Subgroups — #27270829

Order 13568.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272779	C₂	1	1	yes	maximal, characteristic
2	#20272632	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1367	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11719908	—	2	1	no	—
5	C₄×#11719908	C₄	1	1	yes	characteristic
6	C₄×#11719908	C₄	1	1	yes	characteristic
7	C₅₃×C₄⋊C₁₆	—	2	1	no	—
8	C₈²×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#11719908	—	2	1	no	—
10	C₂×#11719908	—	4	1	no	—
11	C₂×#11719908	C₈	1	1	yes	characteristic
12	C₂×#11719908	C₈	1	1	yes	characteristic
13	C₂×C₁₆×C₅₃	—	4	1	no	—
14	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
16	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
17	C₂×C₈×C₅₃	—	2	1	no	—
18	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
19	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
20	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
21	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
22	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
23	C₁₆×C₅₃	—	8	1	no	—
24	#11719908	—	2	2	no	—
25	#11719908	—	2	1	no	—
26	#11719908	—	8	1	no	—
27	#11719908	C₁₆	1	2	yes	complemented
28	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
29	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
30	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
31	C₈×C₅₃	—	2	1	no	—
32	C₈×C₅₃	—	2	1	no	—
33	C₈×C₅₃	—	2	1	no	—
34	C₈×C₅₃	—	4	1	no	—
35	C₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
36	C₈×C₅₃	M₃₂	1	1	yes	characteristic
37	#7099	—	53	1	no	maximal, Sylow
38	C₂²×C₅₃	#334	1	1	yes	characteristic
39	C₄×C₅₃	—	2	1	no	—
40	C₄×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
41	C₄×C₅₃	#364	1	1	yes	characteristic
42	C₄×C₅₃	#365	1	1	yes	characteristic
43	C₄×C₅₃	#365	1	1	yes	characteristic
44	#1367	—	53	1	no	—
45	#1367	D₅₃	1	1	yes	characteristic
46	#1311	—	53	1	no	—
47	C₂×C₅₃	#1367	1	1	yes	characteristic
48	C₂×C₅₃	#1368	1	1	yes	characteristic
49	C₂×C₅₃	#1369	1	1	yes	characteristic
50	C₄⋊C₁₆	—	2	1	no	—
51	C₄⋊C₁₆	—	106	1	no	—
52	C₄×C₁₆	—	53	1	no	—
53	C₄×C₁₆	—	53	1	no	—
54	C₈²	C₂×D₅₃	1	1	yes	characteristic
55	C₅₃	#7099	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₁₆	—	4	1	no	—
57	C₂×C₁₆	—	53	1	no	—
58	C₂×C₁₆	—	53	1	no	—
59	C₂×C₁₆	—	106	1	no	—
60	C₂×C₁₆	—	212	1	no	—
61	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
62	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
63	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
66	C₂×C₈	#11719912	1	1	yes	characteristic
67	C₂×C₈	#11719923	1	1	yes	characteristic
68	C₂×C₈	#11719924	1	1	yes	characteristic
69	C₄²	#11719919	1	1	yes	characteristic
70	C₁₆	—	8	1	no	—
71	C₁₆	—	53	2	no	—
72	C₁₆	—	106	2	no	—
73	C₁₆	—	106	1	no	—
74	C₁₆	—	424	1	no	—
75	C₂×C₄	#13133146	1	1	yes	characteristic
76	C₂×C₄	#13133148	1	1	yes	characteristic
77	C₂×C₄	#13133152	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	2	1	no	—
80	C₈	—	2	1	no	—
81	C₈	—	4	1	no	—
82	C₈	C₁₆×D₅₃	1	1	yes	characteristic
83	C₈	#13133156	1	1	yes	characteristic
84	C₂²	#16810373	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	#16810441	1	1	yes	characteristic
87	C₄	#16810443	1	1	yes	characteristic
88	C₄	#16810445	1	1	yes	characteristic
89	C₄	#16810447	1	1	yes	characteristic
90	C₂	#20272778	1	1	yes	characteristic
91	C₂	#20272780	1	1	yes	characteristic
92	C₂	#20272782	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133148
commutator	C₈×C₅₃	C₂×C₁₆
Frattini	C₈²	C₂×D₅₃
Fitting	C₅₃×#1367	C₂
socle, prime radical	C₂²×C₅₃	#334
cosocle	C₈²×C₅₃	C₂²
hypercenter	#1367	D₅₃
hypocenter	C₅₃	#7099
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1367

Complements

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

N	H ≅ G/N	type
27. #11719908	C₁₆	semidirect
55. C₅₃	#7099	semidirect

Maximal subgroups

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

subgroup	index
#20272779	2
#20272632	2
C₅₃×#1367	2
#7099	53