Subgroups — #27538743

Order 13568.

598 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄₂₄	C₂	1	1	yes	maximal, characteristic
2	#20276869	C₂	1	4	yes	maximal, complemented
3	#20276859	C₂	1	2	yes	maximal, complemented
4	#20276849	C₂	1	2	yes	maximal, complemented
5	#20276504	C₂	1	2	yes	maximal, complemented
6	#20276471	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276430	C₂	1	2	yes	maximal
8	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄₂₄	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₂₁₂	—	2	1	no	—
11	C₂×#13133157	C₂²	1	2	yes	—
12	C₄×D₄₂₄	—	2	1	no	—
13	C₄×D₄₂₄	—	2	1	no	—
14	C₄×D₄₂₄	C₂²	1	2	yes	—
15	C₄×D₄₂₄	C₂²	1	2	yes	—
16	C₂×#13133151	C₂²	1	1	yes	characteristic
17	C₂×#13133139	—	2	1	no	—
18	#16810450	C₂²	1	4	yes	complemented
19	#16810448	C₂²	1	4	yes	complemented
20	C₄⋊#11719908	C₂²	1	2	yes	complemented
21	C₄⋊#11719908	C₂²	1	2	yes	complemented
22	#16810423	C₂²	1	4	yes	—
23	#16810423	C₂²	1	4	yes	—
24	C₅₃×#533	—	2	1	no	—
25	C₅₃×#500	C₂²	1	1	yes	characteristic, complemented
26	C₅₃×#500	C₂²	1	1	yes	characteristic, complemented
27	C₅₃×#499	—	2	1	no	—
28	C₅₃×#499	C₂²	1	2	yes	complemented
29	C₅₃×#498	—	2	1	no	—
30	C₅₃×#427	—	2	1	no	—
31	C₅₃×#427	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₂²×D₂₁₂	—	2	1	no	—
34	C₂²×C₄×D₅₃	—	4	1	no	—
35	C₂×D₄₂₄	—	2	1	no	—
36	C₂×D₄₂₄	—	2	2	no	—
37	C₂×D₄₂₄	Q₈	1	4	yes	complemented
38	C₂×D₄₂₄	C₂³	1	2	yes	—
39	C₄×D₂₁₂	—	2	2	no	—
40	C₄×D₂₁₂	—	2	2	no	—
41	C₄×D₂₁₂	—	4	1	no	—
42	C₄×D₂₁₂	—	4	1	no	—
43	C₂×C₄⋊Dic₅₃	—	2	1	no	—
44	C₂×#11719921	—	4	1	no	—
45	#13133157	—	2	4	no	—
46	C₂×#11719908	C₂³	1	4	yes	—
47	#13133151	—	2	1	no	—
48	#13133151	C₂³	1	2	yes	—
49	#13133139	—	2	2	no	—
50	#13133139	—	2	2	no	—
51	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₅₃	—	2	1	no	—
54	C₄⋊Q₈×C₅₃	—	2	1	no	—
55	#125×C₅₃	—	2	2	no	—
56	C₂²⋊Q₈×C₅₃	—	4	1	no	—
57	C₂²⋊Q₈×C₅₃	—	4	1	no	—
58	C₄×Q₈×C₅₃	—	4	1	no	—
59	C₄×Q₈×C₅₃	—	4	1	no	—
60	#117×C₅₃	—	2	2	no	—
61	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
62	#107×C₅₃	—	2	1	no	—
63	#107×C₅₃	—	2	1	no	—
64	#107×C₅₃	C₂³	1	2	yes	—
65	#107×C₅₃	C₂³	1	2	yes	—
66	#106×C₅₃	—	2	2	no	—
67	#106×C₅₃	—	2	2	no	—
68	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
72	C₂³×D₅₃	—	4	1	no	—
73	C₂×Q₈×C₅₃	—	4	1	no	—
74	C₂×Q₈×C₅₃	—	4	1	no	—
75	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×Dic₅₃	—	4	1	no	—
79	C₂×D₂₁₂	—	2	4	no	—
80	C₂×D₂₁₂	—	2	2	no	—
81	C₂×D₂₁₂	—	4	1	no	—
82	C₂×D₂₁₂	—	4	2	no	—
83	C₂×C₄×D₅₃	—	4	2	no	—
84	C₂×C₄×D₅₃	—	4	2	no	—
85	C₂×C₄×D₅₃	—	8	4	no	—
86	C₂×C₈×C₅₃	—	2	2	no	—
87	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
91	C₂×C₈×C₅₃	#41	1	2	yes	—
92	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₅₃	—	2	2	no	—
94	C₄⋊C₄×C₅₃	—	2	2	no	—
95	C₄⋊C₄×C₅₃	—	4	2	no	—
96	C₄⋊C₄×C₅₃	—	4	2	no	—
97	C₄⋊C₄×C₅₃	—	4	1	no	—
98	C₄⋊C₄×C₅₃	—	4	1	no	—
99	C₂²⋊C₄×C₅₃	—	4	2	no	—
100	C₄²×C₅₃	—	4	2	no	—
101	C₄²×C₅₃	D₈	1	1	yes	characteristic
102	C₄²×C₅₃	D₈	1	1	yes	characteristic
103	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
105	#11719921	—	4	2	no	—
106	#11719921	—	4	2	no	—
107	C₄⋊Dic₅₃	—	2	2	no	—
108	C₄⋊Dic₅₃	—	4	1	no	—
109	D₄₂₄	—	2	4	no	—
110	D₄₂₄	—	2	2	no	—
111	D₄₂₄	C₂×Q₈	1	4	yes	—
112	#11719908	—	2	4	no	—
113	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
114	C₂²×D₅₃	—	4	2	no	—
115	C₂²×D₅₃	—	4	4	no	—
116	C₂²×D₅₃	—	8	2	no	—
