Subgroups — #28301713

Order 13568.

514 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133282	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133271	C₂	1	1	yes	maximal, characteristic, complemented
3	#20290009	C₂	1	1	yes	maximal, characteristic
4	#20289925	C₂	1	1	yes	maximal, characteristic, complemented
5	#20289546	C₂	1	1	yes	maximal, characteristic, complemented
6	#20289542	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×#11719949	C₂	1	1	yes	maximal, characteristic
8	#16811642	C₂²	1	1	yes	characteristic
9	C₂×#13133282	C₄	1	2	yes	—
10	C₄×#11719936	C₂²	1	1	yes	characteristic
11	C₂×#13133271	—	2	1	no	—
12	C₂×#13133271	—	2	1	no	—
13	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₄²×#6157	C₂²	1	1	yes	characteristic
15	#16811493	C₄	1	2	yes	—
16	Q₈⋊#88324	C₄	1	2	yes	complemented
17	#16811471	C₄	1	2	yes	complemented
18	#88324⋊C₈	—	2	1	no	—
19	#16811464	—	2	1	no	—
20	C₄×#11719949	C₂²	1	1	yes	characteristic
21	Q₈×#88322	C₄	1	1	yes	characteristic, complemented
22	C₈×D₂₁₂	—	2	1	no	—
23	C₈×D₂₁₂	C₄	1	1	yes	characteristic, complemented
24	#16810688	—	2	1	no	—
25	#16810688	C₄	1	1	yes	characteristic, complemented
26	#16810675	—	2	1	no	—
27	#16810675	C₂²	1	1	yes	characteristic
28	C₄×C₈×D₅₃	—	2	1	no	—
29	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
30	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
31	C₂×#11719936	C₂×C₄	1	2	yes	—
32	C₂×D₅₃⋊C₈	—	2	1	no	—
33	C₂×C₄×#6157	—	2	1	no	—
34	C₂×C₄×#6157	—	2	1	no	—
35	C₂×C₄×#6157	C₂×C₄	1	2	yes	—
36	C₄⋊#88324	C₂×C₄	1	2	yes	—
37	#13133282	—	2	4	no	—
38	#13133274	—	2	1	no	—
39	#13133272	—	2	1	no	—
40	#13133271	—	2	4	no	—
41	#13133271	—	2	4	no	—
42	C₄×#88324	—	2	1	no	—
43	C₄×#88324	—	2	1	no	—
44	C₄×#88324	—	2	1	no	—
45	C₄×#88324	C₂×C₄	1	2	yes	—
46	C₂×#11719949	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×D₅₃	—	2	1	no	—
48	C₂×C₈×D₅₃	—	2	1	no	—
49	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
50	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
51	Q₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄×D₂₁₂	—	2	1	no	—
53	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
54	#13133179	—	2	1	no	—
55	#13133179	C₂×C₄	1	1	yes	characteristic
56	#13133175	—	2	1	no	—
57	#13133175	C₂×C₄	1	1	yes	characteristic
58	C₄²×D₅₃	—	2	1	no	—
59	C₄²×D₅₃	C₂³	1	1	yes	characteristic
60	C₄⋊#88322	—	2	1	no	—
61	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
62	#13133129	—	2	1	no	—
63	#13133129	—	2	1	no	—
64	#13133129	—	2	1	no	—
65	C₄×#88322	—	2	1	no	—
66	C₄×#88322	C₂×C₄	1	1	yes	characteristic
67	C₈×Dic₅₃	—	2	1	no	—
68	C₈×Dic₅₃	D₄	1	1	yes	characteristic
69	C₈×Dic₅₃	D₄	1	1	yes	characteristic
70	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₄⋊C₈×C₅₃	—	2	1	no	—
72	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₅₃	—	2	1	no	—
74	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
75	C₂²×#6157	—	2	2	no	—
76	C₂×Q₈×C₅₃	C₄²	1	1	yes	characteristic
77	#11719949	—	2	2	no	—
