Subgroups — #35214656

Order 20864.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₁₆₃⋊C₈)	C₂	1	2	yes	maximal
2	#23781420	C₂	1	4	yes	maximal
3	#376×C₁₆₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆₃⋊C₈	C₂²	1	4	yes	—
5	C₄⋊(C₁₆₃⋊C₈)	—	2	4	no	—
6	C₂×C₄²×C₁₆₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
8	#95×C₁₆₃	C₄	1	4	yes	—
9	C₂×C₁₆₃⋊C₈	—	2	4	no	—
10	C₂×C₁₆₃⋊C₈	—	2	8	no	—
11	C₂²×C₄×C₁₆₃	C₂×C₄	1	4	yes	—
12	C₂²×C₄×C₁₆₃	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
14	C₄²×C₁₆₃	—	2	4	no	—
15	C₄²×C₁₆₃	D₄	1	2	yes	—
16	C₄²×C₁₆₃	Q₈	1	2	yes	—
17	C₁₆₃⋊C₈	—	4	4	no	—
18	C₂³×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₆₃	—	2	4	no	—
20	C₂×C₄×C₁₆₃	—	2	4	no	—
21	C₂×C₄×C₁₆₃	—	2	8	no	—
22	C₂×C₄×C₁₆₃	C₄⋊C₄	1	4	yes	—
23	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
25	C₂×C₄×C₁₆₃	#41	1	4	yes	—
26	C₂²×C₁₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₆₃	#117	1	2	yes	—
28	C₂²×C₁₆₃	#131	1	4	yes	—
29	C₄×C₁₆₃	—	2	4	no	—
30	C₄×C₁₆₃	—	4	4	no	—
31	C₄×C₁₆₃	#124	1	2	yes	—
32	C₄×C₁₆₃	#125	1	2	yes	—
33	C₂×C₁₆₃	#382	1	1	yes	characteristic
34	C₂×C₁₆₃	#424	1	2	yes	—
35	C₂×C₁₆₃	#433	1	4	yes	—
36	C₁₆₃	#1832	1	1	yes	characteristic, complemented, Sylow
37	#1832	—	163	1	no	maximal, Sylow
38	C₂×C₄⋊C₈	—	163	2	no	—
39	#376	D₁₆₃	1	1	yes	characteristic
40	#336	—	163	4	no	—
41	C₂²×C₈	—	163	4	no	—
42	C₂×C₄²	C₂×D₁₆₃	1	2	yes	—
43	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
44	C₄⋊C₈	—	326	4	no	—
45	#95	Dic₁₆₃	1	4	yes	—
46	C₂²×C₄	C₂×Dic₁₆₃	1	4	yes	—
47	C₂²×C₄	C₂×Dic₁₆₃	1	2	yes	—
48	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
49	C₂×C₈	—	326	4	no	—
50	C₂×C₈	—	326	8	no	—
51	C₄²	—	2	4	no	—
52	C₄²	D₆₅₂	1	2	yes	—
53	C₄²	Dic₃₂₆	1	2	yes	—
54	C₂³	C₂²×Dic₁₆₃	1	1	yes	characteristic
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	2	4	no	—
57	C₂×C₄	—	2	8	no	—
58	C₂×C₄	C₄⋊Dic₁₆₃	1	4	yes	—
59	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
60	C₂×C₄	C₂×Dic₃₂₆	1	1	yes	characteristic
61	C₂×C₄	#14579699	1	4	yes	—
62	C₈	—	652	4	no	—
63	C₂²	C₂×C₄⋊Dic₁₆₃	1	1	yes	characteristic
64	C₂²	#19104694	1	2	yes	—
65	C₂²	#19104742	1	4	yes	—
66	C₄	—	2	4	no	—
67	C₄	—	4	4	no	—
68	C₄	#19104723	1	2	yes	—
69	C₄	#19104727	1	2	yes	—
70	C₂	#23781531	1	1	yes	characteristic
71	C₂	#23781692	1	2	yes	—
72	C₂	#23781726	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₆₃
commutator, socle, prime radical	C₂³×C₁₆₃	C₂²×C₄
Fitting	#376×C₁₆₃	C₂
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	#376	D₁₆₃
hypocenter	C₁₆₃	#1832
LCS	G ▷ C₂³×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
36. C₁₆₃	#1832	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₁₆₃⋊C₈)	2
#23781420	2
#376×C₁₆₃	2
#1832	163