Subgroups — #27353572

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283824	C₂	1	2	yes	maximal, complemented
2	C₄×#13133239	C₂	1	1	yes	maximal, characteristic
3	#20273526	C₂	1	4	yes	maximal, complemented
4	#20273521	C₂	1	2	yes	maximal, complemented
5	#20273509	C₂	1	2	yes	maximal, complemented
6	#20273100	C₂	1	2	yes	maximal, complemented
7	#20272998	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133239	C₄	1	1	yes	characteristic
10	C₄×#11719945	C₂²	1	2	yes	—
11	#16810747	C₂²	1	4	yes	—
12	C₄×Dic₂₁₂	C₄	1	1	yes	characteristic
13	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
14	C₄×#11719913	C₄	1	2	yes	—
15	C₄×#11719913	C₂²	1	2	yes	—
16	C₄×D₄₂₄	C₄	1	1	yes	characteristic
17	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
18	#16810688	C₂²	1	4	yes	—
19	C₈×C₂²⋊D₅₃	—	2	4	no	—
20	#16810676	C₂²	1	1	yes	characteristic
21	#16810676	C₂²	1	1	yes	characteristic
22	#16810675	—	2	2	no	—
23	#16810661	C₄	1	1	yes	characteristic
24	C₄⋊C₈×D₅₃	—	2	2	no	—
25	#16810620	C₄	1	2	yes	—
26	#16810353	C₂²	1	4	yes	—
27	#16810350	C₂²	1	4	yes	—
28	#16810344	C₂²	1	2	yes	—
29	C₈×#88322	C₂²	1	2	yes	—
30	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#335	C₂²	1	2	yes	—
33	C₅₃×#335	C₂²	1	2	yes	—
34	C₂×#11719945	C₂×C₄	1	2	yes	—
35	#13133239	C₈	1	8	yes	complemented
36	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
37	C₂×#11719913	C₂×C₄	1	2	yes	—
38	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₅₃	—	2	4	no	—
40	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
41	C₄×Dic₁₀₆	C₂³	1	2	yes	—
42	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
43	C₄×D₂₁₂	C₂³	1	2	yes	—
44	C₄×C₂²⋊D₅₃	—	2	2	no	—
45	C₄×C₂²⋊D₅₃	—	2	2	no	—
46	#13133176	C₂×C₄	1	2	yes	—
47	#13133174	—	2	2	no	—
48	C₄²×D₅₃	—	2	2	no	—
49	#13133151	C₂×C₄	1	1	yes	characteristic
50	#13133151	C₂×C₄	1	1	yes	characteristic
51	#13133149	C₂×C₄	1	2	yes	—
52	C₄⋊#88322	—	2	1	no	—
53	C₄⋊#88322	D₄	1	1	yes	characteristic
54	C₄⋊#88322	D₄	1	1	yes	characteristic
55	#13133146	—	2	4	no	—
56	#13133142	C₂×C₄	1	2	yes	—
57	#13133142	C₂×C₄	1	2	yes	—
58	#13133139	C₂×C₄	1	2	yes	—
59	#13133139	C₂×C₄	1	2	yes	—
60	C₂²⋊#88322	—	2	1	no	—
61	C₂²⋊#88322	D₄	1	2	yes	—
62	#13133129	—	2	4	no	—
63	C₄×#88322	C₂³	1	2	yes	—
64	C₄×#88322	C₂³	1	2	yes	—
65	C₈×Dic₅₃	—	2	4	no	—
66	C₂²×C₈×C₅₃	—	2	2	no	—
67	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₈×C₅₃	—	2	2	no	—
71	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
72	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
79	C₂×C₂²⋊D₅₃	—	2	2	no	—
80	#11719945	—	2	4	no	—
81	#11719945	C₂×C₈	1	8	yes	complemented
82	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₅₃	—	2	2	no	—
84	C₂×C₄×D₅₃	—	2	2	no	—
85	C₂×C₄×D₅₃	—	2	2	no	—
86	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₅₃	—	2	2	no	—
88	C₂×C₈×C₅₃	—	2	4	no	—
89	C₂×C₈×C₅₃	—	8	2	no	—
90	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	complemented
91	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₅₃	#41	1	1	yes	characteristic
98	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
99	C₂²⋊Dic₅₃	—	2	2	no	—
100	#11719921	—	2	2	no	—
101	#11719921	—	2	2	no	—
102	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
103	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
104	#11719919	—	2	2	no	—
105	#11719919	—	2	2	no	—
106	C₄×Dic₅₃	—	2	2	no	—
107	C₄×Dic₅₃	—	2	2	no	—
108	C₄×Dic₅₃	—	2	2	no	—
109	C₂×#88322	—	2	2	no	—
110	C₂×#88322	C₂×D₄	1	2	yes	—
111	C₂×#88322	C₂×D₄	1	2	yes	—
112	Dic₂₁₂	C₂×C₈	1	4	yes	complemented
113	D₄₂₄	C₂×C₈	1	4	yes	complemented
114	#11719913	C₂×C₈	1	8	yes	complemented