78	#11719949	M₁₆	1	4	yes	—
79	C₂×D₂₁₂	—	2	1	no	—
80	C₂×D₂₁₂	C₄²	1	1	yes	characteristic
81	C₂×C₄×D₅₃	—	2	1	no	—
82	C₂×C₄×D₅₃	—	2	1	no	—
83	C₂×C₄×D₅₃	—	2	1	no	—
84	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
87	C₂×#88324	—	2	2	no	—
88	C₂×#88324	—	2	2	no	—
89	C₂×#88324	—	2	2	no	—
90	C₄×#6157	—	2	2	no	—
91	C₄×#6157	—	2	2	no	—
92	C₄×#6157	—	2	2	no	—
93	C₄×#6157	—	2	2	no	—
94	C₄×#6157	—	4	2	no	—
95	#11719936	—	2	2	no	—
96	#11719936	—	2	2	no	—
97	D₅₃⋊C₈	—	2	2	no	—
98	D₅₃⋊C₈	—	2	2	no	—
99	C₂×C₈×C₅₃	—	2	1	no	—
100	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
102	C₄⋊C₄×C₅₃	—	2	1	no	—
103	C₄⋊C₄×C₅₃	C₄²	1	1	yes	characteristic
104	C₄²×C₅₃	—	2	1	no	—
105	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	#11719921	—	2	1	no	—
107	#11719921	—	2	1	no	—
108	#11719921	—	2	1	no	—
109	C₄⋊Dic₅₃	—	2	1	no	—
110	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
111	C₄×Dic₅₃	—	2	1	no	—
112	C₄×Dic₅₃	—	2	1	no	—
113	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
115	C₄×Dic₅₃	#41	1	1	yes	characteristic
116	C₂×#88322	—	2	1	no	—
117	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
118	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
119	C₈×D₅₃	—	2	2	no	—
120	C₈×D₅₃	—	2	2	no	—
121	C₈×D₅₃	—	4	2	no	—
122	C₈×D₅₃	—	4	2	no	—
123	C₂²×D₅₃	—	2	1	no	—
124	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
125	C₂×#6157	—	2	4	no	—
126	C₂×#6157	—	4	2	no	—
127	C₂×#6157	—	4	2	no	—
128	Q₈×C₅₃	—	2	1	no	—
129	Q₈×C₅₃	#97	1	2	yes	complemented
130	C₂×C₄×C₅₃	—	2	1	no	—
131	C₂×C₄×C₅₃	—	2	1	no	—
132	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
135	C₂×Dic₅₃	—	2	1	no	—
136	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
137	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
138	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
139	D₂₁₂	—	2	4	no	—
140	D₂₁₂	—	2	1	no	—
141	D₂₁₂	#97	1	2	yes	complemented
142	C₄×D₅₃	—	2	1	no	—
143	C₄×D₅₃	—	2	1	no	—
144	C₄×D₅₃	—	2	4	no	—
145	C₄×D₅₃	—	2	2	no	—
146	C₄×D₅₃	—	4	1	no	—
147	C₄×D₅₃	—	4	1	no	—
148	C₄×D₅₃	—	4	2	no	—
149	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
150	C₄×D₅₃	#131	1	2	yes	—
151	#88324	—	2	4	no	—
152	#88324	—	4	2	no	—
153	#88324	—	4	2	no	—
154	C₈×C₅₃	—	2	1	no	—
155	C₈×C₅₃	—	4	1	no	—
156	C₈×C₅₃	#104	1	2	yes	complemented
157	#88322	—	2	1	no	—
158	#88322	—	4	1	no	—
159	#88322	#104	1	2	yes	complemented
160	#8192	—	53	1	no	maximal, Sylow
161	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×D₅₃	—	2	1	no	—
163	C₂×D₅₃	—	2	1	no	—
164	C₂×D₅₃	—	2	2	no	—
165	C₂×D₅₃	—	4	1	no	—
166	C₂×D₅₃	—	4	1	no	—
167	C₂×D₅₃	#435	1	2	yes	—
168	#6157	—	4	4	no	—
169	C₄×C₅₃	—	2	2	no	—
170	C₄×C₅₃	—	2	1	no	—
171	C₄×C₅₃	—	4	1	no	—
172	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
173	C₄×C₅₃	#405	1	1	yes	characteristic
174	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
175	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
176	Dic₅₃	—	2	1	no	—
177	Dic₅₃	—	4	1	no	—
