Subgroups — #27282943

Order 13568.

554 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11719913	C₂	1	1	yes	maximal, characteristic
2	#20275651	C₂	1	1	yes	maximal, characteristic, complemented
3	#20275509	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810374	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810355	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810352	C₂	1	1	yes	maximal, characteristic
7	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11719913	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
11	C₄×#11719913	—	2	2	no	—
12	C₄×#11719913	—	2	2	no	—
13	C₂×#13133149	C₄	1	1	yes	characteristic
14	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
16	C₂×#13133142	C₄	1	1	yes	characteristic
17	C₂×#13133139	C₄	1	1	yes	characteristic
18	C₂×#13133129	—	2	2	no	—
19	#16810377	—	2	2	no	—
20	#16810374	—	2	2	no	—
21	#16810355	—	2	2	no	—
22	#16810352	—	2	2	no	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₈²×C₅₃	—	2	2	no	—
27	C₂²×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
28	C₂²×D₂₁₂	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×D₅₃	—	2	1	no	—
30	C₂×#11719913	—	2	2	no	—
31	C₂×#11719913	—	2	2	no	—
32	C₂×#11719913	C₈	1	4	yes	complemented
33	C₂²×#88322	—	2	1	no	—
34	C₂²×#88322	—	2	1	no	—
35	C₄×Dic₁₀₆	—	2	1	no	—
36	C₄×Dic₁₀₆	—	2	1	no	—
37	C₄×Dic₁₀₆	—	2	1	no	—
38	C₄×Dic₁₀₆	—	2	1	no	—
39	C₄×D₂₁₂	—	2	1	no	—
40	C₄×D₂₁₂	—	2	1	no	—
41	C₄×D₂₁₂	—	2	1	no	—
42	C₄×D₂₁₂	—	2	1	no	—
43	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×#11719919	—	2	1	no	—
46	C₂×#11719921	—	2	1	no	—
47	C₂×C₄×Dic₅₃	—	2	1	no	—
48	#13133149	—	2	2	no	—
49	C₄⋊#88322	—	2	1	no	—
50	C₄⋊#88322	—	2	1	no	—
51	C₄⋊#88322	—	2	1	no	—
52	C₄⋊#88322	—	2	1	no	—
53	#13133142	—	2	2	no	—
54	#13133139	—	2	2	no	—
55	#13133129	—	4	4	no	—
56	C₂²×C₈×C₅₃	—	2	1	no	—
57	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₈×C₅₃	D₄	1	2	yes	—
60	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₅₃	—	2	1	no	—
62	C₄×C₈×C₅₃	—	2	1	no	—
63	C₄×C₈×C₅₃	—	2	1	no	—
64	C₄×C₈×C₅₃	D₄	1	2	yes	—
65	C₄×C₈×C₅₃	D₄	1	2	yes	—
66	C₄×C₈×C₅₃	D₄	1	2	yes	—
67	C₂³×D₅₃	—	2	1	no	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
71	C₂²×Dic₅₃	—	2	1	no	—
72	C₂²×Dic₅₃	—	2	1	no	—
73	C₂²×Dic₅₃	—	2	1	no	—
74	C₂×D₂₁₂	—	2	1	no	—
75	C₂×D₂₁₂	—	2	1	no	—
76	C₂×D₂₁₂	—	2	1	no	—
77	C₂×D₂₁₂	—	2	1	no	—
78	C₂×D₂₁₂	—	2	1	no	—
79	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
80	C₂×C₄×D₅₃	—	4	1	no	—
81	C₂×C₄×D₅₃	—	4	1	no	—
82	C₂×C₄×D₅₃	—	4	2	no	—
83	C₂×C₄×D₅₃	—	4	2	no	—
84	C₂×Dic₁₀₆	—	2	1	no	—
85	C₂×Dic₁₀₆	—	2	1	no	—
86	C₂×Dic₁₀₆	—	2	1	no	—
87	C₂×Dic₁₀₆	—	2	1	no	—
88	C₂×Dic₁₀₆	—	2	1	no	—
89	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
90	C₂×C₈×C₅₃	—	2	4	no	—
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	—	2	1	no	—
94	C₂×C₈×C₅₃	—	2	1	no	—
95	C₂×C₈×C₅₃	—	2	2	no	—
96	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
97	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
