Subgroups — #27355827

Order 13568.

648 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283847	C₂	1	2	yes	maximal, complemented
2	#20273629	C₂	1	2	yes	maximal, complemented
3	#20273559	C₂	1	4	yes	maximal, complemented
4	#20273534	C₂	1	2	yes	maximal, complemented
5	#20273471	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273154	C₂	1	2	yes	maximal, complemented
7	#20273004	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#135×C₅₃	C₂	1	1	yes	maximal, characteristic
9	Q₈×#88322	C₂²	1	4	yes	—
10	#16810701	—	2	4	no	—
11	#16810701	C₂²	1	4	yes	—
12	#16810676	—	2	2	no	—
13	#16810676	C₂²	1	1	yes	characteristic
14	#16810647	—	2	2	no	—
15	#16810647	C₂²	1	2	yes	—
16	C₄×#11719917	C₂²	1	1	yes	characteristic
17	#16810374	C₂²	1	2	yes	—
18	#16810374	C₂²	1	2	yes	—
19	Q₈⋊#88322	C₂²	1	4	yes	—
20	D₄⋊#88322	C₂²	1	4	yes	—
21	#16810345	C₂²	1	2	yes	—
22	#16810344	C₂²	1	2	yes	—
23	C₂×#135×C₅₃	C₄	1	1	yes	characteristic
24	C₄×#41×C₅₃	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
26	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
27	C₄×SD₁₆×C₅₃	C₄	1	2	yes	—
28	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
29	C₄×D₈×C₅₃	C₄	1	1	yes	characteristic
30	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#427	C₄	1	1	yes	characteristic
32	C₅₃×#416	C₄	1	2	yes	—
33	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×#11719917	—	2	4	no	—
35	C₂×#11719917	—	2	2	no	—
36	C₂×#11719917	—	2	1	no	—
37	C₄⋊#88322	—	2	2	no	—
38	C₄⋊#88322	—	2	4	no	—
39	C₄⋊#88322	D₄	1	2	yes	—
40	C₄⋊#88322	C₂³	1	4	yes	—
41	C₂²⋊#88322	—	2	4	no	—
42	C₂²⋊#88322	D₄	1	2	yes	—
43	#13133128	—	2	1	no	—
44	C₄×#88322	—	2	4	no	—
45	C₄×#88322	C₂³	1	2	yes	—
46	C₄×#88322	C₂³	1	2	yes	—
47	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
48	#135×C₅₃	—	2	4	no	—
49	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
55	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
56	C₄×D₄×C₅₃	—	2	2	no	—
57	C₄×D₄×C₅₃	—	2	2	no	—
58	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
59	C₄×D₄×C₅₃	C₂³	1	2	yes	—
60	#117×C₅₃	—	2	2	no	—
61	#117×C₅₃	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₅₃	—	2	2	no	—
63	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
64	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
65	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
66	#106×C₅₃	C₂×C₄	1	2	yes	—
67	#103×C₅₃	C₂×C₄	1	2	yes	—
68	#103×C₅₃	C₂×C₄	1	2	yes	—
69	#102×C₅₃	C₂×C₄	1	2	yes	—
70	#102×C₅₃	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	#41×C₅₃	—	2	4	no	—
76	#41×C₅₃	—	4	2	no	—
77	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂×D₄×C₅₃	—	2	2	no	—
79	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₅₃	—	2	2	no	—
81	C₂²×C₄×C₅₃	—	2	2	no	—
82	C₂²×C₄×C₅₃	—	2	2	no	—
83	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
86	Q₁₆×C₅₃	M₁₆	1	4	yes	complemented
87	SD₁₆×C₅₃	—	2	4	no	—
88	D₈×C₅₃	—	2	2	no	—
89	C₂×C₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	—	4	1	no	—
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	C₄⋊C₄×C₅₃	—	2	2	no	—
96	C₄⋊C₄×C₅₃	—	2	2	no	—
97	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
98	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
99	C₂²⋊C₄×C₅₃	—	2	2	no	—
100	C₂²⋊C₄×C₅₃	—	2	2	no	—
101	C₂²⋊C₄×C₅₃	—	2	2	no	—
102	C₄²×C₅₃	—	2	2	no	—
103	C₄²×C₅₃	—	2	2	no	—
104	C₄²×C₅₃	—	2	2	no	—
105	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₅₃	#41	1	1	yes	characteristic
