Subgroups — #28328316

Order 13568.

690 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×#16811682	C₂	1	2	yes	maximal, complemented
2	#20291101	C₂	1	1	yes	maximal, characteristic, complemented
3	#20291100	C₂	1	2	yes	maximal, complemented
4	#20291099	C₂	1	1	yes	maximal, characteristic, complemented
5	#20291094	C₂	1	1	yes	maximal, characteristic, complemented
6	#20291093	C₂	1	2	yes	maximal, complemented
7	D₄×#11719910	C₂	1	1	yes	maximal, characteristic, complemented
8	#20291048	C₂	1	1	yes	maximal, characteristic, complemented
9	#20290946	C₂	1	2	yes	maximal, complemented
10	C₄×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	#20283424	C₂	1	1	yes	maximal, characteristic
12	#16811682	—	2	8	no	—
13	C₂×#13133292	C₂²	1	2	yes	—
14	C₂×#13133292	C₂²	1	2	yes	—
15	C₂×#13133291	C₂²	1	2	yes	—
16	C₂×D₅₃⋊C₁₆	C₂²	1	2	yes	—
17	C₂×D₅₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂×D₅₃⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₂²×#11719910	C₂²	1	2	yes	—
20	#16811544	C₂²	1	2	yes	—
21	C₄⋊#11719910	C₂²	1	1	yes	characteristic
22	C₄⋊#11719910	C₂²	1	1	yes	characteristic
23	#16811518	C₂²	1	2	yes	—
24	#16811518	C₂²	1	2	yes	—
25	C₂²⋊#11719910	C₂²	1	2	yes	—
26	C₂²⋊#11719910	C₂²	1	2	yes	—
27	#16811509	C₂²	1	1	yes	characteristic
28	#16811509	C₂²	1	1	yes	characteristic
29	C₄×#11719910	C₂²	1	1	yes	characteristic
30	C₄×#11719910	C₂²	1	1	yes	characteristic
31	C₂×#13133225	C₂²	1	2	yes	—
32	#16811170	C₄	1	1	yes	characteristic
33	#16810702	C₄	1	1	yes	characteristic
34	#16810699	C₄	1	2	yes	—
35	D₄×#88322	C₂²	1	1	yes	characteristic
36	#16810688	C₂²	1	1	yes	characteristic
37	#16810686	C₂²	1	2	yes	—
38	#16810645	C₄	1	1	yes	characteristic
39	#16810588	C₄	1	2	yes	—
40	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#436	C₄	1	1	yes	characteristic
42	#13133292	—	2	4	no	—
43	#13133292	—	2	4	no	—
44	#13133291	—	2	4	no	—
45	D₅₃⋊C₁₆	—	2	2	no	—
46	D₅₃⋊C₁₆	—	2	4	no	—
47	D₅₃⋊C₁₆	D₄	1	4	yes	complemented
48	C₂×#11719910	—	2	4	no	—
49	C₂×#11719910	C₂³	1	2	yes	—
50	C₂×#11719910	C₂³	1	2	yes	—
51	C₂×#11719910	C₂³	1	1	yes	characteristic
52	C₂×#11719910	C₂³	1	1	yes	characteristic
53	C₂×#11719910	C₂³	1	1	yes	characteristic
54	C₂×#11719910	C₂³	1	1	yes	characteristic
55	C₂×#11719945	C₂×C₄	1	2	yes	—
56	#13133225	—	2	4	no	—
57	#13133225	—	2	4	no	—
58	C₂×#11719912	C₂×C₄	1	2	yes	—
59	C₂×#11719912	C₂×C₄	1	2	yes	—
60	C₂²×#88322	C₂³	1	2	yes	—
61	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₈×D₅₃	C₂³	1	2	yes	—
63	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
64	#13133216	C₈	1	1	yes	characteristic
65	#13133213	C₈	1	1	yes	characteristic
66	#13133203	C₈	1	2	yes	—
67	#13133192	C₈	1	1	yes	characteristic
68	#13133190	C₈	1	2	yes	—
69	#13133188	C₈	1	1	yes	characteristic
70	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
71	C₄×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
72	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
73	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
74	#13133146	C₂×C₄	1	2	yes	—
75	C₂²⋊#88322	C₂×C₄	1	2	yes	—
76	#13133129	C₂×C₄	1	2	yes	—
77	#13133129	C₂×C₄	1	2	yes	—
78	C₄×#88322	C₂³	1	1	yes	characteristic
