Subgroups — #27275409

Order 13568.

342 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133153	C₂	1	1	yes	maximal, characteristic
2	#20273005	C₂	1	1	yes	maximal, characteristic
3	#20272996	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810345	C₂	1	1	yes	maximal, characteristic
5	#20272551	C₂	1	1	yes	maximal, characteristic
6	#20272551	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133153	C₄	1	2	yes	—
9	C₄×#11719917	C₂²	1	1	yes	characteristic
10	C₄×#11719917	C₂²	1	1	yes	characteristic
11	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	#16810388	C₄	1	2	yes	—
14	#16810345	—	2	1	no	—
15	#16810345	—	2	1	no	—
16	#16810345	C₄	1	2	yes	—
17	#16810345	C₄	1	2	yes	—
18	#16810344	—	2	1	no	—
19	C₈×#88322	—	2	1	no	—
20	C₅₃×#431	C₂²	1	1	yes	characteristic
21	C₅₃×#431	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#335	C₄	1	2	yes	—
24	C₅₃×#334	C₄	1	2	yes	—
25	C₂×#11719917	C₂×C₄	1	2	yes	—
26	C₂×#11719917	C₂×C₄	1	2	yes	—
27	C₂²×#88322	—	2	1	no	—
28	C₂²×#88322	Q₈	1	2	yes	—
29	#13133153	—	2	4	no	—
30	#13133128	—	2	1	no	—
31	#13133128	—	2	1	no	—
32	C₄×#88322	—	2	1	no	—
33	C₄×#88322	—	2	1	no	—
34	C₄×#88322	—	2	1	no	—
35	C₄×#88322	C₂×C₄	1	2	yes	—
36	C₄×#88322	C₂×C₄	1	2	yes	—
37	C₄×#88322	D₄	1	1	yes	characteristic
38	C₄×#88322	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₂²⋊C₈×C₅₃	—	2	1	no	—
45	C₂²⋊C₈×C₅₃	—	2	1	no	—
46	C₄×C₈×C₅₃	—	2	1	no	—
47	C₄×C₈×C₅₃	—	2	1	no	—
48	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
63	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₅₃	#41	1	1	yes	characteristic
67	#11719917	—	2	2	no	—
68	#11719917	—	2	2	no	—
69	#11719917	—	2	2	no	—
70	#11719917	M₁₆	1	4	yes	—
71	C₂×#88322	—	2	4	no	—
72	C₂×#88322	—	2	2	no	—
73	C₂×#88322	—	2	2	no	—
74	C₂×#88322	—	2	2	no	—
75	C₂×#88322	—	4	2	no	—
76	C₂×#88322	C₄⋊C₄	1	2	yes	—
77	C₂×#88322	C₄⋊C₄	1	2	yes	—
78	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₅₃	#131	1	2	yes	—
89	C₈×C₅₃	—	2	1	no	—
90	C₈×C₅₃	—	2	1	no	—
91	C₈×C₅₃	—	2	1	no	—
92	C₈×C₅₃	—	2	1	no	—
93	C₈×C₅₃	—	4	2	no	—
94	C₈×C₅₃	—	4	2	no	—
95	#88322	—	2	4	no	—
96	#88322	—	2	2	no	—
97	#88322	—	2	2	no	—
98	#88322	—	2	4	no	—
99	#88322	—	4	2	no	—
100	#8296	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	—	2	1	no	—
102	C₂²×C₅₃	—	2	1	no	—
103	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₅₃	#446	1	2	yes	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
108	C₄×C₅₃	#405	1	1	yes	characteristic
109	C₄×C₅₃	#430	1	2	yes	—
110	C₄×C₅₃	#435	1	2	yes	—
111	C₄×C₅₃	#443	1	2	yes	—
112	C₄×#108	—	53	1	no	—
113	#1564	D₅₃	1	1	yes	characteristic
114	#1451	—	53	1	no	—
115	C₈×M₁₆	—	53	1	no	—
116	C₂×#322	—	53	1	no	—
117	#1278	—	53	1	no	—
118	#1278	—	53	1	no	—
119	C₂×C₅₃	—	2	2	no	—
120	C₂×C₅₃	#1771	1	1	yes	characteristic
121	C₂×C₅₃	#1772	1	1	yes	characteristic
122	C₂×C₅₃	#1778	1	1	yes	characteristic
123	#431	C₂×D₅₃	1	1	yes	characteristic
124	#431	C₂×D₅₃	1	1	yes	characteristic
125	C₂×#108	—	53	2	no	—
126	C₄×M₁₆	—	53	1	no	—
127	C₄×M₁₆	—	53	1	no	—
128	C₂×C₄×C₈	—	53	1	no	—
129	C₂×C₄×C₈	—	53	1	no	—
130	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
131	#341	—	53	2	no	—
132	#335	Dic₅₃	1	2	yes	—
133	#334	Dic₅₃	1	2	yes	—
134	#322	—	53	2	no	—
135	#322	—	53	2	no	—
136	#322	—	106	1	no	—
137	#322	—	106	1	no	—
138	#322	—	106	1	no	—
