Subgroups — #27274525

Order 13568.

414 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274461	C₂	1	2	yes	maximal
2	#20272542	C₂	1	4	yes	maximal
3	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#88322	C₂²	1	4	yes	—
5	#16810378	C₄	1	8	yes	—
6	C₂×#97×C₅₃	C₄	1	4	yes	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
8	C₄³×C₅₃	C₂²	1	1	yes	characteristic
9	C₂²×#88322	C₂×C₄	1	8	yes	—
10	C₄×#88322	—	2	8	no	—
11	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
12	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
13	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
15	#97×C₅₃	—	2	8	no	—
16	C₄×C₈×C₅₃	—	2	4	no	—
17	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
18	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
20	C₂×C₈×C₅₃	—	2	4	no	—
21	C₂×C₈×C₅₃	—	2	4	no	—
22	C₂×C₈×C₅₃	M₁₆	1	8	yes	—
23	C₄²×C₅₃	—	2	4	no	—
24	C₄²×C₅₃	—	2	8	no	—
25	C₄²×C₅₃	#41	1	4	yes	—
26	C₂×#88322	—	2	8	no	—
27	C₂×#88322	—	2	16	no	—
28	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
29	C₂×C₄×C₅₃	—	2	4	no	—
30	C₂×C₄×C₅₃	—	2	8	no	—
31	C₂×C₄×C₅₃	#117	1	4	yes	—
32	C₂×C₄×C₅₃	#117	1	2	yes	—
33	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
34	C₂×C₄×C₅₃	#131	1	8	yes	—
35	C₈×C₅₃	—	2	8	no	—
36	#88322	—	4	8	no	—
37	#8134	—	53	1	no	maximal, Sylow
38	C₂²×C₅₃	#376	1	1	yes	characteristic
39	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
40	C₂²×C₅₃	#405	1	4	yes	—
41	C₄×C₅₃	—	4	4	no	—
42	C₄×C₅₃	#432	1	4	yes	—
43	C₄×C₅₃	#433	1	8	yes	—
44	#1744	—	53	2	no	—
45	C₄×#97	D₅₃	1	1	yes	characteristic
46	#1273	—	53	4	no	—
47	C₂×C₅₃	#1453	1	4	yes	—
48	C₂×C₅₃	#1745	1	1	yes	characteristic
49	C₂×C₅₃	#1746	1	2	yes	—
50	C₂×#97	Dic₅₃	1	4	yes	—
51	C₂×C₄×C₈	—	53	4	no	—
52	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
53	C₄³	C₂×D₅₃	1	1	yes	characteristic
54	#336	—	53	8	no	—
55	C₅₃	#8134	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₈	—	53	8	no	—
57	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
58	C₂×C₄²	C₄×D₅₃	1	4	yes	—
59	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
60	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
61	#97	—	2	8	no	—
62	C₄×C₈	—	2	4	no	—
63	C₄×C₈	—	106	8	no	—
64	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
65	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
66	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
67	C₂×C₈	—	2	4	no	—
68	C₂×C₈	—	2	4	no	—
69	C₂×C₈	—	106	8	no	—
70	C₂×C₈	—	106	16	no	—
71	C₂×C₈	#11719917	1	8	yes	—
72	C₄²	—	2	4	no	—
73	C₄²	—	2	8	no	—
74	C₄²	#11719945	1	4	yes	—
75	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
76	C₂×C₄	—	2	4	no	—
77	C₂×C₄	—	2	8	no	—
78	C₂×C₄	#13133174	1	4	yes	—
79	C₂×C₄	#13133176	1	2	yes	—
80	C₂×C₄	C₂×#11719917	1	4	yes	—
81	C₂×C₄	#13133225	1	8	yes	—
82	C₈	—	2	8	no	—
83	C₈	—	212	8	no	—
84	C₂²	#16810469	1	1	yes	characteristic
85	C₂²	C₄×#11719917	1	2	yes	—
86	C₂²	#16810581	1	4	yes	—
87	C₄	—	4	4	no	—
88	C₄	#16810680	1	4	yes	—
89	C₄	#16810685	1	8	yes	—
90	C₂	#20273012	1	4	yes	—
91	C₂	#20274464	1	1	yes	characteristic
92	C₂	#20274468	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂×C₄²
Fitting	C₄×#97×C₅₃	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#97	D₅₃
hypocenter	C₅₃	#8134
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
55. C₅₃	#8134	semidirect

Maximal subgroups

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

subgroup	index
#20274461	2
#20272542	2
C₄×#97×C₅₃	2
#8134	53