115	C₈×D₅₃	—	8	4	no	—
116	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	1	no	—
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	4	1	no	—
121	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₅₃	#131	1	2	yes	—
126	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
129	C₂²⋊D₅₃	—	2	8	no	—
130	C₂×Dic₅₃	—	2	2	no	—
131	C₂×Dic₅₃	—	2	2	no	—
132	C₂×Dic₅₃	—	2	2	no	—
133	D₂₁₂	—	2	2	no	—
134	D₂₁₂	C₂²×C₈	1	4	yes	—
135	C₄×D₅₃	—	2	8	no	—
136	C₄×D₅₃	—	4	2	no	—
137	C₄×D₅₃	—	4	2	no	—
138	C₄×D₅₃	—	8	2	no	—
139	Dic₁₀₆	—	2	2	no	—
140	Dic₁₀₆	C₂²×C₈	1	4	yes	—
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	—	2	1	no	—
143	C₈×C₅₃	—	4	4	no	—
144	C₈×C₅₃	C₂²×C₈	1	2	yes	—
145	C₈×C₅₃	C₂²×C₈	1	2	yes	—
146	#88322	—	2	2	no	—
147	#88322	—	2	2	no	—
148	#88322	—	4	2	no	—
149	#17857	—	53	1	no	maximal, Sylow
150	C₂²×C₅₃	—	2	1	no	—
151	C₂²×C₅₃	—	2	1	no	—
152	C₂²×C₅₃	C₈×D₄	1	2	yes	—
153	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
154	C₂×D₅₃	—	2	4	no	—
155	C₂×D₅₃	—	4	2	no	—
156	C₂×D₅₃	—	4	2	no	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	C₈×D₄	1	2	yes	—
161	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
162	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
163	C₄×C₅₃	#576	1	1	yes	characteristic
164	C₄×C₅₃	#578	1	1	yes	characteristic
165	Dic₅₃	—	2	4	no	—
166	Dic₅₃	—	4	2	no	—
167	Dic₅₃	—	4	2	no	—
168	#2988	—	53	2	no	—
169	C₄×#135	—	53	1	no	—
170	SD₁₆⋊C₈	—	53	4	no	—
171	C₈×Q₁₆	—	53	2	no	—
172	C₈×D₈	—	53	2	no	—
173	C₂×#335	D₅₃	1	1	yes	characteristic, complemented
174	#1477	—	53	2	no	—
175	#1450	—	53	1	no	—
176	C₂×C₅₃	—	2	2	no	—
177	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
178	C₂×C₅₃	#2951	1	1	yes	characteristic
179	C₂×C₅₃	#2957	1	1	yes	characteristic
180	D₅₃	—	4	4	no	—
181	C₂×#135	—	53	1	no	—
182	C₄×#41	—	53	2	no	—
183	C₈×Q₈	—	53	4	no	—
184	C₄×Q₁₆	—	53	1	no	—
185	C₄×Q₁₆	—	53	1	no	—
186	C₄×SD₁₆	—	53	2	no	—
187	C₄×SD₁₆	—	53	2	no	—
188	C₄×D₈	—	53	1	no	—
189	C₄×D₈	—	53	1	no	—
190	C₈×D₄	—	53	4	no	—
191	C₈×D₄	—	106	4	no	—
192	#431	—	53	1	no	—
193	#431	—	53	1	no	—
194	#431	—	106	2	no	—
195	#427	—	53	1	no	—
196	C₂×C₄⋊C₈	—	106	2	no	—
197	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
198	#416	—	53	2	no	—
199	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
200	#335	C₂×D₅₃	1	2	yes	—
201	#335	C₂×D₅₃	1	2	yes	—
202	Q₈⋊C₈	—	53	4	no	—
203	D₄⋊C₈	—	53	4	no	—
204	#322	—	53	2	no	—
205	C₈²	—	53	2	no	—
206	C₅₃	#17857	1	1	yes	characteristic, complemented, Sylow
207	C₂×#41	—	53	2	no	—
208	#135	—	53	8	no	—
209	C₂×Q₁₆	—	53	1	no	—
210	C₂×SD₁₆	—	53	2	no	—
211	C₂×D₈	—	53	1	no	—
212	C₂²×C₈	—	2	2	no	—
213	C₂²×C₈	—	106	4	no	—
214	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
215	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
216	C₄×Q₈	—	53	2	no	—
217	C₄×Q₈	—	53	2	no	—
218	C₄×D₄	—	53	2	no	—
219	C₄×D₄	—	53	2	no	—
220	C₄×D₄	—	106	2	no	—
221	C₄×D₄	—	106	2	no	—
222	#117	—	53	2	no	—
223	#117	—	106	2	no	—
224	C₂×C₄²	—	106	2	no	—
225	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
226	#107	—	53	1	no	—
227	#107	—	53	1	no	—
228	#106	—	53	2	no	—
229	C₄⋊C₈	—	2	2	no	—
230	C₄⋊C₈	—	53	1	no	—
231	C₄⋊C₈	—	53	1	no	—
232	C₄⋊C₈	—	106	4	no	—
233	C₄⋊C₈	—	106	1	no	—
234	C₄⋊C₈	C₂²×D₅₃	1	4	yes	—
235	#103	—	53	2	no	—
236	#103	—	53	2	no	—
237	#102	—	53	2	no	—
238	#102	—	53	2	no	—
239	C₂²⋊C₈	—	53	2	no	—
240	C₂²⋊C₈	—	106	4	no	—
241	C₂²⋊C₈	—	106	1	no	—
242	C₄×C₈	—	53	2	no	—
243	C₄×C₈	—	53	2	no	—
244	C₄×C₈	—	106	4	no	—
245	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
246	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
