Subgroups — #28302930

Order 13568.

568 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290630	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88324	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#88324	C₂	1	2	yes	maximal
4	#20289551	C₂	1	2	yes	maximal
5	C₄×Q₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88324	—	2	1	no	—
7	C₂×C₄⋊#88324	C₂²	1	2	yes	—
8	C₂×C₄×#88324	C₂²	1	2	yes	—
9	#16811489	—	2	1	no	—
10	#16811489	—	2	1	no	—
11	#16811489	—	2	1	no	—
12	Q₈⋊#88324	—	2	4	no	—
13	C₂×Q₈×Dic₅₃	C₄	1	1	yes	characteristic
14	C₂×Q₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₅₃	—	2	1	no	—
16	C₄⋊C₄×Dic₅₃	—	2	1	no	—
17	C₄⋊C₄×Dic₅₃	C₄	1	1	yes	characteristic
18	C₄⋊C₄×Dic₅₃	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊Dic₅₃	—	2	1	no	—
20	C₄×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
21	C₄²×Dic₅₃	—	2	1	no	—
22	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
24	C₂²×#88324	—	2	2	no	—
25	C₂²×#88324	—	2	2	no	—
26	C₄⋊#88324	—	2	4	no	—
27	C₄⋊#88324	—	2	2	no	—
28	C₄⋊#88324	—	2	2	no	—
29	C₄×#88324	—	2	2	no	—
30	C₄×#88324	—	2	2	no	—
31	Q₈×Dic₅₃	—	2	2	no	—
32	Q₈×Dic₅₃	—	2	1	no	—
33	Q₈×Dic₅₃	—	2	1	no	—
34	Q₈×Dic₅₃	—	2	2	no	—
35	Q₈×Dic₅₃	D₄	1	2	yes	—
36	Q₈×Dic₅₃	Q₈	1	2	yes	—
37	C₂×C₄⋊Dic₅₃	—	2	1	no	—
38	C₂×C₄⋊Dic₅₃	—	2	1	no	—
39	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₅₃	—	2	1	no	—
42	C₂×C₄×Dic₅₃	—	2	1	no	—
43	C₂×C₄×Dic₅₃	—	2	1	no	—
44	C₂×C₄×Dic₅₃	—	2	1	no	—
45	C₂×C₄×Dic₅₃	—	2	1	no	—
46	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
50	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
51	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
52	#13133124	—	2	1	no	—
53	#13133124	—	2	1	no	—
54	#13133124	—	2	1	no	—
55	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₄×Q₈×C₅₃	—	2	2	no	—
57	C₄×Q₈×C₅₃	C₈	1	4	yes	complemented
58	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
59	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	—	2	1	no	—
61	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×Q₈×C₅₃	—	2	1	no	—
63	C₂×Q₈×C₅₃	—	2	1	no	—
64	C₂×Q₈×C₅₃	—	2	1	no	—
65	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
66	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
67	C₂×Q₈×C₅₃	M₁₆	1	2	yes	—
68	C₂²×C₄×C₅₃	—	2	1	no	—
69	C₂²×C₄×C₅₃	—	2	1	no	—
70	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×Dic₅₃	—	2	1	no	—
74	C₂²×Dic₅₃	—	2	1	no	—
75	C₂²×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂×#88324	—	2	4	no	—
80	C₂×#88324	—	4	2	no	—
81	C₂×#88324	—	4	2	no	—
82	C₂×#88324	—	4	2	no	—
83	C₂×#88324	—	4	2	no	—
84	C₂×#88324	—	4	2	no	—
85	C₄⋊C₄×C₅₃	—	2	4	no	—
86	C₄⋊C₄×C₅₃	—	2	1	no	—
87	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
88	C₄²×C₅₃	—	2	4	no	—
89	C₄²×C₅₃	C₂×C₈	1	2	yes	—
90	C₄²×C₅₃	M₁₆	1	2	yes	—
91	C₄⋊Dic₅₃	—	2	4	no	—
92	C₄⋊Dic₅₃	—	2	1	no	—
