Subgroups — #27289733

Order 13568.

650 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810394	C₂	1	2	yes	maximal
2	#20274454	C₂	1	1	yes	maximal, characteristic
3	#20274448	C₂	1	1	yes	maximal, characteristic
4	#20274411	C₂	1	2	yes	maximal
5	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	2	yes	—
7	C₂×#13133176	C₂²	1	2	yes	—
8	#16810394	—	2	4	no	—
9	#16810392	C₄	1	4	yes	complemented
10	#16810390	C₄	1	4	yes	—
11	#16810377	—	2	4	no	—
12	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#405	—	2	4	no	—
16	C₂×#97×C₅₃	C₄	1	2	yes	complemented
17	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	complemented
18	C₂×#11719917	—	2	2	no	—
19	C₂×#11719917	—	2	2	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	C₂×C₄	1	4	yes	—
24	C₂²×#88322	—	2	2	no	—
25	#13133176	—	2	2	no	—
26	#13133176	—	2	2	no	—
27	#13133176	—	2	2	no	—
28	#13133176	—	2	2	no	—
29	C₂×C₄⋊Dic₅₃	—	2	2	no	—
30	C₂×C₂²⋊Dic₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₄×Dic₅₃	—	2	2	no	—
32	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₅₃	—	2	2	no	—
34	C₂×M₁₆×C₅₃	—	2	2	no	—
35	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
36	C₂×M₁₆×C₅₃	Q₈	1	2	yes	—
37	C₂²×C₈×C₅₃	—	2	2	no	—
38	C₂²×C₈×C₅₃	—	2	1	no	—
39	C₂²×C₈×C₅₃	—	2	1	no	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₅₃	D₄	1	4	yes	—
44	#117×C₅₃	—	2	2	no	—
45	#117×C₅₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
49	#97×C₅₃	—	2	4	no	—
50	C₄×C₈×C₅₃	—	2	4	no	—
51	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	#41	1	2	yes	—
59	C₂²×C₄×C₅₃	#41	1	2	yes	—
60	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	#41	1	2	yes	—
62	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
63	C₂²×Dic₅₃	—	2	4	no	—
64	M₁₆×C₅₃	—	2	4	no	—
65	M₁₆×C₅₃	—	2	4	no	—
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	—	2	1	no	—
68	C₂×C₈×C₅₃	—	2	1	no	—
69	C₂×C₈×C₅₃	—	2	1	no	—
70	C₂×C₈×C₅₃	—	2	1	no	—
71	C₂×C₈×C₅₃	—	2	2	no	—
72	C₂×C₈×C₅₃	—	2	4	no	—
73	C₂×C₈×C₅₃	—	2	4	no	—
74	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
77	C₄⋊C₄×C₅₃	—	2	4	no	—
78	C₂²⋊C₄×C₅₃	—	2	4	no	—
79	C₄²×C₅₃	—	2	2	no	—
80	C₄²×C₅₃	—	2	2	no	—
81	C₂²⋊Dic₅₃	—	2	4	no	—
82	C₂²⋊Dic₅₃	—	2	4	no	—
83	C₄⋊Dic₅₃	—	4	2	no	—
84	C₄⋊Dic₅₃	—	4	2	no	—
85	C₄×Dic₅₃	—	4	2	no	—
86	C₄×Dic₅₃	—	4	2	no	—
87	#11719917	—	2	4	no	—
88	#11719917	—	2	4	no	—
89	#11719917	—	4	2	no	—
90	#11719917	—	4	2	no	—
91	C₂×#88322	—	2	4	no	—
92	C₂×#88322	—	4	2	no	—
93	C₂×#88322	—	4	2	no	—
94	C₂³×C₅₃	—	4	2	no	—
95	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
96	C₂³×C₅₃	#117	1	2	yes	—
97	C₂³×C₅₃	C₄×D₄	1	2	yes	—
98	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
99	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	—	2	4	no	—
101	C₂×C₄×C₅₃	—	4	4	no	—
102	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#117	1	2	yes	—
105	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#121	1	2	yes	—
