Subgroups — #35212462

Order 20864.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊(C₁₆₃⋊C₁₆)	C₂	1	2	yes	maximal, complemented
2	#408×C₁₆₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆₃⋊C₁₆	—	2	2	no	—
4	C₂×#41×C₁₆₃	C₄	1	1	yes	characteristic
5	C₂×M₁₆×C₁₆₃	C₄	1	1	yes	characteristic
6	C₂²×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
7	C₂²⋊C₈×C₁₆₃	—	2	2	no	—
8	C₁₆₃⋊C₁₆	—	4	2	no	—
9	#41×C₁₆₃	—	4	1	no	—
10	#41×C₁₆₃	—	4	1	no	—
11	C₂×Q₈×C₁₆₃	C₈	1	1	yes	characteristic
12	C₂×D₄×C₁₆₃	—	2	1	no	—
13	C₂×D₄×C₁₆₃	C₈	1	1	yes	characteristic
14	C₂²×C₄×C₁₆₃	—	2	1	no	—
15	C₂²×C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
16	M₁₆×C₁₆₃	—	4	1	no	—
17	C₂×C₈×C₁₆₃	—	2	1	no	—
18	C₂×C₈×C₁₆₃	—	4	1	no	—
19	C₂×C₈×C₁₆₃	D₄	1	2	yes	—
20	C₂³×C₁₆₃	—	2	1	no	—
21	C₂³×C₁₆₃	M₁₆	1	1	yes	characteristic
22	Q₈×C₁₆₃	—	4	1	no	—
23	D₄×C₁₆₃	—	4	1	no	—
24	D₄×C₁₆₃	—	4	2	no	—
25	C₂×C₄×C₁₆₃	—	2	1	no	—
26	C₂×C₄×C₁₆₃	—	2	1	no	—
27	C₂×C₄×C₁₆₃	—	2	1	no	—
28	C₂×C₄×C₁₆₃	—	4	1	no	—
29	C₂×C₄×C₁₆₃	—	4	1	no	—
30	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₆₃	C₂×C₈	1	1	yes	characteristic
32	C₈×C₁₆₃	—	2	2	no	—
33	C₈×C₁₆₃	—	4	1	no	—
34	C₂²×C₁₆₃	—	2	1	no	—
35	C₂²×C₁₆₃	—	2	2	no	—
36	C₂²×C₁₆₃	—	4	1	no	—
37	C₂²×C₁₆₃	—	4	2	no	—
38	C₂²×C₁₆₃	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₁₆₃	—	2	1	no	—
40	C₄×C₁₆₃	—	4	1	no	—
41	C₄×C₁₆₃	#99	1	1	yes	characteristic
42	C₄×C₁₆₃	#100	1	1	yes	characteristic
43	C₂×C₁₆₃	—	4	1	no	—
44	C₂×C₁₆₃	—	8	1	no	—
45	C₂×C₁₆₃	#323	1	1	yes	characteristic
46	C₂×C₁₆₃	#350	1	2	yes	—
47	C₁₆₃	#1315	1	1	yes	characteristic, complemented, Sylow
48	#1315	—	163	1	no	maximal, Sylow
49	#408	D₁₆₃	1	1	yes	characteristic
50	C₂²⋊C₁₆	—	163	2	no	—
51	C₂×#41	Dic₁₆₃	1	1	yes	characteristic
52	C₂×M₁₆	Dic₁₆₃	1	1	yes	characteristic
53	C₂²×C₈	C₂×D₁₆₃	1	1	yes	characteristic
54	C₂×C₁₆	—	326	2	no	—
55	C₂²⋊C₈	—	2	2	no	—
56	#41	—	4	1	no	—
57	#41	—	4	1	no	—
58	C₂×Q₈	C₁₆₃⋊C₈	1	1	yes	characteristic
59	C₂×D₄	—	2	1	no	—
60	C₂×D₄	C₁₆₃⋊C₈	1	1	yes	characteristic
61	C₂²×C₄	—	2	1	no	—
62	C₂²×C₄	C₂×Dic₁₆₃	1	1	yes	characteristic
63	M₁₆	—	4	1	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	—	4	1	no	—
66	C₂×C₈	C₂²⋊D₁₆₃	1	2	yes	—
67	C₁₆	—	652	2	no	—
68	C₂³	—	2	1	no	—
69	C₂³	#14579683	1	1	yes	characteristic
70	Q₈	—	4	1	no	—
71	D₄	—	4	1	no	—
72	D₄	—	4	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	4	1	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	C₂²⋊Dic₁₆₃	1	1	yes	characteristic
79	C₂×C₄	C₂×C₁₆₃⋊C₈	1	1	yes	characteristic
80	C₈	—	2	2	no	—
81	C₈	—	4	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	4	1	no	—
85	C₂²	—	4	2	no	—
86	C₂²	C₂²⋊(C₁₆₃⋊C₈)	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	4	1	no	—
89	C₄	#19104650	1	1	yes	characteristic
90	C₄	#19104652	1	1	yes	characteristic
91	C₂	—	4	1	no	—
92	C₂	—	8	1	no	—
93	C₂	#23781389	1	1	yes	characteristic
94	C₂	#23781449	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₁₆₃
commutator	C₂×C₄×C₁₆₃	C₂×C₈
Frattini	C₂²×C₈	C₂×D₁₆₃
Fitting	#408×C₁₆₃	C₂
socle	C₂²×C₁₆₃	C₂²⋊C₈
prime radical	C₂×#41×C₁₆₃	C₄
cosocle	C₂²×C₈×C₁₆₃	C₂²
hypercenter	#408	D₁₆₃
hypocenter	C₁₆₃	#1315
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #408

Complements

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

N	H ≅ G/N	type
1. C₂²⋊(C₁₆₃⋊C₁₆)	C₂	semidirect
47. C₁₆₃	#1315	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊(C₁₆₃⋊C₁₆)	2
#408×C₁₆₃	2
#1315	163