98	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
99	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
100	C₂×C₈×C₅₃	SD₁₆	1	4	yes	complemented
101	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₅₃	#41	1	1	yes	characteristic
105	#11719921	—	4	1	no	—
106	#11719921	—	4	1	no	—
107	C₄⋊Dic₅₃	—	2	1	no	—
108	C₄⋊Dic₅₃	—	2	1	no	—
109	C₄⋊Dic₅₃	—	2	1	no	—
110	C₄⋊Dic₅₃	—	2	1	no	—
111	#11719919	—	4	1	no	—
112	#11719919	—	4	1	no	—
113	C₄×Dic₅₃	—	4	1	no	—
114	C₄×Dic₅₃	—	4	1	no	—
115	C₂×#88322	—	4	1	no	—
116	C₂×#88322	—	4	1	no	—
117	C₂×#88322	—	4	2	no	—
118	C₂×#88322	—	4	2	no	—
119	#11719913	—	2	8	no	—
120	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
121	C₂²×D₅₃	—	2	2	no	—
122	C₂²×D₅₃	—	4	1	no	—
123	C₂²×D₅₃	—	4	1	no	—
124	C₂²×D₅₃	—	4	2	no	—
125	C₂²×D₅₃	—	4	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
131	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
132	C₂×C₄×C₅₃	#127	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	#131	1	1	yes	characteristic
136	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
138	C₂×C₄×C₅₃	#135	1	2	yes	—
139	C₂×Dic₅₃	—	2	2	no	—
140	C₂×Dic₅₃	—	4	1	no	—
141	C₂×Dic₅₃	—	4	1	no	—
142	C₂×Dic₅₃	—	4	1	no	—
143	C₂×Dic₅₃	—	4	1	no	—
144	C₂×Dic₅₃	—	4	2	no	—
145	C₂×Dic₅₃	—	4	2	no	—
146	D₂₁₂	—	2	2	no	—
147	D₂₁₂	—	2	2	no	—
148	D₂₁₂	—	4	1	no	—
149	D₂₁₂	—	4	1	no	—
150	C₄×D₅₃	—	8	2	no	—
151	C₄×D₅₃	—	8	1	no	—
152	C₄×D₅₃	—	8	1	no	—
153	Dic₁₀₆	—	2	2	no	—
154	Dic₁₀₆	—	2	2	no	—
155	Dic₁₀₆	—	4	1	no	—
156	Dic₁₀₆	—	4	1	no	—
157	C₈×C₅₃	—	2	4	no	—
158	C₈×C₅₃	—	2	2	no	—
159	C₈×C₅₃	—	2	4	no	—
160	C₈×C₅₃	C₂²⋊C₈	1	4	yes	complemented
161	#88322	—	8	1	no	—
162	#88322	—	8	1	no	—
163	#9655	—	53	1	no	maximal, Sylow
164	C₂²×C₅₃	#390	1	1	yes	characteristic
165	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
166	C₂²×C₅₃	#436	1	1	yes	characteristic
167	C₂²×C₅₃	C₄×SD₁₆	1	2	yes	—
168	C₂²×C₅₃	#443	1	2	yes	—
169	C₂×D₅₃	—	4	4	no	—
170	C₂×D₅₃	—	4	2	no	—
171	C₂×D₅₃	—	8	2	no	—
172	C₂×D₅₃	—	8	1	no	—
173	C₂×D₅₃	—	8	1	no	—
174	C₄×C₅₃	—	2	1	no	—
175	C₄×C₅₃	—	2	1	no	—
176	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
177	C₄×C₅₃	#408	1	1	yes	characteristic
178	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
179	C₄×C₅₃	#436	1	1	yes	characteristic
180	C₄×C₅₃	#465	1	2	yes	—
181	C₄×C₅₃	#492	1	1	yes	characteristic
182	C₄×C₅₃	#495	1	1	yes	characteristic
183	Dic₅₃	—	4	2	no	—
184	Dic₅₃	—	8	1	no	—
185	Dic₅₃	—	8	1	no	—
186	C₂×C₄×SD₁₆	—	53	1	no	—
187	#1984	—	53	1	no	—
188	#1954	—	53	1	no	—
189	C₂×#334	—	53	1	no	—
190	C₂×Q₈⋊C₈	—	53	1	no	—
191	C₂×D₄⋊C₈	—	53	1	no	—
192	C₂×C₈²	D₅₃	1	1	yes	characteristic, complemented
193	C₂×C₅₃	C₈×SD₁₆	1	2	yes	—
194	C₂×C₅₃	#1590	1	2	yes	—
195	C₂×C₅₃	#1954	1	1	yes	characteristic
196	C₂×C₅₃	#1968	1	1	yes	characteristic
197	C₂×C₅₃	#1974	1	1	yes	characteristic
198	D₅₃	—	8	2	no	—
199	C₂²×SD₁₆	—	53	1	no	—
200	C₂×C₄×Q₈	—	53	1	no	—
201	C₂×C₄×D₄	—	53	1	no	—
202	C₄×SD₁₆	—	106	2	no	—
203	C₄×SD₁₆	—	106	2	no	—
