Subgroups — #27274529

Order 13568.

478 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133128	C₂	1	2	yes	maximal
2	#20272542	C₂	1	4	yes	maximal
3	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133128	C₄	1	8	yes	—
5	C₂×C₄×#88322	C₂²	1	4	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	#13133128	—	2	16	no	—
11	C₄×#88322	—	2	8	no	—
12	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
13	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
14	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
16	#97×C₅₃	—	2	8	no	—
17	C₄×C₈×C₅₃	—	2	4	no	—
18	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
19	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
20	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	—	2	4	no	—
22	C₂×C₈×C₅₃	—	2	4	no	—
23	C₂×C₈×C₅₃	M₁₆	1	8	yes	—
24	C₄²×C₅₃	—	2	4	no	—
25	C₄²×C₅₃	—	2	8	no	—
26	C₄²×C₅₃	#41	1	4	yes	—
27	C₂×#88322	—	2	8	no	—
28	C₂×#88322	—	2	8	no	—
29	C₂×#88322	M₁₆	1	16	yes	—
30	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
31	C₂×C₄×C₅₃	—	2	8	no	—
32	C₂×C₄×C₅₃	—	2	4	no	—
33	C₂×C₄×C₅₃	#117	1	4	yes	—
34	C₂×C₄×C₅₃	#117	1	2	yes	—
35	C₂×C₄×C₅₃	C₂×M₁₆	1	8	yes	—
36	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
37	C₈×C₅₃	—	2	8	no	—
38	#88322	—	2	16	no	—
39	#8136	—	53	1	no	maximal, Sylow
40	C₂²×C₅₃	#376	1	1	yes	characteristic
41	C₂²×C₅₃	C₄×M₁₆	1	4	yes	—
42	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
43	C₄×C₅₃	—	4	4	no	—
44	C₄×C₅₃	#432	1	8	yes	—
45	C₄×C₅₃	#432	1	4	yes	—
46	C₄×#97	—	53	2	no	—
47	C₄×#97	D₅₃	1	1	yes	characteristic
48	#1273	—	53	4	no	—
49	C₂×C₅₃	#1455	1	4	yes	—
50	C₂×C₅₃	#1745	1	2	yes	—
51	C₂×C₅₃	#1745	1	1	yes	characteristic
52	C₂×#97	—	53	8	no	—
53	C₂×#97	Dic₅₃	1	4	yes	—
54	C₂×C₄×C₈	—	53	4	no	—
55	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
56	C₄³	C₂×D₅₃	1	1	yes	characteristic
57	C₅₃	#8136	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₈	—	53	8	no	—
59	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
60	C₂×C₄²	C₄×D₅₃	1	4	yes	—
61	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
62	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
63	#97	—	2	8	no	—
64	#97	—	106	16	no	—
65	C₄×C₈	—	2	4	no	—
66	C₄×C₈	—	106	8	no	—
67	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
68	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
69	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
70	C₂×C₈	—	2	4	no	—
71	C₂×C₈	—	2	4	no	—
72	C₂×C₈	—	53	16	no	—
73	C₂×C₈	—	106	8	no	—
74	C₂×C₈	—	106	8	no	—
75	C₂×C₈	#11719917	1	8	yes	—
76	C₄²	—	2	8	no	—
77	C₄²	—	2	4	no	—
78	C₄²	#11719945	1	4	yes	—
79	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
80	C₂×C₄	—	2	8	no	—
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	#13133174	1	4	yes	—
83	C₂×C₄	#13133176	1	2	yes	—
84	C₂×C₄	M₁₆×D₅₃	1	8	yes	—
85	C₂×C₄	C₂×#11719917	1	4	yes	—
86	C₈	—	2	8	no	—
87	C₈	—	106	16	no	—
88	C₂²	#16810469	1	1	yes	characteristic
89	C₂²	M₁₆×Dic₅₃	1	4	yes	—
90	C₂²	C₄×#11719917	1	2	yes	—
91	C₄	—	4	4	no	—
92	C₄	#16810678	1	8	yes	—
93	C₄	#16810680	1	4	yes	—
94	C₂	#20273018	1	4	yes	—
95	C₂	#20274463	1	2	yes	—
96	C₂	#20274464	1	1	yes	characteristic

	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	#8136
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
57. C₅₃	#8136	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133128	2
#20272542	2
C₄×#97×C₅₃	2
#8136	53