Subgroups — #27302714

Order 13568.

370 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133157	C₂	1	1	yes	maximal, characteristic
2	#20274872	C₂	1	1	yes	maximal, characteristic
3	#20272820	C₂	1	4	yes	maximal
4	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	2	yes	—
6	C₂×#13133149	—	2	1	no	—
7	C₂×#13133149	C₂²	1	2	yes	—
8	#16810493	—	2	1	no	—
9	#16810399	—	2	1	no	—
10	C₄×#11719908	—	2	1	no	—
11	C₅₃×#533	—	2	1	no	—
12	C₅₃×#500	C₄	1	2	yes	complemented
13	C₅₃×#499	—	2	1	no	—
14	C₅₃×#499	C₄	1	2	yes	complemented
15	C₅₃×#498	—	2	1	no	—
16	C₅₃×#427	—	2	1	no	—
17	C₅₃×#427	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₅₃	—	2	2	no	—
20	C₂×C₄⋊Dic₅₃	—	4	2	no	—
21	#13133157	—	2	2	no	—
22	#13133157	Q₈	1	4	yes	complemented
23	C₂×#11719908	—	2	2	no	—
24	#13133149	—	2	4	no	—
25	#13133149	—	4	2	no	—
26	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	C₄⋊Q₈×C₅₃	—	2	2	no	—
29	#125×C₅₃	—	2	2	no	—
30	C₂²⋊Q₈×C₅₃	—	4	2	no	—
31	C₄×Q₈×C₅₃	—	4	2	no	—
32	#117×C₅₃	—	2	2	no	—
33	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
34	#107×C₅₃	—	2	2	no	—
35	#107×C₅₃	C₂×C₄	1	4	yes	—
36	#106×C₅₃	—	2	2	no	—
37	#106×C₅₃	—	2	2	no	—
38	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₅₃	—	4	2	no	—
43	C₂²×C₄×C₅₃	D₈	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₅₃	—	4	2	no	—
46	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
49	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
50	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₅₃	#41	1	2	yes	—
54	C₄⋊C₄×C₅₃	—	2	4	no	—
55	C₄⋊C₄×C₅₃	—	4	4	no	—
56	C₄⋊C₄×C₅₃	—	4	2	no	—
57	C₂²⋊C₄×C₅₃	—	4	2	no	—
58	C₄²×C₅₃	—	4	2	no	—
59	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₅₃	SD₁₆	1	2	yes	—
62	C₄⋊Dic₅₃	—	4	2	no	—
63	C₄⋊Dic₅₃	—	4	2	no	—
64	C₄⋊Dic₅₃	—	8	4	no	—
65	#11719908	—	2	4	no	—
66	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
67	Q₈×C₅₃	—	8	4	no	—
68	C₂×C₄×C₅₃	—	2	4	no	—
69	C₂×C₄×C₅₃	—	4	4	no	—
70	C₂×C₄×C₅₃	#102	1	2	yes	—
71	C₂×C₄×C₅₃	#102	1	2	yes	—
72	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×Dic₅₃	—	8	4	no	—
79	C₂×Dic₅₃	—	8	2	no	—
80	C₈×C₅₃	#103	1	2	yes	—
81	C₈×C₅₃	#103	1	2	yes	—
82	C₈×C₅₃	C₄×Q₈	1	2	yes	—
83	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
84	#13287	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	—	2	2	no	—
86	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
87	C₂²×C₅₃	#486	1	1	yes	characteristic
88	C₂²×C₅₃	#500	1	1	yes	characteristic
89	C₄×C₅₃	—	2	2	no	—
90	C₄×C₅₃	—	8	4	no	—
91	C₄×C₅₃	#360	1	4	yes	—
92	C₄×C₅₃	#391	1	1	yes	characteristic
93	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
94	C₄×C₅₃	#487	1	1	yes	characteristic
95	C₄×C₅₃	#498	1	1	yes	characteristic
96	Dic₅₃	—	16	2	no	—
97	#3160	D₅₃	1	1	yes	characteristic
98	C₄×M₃₂	—	53	1	no	—
99	#1844	—	53	1	no	—
100	#1386	—	53	4	no	—
101	C₂×C₅₃	—	2	1	no	—
102	C₂×C₅₃	—	2	1	no	—
103	C₂×C₅₃	#1990	1	1	yes	characteristic
104	C₂×C₅₃	C₂×#360	1	2	yes	—
105	#533	—	2	1	no	—
106	C₂×M₃₂	—	53	2	no	—
107	#500	Dic₅₃	1	2	yes	complemented
108	#499	—	2	1	no	—
109	#499	Dic₅₃	1	2	yes	complemented
110	#498	—	2	1	no	—
111	#427	—	2	1	no	—
112	#427	C₂×D₅₃	1	2	yes	—
113	C₂×#106	—	53	2	no	—
114	C₂×#106	—	106	1	no	—
115	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