178	Dic₅₃	#435	1	2	yes	—
179	Dic₅₃	#444	1	2	yes	—
180	C₈×#41	—	53	1	no	—
181	C₄×#104	—	53	1	no	—
182	C₄×#97	—	53	1	no	—
183	#1476	—	53	1	no	—
184	#1451	—	53	1	no	—
185	#1278	—	53	1	no	—
186	#1276	—	53	1	no	—
187	C₂×C₅₃	#1752	1	1	yes	characteristic
188	C₂×C₅₃	C₄×#104	1	1	yes	characteristic
189	C₂×C₅₃	#1765	1	1	yes	characteristic
190	D₅₃	—	2	2	no	—
191	D₅₃	—	4	2	no	—
192	C₄×#41	—	53	1	no	—
193	C₈×Q₈	—	53	1	no	—
194	C₈×Q₈	#6157	1	1	yes	characteristic, complemented
195	C₈×D₄	—	53	1	no	—
196	C₈×D₄	—	53	1	no	—
197	C₈×D₄	—	106	1	no	—
198	C₈×D₄	—	106	1	no	—
199	#431	—	53	1	no	—
200	#431	—	53	1	no	—
201	#431	—	106	1	no	—
202	C₂×#104	—	53	2	no	—
203	C₄×M₁₆	—	53	1	no	—
204	C₂×#97	—	106	1	no	—
205	C₂×#97	—	106	1	no	—
206	C₂×C₄×C₈	—	53	1	no	—
207	C₂×C₄×C₈	—	53	1	no	—
208	C₂×C₄×C₈	—	106	1	no	—
209	C₄³	—	53	1	no	—
210	#339	—	53	2	no	—
211	Q₈⋊C₈	—	53	2	no	—
212	D₄⋊C₈	—	53	2	no	—
213	#322	—	106	1	no	—
214	#322	—	106	1	no	—
215	C₅₃	#8192	1	1	yes	characteristic, complemented, Sylow
216	C₂×#41	—	53	1	no	—
217	C₂×M₁₆	—	53	2	no	—
218	C₂²×C₈	—	53	1	no	—
219	C₂²×C₈	—	53	1	no	—
220	C₂²×C₈	—	106	1	no	—
221	C₂²×C₈	—	106	1	no	—
222	C₂²×C₈	—	106	1	no	—
223	C₄×Q₈	—	53	1	no	—
224	C₄×Q₈	C₂×#6157	1	1	yes	characteristic
225	C₄×D₄	—	53	1	no	—
226	C₄×D₄	—	53	1	no	—
227	C₄×D₄	—	106	1	no	—
228	C₄×D₄	—	106	1	no	—
229	#117	—	53	1	no	—
230	#117	—	106	1	no	—
231	C₂×C₄²	—	53	1	no	—
232	C₂×C₄²	—	53	2	no	—
233	C₂×C₄²	—	106	1	no	—
234	C₂×C₄²	—	106	1	no	—
235	C₂×C₄²	—	106	1	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	53	2	no	—
238	C₄⋊C₈	—	53	1	no	—
239	C₄⋊C₈	—	106	1	no	—
240	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic
241	#104	—	106	4	no	—
242	C₂²⋊C₈	—	106	1	no	—
243	C₂²⋊C₈	—	106	1	no	—
244	C₂²⋊C₈	—	106	1	no	—
245	C₂²⋊C₈	—	106	1	no	—
246	#97	—	106	1	no	—
247	#97	—	106	4	no	—
248	#97	—	106	4	no	—
249	C₄×C₈	—	2	1	no	—
250	C₄×C₈	—	53	1	no	—
251	C₄×C₈	—	53	1	no	—
252	C₄×C₈	—	53	2	no	—
253	C₄×C₈	—	53	1	no	—
254	C₄×C₈	—	106	1	no	—
255	C₄×C₈	—	106	1	no	—
256	C₄×C₈	—	106	1	no	—
257	C₄×C₈	—	106	1	no	—
258	C₄×C₈	—	106	1	no	—
259	C₄×C₈	C₂×#6157	1	1	yes	characteristic
260	#41	—	53	4	no	—
261	#41	—	106	2	no	—
262	C₂×Q₈	C₄×#6157	1	1	yes	characteristic
263	C₂×D₄	—	53	1	no	—
264	C₂×D₄	—	106	1	no	—
265	C₂²×C₄	—	53	1	no	—
266	C₂²×C₄	—	53	1	no	—
267	C₂²×C₄	—	53	1	no	—
268	C₂²×C₄	—	106	1	no	—
269	C₂²×C₄	—	106	1	no	—
270	C₂²×C₄	—	106	2	no	—
271	C₂²×C₄	—	106	1	no	—
272	M₁₆	—	106	2	no	—
273	M₁₆	—	106	2	no	—
274	C₂×C₈	—	2	1	no	—
275	C₂×C₈	—	53	1	no	—
276	C₂×C₈	—	53	1	no	—
277	C₂×C₈	—	106	2	no	—
278	C₂×C₈	—	106	2	no	—
279	C₂×C₈	—	106	2	no	—
280	C₂×C₈	—	106	2	no	—
281	C₂×C₈	—	106	2	no	—
282	C₂×C₈	—	106	2	no	—
283	C₂×C₈	—	106	2	no	—
284	C₂×C₈	—	106	1	no	—
285	C₂×C₈	—	212	2	no	—
286	C₂×C₈	—	212	2	no	—
287	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
288	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	53	1	no	—
291	C₄⋊C₄	—	106	1	no	—
292	C₄⋊C₄	C₄×#6157	1	1	yes	characteristic