113	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₅₃	#123	1	2	yes	—
115	C₂×C₄×C₅₃	#123	1	2	yes	—
116	C₂×C₄×C₅₃	#124	1	2	yes	—
117	C₂×C₄×C₅₃	#125	1	2	yes	—
118	C₂×C₄×C₅₃	#126	1	2	yes	—
119	C₂×C₄×C₅₃	#126	1	2	yes	—
120	C₂×Dic₅₃	—	4	8	no	—
121	C₂×Dic₅₃	—	4	4	no	—
122	C₈×C₅₃	—	2	4	no	—
123	C₈×C₅₃	—	2	2	no	—
124	C₈×C₅₃	—	2	2	no	—
125	#88322	—	4	4	no	—
126	#10803	—	53	1	no	maximal, Sylow
127	C₂²×C₅₃	—	2	4	no	—
128	C₂²×C₅₃	—	2	2	no	—
129	C₂²×C₅₃	—	4	2	no	—
130	C₂²×C₅₃	—	4	1	no	—
131	C₂²×C₅₃	—	4	1	no	—
132	C₂²×C₅₃	#376	1	1	yes	characteristic
133	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₅₃	#387	1	2	yes	—
136	C₂²×C₅₃	#388	1	2	yes	—
137	C₄×C₅₃	—	4	4	no	—
138	C₄×C₅₃	#380	1	1	yes	characteristic
139	C₄×C₅₃	#381	1	1	yes	characteristic
140	C₄×C₅₃	#382	1	1	yes	characteristic
141	C₄×C₅₃	#383	1	1	yes	characteristic
142	C₄×C₅₃	#387	1	2	yes	—
143	C₄×C₅₃	#388	1	2	yes	—
144	Dic₅₃	—	8	4	no	—
145	C₂×#405	D₅₃	1	1	yes	characteristic
146	C₂×#344	—	53	2	no	—
147	#1742	—	53	1	no	—
148	#1740	—	53	1	no	—
149	#1729	—	53	2	no	—
150	C₂×C₅₃	—	2	2	no	—
151	C₂×C₅₃	—	2	1	no	—
152	C₂×C₅₃	—	2	1	no	—
153	C₂×C₅₃	—	2	2	no	—
154	C₂×C₅₃	#1434	1	1	yes	characteristic
155	C₂×C₅₃	#1757	1	2	yes	—
156	C₂²×M₁₆	—	53	2	no	—
157	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
158	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
159	C₂×#117	—	53	2	no	—
160	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
161	#405	—	2	4	no	—
162	C₂×#97	Dic₅₃	1	2	yes	complemented
163	C₂×C₄×C₈	Dic₅₃	1	2	yes	complemented
164	#344	—	106	4	no	—
165	#343	—	53	4	no	—
166	#342	—	53	4	no	—
167	#336	—	106	4	no	—
168	C₅₃	#10803	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	53	4	no	—
173	C₂×M₁₆	—	106	2	no	—
174	C₂×M₁₆	—	106	2	no	—
175	C₂×M₁₆	—	106	2	no	—
176	C₂×M₁₆	—	106	2	no	—
177	C₂×M₁₆	—	106	2	no	—
178	C₂×M₁₆	Dic₁₀₆	1	2	yes	—
179	C₂×M₁₆	D₂₁₂	1	2	yes	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	—	106	2	no	—
184	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
185	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
186	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
187	C₂²×C₈	C₂²⋊D₅₃	1	4	yes	—
188	#117	—	2	2	no	—
189	#117	—	2	2	no	—
190	#117	—	106	2	no	—
191	#117	—	106	2	no	—
192	#117	—	106	2	no	—
193	#117	—	106	2	no	—
194	C₂×C₄⋊C₄	—	106	2	no	—
195	C₂×C₄⋊C₄	C₄×D₅₃	1	2	yes	—
196	C₂×C₂²⋊C₄	—	53	4	no	—
197	C₂×C₂²⋊C₄	C₄×D₅₃	1	2	yes	—
198	C₂×C₄²	—	106	2	no	—
199	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
200	#97	—	2	4	no	—
201	C₄×C₈	—	2	4	no	—
202	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
203	C₂²×C₄	—	106	4	no	—
204	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
205	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
208	C₂²×C₄	#11719945	1	2	yes	—
209	C₂²×C₄	#11719945	1	2	yes	—
210	C₂²×C₄	#11719945	1	2	yes	—
211	C₂²×C₄	#11719945	1	1	yes	characteristic
