Subgroups — #27289858

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810395	C₂	1	2	yes	maximal
2	C₂×#16810388	C₂	1	1	yes	maximal, characteristic
3	#20274432	C₂	1	1	yes	maximal, characteristic
4	#20274424	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	#16810395	C₄	1	8	yes	complemented
9	#16810388	—	2	1	no	—
10	#16810388	—	2	1	no	—
11	#16810386	—	2	2	no	—
12	#16810382	—	2	4	no	—
13	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#405	—	2	2	no	—
17	C₅₃×#405	C₄	1	4	yes	complemented
18	C₂×#97×C₅₃	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	—	2	1	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	—	2	2	no	—
25	C₂×#11719917	C₂×C₄	1	4	yes	—
26	C₂²×#88322	—	2	2	no	—
27	#13133176	—	2	2	no	—
28	#13133176	—	2	2	no	—
29	#13133176	—	2	2	no	—
30	#13133176	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊Dic₅₃	—	2	2	no	—
32	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
33	C₂×C₄×Dic₅₃	—	2	2	no	—
34	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₅₃	—	2	1	no	—
36	C₂×M₁₆×C₅₃	—	2	2	no	—
37	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
38	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₅₃	—	2	1	no	—
40	C₂²×C₈×C₅₃	—	2	2	no	—
41	C₂²×C₈×C₅₃	—	2	2	no	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₅₃	D₄	1	2	yes	—
44	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
46	#117×C₅₃	—	2	1	no	—
47	#117×C₅₃	—	2	1	no	—
48	#117×C₅₃	C₂×C₄	1	2	yes	—
49	#117×C₅₃	D₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₅₃	Q₈	1	2	yes	—
51	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
52	C₂×C₄²×C₅₃	—	2	1	no	—
53	#97×C₅₃	—	2	4	no	—
54	C₄×C₈×C₅₃	—	2	4	no	—
55	C₂⁴×C₅₃	#41	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	#41	1	2	yes	—
65	C₂²×C₄×C₅₃	#41	1	2	yes	—
66	C₂²×C₄×C₅₃	#41	1	2	yes	—
67	C₂²×Dic₅₃	—	4	2	no	—
68	M₁₆×C₅₃	—	2	2	no	—
69	M₁₆×C₅₃	—	2	4	no	—
70	M₁₆×C₅₃	C₄²	1	4	yes	—
71	C₂×C₈×C₅₃	—	2	2	no	—
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	4	no	—
75	C₂×C₈×C₅₃	—	2	4	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
80	C₄⋊C₄×C₅₃	—	2	2	no	—
81	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
82	C₂²⋊C₄×C₅₃	—	2	2	no	—
83	C₂²⋊C₄×C₅₃	—	2	2	no	—
84	C₄²×C₅₃	—	2	2	no	—
85	C₄²×C₅₃	—	2	2	no	—
86	C₂²⋊Dic₅₃	—	2	4	no	—
87	C₂²⋊Dic₅₃	—	4	2	no	—
88	C₄⋊Dic₅₃	—	2	4	no	—
89	C₄⋊Dic₅₃	—	4	2	no	—
90	C₄×Dic₅₃	—	2	4	no	—
91	C₄×Dic₅₃	—	4	2	no	—
92	#11719917	—	2	4	no	—
93	#11719917	—	2	4	no	—
94	#11719917	—	4	2	no	—
95	#11719917	—	4	2	no	—
96	C₂×#88322	—	2	4	no	—
97	C₂×#88322	—	4	2	no	—
98	C₂×#88322	—	4	2	no	—
99	C₂³×C₅₃	—	4	2	no	—
100	C₂³×C₅₃	#117	1	2	yes	—
101	C₂³×C₅₃	#117	1	1	yes	characteristic
102	C₂³×C₅₃	#121	1	1	yes	characteristic
103	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂³×C₅₃	#126	1	2	yes	—
105	C₂×C₄×C₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	—	4	4	no	—
107	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#117	1	2	yes	—
111	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
118	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₅₃	#123	1	2	yes	—
121	C₂×C₄×C₅₃	#123	1	2	yes	—
122	C₂×C₄×C₅₃	#124	1	2	yes	—
123	C₂×C₄×C₅₃	#125	1	2	yes	—
124	C₂×C₄×C₅₃	#126	1	2	yes	—
125	C₂×Dic₅₃	—	4	4	no	—
126	C₂×Dic₅₃	—	8	4	no	—
127	C₈×C₅₃	—	2	4	no	—
128	C₈×C₅₃	—	2	2	no	—
129	C₈×C₅₃	—	2	2	no	—
130	#88322	—	4	4	no	—
131	#10828	—	53	1	no	maximal, Sylow
132	C₂²×C₅₃	—	2	2	no	—
133	C₂²×C₅₃	—	2	2	no	—
134	C₂²×C₅₃	—	2	2	no	—
135	C₂²×C₅₃	—	4	2	no	—
136	C₂²×C₅₃	—	4	2	no	—
137	C₂²×C₅₃	#376	1	1	yes	characteristic
138	C₂²×C₅₃	#380	1	1	yes	characteristic
139	C₂²×C₅₃	#383	1	1	yes	characteristic
140	C₂²×C₅₃	#387	1	2	yes	—
141	C₂²×C₅₃	#388	1	2	yes	—
142	C₄×C₅₃	—	4	4	no	—
143	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
144	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₅₃	#381	1	1	yes	characteristic
146	C₄×C₅₃	#382	1	1	yes	characteristic
147	C₄×C₅₃	#387	1	2	yes	—
148	C₄×C₅₃	#388	1	2	yes	—
149	Dic₅₃	—	8	4	no	—
150	C₂×#405	D₅₃	1	1	yes	characteristic
151	C₂×#344	—	53	2	no	—
152	C₂×#341	—	53	1	no	—
153	#1735	—	53	1	no	—
154	#1733	—	53	2	no	—
155	C₂×C₅₃	—	2	2	no	—
156	C₂×C₅₃	—	2	1	no	—
157	C₂×C₅₃	—	2	1	no	—
158	C₂×C₅₃	—	2	2	no	—
159	C₂×C₅₃	#1434	1	1	yes	characteristic
160	C₂×C₅₃	#1814	1	2	yes	—
161	C₂²×M₁₆	—	53	2	no	—
162	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
163	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
164	C₂×#117	—	53	2	no	—
165	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