79	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
80	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
81	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
82	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
83	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
85	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
88	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
89	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
90	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
91	#11719945	—	2	4	no	—
92	#11719945	—	2	4	no	—
93	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
94	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
95	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
96	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
97	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
98	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
100	M₁₆×C₅₃	—	2	4	no	—
101	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
105	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
106	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂²⋊Dic₅₃	C₂×C₈	1	2	yes	—
108	#11719921	C₂×C₈	1	2	yes	—
109	#11719921	C₂×C₈	1	2	yes	—
110	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
111	#11719919	C₂×C₈	1	2	yes	—
112	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
113	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
114	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×#88322	—	2	2	no	—
116	C₂×#88322	—	2	2	no	—
117	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
118	C₂×#88322	C₂²×C₄	1	2	yes	—
119	C₂×#88322	C₂⁴	1	1	yes	characteristic
120	#11719912	—	2	4	no	—
121	#11719912	—	2	4	no	—
122	C₈×D₅₃	—	2	2	no	—
123	C₈×D₅₃	—	2	2	no	—
124	C₈×D₅₃	—	2	2	no	—
125	C₈×D₅₃	C₂×D₄	1	2	yes	—
126	C₈×D₅₃	#41	1	2	yes	—
127	#11719910	—	2	2	no	—
128	#11719910	—	2	2	no	—
129	#11719910	—	2	1	no	—
130	#11719910	—	2	1	no	—
131	#11719910	C₂×D₄	1	2	yes	—
132	#11719910	C₂×D₄	1	2	yes	—
133	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
134	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
135	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
136	D₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
140	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
143	C₂²⋊D₅₃	—	2	4	no	—
144	C₂²⋊D₅₃	—	2	4	no	—
145	C₂²⋊D₅₃	—	2	4	no	—
146	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
147	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
148	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
149	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
150	D₂₁₂	—	2	4	no	—
151	D₂₁₂	—	2	2	no	—
152	C₄×D₅₃	—	2	2	no	—
153	C₄×D₅₃	—	2	2	no	—
154	C₄×D₅₃	—	2	2	no	—
155	C₄×D₅₃	—	2	2	no	—
156	C₄×D₅₃	C₄×D₄	1	2	yes	—
157	C₄×D₅₃	#131	1	2	yes	—
158	Dic₁₀₆	—	2	4	no	—
159	C₈×C₅₃	—	2	2	no	—
160	C₈×C₅₃	—	2	1	no	—
161	C₈×C₅₃	C₄×D₄	1	2	yes	—
162	#88322	—	2	2	no	—
163	#88322	—	2	1	no	—
164	#88322	C₂²×D₄	1	1	yes	characteristic
165	#88322	C₂×#41	1	1	yes	characteristic
166	#33434	—	53	1	no	maximal, Sylow
167	C₂²×C₅₃	—	2	2	no	—
168	C₂²×C₅₃	—	2	2	no	—
169	C₂²×C₅₃	—	2	2	no	—
170	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
171	C₂×D₅₃	—	2	1	no	—