93	C₄⋊Dic₅₃	—	2	1	no	—
94	C₄⋊Dic₅₃	—	2	2	no	—
95	C₄⋊Dic₅₃	—	2	2	no	—
96	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
97	C₄⋊Dic₅₃	M₁₆	1	2	yes	—
98	C₄×Dic₅₃	—	2	4	no	—
99	C₄×Dic₅₃	—	2	1	no	—
100	C₄×Dic₅₃	—	2	1	no	—
101	C₄×Dic₅₃	—	2	2	no	—
102	C₄×Dic₅₃	—	2	2	no	—
103	C₄×Dic₅₃	—	2	2	no	—
104	C₄×Dic₅₃	—	2	2	no	—
105	C₄×Dic₅₃	—	2	2	no	—
106	C₄×Dic₅₃	—	4	1	no	—
107	C₄×Dic₅₃	—	4	2	no	—
108	C₄×Dic₅₃	—	4	2	no	—
109	C₄×Dic₅₃	—	4	2	no	—
110	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
111	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
112	C₄×Dic₅₃	#41	1	1	yes	characteristic
113	C₄×Dic₅₃	#41	1	1	yes	characteristic
114	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	Q₈×C₅₃	—	2	2	no	—
116	Q₈×C₅₃	—	2	2	no	—
117	Q₈×C₅₃	—	2	2	no	—
118	Q₈×C₅₃	C₄⋊C₈	1	4	yes	complemented
119	C₂×C₄×C₅₃	—	2	2	no	—
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	2	1	no	—
123	C₂×C₄×C₅₃	—	2	1	no	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	2	1	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	4	1	no	—
128	C₂×C₄×C₅₃	—	4	1	no	—
129	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
130	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
131	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
137	C₂×Dic₅₃	—	2	1	no	—
138	C₂×Dic₅₃	—	2	1	no	—
139	C₂×Dic₅₃	—	2	1	no	—
140	C₂×Dic₅₃	—	2	1	no	—
141	C₂×Dic₅₃	—	2	2	no	—
142	C₂×Dic₅₃	—	2	2	no	—
143	C₂×Dic₅₃	—	2	1	no	—
144	C₂×Dic₅₃	—	2	1	no	—
145	C₂×Dic₅₃	—	2	1	no	—
146	C₂×Dic₅₃	—	4	1	no	—
147	C₂×Dic₅₃	—	4	1	no	—
148	C₂×Dic₅₃	—	4	1	no	—
149	C₂×Dic₅₃	—	4	1	no	—
150	C₂×Dic₅₃	#104	1	2	yes	—
151	C₂×Dic₅₃	#121	1	1	yes	characteristic
152	C₂×Dic₅₃	#121	1	1	yes	characteristic
153	C₂×Dic₅₃	C₂²⋊Q₈	1	1	yes	characteristic
154	C₂×Dic₅₃	C₂²⋊Q₈	1	1	yes	characteristic
155	C₂×Dic₅₃	#135	1	2	yes	—
156	C₂×Dic₅₃	#137	1	2	yes	—
157	#88324	—	4	4	no	—
158	#88324	—	8	2	no	—
159	#8566	—	53	1	no	maximal, Sylow
160	C₂²×C₅₃	#380	1	1	yes	characteristic
161	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
162	C₂²×C₅₃	#407	1	1	yes	characteristic
163	C₂²×C₅₃	#416	1	1	yes	characteristic
164	C₂²×C₅₃	#419	1	1	yes	characteristic
165	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
166	C₂²×C₅₃	#421	1	1	yes	characteristic
167	C₄×C₅₃	—	2	4	no	—
168	C₄×C₅₃	—	2	2	no	—
169	C₄×C₅₃	—	2	2	no	—
170	C₄×C₅₃	—	4	2	no	—
171	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
172	C₄×C₅₃	#423	1	1	yes	characteristic
173	C₄×C₅₃	#431	1	1	yes	characteristic
174	C₄×C₅₃	#432	1	1	yes	characteristic
175	Dic₅₃	—	2	1	no	—
176	Dic₅₃	—	2	1	no	—
177	Dic₅₃	—	2	2	no	—
178	Dic₅₃	—	4	2	no	—
179	Dic₅₃	—	4	1	no	—
180	Dic₅₃	—	4	2	no	—
181	Dic₅₃	#464	1	2	yes	—
182	Dic₅₃	#479	1	2	yes	—
183	C₄²×Q₈	—	53	1	no	—
184	#1831	—	53	1	no	—
185	C₄×C₄⋊C₈	—	53	1	no	—
186	C₂×Q₈⋊C₈	—	53	2	no	—
187	#1279	—	53	2	no	—
188	C₂×C₅₃	#1476	1	1	yes	characteristic
