Subgroups — #27355591

Order 13568.

630 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11719917	C₂	1	2	yes	maximal
2	#20273624	C₂	1	2	yes	maximal, complemented
3	#20273587	C₂	1	4	yes	maximal, complemented
4	#20273534	C₂	1	2	yes	maximal, complemented
5	#20273471	C₂	1	1	yes	maximal, characteristic
6	#20273144	C₂	1	2	yes	maximal
7	#20273004	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
9	#16810754	—	2	4	no	—
10	#16810754	C₂²	1	4	yes	—
11	Q₈×#88322	C₂²	1	4	yes	—
12	#16810676	C₂²	1	1	yes	characteristic
13	#16810647	—	2	2	no	—
14	#16810647	C₂²	1	2	yes	—
15	C₄×#11719917	—	2	2	no	—
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	Q₈⋊#88322	C₂²	1	4	yes	—
21	#16810345	C₂²	1	2	yes	—
22	#16810344	C₂²	1	2	yes	—
23	C₂²×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₅₃	C₄	1	2	yes	—
26	C₄×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
27	C₄×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
28	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
29	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
30	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
32	C₂×#103×C₅₃	C₄	1	2	yes	—
33	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×#11719917	—	2	2	no	—
35	C₂×#11719917	D₄	1	2	yes	—
36	C₄⋊#88322	—	2	2	no	—
37	C₄⋊#88322	—	2	1	no	—
38	C₄⋊#88322	—	2	4	no	—
39	C₄⋊#88322	—	2	4	no	—
40	C₄⋊#88322	C₂³	1	4	yes	—
41	C₂²⋊#88322	—	2	1	no	—
42	#13133128	—	2	4	no	—
43	#13133128	D₄	1	2	yes	—
44	C₄×#88322	—	2	4	no	—
45	C₄×#88322	C₂³	1	2	yes	—
46	C₄×#88322	C₂³	1	2	yes	—
47	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
48	C₂×Q₁₆×C₅₃	—	2	4	no	—
49	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×Q₁₆×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	—	2	2	no	—
55	C₄×Q₈×C₅₃	—	2	2	no	—
56	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
57	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
58	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
59	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
61	C₂×C₄⋊C₄×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	2	yes	—
65	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
66	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
67	#103×C₅₃	C₂×C₄	1	2	yes	—
68	#103×C₅₃	C₂×C₄	1	2	yes	—
69	#103×C₅₃	C₂×C₄	1	2	yes	—
70	#103×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	C₂×Q₈×C₅₃	—	2	2	no	—
76	C₂×Q₈×C₅₃	—	2	4	no	—
77	C₂×Q₈×C₅₃	—	4	2	no	—
78	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₅₃	—	2	2	no	—
81	C₂²×C₄×C₅₃	—	2	2	no	—
82	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
85	Q₁₆×C₅₃	—	2	4	no	—
86	Q₁₆×C₅₃	—	2	2	no	—
87	Q₁₆×C₅₃	M₁₆	1	4	yes	complemented
88	C₂×C₈×C₅₃	—	2	2	no	—
89	C₂×C₈×C₅₃	—	4	1	no	—
90	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	C₄⋊C₄×C₅₃	—	2	2	no	—
95	C₄⋊C₄×C₅₃	—	2	2	no	—
96	C₄⋊C₄×C₅₃	—	2	2	no	—
97	C₄⋊C₄×C₅₃	—	2	2	no	—
98	C₄⋊C₄×C₅₃	—	2	2	no	—
99	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
100	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
101	C₄²×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	—	2	4	no	—
110	#11719917	—	4	4	no	—
111	C₂×#88322	—	2	4	no	—
112	C₂×#88322	—	2	2	no	—
113	C₂×#88322	C₂×D₄	1	4	yes	—
114	C₂³×C₅₃	C₄×D₄	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	—	4	4	no	—
119	Q₈×C₅₃	C₂×M₁₆	1	4	yes	—
120	C₂×C₄×C₅₃	—	2	2	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	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	4	1	no	—
127	C₂×C₄×C₅₃	—	4	4	no	—
128	C₂×C₄×C₅₃	—	8	2	no	—
129	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	#137	1	4	yes	—
133	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
136	C₈×C₅₃	—	2	1	no	—
137	C₈×C₅₃	—	2	1	no	—
138	C₈×C₅₃	—	2	1	no	—
139	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
140	#88322	—	2	4	no	—
141	#88322	—	4	2	no	—
142	#88322	—	4	4	no	—
143	#18042	—	53	1	no	maximal, Sylow
144	C₂²×C₅₃	—	2	1	no	—
145	C₂²×C₅₃	—	2	1	no	—
146	C₂²×C₅₃	#441	1	2	yes	—
147	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	2	4	no	—
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	—	2	1	no	—
152	C₄×C₅₃	—	4	2	no	—
153	C₄×C₅₃	—	4	2	no	—
154	C₄×C₅₃	—	4	2	no	—
155	C₄×C₅₃	#441	1	2	yes	—
156	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
157	C₄×C₅₃	#568	1	1	yes	characteristic
158	C₄×C₅₃	C₂×#137	1	2	yes	—
159	Q₈×M₁₆	—	53	2	no	—
160	C₂×C₄×Q₁₆	D₅₃	1	1	yes	characteristic
161	#1593	—	53	2	no	—
162	#1587	—	53	4	no	—
163	#1579	—	53	2	no	—
164	#1566	—	53	1	no	—
165	#1487	—	53	2	no	—
166	#1451	—	53	1	no	—
167	C₂×C₅₃	—	2	2	no	—
168	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
169	C₂×C₅₃	C₂×#441	1	1	yes	characteristic
170	C₂×C₅₃	#2957	1	1	yes	characteristic
171	C₂²×Q₁₆	Dic₅₃	1	1	yes	characteristic
172	C₂×C₄×Q₈	C₂×D₅₃	1	2	yes	—