204	C₂×#106	—	53	1	no	—
205	C₂×C₄⋊C₈	—	53	1	no	—
206	C₂×C₄⋊C₈	—	53	1	no	—
207	C₂×#103	—	53	1	no	—
208	C₂×#102	—	53	1	no	—
209	C₂×C₂²⋊C₈	—	106	2	no	—
210	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
211	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
212	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
213	#336	—	106	2	no	—
214	#334	—	106	2	no	—
215	Q₈⋊C₈	—	106	2	no	—
216	D₄⋊C₈	—	106	2	no	—
217	C₈²	—	2	2	no	—
218	C₅₃	#9655	1	1	yes	characteristic, complemented, Sylow
219	C₂²×Q₈	—	53	1	no	—
220	C₂²×D₄	—	53	1	no	—
221	C₂³×C₄	—	106	1	no	—
222	C₂×SD₁₆	—	53	4	no	—
223	C₂×SD₁₆	—	106	2	no	—
224	C₂×SD₁₆	—	106	2	no	—
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	106	1	no	—
227	C₂²×C₈	—	106	1	no	—
228	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
229	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
230	C₂²×C₈	D₂₁₂	1	2	yes	—
231	C₄×Q₈	—	106	1	no	—
232	C₄×Q₈	—	106	1	no	—
233	C₄×Q₈	—	106	1	no	—
234	C₄×Q₈	—	106	1	no	—
235	C₄×D₄	—	106	1	no	—
236	C₄×D₄	—	106	1	no	—
237	C₄×D₄	—	106	1	no	—
238	C₄×D₄	—	106	1	no	—
239	C₂×C₄⋊C₄	—	53	1	no	—
240	C₂×C₄⋊C₄	—	53	1	no	—
241	C₂×C₄⋊C₄	—	106	1	no	—
242	C₂×C₂²⋊C₄	—	106	1	no	—
243	C₂×C₄²	—	106	1	no	—
244	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
245	#106	—	106	2	no	—
246	C₄⋊C₈	—	106	1	no	—
247	C₄⋊C₈	—	106	1	no	—
248	C₄⋊C₈	—	106	1	no	—
249	C₄⋊C₈	—	106	1	no	—
250	#103	—	106	2	no	—
251	#102	—	106	2	no	—
252	C₂²⋊C₈	—	212	4	no	—
253	C₄×C₈	—	2	1	no	—
254	C₄×C₈	—	2	1	no	—
255	C₄×C₈	—	2	1	no	—
256	C₄×C₈	D₂₁₂	1	2	yes	—
257	C₄×C₈	D₂₁₂	1	2	yes	—
258	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
259	C₂⁴	—	106	1	no	—
260	C₂×Q₈	—	53	2	no	—
261	C₂×Q₈	—	106	1	no	—
262	C₂×Q₈	—	106	1	no	—
263	C₂×Q₈	—	106	1	no	—
264	C₂×Q₈	—	106	1	no	—
265	C₂×Q₈	—	106	1	no	—
266	C₂×D₄	—	53	2	no	—
267	C₂×D₄	—	106	1	no	—
268	C₂×D₄	—	106	1	no	—
269	C₂×D₄	—	106	1	no	—
270	C₂×D₄	—	106	1	no	—
271	C₂×D₄	—	106	1	no	—
272	C₂²×C₄	—	106	1	no	—
273	C₂²×C₄	—	106	1	no	—
274	C₂²×C₄	—	106	1	no	—
275	C₂²×C₄	—	212	1	no	—
276	C₂²×C₄	—	212	1	no	—
277	C₂²×C₄	—	212	2	no	—
278	C₂²×C₄	—	212	2	no	—
279	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
280	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
281	C₂²×C₄	#11719945	1	1	yes	characteristic
282	SD₁₆	—	106	8	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	212	1	no	—
290	C₂×C₈	—	212	1	no	—
291	C₂×C₈	—	212	2	no	—
292	C₂×C₈	—	212	2	no	—
293	C₂×C₈	C₈×D₅₃	1	2	yes	—
294	C₂×C₈	#11719912	1	2	yes	—
295	C₂×C₈	#11719913	1	4	yes	complemented
296	C₂×C₈	#11719921	1	2	yes	—
297	C₂×C₈	#11719921	1	2	yes	—
298	C₄⋊C₄	—	106	1	no	—
299	C₄⋊C₄	—	106	1	no	—
300	C₄⋊C₄	—	106	1	no	—
301	C₄⋊C₄	—	106	1	no	—
302	C₄⋊C₄	—	212	1	no	—
303	C₄⋊C₄	—	212	1	no	—
304	C₂²⋊C₄	—	212	1	no	—
305	C₂²⋊C₄	—	212	1	no	—
306	C₄²	—	212	1	no	—
307	C₄²	—	212	1	no	—
308	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
309	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
310	C₄²	#11719945	1	1	yes	characteristic