172	C₂×D₅₃	—	2	1	no	—
173	C₂×D₅₃	—	2	2	no	—
174	C₂×D₅₃	—	2	2	no	—
175	C₂×D₅₃	—	2	2	no	—
176	C₂×D₅₃	C₈×D₄	1	2	yes	—
177	C₄×C₅₃	—	2	1	no	—
178	C₄×C₅₃	—	2	2	no	—
179	C₄×C₅₃	#504	1	2	yes	complemented
180	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
181	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
182	Dic₅₃	—	2	1	no	—
183	Dic₅₃	—	2	2	no	—
184	Dic₅₃	C₈×D₄	1	2	yes	—
185	Dic₅₃	#504	1	2	yes	complemented
186	C₂×#504	—	53	2	no	—
187	#2931	—	53	1	no	—
188	#2169	—	53	1	no	—
189	#2169	—	53	2	no	—
190	#2169	—	53	1	no	—
191	D₄×C₁₆	—	53	1	no	—
192	D₄×C₁₆	—	53	2	no	—
193	D₄×C₁₆	—	53	1	no	—
194	#2150	—	53	1	no	—
195	#2113	—	53	2	no	—
196	C₂×C₄×C₁₆	—	53	1	no	—
197	C₂×C₅₃	—	4	2	no	—
198	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
199	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
200	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
201	D₅₃	—	2	2	no	—
202	D₅₃	—	4	2	no	—
203	C₂×#131	—	53	2	no	—
204	#532	—	53	1	no	—
205	#504	—	106	8	no	—
206	C₂×M₃₂	—	53	2	no	—
207	C₂×M₃₂	—	53	2	no	—
208	C₂×M₃₂	—	53	2	no	—
209	C₂²×C₁₆	—	53	1	no	—
210	C₂²×C₁₆	—	53	1	no	—
211	C₂²×C₁₆	—	53	2	no	—
212	C₂²×C₁₆	—	53	2	no	—
213	#436	—	53	2	no	—
214	#436	—	53	1	no	—
215	#436	#6157	1	1	yes	characteristic
216	C₈×D₄	—	53	1	no	—
217	C₈×D₄	—	53	1	no	—
218	C₈×D₄	—	53	2	no	—
219	#423	—	53	1	no	—
220	#408	—	53	2	no	—
221	C₂×C₄×C₈	—	53	1	no	—
222	C₄⋊C₁₆	—	53	1	no	—
223	C₄⋊C₁₆	—	53	1	no	—
224	C₄⋊C₁₆	—	53	2	no	—
225	C₂²⋊C₁₆	—	53	2	no	—
226	C₂²⋊C₁₆	—	53	2	no	—
227	C₂²⋊C₁₆	—	53	2	no	—
228	C₂²⋊C₁₆	—	53	2	no	—
229	C₄×C₁₆	—	53	1	no	—
230	C₄×C₁₆	—	53	1	no	—
231	C₄×C₁₆	—	53	1	no	—
232	C₄×C₁₆	—	53	1	no	—
233	C₅₃	#33434	1	1	yes	characteristic, complemented, Sylow
234	C₂×#41	—	53	2	no	—
235	#131	—	106	4	no	—
236	#131	—	106	4	no	—
237	C₂×M₁₆	—	53	2	no	—
238	C₂×M₁₆	—	53	2	no	—
239	C₂×M₁₆	C₂×#6157	1	2	yes	—
240	C₂²×C₈	—	53	1	no	—
241	C₂²×C₈	—	53	2	no	—
242	C₂²×C₈	—	53	2	no	—
243	C₂²×C₈	—	53	1	no	—
244	C₄⋊Q₈	—	53	1	no	—
245	#127	—	53	1	no	—
246	#124	—	53	2	no	—
247	#121	—	53	1	no	—
248	#121	—	53	1	no	—
249	#121	—	53	2	no	—
250	C₄×D₄	—	53	2	no	—
251	C₄×D₄	—	53	1	no	—
252	C₄×D₄	C₂×#6157	1	1	yes	characteristic
253	C₂×C₄²	—	53	1	no	—
254	M₃₂	—	106	4	no	—
255	M₃₂	—	106	4	no	—
256	M₃₂	—	106	4	no	—
257	C₂×C₁₆	—	53	1	no	—
258	C₂×C₁₆	—	53	1	no	—
259	C₂×C₁₆	—	53	4	no	—
260	C₂×C₁₆	—	53	1	no	—
261	C₂×C₁₆	—	53	1	no	—
262	C₂×C₁₆	—	53	2	no	—
263	C₂×C₁₆	—	53	2	no	—
264	C₂×C₁₆	—	106	4	no	—
265	C₂×C₁₆	—	106	4	no	—
266	C₂×C₁₆	—	106	2	no	—
267	C₄⋊C₈	—	53	2	no	—
268	C₄⋊C₈	—	53	1	no	—
269	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic
270	C₂²⋊C₈	—	53	2	no	—
271	C₂²⋊C₈	—	53	2	no	—
272	C₂²⋊C₈	—	53	2	no	—
273	C₂²⋊C₈	C₂×#6157	1	2	yes	—
274	C₄×C₈	—	53	1	no	—
275	C₄×C₈	—	53	1	no	—
276	C₄×C₈	—	53	1	no	—
277	C₄×C₈	C₂×#6157	1	1	yes	characteristic
278	#41	—	106	4	no	—
279	#41	—	106	4	no	—
280	C₂×Q₈	—	53	2	no	—
281	C₂×D₄	—	53	2	no	—
282	C₂×D₄	—	53	2	no	—
283	C₂×D₄	—	53	2	no	—
284	C₂×D₄	—	53	1	no	—
285	C₂×D₄	—	53	2	no	—
286	C₂×D₄	D₅₃⋊C₈	1	1	yes	characteristic
287	C₂²×C₄	—	53	1	no	—
288	C₂²×C₄	—	53	1	no	—
289	C₂²×C₄	—	53	1	no	—
290	C₂²×C₄	—	53	2	no	—
291	C₂²×C₄	C₂²×#6157	1	2	yes	—
292	M₁₆	—	2	4	no	—