212	C₂²×C₄	#11719945	1	1	yes	characteristic
213	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	106	4	no	—
218	M₁₆	—	106	4	no	—
219	M₁₆	—	212	2	no	—
220	M₁₆	—	212	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	106	4	no	—
230	C₂×C₈	—	212	2	no	—
231	C₂×C₈	—	212	2	no	—
232	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
233	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
234	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	212	2	no	—
237	C₄⋊C₄	—	212	2	no	—
238	C₂²⋊C₄	—	2	4	no	—
239	C₂²⋊C₄	—	106	4	no	—
240	C₂²⋊C₄	—	106	4	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	212	2	no	—
244	C₄²	—	212	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
247	C₂³	#13133174	1	2	yes	—
248	C₂³	C₄×C₂²⋊D₅₃	1	2	yes	—
249	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
250	C₂³	C₄×Dic₁₀₆	1	1	yes	characteristic
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	212	8	no	—
254	C₂×C₄	—	212	4	no	—
255	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
257	C₂×C₄	#13133174	1	2	yes	—
258	C₂×C₄	#13133176	1	1	yes	characteristic
259	C₂×C₄	#13133176	1	1	yes	characteristic
260	C₂×C₄	#13133176	1	1	yes	characteristic
261	C₂×C₄	#13133176	1	1	yes	characteristic
262	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
263	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
264	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
265	C₂×C₄	#13133191	1	2	yes	—
266	C₂×C₄	#13133196	1	2	yes	—
267	C₂×C₄	#13133200	1	2	yes	—
268	C₂×C₄	#13133200	1	2	yes	—
269	C₂×C₄	#13133205	1	2	yes	—
270	C₂×C₄	#13133209	1	2	yes	—
271	C₂×C₄	#13133211	1	2	yes	—
272	C₂×C₄	#13133211	1	2	yes	—
273	C₈	—	2	4	no	—
274	C₈	—	2	2	no	—
275	C₈	—	2	2	no	—
276	C₈	—	212	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	#16810469	1	1	yes	characteristic
283	C₂²	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
284	C₂²	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
285	C₂²	#16810507	1	2	yes	—
286	C₂²	#16810514	1	2	yes	—
287	C₄	—	4	4	no	—
288	C₄	—	424	4	no	—
289	C₄	#16810483	1	1	yes	characteristic
290	C₄	#16810486	1	1	yes	characteristic
291	C₄	#16810489	1	1	yes	characteristic
292	C₄	#16810491	1	1	yes	characteristic
293	C₄	#16810507	1	2	yes	—
294	C₄	#16810514	1	2	yes	—
295	C₂	—	2	2	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	#20272942	1	1	yes	characteristic
300	C₂	#20274515	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅₃
commutator	C₂³×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#10803
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
9. #16810392	C₄	semidirect
16. C₂×#97×C₅₃	C₄	semidirect
17. C₂×C₄×C₈×C₅₃	C₄	semidirect
162. C₂×#97	Dic₅₃	semidirect
163. C₂×C₄×C₈	Dic₅₃	semidirect
168. C₅₃	#10803	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810394	2
#20274454	2
#20274448	2
#20274411	2
C₂×C₅₃×#405	2
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