116	#384	—	106	1	no	—
117	#346	—	106	1	no	—
118	C₄×C₁₆	—	106	1	no	—
119	C₅₃	#13287	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
122	C₄⋊Q₈	—	2	2	no	—
123	#125	—	2	2	no	—
124	C₂²⋊Q₈	—	4	2	no	—
125	C₄×Q₈	—	4	2	no	—
126	#117	—	2	2	no	—
127	C₂×C₄⋊C₄	—	106	2	no	—
128	C₂×C₄⋊C₄	—	212	2	no	—
129	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
130	M₃₂	—	53	4	no	—
131	M₃₂	—	106	2	no	—
132	C₂×C₁₆	—	106	2	no	—
133	#107	—	2	2	no	—
134	#107	C₂×Dic₅₃	1	4	yes	—
135	#106	—	2	2	no	—
136	#106	—	2	2	no	—
137	#106	—	106	4	no	—
138	#106	—	212	2	no	—
139	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
140	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
141	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
143	C₂×Q₈	—	4	2	no	—
144	C₂²×C₄	—	212	2	no	—
145	C₂²×C₄	#11719922	1	2	yes	—
146	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
147	C₂×C₈	#11719924	1	2	yes	—
148	C₂×C₈	#11719925	1	2	yes	—
149	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
150	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
151	C₂×C₈	Q₈×D₅₃	1	2	yes	—
152	C₂×C₈	#11719949	1	2	yes	—
153	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
154	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	4	4	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	424	4	no	—
159	C₄⋊C₄	—	212	2	no	—
160	C₄⋊C₄	—	212	2	no	—
161	C₂²⋊C₄	—	4	2	no	—
162	C₄²	—	4	2	no	—
163	C₄²	#11719923	1	2	yes	—
164	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
165	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
166	C₁₆	—	106	4	no	—
167	C₂³	C₂×#11719922	1	1	yes	characteristic
168	Q₈	—	8	4	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	424	4	no	—
172	C₂×C₄	—	424	2	no	—
173	C₂×C₄	#13133138	1	2	yes	—
174	C₂×C₄	#13133138	1	2	yes	—
175	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
176	C₂×C₄	#13133206	1	1	yes	characteristic
177	C₂×C₄	#13133215	1	1	yes	characteristic
178	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
179	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
180	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
181	C₈	#13133141	1	2	yes	—
182	C₈	#13133141	1	2	yes	—
183	C₈	Q₈×Dic₅₃	1	2	yes	—
184	C₈	#13133193	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₂×#13133138	1	1	yes	characteristic
187	C₂²	#16810971	1	1	yes	characteristic
188	C₂²	#16811026	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	8	4	no	—
191	C₄	—	848	2	no	—
192	C₄	#16810433	1	4	yes	—
193	C₄	#16810530	1	1	yes	characteristic
194	C₄	C₂×#13133141	1	1	yes	characteristic
195	C₄	#16810975	1	1	yes	characteristic
196	C₄	#16811019	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#20275677	1	1	yes	characteristic
200	C₂	C₂×#16810433	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13133138
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#3160	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#3160	D₅₃
hypocenter	C₅₃	#13287
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
12. C₅₃×#500	C₄	semidirect
14. C₅₃×#499	C₄	semidirect
22. #13133157	Q₈	semidirect
107. #500	Dic₅₃	semidirect
109. #499	Dic₅₃	semidirect
119. C₅₃	#13287	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133157	2
#20274872	2
#20272820	2
C₅₃×#3160	2
#13287	53