Subgroups — #27355790

Order 13568.

656 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283844	C₂	1	2	yes	maximal, complemented
2	C₄×#13133239	C₂	1	1	yes	maximal, characteristic
3	#20273601	C₂	1	2	yes	maximal, complemented
4	#20273595	C₂	1	4	yes	maximal, complemented
5	C₈×Dic₂₁₂	C₂	1	2	yes	maximal, complemented
6	#20273474	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273155	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133239	C₄	1	1	yes	characteristic
10	C₄×#11719945	C₂²	1	2	yes	—
11	C₈×Dic₁₀₆	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	#16810702	C₂²	1	4	yes	—
19	#16810698	—	2	4	no	—
20	#16810675	—	2	2	no	—
21	#16810661	C₄	1	1	yes	characteristic
22	#16810647	C₂²	1	2	yes	—
23	#16810645	—	2	2	no	—
24	#16810620	C₄	1	2	yes	—
25	#16810376	C₂²	1	2	yes	—
26	#16810376	C₂²	1	2	yes	—
27	#16810355	C₂²	1	4	yes	—
28	#16810352	C₂²	1	4	yes	—
29	C₅₃×#431	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#322	C₂²	1	2	yes	—
33	C₈²×C₅₃	C₂²	1	2	yes	—
34	C₂×#11719945	C₂×C₄	1	2	yes	—
35	#13133239	—	2	4	no	—
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×#11719917	—	2	2	no	—
40	C₂×#11719912	—	2	4	no	—
41	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₀₆	C₂³	1	2	yes	—
43	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
44	C₄×D₂₁₂	C₂³	1	2	yes	—
45	C₄×C₂²⋊D₅₃	—	2	2	no	—
46	C₄×C₂²⋊D₅₃	—	2	2	no	—
47	#13133176	C₂×C₄	1	2	yes	—
48	#13133174	—	2	2	no	—
49	C₄²×D₅₃	—	2	2	no	—
50	#13133151	C₂×C₄	1	1	yes	characteristic
51	#13133151	C₂×C₄	1	1	yes	characteristic
52	#13133149	C₂×C₄	1	2	yes	—
53	C₄⋊#88322	—	2	2	no	—
54	C₄⋊#88322	C₂³	1	4	yes	—
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	#13133129	—	2	4	no	—
61	C₈×Dic₅₃	—	2	4	no	—
62	C₂×M₁₆×C₅₃	—	2	1	no	—
63	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
67	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
68	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
69	#97×C₅₃	—	2	1	no	—
70	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₅₃	C₂³	1	2	yes	—
73	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₅₃	C₂³	1	2	yes	—
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	—	4	2	no	—
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	M₁₆×C₅₃	—	2	4	no	—
88	C₂×C₈×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	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
97	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₅₃	#41	1	1	yes	characteristic
100	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
101	C₂²⋊Dic₅₃	—	2	2	no	—
102	#11719921	—	2	2	no	—
103	#11719921	—	2	2	no	—
104	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
105	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
106	#11719919	—	2	2	no	—
107	#11719919	—	2	2	no	—
108	C₄×Dic₅₃	—	2	2	no	—
109	C₄×Dic₅₃	—	2	2	no	—
110	C₄×Dic₅₃	—	2	2	no	—
111	#11719917	—	8	2	no	—
112	C₂×#88322	—	2	4	no	—
113	Dic₂₁₂	M₁₆	1	4	yes	complemented
114	D₄₂₄	—	2	2	no	—
115	#11719913	—	2	4	no	—
116	#11719912	—	8	4	no	—
117	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
118	C₂²×D₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	2	1	no	—
122	C₂×C₄×C₅₃	—	2	1	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
130	C₂²⋊D₅₃	—	4	4	no	—
131	C₂×Dic₅₃	—	2	2	no	—
132	C₂×Dic₅₃	—	2	2	no	—
133	C₂×Dic₅₃	—	2	2	no	—