189	C₂×C₅₃	#1479	1	1	yes	characteristic
190	C₂×C₅₃	#1488	1	1	yes	characteristic
191	C₂×C₅₃	#1491	1	1	yes	characteristic
192	C₂×C₅₃	#1797	1	1	yes	characteristic
193	C₂×C₅₃	#1805	1	1	yes	characteristic
194	C₂×C₅₃	#1806	1	1	yes	characteristic
195	C₂×C₄×Q₈	—	53	1	no	—
196	C₂×C₄×Q₈	—	53	1	no	—
197	C₂×C₄×Q₈	#6157	1	1	yes	characteristic
198	C₂×C₄⋊C₈	—	53	2	no	—
199	C₂×C₄⋊C₈	—	106	1	no	—
200	C₂×C₄×C₈	—	53	2	no	—
201	C₄×C₄⋊C₄	—	53	1	no	—
202	C₄×C₄⋊C₄	—	53	1	no	—
203	C₄×C₄⋊C₄	—	53	1	no	—
204	C₄×C₄⋊C₄	—	106	1	no	—
205	C₄×C₄⋊C₄	—	106	1	no	—
206	C₄×C₄⋊C₄	—	106	1	no	—
207	C₄³	—	53	1	no	—
208	C₄³	—	106	1	no	—
209	#336	—	106	1	no	—
210	#336	—	106	1	no	—
211	#336	—	106	1	no	—
212	Q₈⋊C₈	—	106	4	no	—
213	C₅₃	#8566	1	1	yes	characteristic, complemented, Sylow
214	C₂²×Q₈	C₂×#6157	1	1	yes	characteristic
215	C₂²×C₈	—	106	2	no	—
216	C₂²×C₈	—	106	2	no	—
217	C₄×Q₈	—	2	2	no	—
218	C₄×Q₈	—	53	2	no	—
219	C₄×Q₈	—	53	2	no	—
220	C₄×Q₈	—	106	1	no	—
221	C₄×Q₈	—	106	1	no	—
222	C₄×Q₈	—	106	2	no	—
223	C₄×Q₈	—	106	2	no	—
224	C₄×Q₈	#88324	1	4	yes	complemented
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄⋊C₄	—	53	1	no	—
227	C₂×C₄⋊C₄	—	53	1	no	—
228	C₂×C₄⋊C₄	—	106	1	no	—
229	C₂×C₄⋊C₄	—	106	1	no	—
230	C₂×C₄⋊C₄	C₂×#6157	1	1	yes	characteristic
231	C₂×C₄²	—	2	1	no	—
232	C₂×C₄²	—	53	1	no	—
233	C₂×C₄²	—	53	1	no	—
234	C₂×C₄²	—	53	1	no	—
235	C₂×C₄²	—	53	1	no	—
236	C₂×C₄²	—	53	1	no	—
237	C₂×C₄²	—	53	1	no	—
238	C₂×C₄²	—	106	1	no	—
239	C₂×C₄²	—	106	1	no	—
240	C₂×C₄²	—	106	1	no	—
241	C₂×C₄²	—	106	1	no	—
242	C₂×C₄²	—	106	1	no	—
243	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
244	C₄⋊C₈	—	106	2	no	—
245	C₄⋊C₈	—	106	2	no	—
246	C₄⋊C₈	—	106	4	no	—
247	C₄×C₈	—	106	2	no	—
248	C₄×C₈	—	106	2	no	—
249	#95	—	106	1	no	—
250	#95	—	106	1	no	—
251	#95	—	106	1	no	—
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	—	2	1	no	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×Q₈	#11719936	1	2	yes	—
256	C₂×Q₈	C₄⋊#6157	1	2	yes	—
257	C₂×Q₈	C₂×#88324	1	2	yes	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	53	1	no	—
261	C₂²×C₄	—	53	1	no	—
262	C₂²×C₄	—	53	1	no	—
263	C₂²×C₄	—	53	1	no	—
264	C₂²×C₄	—	106	1	no	—
265	C₂²×C₄	—	106	1	no	—
266	C₂²×C₄	C₂²⋊#6157	1	1	yes	characteristic
267	C₂²×C₄	C₂²⋊#6157	1	1	yes	characteristic
268	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
269	C₂×C₈	—	106	4	no	—
270	C₂×C₈	—	212	2	no	—
271	C₂×C₈	—	212	2	no	—
272	C₂×C₈	—	212	2	no	—
273	C₂×C₈	—	212	2	no	—
274	C₂×C₈	—	212	2	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	2	1	no	—
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	C₄⋊C₄	—	106	4	no	—
282	C₄⋊C₄	—	106	1	no	—
283	C₄⋊C₄	—	106	1	no	—
284	C₄⋊C₄	C₂×#88324	1	2	yes	—
285	C₄²	—	2	4	no	—
286	C₄²	—	53	1	no	—
287	C₄²	—	53	1	no	—
288	C₄²	—	53	1	no	—
289	C₄²	—	53	1	no	—
290	C₄²	—	106	2	no	—
291	C₄²	—	106	2	no	—
