Subgroups — #16810971

Order 3392.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133214	C₂	1	1	yes	maximal, characteristic
2	#13133137	C₂	1	4	yes	maximal, complemented
3	C₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₈×C₅₃	—	2	1	no	—
6	C₂×D₂₁₂	—	2	1	no	—
7	C₂×D₂₁₂	—	2	1	no	—
8	C₂×D₂₁₂	C₂²	1	2	yes	—
9	C₄⋊C₄×C₅₃	—	2	1	no	—
10	C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
11	C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#88322	C₂²	1	2	yes	—
13	C₂²×D₅₃	—	2	2	no	—
14	Q₈×C₅₃	—	4	2	no	—
15	C₂×C₄×C₅₃	—	2	2	no	—
16	C₂×C₄×C₅₃	D₄	1	1	yes	characteristic
17	C₂×C₄×C₅₃	D₄	1	1	yes	characteristic
18	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	D₂₁₂	—	2	4	no	—
20	D₂₁₂	—	4	2	no	—
21	D₂₁₂	—	4	2	no	—
22	#88322	—	2	2	no	—
23	C₂²×C₅₃	C₂×D₄	1	1	yes	characteristic
24	C₂×D₅₃	—	4	2	no	—
25	C₂×D₅₃	—	4	2	no	—
26	C₂×D₅₃	—	4	2	no	—
27	C₄×C₅₃	—	4	2	no	—
28	C₄×C₅₃	D₈	1	2	yes	—
29	C₄×C₅₃	SD₁₆	1	2	yes	—
30	C₄×C₅₃	#41	1	2	yes	—
31	C₂×C₅₃	#124	1	1	yes	characteristic
32	C₂×C₅₃	C₂×D₈	1	1	yes	characteristic
33	C₂×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
34	D₅₃	—	8	2	no	—
35	#486	—	53	1	no	maximal, Sylow
36	C₅₃	#486	1	1	yes	characteristic, complemented, Sylow
37	C₄⋊Q₈	D₅₃	1	1	yes	characteristic, complemented
38	#127	—	53	1	no	—
39	#102	—	53	4	no	—
40	C₄×C₈	—	53	1	no	—
41	C₂×Q₈	—	2	1	no	—
42	C₂×D₄	—	53	2	no	—
43	C₂×D₄	—	106	1	no	—
44	C₂×D₄	—	106	1	no	—
45	C₂×C₈	—	53	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	C₂×D₅₃	1	2	yes	—
48	C₄²	C₂×D₅₃	1	1	yes	characteristic
49	C₂³	—	106	2	no	—
50	Q₈	—	4	2	no	—
51	D₄	—	106	4	no	—
52	D₄	—	212	2	no	—
53	D₄	—	212	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
57	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
58	C₈	—	106	2	no	—
59	C₂²	—	212	2	no	—
60	C₂²	—	212	2	no	—
61	C₂²	—	212	2	no	—
62	C₂²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
63	C₄	—	4	2	no	—
64	C₄	#11719922	1	2	yes	—
65	C₄	#11719924	1	2	yes	—
66	C₄	#11719949	1	2	yes	—
67	C₂	—	424	2	no	—
68	C₂	#13133206	1	1	yes	characteristic
69	C₂	C₂×#11719922	1	1	yes	characteristic
70	C₂	C₂×#11719924	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂³
Frattini	C₂×C₄	C₂²×D₅₃
Fitting	C₄⋊Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₄
prime radical	#13133214	C₂
hypercenter	C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#486
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #13133137	C₂	semidirect
3. C₄×#88322	C₂	semidirect
4. C₄⋊Q₈×C₅₃	C₂	semidirect
36. C₅₃	#486	semidirect
37. C₄⋊Q₈	D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#13133214	2
#13133137	2
C₄×#88322	2
C₄⋊Q₈×C₅₃	2
#486	53