Subgroups — #16811234

Order 3392.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133212	C₂	1	4	yes	maximal
2	#13133209	C₂	1	1	yes	maximal, characteristic, complemented
3	#13133207	C₂	1	4	yes	maximal, complemented
4	#13133193	C₂	1	2	yes	maximal
5	Q₈×Dic₅₃	C₂	1	2	yes	maximal, complemented
6	#13133174	C₂	1	1	yes	maximal, characteristic
7	C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄×D₅₃	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
11	C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	#11719921	C₂²	1	2	yes	—
13	#11719921	C₂²	1	4	yes	—
14	C₄⋊Dic₅₃	C₂²	1	2	yes	—
15	C₄⋊Dic₅₃	C₂²	1	4	yes	—
16	#11719919	C₂²	1	2	yes	—
17	#11719919	C₂²	1	4	yes	—
18	#11719919	C₂²	1	4	yes	—
19	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
20	C₄×Dic₅₃	C₂²	1	4	yes	—
21	C₂²×D₅₃	C₂³	1	1	yes	characteristic
22	Q₈×C₅₃	—	2	4	no	—
23	C₂×C₄×C₅₃	C₂³	1	4	yes	—
24	C₂×C₄×C₅₃	C₂³	1	2	yes	—
25	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×Dic₅₃	C₂³	1	1	yes	characteristic
27	C₂×Dic₅₃	C₂³	1	2	yes	—
28	C₂×Dic₅₃	C₂³	1	4	yes	—
29	C₄×D₅₃	—	2	2	no	—
30	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
31	C₂×D₅₃	—	2	1	no	—
32	C₂×D₅₃	—	2	2	no	—
33	C₄×C₅₃	—	2	4	no	—
34	C₄×C₅₃	—	2	2	no	—
35	C₄×C₅₃	#41	1	2	yes	—
36	Dic₅₃	—	2	1	no	—
37	Dic₅₃	—	2	2	no	—
38	Dic₅₃	—	2	4	no	—
39	C₂×C₅₃	C₂×#41	1	1	yes	characteristic
40	C₂×C₅₃	2₋¹⁺⁴	1	2	yes	—
41	D₅₃	—	4	1	no	—
42	#541	—	53	1	no	maximal, Sylow
43	C₅₃	#541	1	1	yes	characteristic, complemented, Sylow
44	C₄⋊Q₈	D₅₃	1	1	yes	characteristic, complemented
45	#126	—	53	4	no	—
46	#125	—	53	4	no	—
47	#125	—	53	1	no	—
48	C₂²⋊Q₈	—	53	2	no	—
49	C₄×Q₈	—	53	2	no	—
50	#117	—	53	1	no	—
51	C₂×Q₈	C₂×D₅₃	1	2	yes	—
52	C₂²×C₄	—	53	1	no	—
53	C₄⋊C₄	—	53	2	no	—
54	C₄⋊C₄	—	53	2	no	—
55	C₄⋊C₄	—	53	4	no	—
56	C₄⋊C₄	—	53	4	no	—
57	C₄⋊C₄	—	53	4	no	—
58	C₄⋊C₄	C₂×D₅₃	1	4	yes	—
59	C₂²⋊C₄	—	53	2	no	—
60	C₂²⋊C₄	—	53	4	no	—
61	C₄²	—	53	1	no	—
62	C₄²	—	53	4	no	—
63	C₄²	C₂×D₅₃	1	1	yes	characteristic
64	C₂³	—	53	1	no	—
65	Q₈	—	2	4	no	—
66	C₂×C₄	—	53	2	no	—
67	C₂×C₄	—	53	4	no	—
68	C₂×C₄	—	53	1	no	—
69	C₂×C₄	—	106	2	no	—
70	C₂×C₄	C₂²×D₅₃	1	4	yes	—
71	C₂×C₄	C₂²×D₅₃	1	2	yes	—
72	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
73	C₂²	—	106	1	no	—
74	C₂²	—	106	2	no	—
75	C₂²	C₂³×D₅₃	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	2	2	no	—
78	C₄	—	106	1	no	—
79	C₄	—	106	2	no	—
80	C₄	—	106	4	no	—
81	C₄	#11719947	1	2	yes	—
82	C₂	—	212	1	no	—
83	C₂	C₂×#11719947	1	1	yes	characteristic
84	C₂	#13133265	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁴
Fitting	C₄⋊Q₈×C₅₃	C₂
prime radical	C₂²×D₅₃	C₂³
hypercenter	C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#541
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #13133209	C₂	semidirect
3. #13133207	C₂	semidirect
5. Q₈×Dic₅₃	C₂	semidirect
7. C₄⋊Q₈×C₅₃	C₂	semidirect
43. C₅₃	#541	semidirect
44. 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
#13133212	2
#13133209	2
#13133207	2
#13133193	2
Q₈×Dic₅₃	2
#13133174	2
C₄⋊Q₈×C₅₃	2
#541	53