139	C₈²	—	106	1	no	—
140	C₅₃	#8296	1	1	yes	characteristic, complemented, Sylow
141	C₂×M₁₆	—	53	2	no	—
142	C₂×M₁₆	—	53	2	no	—
143	C₂²×C₈	—	53	2	no	—
144	C₂²×C₈	—	106	1	no	—
145	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
146	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
147	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
148	#108	—	106	4	no	—
149	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
150	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
151	C₂²⋊C₈	—	2	1	no	—
152	C₂²⋊C₈	—	2	1	no	—
153	#97	—	106	1	no	—
154	#97	—	106	1	no	—
155	C₄×C₈	—	2	1	no	—
156	C₄×C₈	—	2	1	no	—
157	C₄×C₈	—	53	2	no	—
158	C₄×C₈	—	53	2	no	—
159	C₄×C₈	—	53	1	no	—
160	C₄×C₈	—	53	1	no	—
161	C₄×C₈	—	106	1	no	—
162	C₄×C₈	—	106	1	no	—
163	C₄×C₈	—	106	1	no	—
164	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
165	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
166	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
167	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
169	C₂²×C₄	#11719947	1	1	yes	characteristic
170	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
171	M₁₆	—	53	4	no	—
172	M₁₆	—	106	2	no	—
173	M₁₆	—	106	2	no	—
174	M₁₆	—	106	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	53	2	no	—
180	C₂×C₈	—	53	2	no	—
181	C₂×C₈	—	106	4	no	—
182	C₂×C₈	—	106	2	no	—
183	C₂×C₈	—	106	2	no	—
184	C₂×C₈	—	106	2	no	—
185	C₂×C₈	—	212	2	no	—
186	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
187	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
188	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
189	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
190	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
191	C₄²	D₄×D₅₃	1	1	yes	characteristic
192	C₄²	#11719949	1	1	yes	characteristic
193	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
194	C₂³	#13133182	1	1	yes	characteristic
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
198	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
199	C₂×C₄	#13133175	1	1	yes	characteristic
200	C₂×C₄	#13133179	1	1	yes	characteristic
201	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
202	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
203	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
204	C₂×C₄	#13133223	1	2	yes	—
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	4	2	no	—
210	C₈	—	4	2	no	—
211	C₈	—	106	4	no	—
212	C₈	—	106	2	no	—
213	C₈	—	106	2	no	—
214	C₈	—	106	4	no	—
215	C₈	—	212	2	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
219	C₂²	#16810752	1	2	yes	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	M₁₆×Dic₅₃	1	1	yes	characteristic
223	C₄	#16810578	1	1	yes	characteristic
224	C₄	#16810670	1	2	yes	—
225	C₄	#16810693	1	2	yes	—
226	C₄	#16810740	1	2	yes	—
227	C₂	—	2	2	no	—
228	C₂	#20274592	1	1	yes	characteristic
229	C₂	#20274596	1	1	yes	characteristic
230	C₂	#20274623	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1564	C₂
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂³×C₅₃	C₄×Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1564	D₅₃
hypocenter	C₅₃	#8296
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
140. C₅₃	#8296	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133153	2
#20273005	2
#20272996	2
C₂×#16810345	2
#20272551	2
C₅₃×#1564	2
#8296	53