Subgroups — #16811658

Order 3392.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719940	C₂	1	2	yes	maximal, complemented
2	C₄⋊#88324	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#88324	C₂	1	2	yes	maximal, complemented
4	C₄×#88324	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×D₄×C₅₃	C₄	1	1	yes	characteristic
7	C₂²×Dic₅₃	C₂²	1	2	yes	—
8	#11719940	—	2	4	no	—
9	C₂×#88324	C₂²	1	1	yes	characteristic
10	C₂×#88324	C₂²	1	2	yes	—
11	C₂×#88324	C₂²	1	1	yes	characteristic
12	C₂²⋊Dic₅₃	C₄	1	2	yes	—
13	C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
14	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂³×C₅₃	C₂×C₄	1	2	yes	—
16	D₄×C₅₃	—	2	2	no	—
17	C₂×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₅₃	—	2	2	no	—
19	C₂×Dic₅₃	—	2	2	no	—
20	C₂×Dic₅₃	C₂×C₄	1	2	yes	—
21	C₂×Dic₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₅₃	C₂³	1	1	yes	characteristic
23	#88324	—	2	2	no	—
24	#88324	—	2	1	no	—
25	#88324	D₄	1	2	yes	complemented
26	C₂²×C₅₃	—	2	2	no	—
27	C₂²×C₅₃	—	2	2	no	—
28	C₂²×C₅₃	—	2	2	no	—
29	C₂²×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₅₃	M₁₆	1	2	yes	complemented
31	Dic₅₃	—	2	2	no	—
32	Dic₅₃	—	2	1	no	—
33	Dic₅₃	C₂×D₄	1	1	yes	characteristic
34	Dic₅₃	#41	1	1	yes	characteristic
35	C₂×C₅₃	—	4	2	no	—
36	C₂×C₅₃	C₄×D₄	1	1	yes	characteristic
37	C₂×C₅₃	C₂×M₁₆	1	1	yes	characteristic
38	C₂×C₅₃	#131	1	1	yes	characteristic
39	#436	—	53	1	no	maximal, Sylow
40	C₅₃	#436	1	1	yes	characteristic, complemented, Sylow
41	C₂×M₁₆	—	53	2	no	—
42	C₄×D₄	—	53	1	no	—
43	C₄⋊C₈	—	53	1	no	—
44	C₂²⋊C₈	—	53	2	no	—
45	C₄×C₈	—	53	1	no	—
46	C₂×D₄	#6157	1	1	yes	characteristic
47	C₂²×C₄	—	53	2	no	—
48	M₁₆	—	106	4	no	—
49	C₂×C₈	—	53	1	no	—
50	C₂×C₈	—	53	1	no	—
51	C₂×C₈	—	53	2	no	—
52	C₄⋊C₄	—	53	1	no	—
53	C₂²⋊C₄	—	53	2	no	—
54	C₄²	—	53	1	no	—
55	C₂³	C₂×#6157	1	2	yes	—
56	D₄	—	2	2	no	—
57	C₂×C₄	—	53	1	no	—
58	C₂×C₄	—	53	1	no	—
59	C₂×C₄	—	53	2	no	—
60	C₂×C₄	—	106	2	no	—
61	C₂×C₄	—	106	2	no	—
62	C₂×C₄	C₂×#6157	1	1	yes	characteristic
63	C₈	—	53	2	no	—
64	C₈	—	106	1	no	—
65	C₈	—	106	2	no	—
66	C₂²	—	2	2	no	—
67	C₂²	—	2	2	no	—
68	C₂²	—	2	2	no	—
69	C₂²	C₂²×#6157	1	1	yes	characteristic
70	C₄	—	53	1	no	—
71	C₄	—	53	1	no	—
72	C₄	—	106	2	no	—
73	C₄	—	106	1	no	—
74	C₄	#11719940	1	2	yes	complemented
75	C₂	—	4	2	no	—
76	C₂	D₄×#6157	1	1	yes	characteristic
77	C₂	C₂×#11719940	1	1	yes	characteristic
78	C₂	#13133304	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂²×C₄
Fitting, prime radical	C₂×D₄×C₅₃	C₄
cosocle	C₂×Dic₅₃	C₂³
hypercenter	C₂×D₄	#6157
hypocenter	C₅₃	#436
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#11719940	C₂	semidirect
2. C₄⋊#88324	C₂	semidirect
3. C₂²⋊#88324	C₂	semidirect
4. C₄×#88324	C₂	semidirect
25. #88324	D₄	semidirect
30. C₄×C₅₃	M₁₆	semidirect
40. C₅₃	#436	semidirect
74. C₄	#11719940	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719940	2
C₄⋊#88324	2
C₂²⋊#88324	2
C₄×#88324	2
D₄×Dic₅₃	2
#436	53