108	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
109	#11719917	—	4	2	no	—
110	#11719917	—	8	4	no	—
111	#11719917	—	8	2	no	—
112	C₂×#88322	—	2	4	no	—
113	C₂×#88322	—	2	2	no	—
114	C₂×#88322	C₂×D₄	1	4	yes	—
115	C₂³×C₅₃	—	2	2	no	—
116	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
117	Q₈×C₅₃	—	2	2	no	—
118	Q₈×C₅₃	C₂×M₁₆	1	4	yes	—
119	D₄×C₅₃	—	2	2	no	—
120	D₄×C₅₃	—	2	2	no	—
121	D₄×C₅₃	—	4	4	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	1	no	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	4	4	no	—
130	C₂×C₄×C₅₃	—	4	2	no	—
131	C₂×C₄×C₅₃	—	4	2	no	—
132	C₂×C₄×C₅₃	—	4	4	no	—
133	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
139	C₈×C₅₃	—	2	1	no	—
140	C₈×C₅₃	—	2	1	no	—
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
143	#88322	—	2	4	no	—
144	#88322	—	4	2	no	—
145	#88322	—	4	4	no	—
146	#18063	—	53	1	no	maximal, Sylow
147	C₂²×C₅₃	—	2	1	no	—
148	C₂²×C₅₃	—	2	1	no	—
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	4	2	no	—
151	C₂²×C₅₃	—	4	2	no	—
152	C₂²×C₅₃	—	4	2	no	—
153	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	2	4	no	—
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	—	4	2	no	—
159	C₄×C₅₃	—	4	2	no	—
160	C₄×C₅₃	#444	1	2	yes	—
161	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
162	C₄×C₅₃	#568	1	1	yes	characteristic
163	C₄×C₅₃	#577	1	2	yes	—
164	#2991	—	53	2	no	—
165	C₄×#135	D₅₃	1	1	yes	characteristic
166	#1594	—	53	2	no	—
167	#1583	—	53	4	no	—
168	#1579	—	53	2	no	—
169	#1566	—	53	1	no	—
170	#1489	—	53	2	no	—
171	#1451	—	53	1	no	—
172	C₂×C₅₃	—	4	1	no	—
173	C₂×C₅₃	—	8	2	no	—
174	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
175	C₂×C₅₃	#2952	1	1	yes	characteristic
176	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
177	C₂×#135	Dic₅₃	1	1	yes	characteristic
178	C₄×#41	C₂×D₅₃	1	2	yes	—
179	C₈×Q₈	—	53	4	no	—
180	C₄×Q₁₆	Dic₅₃	1	1	yes	characteristic
181	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
182	C₄×SD₁₆	Dic₅₃	1	2	yes	—
183	C₄×SD₁₆	C₂×D₅₃	1	2	yes	—
184	C₄×D₈	Dic₅₃	1	1	yes	characteristic
185	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
186	#436	—	53	4	no	—
187	#436	—	106	4	no	—
188	#431	—	53	1	no	—
189	#431	—	106	2	no	—
190	#427	Dic₅₃	1	1	yes	characteristic
191	#423	—	53	2	no	—
192	#423	—	106	2	no	—
193	#416	Dic₅₃	1	2	yes	—
194	C₄×M₁₆	—	53	1	no	—
195	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
196	#334	—	53	2	no	—
197	#334	—	53	2	no	—
198	Q₈⋊C₈	—	53	4	no	—
199	D₄⋊C₈	—	53	4	no	—
200	#322	—	53	2	no	—
201	#322	—	53	2	no	—
202	C₅₃	#18063	1	1	yes	characteristic, complemented, Sylow
203	C₂×#41	C₂×Dic₅₃	1	2	yes	—
204	#135	—	2	4	no	—
205	C₂×Q₁₆	C₂×Dic₅₃	1	1	yes	characteristic
206	C₂×SD₁₆	C₂×Dic₅₃	1	2	yes	—
207	C₂×D₈	C₂×Dic₅₃	1	1	yes	characteristic
208	C₂×M₁₆	—	106	4	no	—
209	C₂×M₁₆	—	106	2	no	—
210	C₂×M₁₆	—	106	1	no	—
211	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
212	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
213	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
214	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
215	C₄×D₄	—	2	2	no	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
218	C₄×D₄	C₂²×D₅₃	1	2	yes	—
219	#117	—	2	2	no	—
220	#117	C₂×Dic₅₃	1	2	yes	—
221	C₂×C₄²	—	2	2	no	—
222	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
223	#107	C₂×Dic₅₃	1	1	yes	characteristic