292	C₄²	—	106	2	no	—
293	C₄²	—	106	2	no	—
294	C₄²	—	106	2	no	—
295	C₄²	—	106	4	no	—
296	C₄²	—	106	1	no	—
297	C₄²	—	106	1	no	—
298	C₄²	—	212	1	no	—
299	C₄²	—	212	2	no	—
300	C₄²	—	212	2	no	—
301	C₄²	—	212	2	no	—
302	C₄²	C₂×#88324	1	2	yes	—
303	C₄²	#11719940	1	2	yes	—
304	C₂³	C₂×C₂²⋊#6157	1	1	yes	characteristic
305	Q₈	—	2	2	no	—
306	Q₈	—	2	2	no	—
307	Q₈	—	2	2	no	—
308	Q₈	C₄⋊#88324	1	4	yes	complemented
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	53	1	no	—
320	C₂×C₄	—	53	1	no	—
321	C₂×C₄	—	53	2	no	—
322	C₂×C₄	—	53	2	no	—
323	C₂×C₄	—	53	1	no	—
324	C₂×C₄	—	53	1	no	—
325	C₂×C₄	—	53	2	no	—
326	C₂×C₄	—	106	1	no	—
327	C₂×C₄	—	106	1	no	—
328	C₂×C₄	—	106	1	no	—
329	C₂×C₄	—	106	1	no	—
330	C₂×C₄	—	106	2	no	—
331	C₂×C₄	—	106	2	no	—
332	C₂×C₄	—	106	1	no	—
333	C₂×C₄	—	106	1	no	—
334	C₂×C₄	—	106	1	no	—
335	C₂×C₄	—	212	1	no	—
336	C₂×C₄	—	212	1	no	—
337	C₂×C₄	—	212	1	no	—
338	C₂×C₄	—	212	1	no	—
339	C₂×C₄	C₂²⋊#88324	1	2	yes	—
340	C₂×C₄	C₂²⋊#88324	1	2	yes	—
341	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
342	C₂×C₄	#13133296	1	1	yes	characteristic
343	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
344	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
345	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
346	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
347	C₈	—	424	2	no	—
348	C₈	—	212	4	no	—
349	C₂²	#16811565	1	1	yes	characteristic
350	C₂²	C₂×C₂²⋊#88324	1	1	yes	characteristic
351	C₂²	#16811594	1	1	yes	characteristic
352	C₂²	#16811613	1	1	yes	characteristic
353	C₂²	#16811616	1	1	yes	characteristic
354	C₂²	D₅₃⋊#104	1	1	yes	characteristic
355	C₂²	#16811622	1	1	yes	characteristic
356	C₄	—	2	4	no	—
357	C₄	—	2	2	no	—
358	C₄	—	2	2	no	—
359	C₄	—	4	2	no	—
360	C₄	—	53	2	no	—
361	C₄	—	53	2	no	—
362	C₄	—	106	2	no	—
363	C₄	—	106	1	no	—
364	C₄	—	106	1	no	—
365	C₄	—	212	1	no	—
366	C₄	—	212	2	no	—
367	C₄	—	212	2	no	—
368	C₄	C₂×C₄⋊#88324	1	1	yes	characteristic
369	C₄	#16811626	1	1	yes	characteristic
370	C₄	#16811639	1	1	yes	characteristic
371	C₄	#16811644	1	1	yes	characteristic
372	C₂	#20290006	1	1	yes	characteristic
373	C₂	#20290017	1	1	yes	characteristic
374	C₂	#20290049	1	1	yes	characteristic
375	C₂	#20290058	1	1	yes	characteristic
376	C₂	#20290569	1	1	yes	characteristic
377	C₂	#20290586	1	1	yes	characteristic
378	C₂	#20290590	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×C₄×Q₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×C₄×Q₈	#6157
hypocenter	C₅₃	#8566
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
57. C₄×Q₈×C₅₃	C₈	semidirect
118. Q₈×C₅₃	C₄⋊C₈	semidirect
213. C₅₃	#8566	semidirect
224. C₄×Q₈	#88324	semidirect
308. Q₈	C₄⋊#88324	semidirect

Maximal subgroups

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

subgroup	index
#20290630	2
C₄×C₄⋊#88324	2
C₂×Q₈⋊#88324	2
#20289551	2
C₄×Q₈×Dic₅₃	2
#8566	53