Subgroups — #27284232

Order 13568.

430 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274832	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	#20272598	C₂	1	2	yes	maximal
4	#20272598	C₂	1	2	yes	maximal
5	C₅₃×#2859	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×C₄×#88322	—	2	1	no	—
9	#16810378	—	2	1	no	—
10	#16810378	—	2	1	no	—
11	#16810378	—	2	1	no	—
12	C₂×#125×C₅₃	C₄	1	2	yes	—
13	C₅₃×#389	—	2	1	no	—
14	C₅₃×#389	—	2	1	no	—
15	C₅₃×#389	—	2	1	no	—
16	C₅₃×#389	—	2	1	no	—
17	C₅₃×#384	C₂²	1	1	yes	characteristic
18	C₅₃×#384	C₂²	1	1	yes	characteristic
19	C₅₃×#382	C₄	1	2	yes	—
20	C₄³×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×#88322	—	2	2	no	—
22	C₂²×#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	4	no	—
25	C₄⋊#88322	Q₈	1	4	yes	—
26	C₄×#88322	—	4	2	no	—
27	#125×C₅₃	—	4	4	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	#95×C₅₃	—	2	2	no	—
44	#95×C₅₃	—	2	2	no	—
45	C₂²×C₄×C₅₃	—	2	2	no	—
46	C₂²×C₄×C₅₃	—	2	2	no	—
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₄⋊C₄×C₅₃	—	4	4	no	—
55	C₄⋊C₄×C₅₃	—	4	2	no	—
56	C₄⋊C₄×C₅₃	—	4	2	no	—
57	C₄⋊C₄×C₅₃	—	4	4	no	—
58	C₄⋊C₄×C₅₃	—	4	2	no	—
59	C₄⋊C₄×C₅₃	—	4	2	no	—
60	C₄⋊C₄×C₅₃	—	4	2	no	—
61	C₄⋊C₄×C₅₃	—	4	1	no	—
62	C₄⋊C₄×C₅₃	—	4	1	no	—
63	C₄²×C₅₃	—	2	2	no	—
64	C₄²×C₅₃	—	2	2	no	—
65	C₄²×C₅₃	—	2	2	no	—
66	C₄²×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	—	2	2	no	—
68	C₄²×C₅₃	—	2	1	no	—
69	C₄²×C₅₃	—	2	1	no	—
70	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
71	C₄²×C₅₃	#41	1	2	yes	—
72	C₂×#88322	—	2	4	no	—
73	C₂×#88322	—	4	2	no	—
74	C₂×#88322	—	4	2	no	—
75	C₂×#88322	—	4	2	no	—
76	C₂×#88322	—	4	2	no	—
77	C₂×#88322	—	4	2	no	—
78	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	4	2	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	—	4	2	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	#100	1	2	yes	—
98	C₂×C₄×C₅₃	#101	1	2	yes	—
99	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
100	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
101	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#135	1	2	yes	—
104	C₂×C₄×C₅₃	#135	1	2	yes	—
105	C₂×C₄×C₅₃	#135	1	2	yes	—
106	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
108	#88322	—	4	4	no	—
109	#88322	—	8	2	no	—
110	#9901	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	#391	1	1	yes	characteristic
112	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
113	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
114	C₂²×C₅₃	#416	1	1	yes	characteristic
115	C₂²×C₅₃	#416	1	1	yes	characteristic
116	C₂²×C₅₃	#416	1	1	yes	characteristic
117	C₂²×C₅₃	#417	1	1	yes	characteristic
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	2	2	no	—
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	4	1	no	—
122	C₄×C₅₃	—	4	1	no	—
123	C₄×C₅₃	—	8	2	no	—
124	C₄×C₅₃	—	8	2	no	—
125	C₄×C₅₃	#478	1	2	yes	—
126	C₄×C₅₃	#479	1	2	yes	—
127	C₄×C₅₃	#488	1	2	yes	—
128	C₄×C₅₃	#499	1	2	yes	—
129	#2859	D₅₃	1	1	yes	characteristic
130	#1831	—	53	1	no	—
131	C₄×C₄⋊C₈	—	53	1	no	—
132	#1296	—	53	2	no	—
133	#1296	—	53	2	no	—
134	C₂×C₅₃	#1543	1	1	yes	characteristic
135	C₂×C₅₃	#1544	1	1	yes	characteristic
136	C₂×C₅₃	#1551	1	1	yes	characteristic
137	C₂×C₅₃	#1552	1	1	yes	characteristic
138	C₂×C₅₃	#1989	1	1	yes	characteristic
139	C₂×C₅₃	#1991	1	1	yes	characteristic
140	C₂×C₅₃	#1997	1	1	yes	characteristic
141	C₂×#125	Dic₅₃	1	2	yes	—
142	C₂×C₄⋊C₈	—	53	2	no	—
143	C₂×C₄⋊C₈	—	53	2	no	—