117	C₂²×D₅₃	—	8	2	no	—
118	Q₈×C₅₃	—	8	2	no	—
119	Q₈×C₅₃	—	8	2	no	—
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	4	1	no	—
122	C₂×C₄×C₅₃	—	4	2	no	—
123	C₂×C₄×C₅₃	—	4	2	no	—
124	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
125	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₅₃	#137	1	2	yes	—
131	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×Dic₅₃	—	4	2	no	—
133	C₂×Dic₅₃	—	8	2	no	—
134	D₂₁₂	—	2	4	no	—
135	D₂₁₂	—	4	2	no	—
136	D₂₁₂	—	4	4	no	—
137	C₄×D₅₃	—	8	4	no	—
138	C₄×D₅₃	—	8	4	no	—
139	C₈×C₅₃	—	2	1	no	—
140	C₈×C₅₃	—	2	1	no	—
141	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
142	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
143	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
144	#33627	—	53	1	no	maximal, Sylow
145	C₂²×C₅₃	—	2	1	no	—
146	C₂²×C₅₃	—	2	1	no	—
147	C₂²×C₅₃	#474	1	2	yes	—
148	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₅₃	—	4	4	no	—
150	C₂×D₅₃	—	8	8	no	—
151	C₂×D₅₃	—	8	2	no	—
152	C₂×D₅₃	—	8	2	no	—
153	C₄×C₅₃	—	2	1	no	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	8	2	no	—
157	C₄×C₅₃	—	8	2	no	—
158	C₄×C₅₃	#474	1	2	yes	—
159	C₄×C₅₃	#509	1	2	yes	—
160	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
161	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
162	Dic₅₃	—	8	2	no	—
163	#3160	D₅₃	1	1	yes	characteristic, complemented
164	C₂×C₄×D₈	—	53	1	no	—
165	#2228	—	53	4	no	—
166	#2226	—	53	2	no	—
167	#2224	—	53	2	no	—
168	#2159	—	53	2	no	—
169	#2150	—	53	1	no	—
170	#2140	—	53	2	no	—
171	C₂×C₅₃	—	2	2	no	—
172	C₂×C₅₃	C₂×#474	1	1	yes	characteristic
173	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
174	C₂×C₅₃	#3414	1	1	yes	characteristic
175	D₅₃	—	8	4	no	—
176	C₂²×D₈	—	53	1	no	—
177	#533	—	2	1	no	—
178	C₂×C₄×D₄	—	106	1	no	—
179	C₂×M₃₂	—	53	2	no	—
180	#500	C₂×D₅₃	1	1	yes	characteristic, complemented
181	#500	C₂×D₅₃	1	1	yes	characteristic, complemented
182	#499	—	2	1	no	—
183	#499	C₂×D₅₃	1	2	yes	complemented
184	#498	—	2	1	no	—
185	C₄×D₈	—	53	2	no	—
186	C₄×D₈	—	53	2	no	—
187	C₄×D₈	—	106	1	no	—
188	C₄×D₈	—	106	1	no	—
189	#427	—	2	1	no	—
190	#427	C₂×D₅₃	1	2	yes	—
191	C₂×#107	—	53	1	no	—
192	C₂×#102	—	106	1	no	—
193	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
194	#367	—	53	4	no	—
195	#366	—	53	4	no	—
196	C₄⋊C₁₆	—	53	2	no	—
197	C₄⋊C₁₆	—	53	2	no	—
198	#357	—	53	4	no	—
199	#357	—	53	4	no	—
200	C₅₃	#33627	1	1	yes	characteristic, complemented, Sylow
201	C₂²×D₄	—	106	1	no	—
202	C₂³×C₄	—	212	1	no	—
203	C₂×D₈	—	53	4	no	—
204	C₂×D₈	—	53	2	no	—
205	C₂×D₈	—	106	2	no	—
206	C₂×D₈	—	106	1	no	—
207	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
208	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	#125	—	2	2	no	—
212	C₂²⋊Q₈	—	4	1	no	—
213	C₂²⋊Q₈	—	4	1	no	—
214	C₄×Q₈	—	4	1	no	—
215	C₄×Q₈	—	4	1	no	—
216	C₄×D₄	—	106	2	no	—
217	C₄×D₄	—	106	2	no	—
218	C₄×D₄	—	212	1	no	—
219	C₄×D₄	—	212	1	no	—
220	#117	—	2	2	no	—
221	C₂×C₄⋊C₄	—	106	1	no	—
222	C₂×C₂²⋊C₄	—	212	1	no	—
223	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
224	M₃₂	—	106	4	no	—
225	C₂×C₁₆	—	53	4	no	—
226	#107	—	2	1	no	—
227	#107	—	2	1	no	—
228	#107	—	53	2	no	—
229	#107	—	106	1	no	—
230	#107	C₂²×D₅₃	1	2	yes	—
231	#107	C₂²×D₅₃	1	2	yes	—
232	#106	—	2	2	no	—
233	#106	—	2	2	no	—
234	#102	—	106	2	no	—
235	#102	—	106	2	no	—
236	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