311	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
312	C₂³	—	106	2	no	—
313	C₂³	—	212	1	no	—
314	C₂³	—	212	1	no	—
315	C₂³	—	212	2	no	—
316	C₂³	—	212	2	no	—
317	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
318	Q₈	—	106	2	no	—
319	Q₈	—	106	2	no	—
320	Q₈	—	212	1	no	—
321	Q₈	—	212	1	no	—
322	D₄	—	106	2	no	—
323	D₄	—	106	2	no	—
324	D₄	—	212	1	no	—
325	D₄	—	212	1	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	106	2	no	—
328	C₂×C₄	—	424	2	no	—
329	C₂×C₄	—	424	1	no	—
330	C₂×C₄	—	424	1	no	—
331	C₂×C₄	—	212	1	no	—
332	C₂×C₄	—	212	1	no	—
333	C₂×C₄	—	212	2	no	—
334	C₂×C₄	—	212	2	no	—
335	C₂×C₄	—	212	1	no	—
336	C₂×C₄	—	212	1	no	—
337	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
338	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
339	C₂×C₄	#13133191	1	1	yes	characteristic
340	C₂×C₄	#13133191	1	1	yes	characteristic
341	C₂×C₄	#13133205	1	1	yes	characteristic
342	C₂×C₄	#13133214	1	1	yes	characteristic
343	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
344	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
345	C₂×C₄	#13133223	1	1	yes	characteristic
346	C₂×C₄	#13133223	1	1	yes	characteristic
347	C₂×C₄	C₂×#11719913	1	2	yes	—
348	C₂×C₄	#13133239	1	2	yes	—
349	C₈	—	2	4	no	—
350	C₈	—	2	2	no	—
351	C₈	—	2	4	no	—
352	C₈	—	424	1	no	—
353	C₈	—	424	1	no	—
354	C₈	#13133129	1	4	yes	complemented
355	C₂²	—	424	1	no	—
356	C₂²	—	424	1	no	—
357	C₂²	—	424	2	no	—
358	C₂²	—	212	2	no	—
359	C₂²	—	212	4	no	—
360	C₂²	#16810525	1	1	yes	characteristic
361	C₂²	C₈×D₂₁₂	1	1	yes	characteristic
362	C₂²	#16810702	1	1	yes	characteristic
363	C₂²	C₄×#11719913	1	2	yes	—
364	C₂²	#16810741	1	2	yes	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	424	1	no	—
368	C₄	—	424	1	no	—
369	C₄	—	212	2	no	—
370	C₄	C₂×#13133129	1	1	yes	characteristic
371	C₄	#16810589	1	1	yes	characteristic
372	C₄	C₈×D₂₁₂	1	1	yes	characteristic
373	C₄	#16810702	1	1	yes	characteristic
374	C₄	#16810852	1	2	yes	—
375	C₄	#16810996	1	1	yes	characteristic
376	C₄	#16811007	1	1	yes	characteristic
377	C₂	—	424	2	no	—
378	C₂	C₈×#11719913	1	2	yes	—
379	C₂	#20273607	1	2	yes	—
380	C₂	#20275509	1	1	yes	characteristic
381	C₂	#20275569	1	1	yes	characteristic
382	C₂	#20275603	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₈²×C₅₃	C₂
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂²×D₂₁₂	C₂×C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₈²	D₅₃
hypocenter	C₅₃	#9655
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. #20275651	C₂	semidirect
4. C₂×#16810374	C₂	semidirect
5. C₂×#16810355	C₂	semidirect
7. C₂×C₈²×C₅₃	C₂	semidirect
32. C₂×#11719913	C₈	semidirect
100. C₂×C₈×C₅₃	SD₁₆	semidirect
160. C₈×C₅₃	C₂²⋊C₈	semidirect
192. C₂×C₈²	D₅₃	semidirect
218. C₅₃	#9655	semidirect
295. C₂×C₈	#11719913	semidirect
354. C₈	#13133129	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11719913	2
#20275651	2
#20275509	2
C₂×#16810374	2
C₂×#16810355	2
C₂×#16810352	2
C₂×C₈²×C₅₃	2
#9655	53