293	M₁₆	—	106	4	no	—
294	M₁₆	—	106	4	no	—
295	C₂×C₈	—	53	1	no	—
296	C₂×C₈	—	53	2	no	—
297	C₂×C₈	—	53	2	no	—
298	C₂×C₈	—	53	1	no	—
299	C₂×C₈	—	53	2	no	—
300	C₂×C₈	—	106	2	no	—
301	C₂×C₈	—	106	2	no	—
302	C₂×C₈	—	106	2	no	—
303	C₂×C₈	—	106	2	no	—
304	C₂×C₈	—	106	2	no	—
305	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
306	C₂×C₈	C₂²×#6157	1	2	yes	—
307	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
308	C₄⋊C₄	—	53	2	no	—
309	C₄⋊C₄	—	53	1	no	—
310	C₄⋊C₄	D₅₃⋊C₈	1	1	yes	characteristic
311	C₂²⋊C₄	—	53	2	no	—
312	C₂²⋊C₄	—	53	2	no	—
313	C₂²⋊C₄	—	53	2	no	—
314	C₂²⋊C₄	D₅₃⋊C₈	1	2	yes	—
315	C₄²	—	53	1	no	—
316	C₄²	—	53	1	no	—
317	C₄²	—	53	1	no	—
318	C₄²	C₂²×#6157	1	1	yes	characteristic
319	C₁₆	—	53	2	no	—
320	C₁₆	—	53	2	no	—
321	C₁₆	—	106	2	no	—
322	C₁₆	—	106	2	no	—
323	C₁₆	—	106	1	no	—
324	C₁₆	—	106	1	no	—
325	C₂³	—	53	1	no	—
326	C₂³	—	53	2	no	—
327	C₂³	C₂×D₅₃⋊C₈	1	2	yes	—
328	Q₈	—	106	4	no	—
329	D₄	—	2	2	no	—
330	D₄	—	106	4	no	—
331	D₄	—	106	4	no	—
332	D₄	—	106	4	no	—
333	D₄	—	106	4	no	—
334	D₄	—	106	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	53	1	no	—
338	C₂×C₄	—	53	1	no	—
339	C₂×C₄	—	53	2	no	—
340	C₂×C₄	—	53	2	no	—
341	C₂×C₄	—	53	1	no	—
342	C₂×C₄	—	53	2	no	—
343	C₂×C₄	—	106	2	no	—
344	C₂×C₄	—	106	2	no	—
345	C₂×C₄	—	106	2	no	—
346	C₂×C₄	—	106	2	no	—
347	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
348	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
349	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
350	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
351	C₈	—	2	2	no	—
352	C₈	—	2	1	no	—
353	C₈	—	53	1	no	—
354	C₈	—	53	1	no	—
355	C₈	—	106	2	no	—
356	C₈	—	106	1	no	—
357	C₈	D₄×#6157	1	2	yes	—
358	C₂²	—	2	2	no	—
359	C₂²	—	2	2	no	—
360	C₂²	—	2	2	no	—
361	C₂²	—	53	2	no	—
362	C₂²	—	106	1	no	—
363	C₂²	—	106	1	no	—
364	C₂²	—	106	2	no	—
365	C₂²	—	106	2	no	—
366	C₂²	—	106	2	no	—
367	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
368	C₄	—	2	2	no	—
369	C₄	—	2	1	no	—
370	C₄	—	53	2	no	—
371	C₄	—	53	2	no	—
372	C₄	—	106	2	no	—
373	C₄	—	106	1	no	—
374	C₄	#16811682	1	2	yes	complemented
375	C₄	C₂×D₄×#6157	1	1	yes	characteristic
376	C₄	D₅₃⋊#131	1	1	yes	characteristic
377	C₂	—	4	2	no	—
378	C₂	—	106	2	no	—
379	C₂	—	212	2	no	—
380	C₂	D₄×D₅₃⋊C₈	1	1	yes	characteristic
381	C₂	D₅₃⋊#504	1	1	yes	characteristic
382	C₂	C₂×#16811682	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#436	C₄
prime radical	#16811170	C₄
cosocle	C₂×#88322	C₂⁴
hypercenter	#436	#6157
hypocenter	C₅₃	#33434
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
1. C₂×#16811682	C₂	semidirect
2. #20291101	C₂	semidirect
3. #20291100	C₂	semidirect
4. #20291099	C₂	semidirect
5. #20291094	C₂	semidirect
6. #20291093	C₂	semidirect
7. D₄×#11719910	C₂	semidirect
8. #20291048	C₂	semidirect
9. #20290946	C₂	semidirect
10. C₄×D₅₃⋊C₁₆	C₂	semidirect
47. D₅₃⋊C₁₆	D₄	semidirect
179. C₄×C₅₃	#504	semidirect
185. Dic₅₃	#504	semidirect
233. C₅₃	#33434	semidirect
374. C₄	#16811682	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811682	2
#20291101	2
#20291100	2
#20291099	2
#20291094	2
#20291093	2
D₄×#11719910	2
#20291048	2
#20290946	2
C₄×D₅₃⋊C₁₆	2
#20283424	2
#33434	53