144	C₂×C₄×C₈	—	106	1	no	—
145	#389	—	2	1	no	—
146	#389	—	2	1	no	—
147	#389	—	2	1	no	—
148	#389	—	2	1	no	—
149	#384	C₂×D₅₃	1	1	yes	characteristic
150	#384	C₂×D₅₃	1	1	yes	characteristic
151	#382	Dic₅₃	1	2	yes	—
152	C₄³	C₂×D₅₃	1	1	yes	characteristic
153	#336	—	106	1	no	—
154	#336	—	106	1	no	—
155	#336	—	106	1	no	—
156	C₅₃	#9901	1	1	yes	characteristic, complemented, Sylow
157	C₂²×C₈	—	106	2	no	—
158	C₂²×C₈	—	106	2	no	—
159	#125	—	4	4	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
167	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
168	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
169	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
170	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
171	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
172	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
173	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
175	C₄⋊C₈	—	53	4	no	—
176	C₄⋊C₈	—	106	2	no	—
177	C₄⋊C₈	—	106	4	no	—
178	C₄×C₈	—	212	2	no	—
179	#95	—	2	2	no	—
180	#95	—	2	2	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
184	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
185	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
186	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
187	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
190	C₂×C₈	—	106	4	no	—
191	C₂×C₈	—	212	2	no	—
192	C₂×C₈	—	212	2	no	—
193	C₂×C₈	—	212	2	no	—
194	C₂×C₈	—	212	2	no	—
195	C₂×C₈	—	212	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₄⋊C₄	—	4	4	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	1	no	—
204	C₄⋊C₄	—	4	1	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	Q₈×D₅₃	1	2	yes	—
213	C₄²	#11719949	1	2	yes	—
214	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	#13133134	1	2	yes	—
234	C₂×C₄	#13133136	1	2	yes	—
235	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
236	C₂×C₄	#13133193	1	2	yes	—
237	C₂×C₄	#13133206	1	1	yes	characteristic
238	C₂×C₄	#13133215	1	1	yes	characteristic
239	C₂×C₄	#13133237	1	2	yes	—
240	C₂×C₄	#13133237	1	2	yes	—
241	C₂×C₄	#13133237	1	2	yes	—
242	C₂×C₄	#13133242	1	1	yes	characteristic
243	C₂×C₄	#13133247	1	1	yes	characteristic
244	C₈	—	424	2	no	—
245	C₈	—	212	4	no	—
246	C₂²	#16810530	1	1	yes	characteristic
247	C₂²	C₂×#13133134	1	1	yes	characteristic
248	C₂²	C₂×#13133136	1	1	yes	characteristic
249	C₂²	#16810619	1	1	yes	characteristic
250	C₂²	#16810619	1	1	yes	characteristic
251	C₂²	#16810619	1	1	yes	characteristic
252	C₂²	#16810624	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	1	no	—
258	C₄	—	8	2	no	—
259	C₄	—	8	2	no	—
260	C₄	#16810920	1	2	yes	—
261	C₄	#16810927	1	2	yes	—
262	C₄	#16810980	1	2	yes	—
263	C₄	#16811022	1	2	yes	—
264	C₂	#20273377	1	1	yes	characteristic
265	C₂	#20273380	1	1	yes	characteristic
266	C₂	#20273410	1	1	yes	characteristic
267	C₂	#20273415	1	1	yes	characteristic
268	C₂	#20275673	1	1	yes	characteristic
269	C₂	#20275681	1	1	yes	characteristic
270	C₂	#20275705	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2859	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2859	D₅₃
hypocenter	C₅₃	#9901
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

Complements

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

N	H ≅ G/N	type
156. C₅₃	#9901	semidirect

Maximal subgroups

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

subgroup	index
#20274832	2
C₄×C₄⋊#88322	2
#20272598	2
C₅₃×#2859	2
#9901	53