293	C₂²⋊C₄	—	106	1	no	—
294	C₂²⋊C₄	—	106	1	no	—
295	C₂²⋊C₄	—	106	1	no	—
296	C₄²	—	2	1	no	—
297	C₄²	—	53	1	no	—
298	C₄²	—	53	1	no	—
299	C₄²	—	53	1	no	—
300	C₄²	—	106	1	no	—
301	C₄²	—	106	1	no	—
302	C₄²	—	106	2	no	—
303	C₄²	—	106	2	no	—
304	C₄²	—	106	2	no	—
305	C₄²	—	106	2	no	—
306	C₄²	—	212	2	no	—
307	C₄²	C₂²×#6157	1	1	yes	characteristic
308	C₂³	—	53	1	no	—
309	C₂³	—	106	1	no	—
310	Q₈	—	2	1	no	—
311	Q₈	#13133271	1	2	yes	complemented
312	D₄	—	53	2	no	—
313	D₄	—	106	4	no	—
314	D₄	—	106	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	53	1	no	—
318	C₂×C₄	—	53	1	no	—
319	C₂×C₄	—	53	2	no	—
320	C₂×C₄	—	53	2	no	—
321	C₂×C₄	—	53	1	no	—
322	C₂×C₄	—	106	2	no	—
323	C₂×C₄	—	106	4	no	—
324	C₂×C₄	—	106	1	no	—
325	C₂×C₄	—	106	1	no	—
326	C₂×C₄	—	106	4	no	—
327	C₂×C₄	—	106	1	no	—
328	C₂×C₄	—	212	2	no	—
329	C₂×C₄	—	212	2	no	—
330	C₂×C₄	—	212	2	no	—
331	C₂×C₄	—	212	1	no	—
332	C₂×C₄	—	212	1	no	—
333	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
334	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
335	C₂×C₄	#13133296	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	4	1	no	—
338	C₈	—	53	2	no	—
339	C₈	—	106	4	no	—
340	C₈	—	106	1	no	—
341	C₈	—	212	2	no	—
342	C₈	—	212	2	no	—
343	C₈	—	212	1	no	—
344	C₈	#13133282	1	2	yes	complemented
345	C₂²	—	53	2	no	—
346	C₂²	—	106	1	no	—
347	C₂²	—	106	1	no	—
348	C₂²	—	106	2	no	—
349	C₂²	—	212	1	no	—
350	C₂²	—	212	1	no	—
351	C₂²	C₄×C₂²⋊#6157	1	1	yes	characteristic
352	C₄	—	2	2	no	—
353	C₄	—	2	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	53	2	no	—
356	C₄	—	53	2	no	—
357	C₄	—	106	1	no	—
358	C₄	—	212	4	no	—
359	C₄	—	212	1	no	—
360	C₄	C₂×#13133271	1	1	yes	characteristic
361	C₄	#16811586	1	1	yes	characteristic
362	C₄	D₅₃⋊#104	1	1	yes	characteristic
363	C₄	C₂×#13133282	1	1	yes	characteristic
364	C₂	—	106	2	no	—
365	C₂	—	212	2	no	—
366	C₂	#20290451	1	1	yes	characteristic
367	C₂	C₄×#13133282	1	1	yes	characteristic
368	C₂	#20290493	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#6157
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₄²	C₂²×#6157
Fitting	C₈×Q₈×C₅₃	C₄
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂×#11719949	C₂×C₄
cosocle	C₄²×D₅₃	C₂³
hypercenter	C₈×Q₈	#6157
hypocenter	C₅₃	#8192
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. C₄×#13133271	C₂	semidirect
4. #20289925	C₂	semidirect
5. #20289546	C₂	semidirect
6. #20289542	C₂	semidirect
16. Q₈⋊#88324	C₄	semidirect
17. #16811471	C₄	semidirect
21. Q₈×#88322	C₄	semidirect
23. C₈×D₂₁₂	C₄	semidirect
25. #16810688	C₄	semidirect
30. C₈×Q₈×C₅₃	C₄	semidirect
129. Q₈×C₅₃	#97	semidirect
141. D₂₁₂	#97	semidirect
156. C₈×C₅₃	#104	semidirect
159. #88322	#104	semidirect
194. C₈×Q₈	#6157	semidirect
215. C₅₃	#8192	semidirect
311. Q₈	#13133271	semidirect
344. C₈	#13133282	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133282	2
C₄×#13133271	2
#20290009	2
#20289925	2
#20289546	2
#20289542	2
C₈×#11719949	2
#8192	53