249	#41	—	53	8	no	—
250	#41	—	106	4	no	—
251	C₂×Q₈	—	53	2	no	—
252	C₂×D₄	—	53	2	no	—
253	C₂×D₄	—	106	2	no	—
254	C₂²×C₄	—	106	2	no	—
255	C₂²×C₄	—	106	2	no	—
256	C₂²×C₄	—	106	2	no	—
257	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
258	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
259	C₂²×C₄	#11719949	1	1	yes	characteristic
260	Q₁₆	—	53	4	no	—
261	SD₁₆	—	53	8	no	—
262	D₈	—	53	4	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	8	2	no	—
266	C₂×C₈	—	53	2	no	—
267	C₂×C₈	—	53	2	no	—
268	C₂×C₈	—	106	2	no	—
269	C₂×C₈	—	424	4	no	—
270	C₂×C₈	C₈×D₅₃	1	4	yes	complemented
271	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
272	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
273	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
274	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
275	C₄⋊C₄	—	53	2	no	—
276	C₄⋊C₄	—	53	2	no	—
277	C₄⋊C₄	—	106	2	no	—
278	C₄⋊C₄	—	106	2	no	—
279	C₂²⋊C₄	—	106	2	no	—
280	C₂²⋊C₄	—	106	2	no	—
281	C₂²⋊C₄	—	106	2	no	—
282	C₄²	—	106	2	no	—
283	C₄²	—	106	2	no	—
284	C₄²	—	106	2	no	—
285	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
286	C₄²	D₄×D₅₃	1	1	yes	characteristic
287	C₄²	#11719949	1	1	yes	characteristic
288	C₄²	C₂³×D₅₃	1	1	yes	characteristic
289	C₂³	—	106	2	no	—
290	C₂³	#13133179	1	1	yes	characteristic
291	Q₈	—	53	4	no	—
292	Q₈	—	106	2	no	—
293	D₄	—	53	4	no	—
294	D₄	—	106	8	no	—
295	D₄	—	106	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	106	2	no	—
300	C₂×C₄	—	106	8	no	—
301	C₂×C₄	—	106	2	no	—
302	C₂×C₄	—	106	2	no	—
303	C₂×C₄	—	424	2	no	—
304	C₂×C₄	—	212	2	no	—
305	C₂×C₄	—	212	2	no	—
306	C₂×C₄	#13133179	1	1	yes	characteristic
307	C₂×C₄	#13133179	1	1	yes	characteristic
308	C₂×C₄	#13133179	1	1	yes	characteristic
309	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
310	C₂×C₄	#13133225	1	2	yes	—
311	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
313	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	4	4	no	—
317	C₈	—	106	2	no	—
318	C₈	—	106	2	no	—
319	C₈	—	212	2	no	—
320	C₈	C₂×C₈×D₅₃	1	2	yes	—
321	C₈	C₂×C₈×D₅₃	1	2	yes	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	106	4	no	—
325	C₂²	—	212	2	no	—
326	C₂²	—	212	2	no	—
327	C₂²	#16810688	1	2	yes	—
328	C₂²	C₂×#13133179	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	106	4	no	—
333	C₄	—	212	2	no	—
334	C₄	—	212	2	no	—
335	C₄	#16810688	1	2	yes	—
336	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
337	C₄	C₂×#13133225	1	1	yes	characteristic
338	C₄	#16811424	1	1	yes	characteristic
339	C₄	#16811436	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	212	4	no	—
342	C₂	C₂×#16810688	1	1	yes	characteristic
343	C₂	#20283589	1	1	yes	characteristic
344	C₂	#20283631	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#335	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133239	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#335	D₅₃
hypocenter	C₅₃	#17857
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
1. #20283824	C₂	semidirect
3. #20273526	C₂	semidirect
4. #20273521	C₂	semidirect
5. #20273509	C₂	semidirect
6. #20273100	C₂	semidirect
7. #20272998	C₂	semidirect
8. C₂×C₅₃×#335	C₂	semidirect
35. #13133239	C₈	semidirect
81. #11719945	C₂×C₈	semidirect
90. C₂×C₈×C₅₃	C₂×C₈	semidirect
112. Dic₂₁₂	C₂×C₈	semidirect
113. D₄₂₄	C₂×C₈	semidirect
114. #11719913	C₂×C₈	semidirect
173. C₂×#335	D₅₃	semidirect
206. C₅₃	#17857	semidirect
270. C₂×C₈	C₈×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20283824	2
C₄×#13133239	2
#20273526	2
#20273521	2
#20273509	2
#20273100	2
#20272998	2
C₂×C₅₃×#335	2
#17857	53