134	D₂₁₂	—	2	2	no	—
135	D₂₁₂	—	2	2	no	—
136	C₄×D₅₃	—	4	4	no	—
137	C₄×D₅₃	—	4	2	no	—
138	C₄×D₅₃	—	4	2	no	—
139	C₄×D₅₃	—	4	4	no	—
140	Dic₁₀₆	—	2	2	no	—
141	Dic₁₀₆	C₂×M₁₆	1	4	yes	—
142	C₈×C₅₃	—	2	4	no	—
143	C₈×C₅₃	—	2	1	no	—
144	C₈×C₅₃	—	2	1	no	—
145	C₈×C₅₃	—	2	1	no	—
146	C₈×C₅₃	—	2	2	no	—
147	C₈×C₅₃	—	2	2	no	—
148	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
149	#88322	—	4	4	no	—
150	#18059	—	53	1	no	maximal, Sylow
151	C₂²×C₅₃	—	2	1	no	—
152	C₂²×C₅₃	—	2	1	no	—
153	C₂²×C₅₃	—	2	1	no	—
154	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
155	C₂×D₅₃	—	4	2	no	—
156	C₂×D₅₃	—	4	2	no	—
157	C₂×D₅₃	—	4	2	no	—
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	—	2	1	no	—
161	C₄×C₅₃	#443	1	2	yes	—
162	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
163	C₄×C₅₃	#568	1	1	yes	characteristic
164	C₄×C₅₃	#576	1	1	yes	characteristic
165	C₄×C₅₃	#578	1	1	yes	characteristic
166	Dic₅₃	—	2	4	no	—
167	Dic₅₃	—	4	2	no	—
168	Dic₅₃	—	4	2	no	—
169	#2991	—	53	2	no	—
170	C₄×#135	—	53	1	no	—
171	#1589	—	53	2	no	—
172	#1588	—	53	4	no	—
173	C₈×Q₁₆	—	53	2	no	—
174	#1567	—	53	1	no	—
175	#1489	—	53	2	no	—
176	C₈×M₁₆	D₅₃	1	1	yes	characteristic, complemented
177	C₂×C₅₃	—	4	1	no	—
178	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
179	C₂×C₅₃	#2951	1	1	yes	characteristic
180	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
181	D₅₃	—	8	2	no	—
182	C₂×#135	—	53	1	no	—
183	C₄×#41	—	53	2	no	—
184	C₈×Q₈	—	53	4	no	—
185	C₄×Q₁₆	—	53	1	no	—
186	C₄×Q₁₆	—	53	1	no	—
187	C₄×SD₁₆	—	53	2	no	—
188	C₄×SD₁₆	—	53	2	no	—
189	C₄×D₈	—	53	1	no	—
190	C₄×D₈	—	53	1	no	—
191	#436	—	53	4	no	—
192	#436	—	106	4	no	—
193	#431	—	106	2	no	—
194	#431	C₂×D₅₃	1	1	yes	characteristic
195	#427	—	53	1	no	—
196	#423	—	53	2	no	—
197	#423	—	106	2	no	—
198	#416	—	53	2	no	—
199	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
200	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
201	#335	—	53	2	no	—
202	#335	—	53	2	no	—
203	Q₈⋊C₈	—	53	4	no	—
204	D₄⋊C₈	—	53	4	no	—
205	#322	C₂×D₅₃	1	2	yes	—
206	C₈²	C₂×D₅₃	1	2	yes	—
207	C₅₃	#18059	1	1	yes	characteristic, complemented, Sylow
208	C₂×#41	—	53	2	no	—
209	#135	—	106	4	no	—
210	C₂×Q₁₆	—	53	1	no	—
211	C₂×SD₁₆	—	53	2	no	—
212	C₂×D₈	—	53	1	no	—
213	C₂×M₁₆	—	2	1	no	—
214	C₂×M₁₆	—	106	2	no	—
215	C₂×M₁₆	—	106	4	no	—
216	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
217	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
218	C₄×Q₈	—	53	2	no	—
219	C₄×Q₈	—	53	2	no	—
220	C₄×D₄	—	53	2	no	—
221	C₄×D₄	—	53	2	no	—
222	C₄×D₄	—	106	2	no	—
223	C₄×D₄	—	106	2	no	—
224	#117	—	53	2	no	—
225	#117	—	106	2	no	—
226	C₂×C₄²	—	106	2	no	—
227	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
228	#107	—	53	1	no	—
229	#107	—	53	1	no	—
230	#106	—	53	2	no	—
231	C₄⋊C₈	—	53	4	no	—
232	C₄⋊C₈	—	106	4	no	—
233	C₄⋊C₈	—	106	2	no	—
234	C₄⋊C₈	D₂₁₂	1	1	yes	characteristic
235	C₄⋊C₈	D₂₁₂	1	1	yes	characteristic
236	#103	—	53	2	no	—
237	#103	—	53	2	no	—
238	#102	—	53	2	no	—
239	#102	—	53	2	no	—
240	C₂²⋊C₈	—	106	4	no	—
241	C₂²⋊C₈	D₂₁₂	1	2	yes	—
242	#97	—	2	1	no	—
243	C₄×C₈	—	106	4	no	—
244	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
245	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₅₃	1	2	yes	—
247	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₅₃	1	2	yes	—
