
29	C₂×C₄×C₁₆₃	C₂×D₄	1	4	yes	—
30	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
31	C₂×C₄×C₁₆₃	C₂×Q₈	1	8	yes	—
32	C₂×C₄×C₁₆₃	C₂×Q₈	1	4	yes	—
33	C₂×C₄×C₁₆₃	#41	1	8	yes	—
34	C₂²×C₁₆₃	C₂²×D₄	1	1	yes	characteristic
35	C₂²×C₁₆₃	C₂²×Q₈	1	2	yes	—
36	C₂²×C₁₆₃	C₂×#41	1	4	yes	—
37	C₄×C₁₆₃	—	2	8	no	—
38	C₄×C₁₆₃	#124	1	8	yes	—
39	C₄×C₁₆₃	C₄⋊Q₈	1	4	yes	—
40	Dic₁₆₃	—	4	8	no	—
41	C₂×C₁₆₃	C₂×#124	1	2	yes	—
42	C₂×C₁₆₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
43	C₂×C₁₆₃	#533	1	4	yes	—
44	C₁₆₃	#2862	1	1	yes	characteristic, complemented, Sylow
45	#2862	—	163	1	no	maximal, Sylow
46	C₂×C₄⋊Q₈	—	163	2	no	—
47	#391	—	163	8	no	—
48	#382	—	163	4	no	—
49	C₄³	D₁₆₃	1	1	yes	characteristic
50	C₂²×Q₈	—	163	4	no	—
51	C₄⋊Q₈	—	326	8	no	—
52	C₂×C₄⋊C₄	—	163	4	no	—
53	C₂×C₄⋊C₄	—	163	4	no	—
54	C₂×C₄²	C₂×D₁₆₃	1	4	yes	—
55	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
56	C₂×C₄²	C₂×D₁₆₃	1	2	yes	—
57	#95	—	163	16	no	—
58	C₂×Q₈	—	326	8	no	—
59	C₂×Q₈	—	326	16	no	—
60	C₂²×C₄	—	163	8	no	—
61	C₂²×C₄	C₂²×D₁₆₃	1	4	yes	—
62	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
63	C₂²×C₄	C₂²×D₁₆₃	1	2	yes	—
64	C₄⋊C₄	—	326	8	no	—
65	C₄⋊C₄	—	326	8	no	—
66	C₄²	—	2	8	no	—
67	C₄²	D₆₅₂	1	4	yes	—
68	C₄²	Dic₃₂₆	1	8	yes	complemented
69	C₂³	C₂³×D₁₆₃	1	1	yes	characteristic
70	Q₈	—	652	16	no	—
71	C₂×C₄	—	2	8	no	—
72	C₂×C₄	—	326	8	no	—
73	C₂×C₄	—	326	16	no	—
74	C₂×C₄	C₂×D₆₅₂	1	4	yes	—
75	C₂×C₄	C₂×D₆₅₂	1	2	yes	—
76	C₂×C₄	C₂×Dic₃₂₆	1	8	yes	—
77	C₂×C₄	C₂×Dic₃₂₆	1	4	yes	—
78	C₂×C₄	#14579699	1	8	yes	—
79	C₂²	C₂²×D₆₅₂	1	1	yes	characteristic
80	C₂²	C₂²×Dic₃₂₆	1	2	yes	—
81	C₂²	C₂×#14579699	1	4	yes	—
82	C₄	—	2	8	no	—
83	C₄	—	652	8	no	—
84	C₄	#19104723	1	8	yes	—
85	C₄	#19104735	1	4	yes	—
86	C₂	C₂×#19104723	1	2	yes	—
87	C₂	C₂×#19104735	1	1	yes	characteristic
88	C₂	#23782222	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₆₃
commutator, socle, prime radical, cosocle	C₂³×C₁₆₃	C₂⁴
Fitting	C₄³×C₁₆₃	C₂
hypercenter	C₄³	D₁₆₃
hypocenter	C₁₆₃	#2862
LCS	G ▷ C₂³×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
23. C₄²×C₁₆₃	Q₈	semidirect
44. C₁₆₃	#2862	semidirect
68. C₄²	Dic₃₂₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19104735	2
#23781571	2
#23781531	2
C₄³×C₁₆₃	2
#2862	163