Subgroups — #16811348

Order 3392.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133215	C₂	1	2	yes	maximal
2	#13133208	C₂	1	2	yes	maximal
3	#13133207	C₂	1	4	yes	maximal
4	Q₈×Dic₅₃	C₂	1	2	yes	maximal
5	C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
7	#13133182	C₂	1	2	yes	maximal
8	C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₈×C₅₃	C₂²	1	2	yes	—
10	C₂×Dic₁₀₆	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₀₆	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
13	C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
14	C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₅₃	C₂²	1	4	yes	—
18	#11719919	C₂²	1	4	yes	—
19	#11719919	C₂²	1	2	yes	—
20	#11719919	C₂²	1	2	yes	—
21	#11719919	C₂²	1	4	yes	—
22	C₄×Dic₅₃	C₂²	1	2	yes	—
23	C₄×Dic₅₃	C₂²	1	4	yes	—
24	C₄×Dic₅₃	C₂²	1	2	yes	—
25	Q₈×C₅₃	—	2	4	no	—
26	C₂×C₄×C₅₃	C₂³	1	4	yes	—
27	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
30	C₂×Dic₅₃	C₂³	1	2	yes	—
31	C₂×Dic₅₃	C₂³	1	2	yes	—
32	C₂×Dic₅₃	C₂³	1	2	yes	—
33	C₂×Dic₅₃	C₂³	1	2	yes	—
34	Dic₁₀₆	—	2	2	no	—
35	Dic₁₀₆	Q₈	1	4	yes	complemented
36	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
37	C₄×C₅₃	—	2	4	no	—
38	C₄×C₅₃	—	2	1	no	—
39	C₄×C₅₃	C₂×Q₈	1	2	yes	—
40	C₄×C₅₃	#41	1	2	yes	—
41	Dic₅₃	—	2	2	no	—
42	Dic₅₃	—	2	2	no	—
43	Dic₅₃	—	2	2	no	—
44	Dic₅₃	C₂×Q₈	1	4	yes	—
45	C₂×C₅₃	C₂²×Q₈	1	1	yes	characteristic
46	C₂×C₅₃	C₂×#41	1	1	yes	characteristic
47	C₂×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
48	#557	—	53	1	no	maximal, Sylow
49	C₅₃	#557	1	1	yes	characteristic, complemented, Sylow
50	C₄⋊Q₈	—	53	2	no	—
51	C₄⋊Q₈	D₅₃	1	1	yes	characteristic
52	#125	—	53	4	no	—
53	#125	—	53	2	no	—
54	C₄×Q₈	—	53	2	no	—
55	C₄×Q₈	—	53	2	no	—
56	C₄×Q₈	—	53	1	no	—
57	C₄×Q₈	—	53	1	no	—
58	C₂×Q₈	—	53	1	no	—
59	C₂×Q₈	—	53	1	no	—
60	C₂×Q₈	C₂×D₅₃	1	2	yes	—
61	C₄⋊C₄	—	53	1	no	—
62	C₄⋊C₄	—	53	2	no	—
63	C₄⋊C₄	—	53	2	no	—
64	C₄⋊C₄	—	53	4	no	—
65	C₄⋊C₄	—	53	1	no	—
66	C₄⋊C₄	—	53	4	no	—
67	C₄⋊C₄	—	53	4	no	—
68	C₄⋊C₄	C₂×D₅₃	1	2	yes	—
69	C₄⋊C₄	C₂×D₅₃	1	2	yes	—
70	C₄²	—	53	4	no	—
71	C₄²	—	53	2	no	—
72	C₄²	—	53	2	no	—
73	C₄²	C₂×D₅₃	1	1	yes	characteristic
74	Q₈	—	2	4	no	—
75	Q₈	—	53	4	no	—
76	Q₈	—	106	2	no	—
77	C₂×C₄	—	53	2	no	—
78	C₂×C₄	—	53	2	no	—
79	C₂×C₄	—	53	2	no	—
80	C₂×C₄	—	53	2	no	—
81	C₂×C₄	C₂²×D₅₃	1	4	yes	—
82	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
83	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
84	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
85	C₂²	C₂³×D₅₃	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	—	2	1	no	—
88	C₄	—	53	4	no	—
89	C₄	—	106	2	no	—
90	C₄	—	106	2	no	—
91	C₄	—	106	2	no	—
92	C₄	#11719947	1	2	yes	—
93	C₄	Q₈×D₅₃	1	2	yes	—
94	C₂	C₂×Q₈×D₅₃	1	1	yes	characteristic
95	C₂	C₂×#11719947	1	1	yes	characteristic
96	C₂	#13133265	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, prime radical, cosocle	C₂²×C₅₃	C₂⁴
Fitting	C₄⋊Q₈×C₅₃	C₂
hypercenter	C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#557
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
35. Dic₁₀₆	Q₈	semidirect
49. C₅₃	#557	semidirect

Maximal subgroups

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

subgroup	index
#13133215	2
#13133208	2
#13133207	2
Q₈×Dic₅₃	2
C₄×Dic₁₀₆	2
#13133182	2
C₄⋊Q₈×C₅₃	2
#557	53