250	#41	—	106	4	no	—
251	#41	—	212	2	no	—
252	C₂×Q₈	—	53	2	no	—
253	C₂×D₄	—	53	2	no	—
254	C₂×D₄	—	106	2	no	—
255	C₂²×C₄	—	106	2	no	—
256	C₂²×C₄	—	106	2	no	—
257	C₂²×C₄	—	106	2	no	—
258	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
259	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
260	C₂²×C₄	#11719945	1	1	yes	characteristic
261	Q₁₆	—	53	4	no	—
262	SD₁₆	—	106	4	no	—
263	D₈	—	106	2	no	—
264	M₁₆	—	2	4	no	—
265	M₁₆	—	424	4	no	—
266	M₁₆	—	424	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	106	4	no	—
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	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
276	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
277	C₄⋊C₄	—	53	2	no	—
278	C₄⋊C₄	—	53	2	no	—
279	C₄⋊C₄	—	106	2	no	—
280	C₄⋊C₄	—	106	2	no	—
281	C₂²⋊C₄	—	106	2	no	—
282	C₂²⋊C₄	—	106	2	no	—
283	C₂²⋊C₄	—	106	2	no	—
284	C₄²	—	106	2	no	—
285	C₄²	—	106	2	no	—
286	C₄²	—	106	2	no	—
287	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
288	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
289	C₄²	#11719945	1	1	yes	characteristic
290	C₄²	C₂³×D₅₃	1	1	yes	characteristic
291	C₂³	—	106	2	no	—
292	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
293	Q₈	—	53	4	no	—
294	Q₈	—	106	2	no	—
295	D₄	—	106	2	no	—
296	D₄	—	106	2	no	—
297	D₄	—	212	4	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	106	2	no	—
304	C₂×C₄	—	106	2	no	—
305	C₂×C₄	—	106	2	no	—
306	C₂×C₄	—	212	4	no	—
307	C₂×C₄	—	212	2	no	—
308	C₂×C₄	—	212	2	no	—
309	C₂×C₄	—	212	4	no	—
310	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
311	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
312	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
313	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
315	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
316	C₈	—	2	4	no	—
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	2	2	no	—
321	C₈	—	2	2	no	—
322	C₈	—	212	4	no	—
323	C₈	M₁₆×D₅₃	1	2	yes	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	212	2	no	—
328	C₂²	—	212	2	no	—
329	C₂²	—	212	2	no	—
330	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	106	4	no	—
335	C₄	—	212	2	no	—
336	C₄	—	212	2	no	—
337	C₄	#16810741	1	2	yes	—
338	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
339	C₄	#16811389	1	1	yes	characteristic
340	C₄	#16811428	1	1	yes	characteristic
341	C₄	#16811440	1	1	yes	characteristic
342	C₂	—	4	1	no	—
343	C₂	—	424	2	no	—
344	C₂	M₁₆×D₂₁₂	1	1	yes	characteristic
345	C₂	#20283592	1	1	yes	characteristic
346	C₂	C₂×#16810741	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₈×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133239	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₈×M₁₆	D₅₃
hypocenter	C₅₃	#18059
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
1. #20283844	C₂	semidirect
3. #20273601	C₂	semidirect
4. #20273595	C₂	semidirect
5. C₈×Dic₂₁₂	C₂	semidirect
6. #20273474	C₂	semidirect
7. #20273155	C₂	semidirect
8. C₈×M₁₆×C₅₃	C₂	semidirect
113. Dic₂₁₂	M₁₆	semidirect
176. C₈×M₁₆	D₅₃	semidirect
207. C₅₃	#18059	semidirect

Maximal subgroups

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

subgroup	index
#20283844	2
C₄×#13133239	2
#20273601	2
#20273595	2
C₈×Dic₂₁₂	2
#20273474	2
#20273155	2
C₈×M₁₆×C₅₃	2
#18059	53