Subgroups — C₂×#22554405 #30681983

Order 16832.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16862360	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₆₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22554405	C₂	1	4	yes	maximal, direct product factor
4	C₂×M₁₆×C₂₆₃	C₂	1	1	yes	maximal, characteristic
5	#16862360	—	2	1	no	—
6	#16862360	C₂²	1	2	yes	—
7	C₂×C₂₆₃⋊C₈	—	2	1	no	—
8	C₂×C₂₆₃⋊C₈	C₄	1	2	yes	—
9	C₂×C₂₆₃⋊C₈	C₄	1	2	yes	—
10	C₂×C₂₆₃⋊C₈	C₂²	1	2	yes	—
11	C₂²×C₄×C₂₆₃	C₂²	1	1	yes	characteristic
12	M₁₆×C₂₆₃	—	2	1	no	—
13	M₁₆×C₂₆₃	C₂²	1	2	yes	—
14	C₂×C₈×C₂₆₃	—	2	1	no	—
15	C₂₆₃⋊C₈	—	2	2	no	—
16	C₂₆₃⋊C₈	C₂×C₄	1	4	yes	—
17	C₂³×C₂₆₃	Q₈	1	1	yes	characteristic
18	C₂×C₄×C₂₆₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₆₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₆₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₆₃	D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₆₃	Q₈	1	1	yes	characteristic
23	C₂×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
24	C₈×C₂₆₃	—	2	2	no	—
25	C₂²×C₂₆₃	—	2	2	no	—
26	C₂²×C₂₆₃	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₂₆₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂²×C₂₆₃	C₂×Q₈	1	1	yes	characteristic
29	C₄×C₂₆₃	C₄⋊C₄	1	1	yes	characteristic
30	C₄×C₂₆₃	C₄⋊C₄	1	1	yes	characteristic
31	C₄×C₂₆₃	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₂₆₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₂₆₃	—	2	1	no	—
34	C₂×C₂₆₃	—	2	1	no	—
35	C₂×C₂₆₃	#108	1	2	yes	complemented
36	C₂×C₂₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
37	C₂₆₃	C₂×#108	1	1	yes	characteristic, complemented, Sylow
38	C₂×#108	—	263	1	no	maximal, Sylow
39	C₂×M₁₆	—	263	1	no	—
40	C₂×M₁₆	D₂₆₃	1	1	yes	characteristic
41	C₂²×C₈	—	263	1	no	—
42	#108	—	263	4	no	—
43	C₂²×C₄	C₂×D₂₆₃	1	1	yes	characteristic
44	M₁₆	—	2	1	no	—
45	M₁₆	—	263	2	no	—
46	M₁₆	—	526	1	no	—
47	M₁₆	C₂×D₂₆₃	1	2	yes	—
48	C₂×C₈	—	2	1	no	—
49	C₂×C₈	—	263	2	no	—
50	C₂×C₈	—	263	2	no	—
51	C₂×C₈	—	263	2	no	—
52	C₂×C₈	—	526	1	no	—
53	C₂³	Dic₅₂₆	1	1	yes	characteristic
54	C₂×C₄	C₄×D₂₆₃	1	1	yes	characteristic
55	C₂×C₄	C₄×D₂₆₃	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₂₆₃	1	1	yes	characteristic
57	C₂×C₄	C₂²⋊D₂₆₃	1	1	yes	characteristic
58	C₂×C₄	Dic₅₂₆	1	1	yes	characteristic
59	C₂×C₄	C₂²×D₂₆₃	1	1	yes	characteristic
60	C₈	—	2	2	no	—
61	C₈	—	263	4	no	—
62	C₈	—	526	2	no	—
63	C₂²	—	2	2	no	—
64	C₂²	Dic₂₆₃⋊C₄	1	1	yes	characteristic
65	C₂²	Dic₂₆₃⋊C₄	1	1	yes	characteristic
66	C₂²	C₂×Dic₅₂₆	1	1	yes	characteristic
67	C₄	Dic₂₆₃⋊C₄	1	1	yes	characteristic
68	C₄	Dic₂₆₃⋊C₄	1	1	yes	characteristic
69	C₄	C₂×C₄×D₂₆₃	1	1	yes	characteristic
70	C₄	C₂×C₂²⋊D₂₆₃	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	—	2	1	no	—
73	C₂	#22554405	1	2	yes	direct product factor
74	C₂	C₂×Dic₂₆₃⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₂₆₃
commutator	C₄×C₂₆₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₆₃
Fitting	C₂×M₁₆×C₂₆₃	C₂
socle	C₂²×C₂₆₃	C₄⋊C₄
prime radical	C₂³×C₂₆₃	Q₈
cosocle	C₂×C₄×C₂₆₃	C₂³
hypercenter	C₂×M₁₆	D₂₆₃
hypocenter	C₂₆₃	C₂×#108
LCS	G ▷ C₄×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22554405	C₂	direct (N × H)
35. C₂×C₂₆₃	#108	semidirect
37. C₂₆₃	C₂×#108	semidirect
73. C₂	#22554405	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16862360	2
C₂²×C₂₆₃⋊C₈	2
#22554405	2
C₂×M₁₆×C₂₆₃	2
C₂×#108	263

Subgroups — C2×#22554405 #30681983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22554405 #30681983