Subgroups — C₂²⋊#16810342 #27271901

Order 13568.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810342	C₂	1	2	yes	maximal, complemented
2	C₂²×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810342	—	2	2	no	—
4	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
5	C₂×C₃₂×C₅₃	—	2	2	no	—
6	C₂×C₃₂×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₅₃	C₈	1	1	yes	characteristic
9	C₂×C₁₆×C₅₃	—	2	1	no	—
10	C₂×C₁₆×C₅₃	—	2	1	no	—
11	C₂×C₁₆×C₅₃	C₈	1	1	yes	characteristic
12	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₃₂×C₅₃	—	2	1	no	—
14	C₃₂×C₅₃	D₄	1	2	yes	—
15	C₂²×C₄×C₅₃	C₁₆	1	1	yes	characteristic
16	C₂×C₈×C₅₃	—	2	1	no	—
17	C₂×C₈×C₅₃	—	2	1	no	—
18	C₂×C₈×C₅₃	C₁₆	1	1	yes	characteristic
19	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
20	C₁₆×C₅₃	—	2	1	no	—
21	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
23	C₂³×C₅₃	C₃₂	1	1	yes	characteristic
24	C₂×C₄×C₅₃	—	2	1	no	—
25	C₂×C₄×C₅₃	—	2	1	no	—
26	C₂×C₄×C₅₃	C₃₂	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
28	C₈×C₅₃	—	2	1	no	—
29	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
30	C₈×C₅₃	M₃₂	1	1	yes	characteristic
31	C₂²⋊C₆₄	—	53	1	no	maximal, Sylow
32	C₂²×C₅₃	—	2	1	no	—
33	C₂²×C₅₃	—	2	1	no	—
34	C₂²×C₅₃	C₆₄	1	2	yes	complemented
35	C₂²×C₅₃	C₂×C₃₂	1	1	yes	characteristic
36	C₄×C₅₃	—	2	1	no	—
37	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
38	C₄×C₅₃	M₆₄	1	1	yes	characteristic
39	C₂²×C₃₂	D₅₃	1	1	yes	characteristic
40	C₂×C₆₄	—	53	2	no	—
41	C₂×C₅₃	—	2	2	no	—
42	C₂×C₅₃	C₂²⋊C₃₂	1	1	yes	characteristic
43	C₂×C₅₃	C₂×C₆₄	1	1	yes	characteristic
44	C₂×C₅₃	M₁₂₈	1	1	yes	characteristic
45	C₂²×C₁₆	Dic₅₃	1	1	yes	characteristic
46	C₂×C₃₂	—	2	2	no	—
47	C₂×C₃₂	Dic₅₃	1	1	yes	characteristic
48	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
49	C₆₄	—	106	2	no	—
50	C₅₃	C₂²⋊C₆₄	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₈	#88322	1	1	yes	characteristic
52	C₂×C₁₆	—	2	1	no	—
53	C₂×C₁₆	—	2	1	no	—
54	C₂×C₁₆	#88322	1	1	yes	characteristic
55	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
56	C₃₂	—	2	1	no	—
57	C₃₂	C₂²⋊D₅₃	1	2	yes	—
58	C₂²×C₄	#11719908	1	1	yes	characteristic
59	C₂×C₈	—	2	1	no	—
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	#11719908	1	1	yes	characteristic
62	C₂×C₈	C₂×#88322	1	1	yes	characteristic
63	C₁₆	—	2	1	no	—
64	C₁₆	#11719917	1	1	yes	characteristic
65	C₁₆	C₂²⋊Dic₅₃	1	1	yes	characteristic
66	C₂³	#13133123	1	1	yes	characteristic
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	#13133123	1	1	yes	characteristic
70	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
71	C₈	—	2	1	no	—
72	C₈	C₂²⋊#88322	1	1	yes	characteristic
73	C₈	#13133157	1	1	yes	characteristic
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	#16810342	1	2	yes	complemented
77	C₂²	C₂×#13133123	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	C₂²⋊#11719908	1	1	yes	characteristic
80	C₄	#16810457	1	1	yes	characteristic
81	C₂	—	2	2	no	—
82	C₂	C₂²⋊#13133123	1	1	yes	characteristic
83	C₂	C₂×#16810342	1	1	yes	characteristic
84	C₂	#20272926	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₃₂	C₂×D₅₃
commutator	C₂×C₅₃	C₂×C₆₄
Fitting	C₂²×C₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₃₂
prime radical	C₂³×C₅₃	C₃₂
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	C₂²×C₃₂	D₅₃
hypocenter	C₅₃	C₂²⋊C₆₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₃₂ ◁ C₂²×C₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#16810342	C₂	semidirect
34. C₂²×C₅₃	C₆₄	semidirect
50. C₅₃	C₂²⋊C₆₄	semidirect
76. C₂²	#16810342	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810342	2
C₂²×C₃₂×C₅₃	2
C₂²⋊C₆₄	53

Subgroups — C22⋊#16810342 #27271901

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊#16810342 #27271901