166	#405	—	2	2	no	—
167	#405	Dic₅₃	1	4	yes	complemented
168	C₂×#97	—	2	1	no	—
169	C₂×C₄×C₈	—	2	1	no	—
170	#344	—	53	8	no	—
171	#341	—	106	1	no	—
172	#341	—	106	1	no	—
173	#340	—	106	2	no	—
174	#339	—	106	4	no	—
175	C₅₃	#10828	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	53	4	no	—
180	C₂×M₁₆	—	106	2	no	—
181	C₂×M₁₆	—	106	2	no	—
182	C₂×M₁₆	—	106	2	no	—
183	C₂×M₁₆	—	106	2	no	—
184	C₂×M₁₆	—	106	2	no	—
185	C₂×M₁₆	C₄×D₅₃	1	4	yes	—
186	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	106	2	no	—
191	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
192	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
193	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
194	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
195	#117	—	2	1	no	—
196	#117	—	2	1	no	—
197	#117	—	53	4	no	—
198	#117	—	106	2	no	—
199	#117	—	106	2	no	—
200	#117	—	106	2	no	—
201	#117	D₂₁₂	1	2	yes	—
202	#117	C₂×Dic₅₃	1	2	yes	—
203	C₂×C₄⋊C₄	—	106	2	no	—
204	C₂×C₄⋊C₄	Dic₁₀₆	1	2	yes	—
205	C₂×C₂²⋊C₄	—	2	1	no	—
206	C₂×C₂²⋊C₄	—	106	2	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	—	106	2	no	—
209	#97	—	2	4	no	—
210	C₄×C₈	—	2	4	no	—
211	C₂⁴	#11719945	1	1	yes	characteristic
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	212	2	no	—
214	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
216	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
217	C₂²×C₄	#11719945	1	1	yes	characteristic
218	C₂²×C₄	#11719945	1	2	yes	—
219	C₂²×C₄	#11719945	1	2	yes	—
220	C₂²×C₄	#11719945	1	2	yes	—
221	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
224	M₁₆	—	2	2	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	106	4	no	—
227	M₁₆	—	106	4	no	—
228	M₁₆	—	212	2	no	—
229	M₁₆	—	212	2	no	—
230	M₁₆	C₄×Dic₅₃	1	4	yes	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	106	4	no	—
239	C₂×C₈	—	212	2	no	—
240	C₂×C₈	—	212	2	no	—
241	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
242	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	106	4	no	—
245	C₄⋊C₄	—	212	2	no	—
246	C₄⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₂²⋊C₄	—	106	4	no	—
250	C₂²⋊C₄	—	212	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	106	4	no	—
254	C₄²	—	212	2	no	—
255	C₂³	—	4	2	no	—
256	C₂³	#13133174	1	2	yes	—
257	C₂³	#13133176	1	1	yes	characteristic
258	C₂³	#13133191	1	1	yes	characteristic
259	C₂³	#13133196	1	1	yes	characteristic
260	C₂³	#13133211	1	2	yes	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	424	4	no	—
264	C₂×C₄	—	212	4	no	—
265	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
266	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
268	C₂×C₄	#13133174	1	2	yes	—
269	C₂×C₄	#13133176	1	1	yes	characteristic
270	C₂×C₄	#13133176	1	1	yes	characteristic
271	C₂×C₄	#13133176	1	1	yes	characteristic
272	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
273	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
274	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
275	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
276	C₂×C₄	#13133191	1	1	yes	characteristic
277	C₂×C₄	#13133196	1	1	yes	characteristic
278	C₂×C₄	#13133200	1	2	yes	—
279	C₂×C₄	#13133200	1	2	yes	—
280	C₂×C₄	#13133205	1	2	yes	—
281	C₂×C₄	#13133209	1	2	yes	—
282	C₂×C₄	#13133211	1	2	yes	—
283	C₈	—	2	4	no	—
284	C₈	—	2	2	no	—
285	C₈	—	2	2	no	—
286	C₈	—	212	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	2	no	—
292	C₂²	#16810469	1	1	yes	characteristic
293	C₂²	#16810483	1	1	yes	characteristic
294	C₂²	#16810491	1	1	yes	characteristic
295	C₂²	#16810507	1	2	yes	—
296	C₂²	#16810514	1	2	yes	—
297	C₄	—	4	4	no	—
298	C₄	—	424	4	no	—
299	C₄	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
300	C₄	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
301	C₄	#16810486	1	1	yes	characteristic
302	C₄	#16810489	1	1	yes	characteristic
303	C₄	#16810507	1	2	yes	—
304	C₄	#16810514	1	2	yes	—
305	C₂	—	2	2	no	—
306	C₂	—	2	2	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	#20272942	1	1	yes	characteristic
310	C₂	#20274769	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133191
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#10828
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
8. #16810395	C₄	semidirect
17. C₅₃×#405	C₄	semidirect
167. #405	Dic₅₃	semidirect
175. C₅₃	#10828	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810395	2
C₂×#16810388	2
#20274432	2
#20274424	2
C₂×C₅₃×#405	2
#10828	53