237	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
240	C₂⁴	—	212	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×D₄	—	106	4	no	—
244	C₂×D₄	—	106	2	no	—
245	C₂×D₄	—	212	1	no	—
246	C₂×D₄	—	212	2	no	—
247	C₂²×C₄	—	424	4	no	—
248	C₂²×C₄	—	212	1	no	—
249	C₂²×C₄	—	212	2	no	—
250	C₂²×C₄	—	212	2	no	—
251	C₂²×C₄	#11719922	1	1	yes	characteristic
252	C₂²×C₄	#11719922	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
254	D₈	—	53	4	no	—
255	D₈	—	106	4	no	—
256	D₈	—	106	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	Q₈×D₅₃	1	2	yes	—
259	C₂×C₈	#11719949	1	2	yes	—
260	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	106	2	no	—
271	C₄⋊C₄	—	212	1	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	212	2	no	—
274	C₂²⋊C₄	—	212	2	no	—
275	C₄²	—	4	2	no	—
276	C₄²	#11719922	1	1	yes	characteristic
277	C₄²	#11719922	1	1	yes	characteristic
278	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
279	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
280	C₁₆	—	106	4	no	—
281	C₂³	—	424	2	no	—
282	C₂³	—	424	2	no	—
283	C₂³	—	212	2	no	—
284	C₂³	—	212	4	no	—
285	C₂³	C₂×#11719922	1	1	yes	characteristic
286	Q₈	—	8	2	no	—
287	Q₈	—	8	2	no	—
288	D₄	—	106	4	no	—
289	D₄	—	212	4	no	—
290	D₄	—	212	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	424	4	no	—
296	C₂×C₄	—	424	2	no	—
297	C₂×C₄	—	424	4	no	—
298	C₂×C₄	—	212	2	no	—
299	C₂×C₄	#13133193	1	2	yes	—
300	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
301	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
302	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
303	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
304	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
305	C₂×C₄	#13133248	1	2	yes	—
306	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	#13133193	1	2	yes	—
310	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
311	C₈	C₂×#11719949	1	1	yes	characteristic
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	424	8	no	—
315	C₂²	—	424	2	no	—
316	C₂²	—	424	2	no	—
317	C₂²	—	212	4	no	—
318	C₂²	#16810900	1	2	yes	—
319	C₂²	C₂²×#11719922	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	424	2	no	—
326	C₄	#16810900	1	2	yes	—
327	C₄	#16811058	1	2	yes	—
328	C₄	C₂×#13133193	1	1	yes	characteristic
329	C₄	C₂×#13133248	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	424	4	no	—
332	C₂	C₂×#16810900	1	1	yes	characteristic
333	C₂	C₂×#16811058	1	1	yes	characteristic
334	C₂	#20287849	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3160	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂²×D₄₂₄	C₂²
hypercenter	#3160	D₅₃
hypocenter	C₅₃	#33627
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
2. #20276869	C₂	semidirect
3. #20276859	C₂	semidirect
4. #20276849	C₂	semidirect
5. #20276504	C₂	semidirect
6. #20276471	C₂	semidirect
8. C₅₃×#3160	C₂	semidirect
18. #16810450	C₂²	semidirect
19. #16810448	C₂²	semidirect
20. C₄⋊#11719908	C₂²	semidirect
21. C₄⋊#11719908	C₂²	semidirect
25. C₅₃×#500	C₂²	semidirect
26. C₅₃×#500	C₂²	semidirect
28. C₅₃×#499	C₂²	semidirect
37. C₂×D₄₂₄	Q₈	semidirect
163. #3160	D₅₃	semidirect
180. #500	C₂×D₅₃	semidirect
181. #500	C₂×D₅₃	semidirect
183. #499	C₂×D₅₃	semidirect
200. C₅₃	#33627	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄₂₄	2
#20276869	2
#20276859	2
#20276849	2
#20276504	2
#20276471	2
#20276430	2
C₅₃×#3160	2
#33627	53