173	#446	—	53	4	no	—
174	#446	—	106	4	no	—
175	C₈×Q₈	—	53	4	no	—
176	C₄×Q₁₆	Dic₅₃	1	2	yes	—
177	C₄×Q₁₆	Dic₅₃	1	1	yes	characteristic
178	C₄×Q₁₆	Dic₅₃	1	1	yes	characteristic
179	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
180	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
181	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
182	#431	—	53	1	no	—
183	C₂×#107	Dic₅₃	1	1	yes	characteristic
184	#423	—	53	2	no	—
185	#423	—	106	2	no	—
186	C₂×#103	Dic₅₃	1	2	yes	—
187	C₄×M₁₆	—	53	1	no	—
188	C₄×M₁₆	—	106	2	no	—
189	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
190	#334	—	53	2	no	—
191	#334	—	53	2	no	—
192	Q₈⋊C₈	—	53	4	no	—
193	Q₈⋊C₈	—	53	4	no	—
194	#322	—	53	2	no	—
195	#322	—	53	2	no	—
196	C₅₃	#18042	1	1	yes	characteristic, complemented, Sylow
197	C₂²×Q₈	C₂×Dic₅₃	1	2	yes	—
198	C₂×Q₁₆	—	2	4	no	—
199	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
200	C₂×Q₁₆	C₂×Dic₅₃	1	1	yes	characteristic
201	C₂×Q₁₆	C₂×Dic₅₃	1	1	yes	characteristic
202	C₂×M₁₆	—	53	2	no	—
203	C₂×M₁₆	—	106	2	no	—
204	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
205	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
206	C₄×Q₈	—	2	2	no	—
207	C₄×Q₈	—	2	2	no	—
208	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
209	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
210	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
211	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
216	#107	C₂×Dic₅₃	1	2	yes	—
217	#107	C₂×Dic₅₃	1	1	yes	characteristic
218	#107	C₂×Dic₅₃	1	1	yes	characteristic
219	C₄⋊C₈	—	53	4	no	—
220	C₄⋊C₈	—	106	4	no	—
221	C₄⋊C₈	—	106	4	no	—
222	C₄⋊C₈	—	106	2	no	—
223	C₄⋊C₈	—	106	1	no	—
224	#103	C₂×Dic₅₃	1	2	yes	—
225	#103	C₂×Dic₅₃	1	2	yes	—
226	#103	C₂×Dic₅₃	1	2	yes	—
227	#103	C₂×Dic₅₃	1	2	yes	—
228	C₂²⋊C₈	—	106	1	no	—
229	#97	—	53	2	no	—
230	#97	—	106	4	no	—
231	C₄×C₈	—	53	2	no	—
232	C₄×C₈	—	53	2	no	—
233	C₄×C₈	—	106	4	no	—
234	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
235	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	4	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×Q₈	C₂²×Dic₅₃	1	2	yes	—
242	C₂×Q₈	C₂²×Dic₅₃	1	2	yes	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
246	C₂²×C₄	#11719947	1	1	yes	characteristic
247	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
248	Q₁₆	—	2	4	no	—
249	Q₁₆	—	2	2	no	—
250	Q₁₆	#11719917	1	4	yes	complemented
251	M₁₆	—	106	4	no	—
252	M₁₆	—	212	4	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	53	4	no	—
256	C₂×C₈	—	106	4	no	—
257	C₂×C₈	—	106	2	no	—
258	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
259	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
260	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
261	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
268	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	D₄×D₅₃	1	1	yes	characteristic
274	C₄²	#11719947	1	1	yes	characteristic
275	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
276	C₄²	C₂³×D₅₃	1	1	yes	characteristic
277	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
278	Q₈	—	2	2	no	—
279	Q₈	—	2	2	no	—
280	Q₈	—	2	2	no	—
281	Q₈	—	4	4	no	—
282	Q₈	C₂×#11719917	1	4	yes	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	8	2	no	—
292	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
293	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
294	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
295	C₂×C₄	#13133249	1	4	yes	—
296	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
297	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
298	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	106	4	no	—
303	C₈	—	212	2	no	—
304	C₈	—	212	4	no	—
305	C₈	C₂×#11719917	1	2	yes	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	#16810730	1	2	yes	—
309	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	4	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	4	2	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	2	no	—
317	C₄	#16810730	1	2	yes	—
318	C₄	C₂²×#11719917	1	1	yes	characteristic
319	C₄	#16811390	1	1	yes	characteristic
320	C₄	C₂×#13133249	1	2	yes	—
321	C₂	—	2	2	no	—
322	C₂	D₄×#11719917	1	1	yes	characteristic
323	C₂	C₂×#16810730	1	1	yes	characteristic
324	C₂	#20283632	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₄×Q₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₅₃
hypocenter	C₅₃	#18042
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #20273624	C₂	semidirect
3. #20273587	C₂	semidirect
4. #20273534	C₂	semidirect
87. Q₁₆×C₅₃	M₁₆	semidirect
196. C₅₃	#18042	semidirect
250. Q₁₆	#11719917	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11719917	2
#20273624	2
#20273587	2
#20273534	2
#20273471	2
#20273144	2
#20273004	2
C₂×C₄×Q₁₆×C₅₃	2
#18042	53