224	#107	C₂×Dic₅₃	1	1	yes	characteristic
225	#106	C₂×Dic₅₃	1	2	yes	—
226	C₄⋊C₈	—	53	2	no	—
227	C₄⋊C₈	—	53	4	no	—
228	C₄⋊C₈	—	106	4	no	—
229	C₄⋊C₈	—	106	2	no	—
230	#103	C₂×Dic₅₃	1	2	yes	—
231	#103	C₂×Dic₅₃	1	2	yes	—
232	#102	C₂×Dic₅₃	1	2	yes	—
233	#102	C₂×Dic₅₃	1	2	yes	—
234	C₂²⋊C₈	—	53	2	no	—
235	C₂²⋊C₈	—	106	4	no	—
236	#97	—	106	1	no	—
237	C₄×C₈	—	53	2	no	—
238	C₄×C₈	—	53	2	no	—
239	C₄×C₈	—	106	4	no	—
240	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
241	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
244	#41	—	2	4	no	—
245	#41	—	4	2	no	—
246	C₂×Q₈	C₂²×Dic₅₃	1	2	yes	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	C₂²×Dic₅₃	1	2	yes	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
253	C₂²×C₄	#11719947	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
255	Q₁₆	#11719917	1	4	yes	complemented
256	SD₁₆	—	2	4	no	—
257	D₈	—	2	2	no	—
258	M₁₆	—	424	4	no	—
259	M₁₆	—	424	2	no	—
260	M₁₆	—	212	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	53	4	no	—
264	C₂×C₈	—	106	4	no	—
265	C₂×C₈	—	106	2	no	—
266	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
267	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
268	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
269	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
273	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
274	C₂²⋊C₄	—	2	2	no	—
275	C₂²⋊C₄	—	2	2	no	—
276	C₂²⋊C₄	—	2	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	D₄×D₅₃	1	1	yes	characteristic
281	C₄²	#11719947	1	1	yes	characteristic
282	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
283	C₄²	C₂³×D₅₃	1	1	yes	characteristic
284	C₂³	—	2	2	no	—
285	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
286	Q₈	—	2	2	no	—
287	Q₈	C₂×#11719917	1	4	yes	—
288	D₄	—	2	2	no	—
289	D₄	—	2	2	no	—
290	D₄	—	4	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
303	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
304	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
305	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
306	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
307	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	106	4	no	—
312	C₈	—	212	2	no	—
313	C₈	—	212	4	no	—
314	C₈	C₂×#11719917	1	2	yes	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	4	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	2	no	—
328	C₄	#16810745	1	2	yes	—
329	C₄	C₂²×#11719917	1	1	yes	characteristic
330	C₄	#16811390	1	1	yes	characteristic
331	C₄	#16811430	1	2	yes	—
332	C₂	—	4	1	no	—
333	C₂	—	8	2	no	—
334	C₂	D₄×#11719917	1	1	yes	characteristic
335	C₂	#20283599	1	1	yes	characteristic
336	C₂	C₂×#16810745	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₄×#135×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#135×C₅₃	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₄×#135	D₅₃
hypocenter	C₅₃	#18063
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #20283847	C₂	semidirect
2. #20273629	C₂	semidirect
3. #20273559	C₂	semidirect
4. #20273534	C₂	semidirect
5. #20273471	C₂	semidirect
6. #20273154	C₂	semidirect
7. #20273004	C₂	semidirect
86. Q₁₆×C₅₃	M₁₆	semidirect
202. C₅₃	#18063	semidirect
255. Q₁₆	#11719917	semidirect

Maximal subgroups

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

subgroup	index
#20283847	2
#20273629	2
#20273559	2
#20273534	2
#20273471	2
#20273154	2
#20273004	2
C₄×#135×C₅₃	2
#18063	53