Subgroups — #28310534

Order 13568.

474 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅₃⋊M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811510	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₆×#6157	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×C₂²⋊#6157	C₂	1	1	yes	maximal, characteristic
5	(C₂×#6157)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×#6157)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133292	C₄	1	2	yes	complemented
9	D₅₃⋊M₃₂	—	2	2	no	—
10	D₅₃⋊M₃₂	—	2	2	no	—
11	C₂×#13133291	C₄	1	2	yes	complemented
12	C₂×D₅₃⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×D₅₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	D₅₃⋊(C₂²⋊C₈)	C₄	1	1	yes	characteristic
15	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
16	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
17	C₄×C₂²⋊#6157	C₄	1	1	yes	characteristic
18	#16811510	—	2	2	no	—
19	C₁₆×#6157	—	2	2	no	—
20	(C₂×#6157)⋊C₈	C₄	1	1	yes	characteristic
21	(C₂×#6157)⋊C₈	C₄	1	1	yes	characteristic
22	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
25	C₂²⋊#11719908	C₄	1	1	yes	characteristic, complemented
26	#16810402	C₄	1	1	yes	characteristic, complemented
27	#16810402	C₄	1	1	yes	characteristic, complemented
28	C₅₃×C₂²⋊C₁₆	C₄	1	1	yes	characteristic, complemented
29	C₂×D₅₃⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×D₅₃⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₂²⋊#6157	C₈	1	2	yes	—
32	C₂×C₄×#6157	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×#6157	C₂×C₄	1	1	yes	characteristic
34	#13133292	—	2	4	no	—
35	#13133291	—	2	4	no	—
36	D₅₃⋊C₁₆	—	2	1	no	—
37	D₅₃⋊C₁₆	—	2	1	no	—
38	D₅₃⋊C₁₆	D₄	1	2	yes	—
39	D₅₃⋊C₁₆	D₄	1	2	yes	—
40	C₂×#11719910	C₂×C₄	1	2	yes	—
41	C₂×#11719910	C₂×C₄	1	2	yes	—
42	#13133274	C₈	1	2	yes	—
43	C₂²⋊#88324	C₈	1	2	yes	—
44	C₈×#6157	—	2	1	no	—
45	C₈×#6157	—	2	1	no	—
46	C₈×#6157	—	2	1	no	—
47	C₈×#6157	—	2	1	no	—
48	(C₂×#6157)⋊C₄	C₈	1	2	yes	—
49	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₅₃	—	2	1	no	—
52	C₂×C₈×D₅₃	—	2	1	no	—
53	C₂×C₈×D₅₃	—	2	1	no	—
54	C₂×C₈×D₅₃	—	2	1	no	—
55	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
59	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
60	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
61	C₁₆×D₅₃	—	2	1	no	—
62	C₁₆×D₅₃	—	2	1	no	—
63	C₁₆×D₅₃	—	2	1	no	—
64	C₁₆×D₅₃	—	2	1	no	—
65	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
69	C₂²×#6157	C₂×C₈	1	2	yes	—
70	C₂²×#6157	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
72	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
73	C₂×C₄×D₅₃	—	2	1	no	—
74	C₂×C₄×D₅₃	—	2	1	no	—
75	C₂×C₄×D₅₃	—	2	1	no	—
76	C₂×C₄×D₅₃	—	2	1	no	—
77	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
78	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
79	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²⋊#6157	—	2	4	no	—
82	C₂×#88324	C₂×C₈	1	2	yes	—
83	C₂×#88324	C₂×C₈	1	2	yes	—
84	C₄×#6157	—	2	1	no	—
85	C₄×#6157	—	2	1	no	—
86	C₄×#6157	—	2	1	no	—
87	C₄×#6157	—	2	1	no	—
88	D₅₃⋊C₈	—	2	1	no	—
89	D₅₃⋊C₈	—	2	1	no	—
90	D₅₃⋊C₈	—	2	1	no	—
91	D₅₃⋊C₈	—	2	1	no	—
92	C₂×C₈×C₅₃	—	2	1	no	—
93	C₂×C₈×C₅₃	—	2	1	no	—
94	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×#88322	—	2	1	no	—
97	C₂×#88322	—	2	1	no	—
98	C₂×#88322	C₄²	1	1	yes	characteristic
99	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
100	C₈×D₅₃	—	2	1	no	—
101	C₈×D₅₃	—	2	1	no	—
102	C₈×D₅₃	—	2	1	no	—
103	C₈×D₅₃	—	2	1	no	—
104	C₈×D₅₃	—	2	1	no	—
105	C₈×D₅₃	—	2	1	no	—
106	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
107	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
108	C₈×D₅₃	#41	1	1	yes	characteristic
109	C₈×D₅₃	#41	1	1	yes	characteristic
110	#11719910	—	2	2	no	—
111	#11719910	C₂²⋊C₄	1	4	yes	complemented
112	C₁₆×C₅₃	—	2	1	no	—
113	C₁₆×C₅₃	—	2	1	no	—
114	#11719908	—	2	1	no	—
115	#11719908	—	2	1	no	—
116	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
117	C₂²×D₅₃	—	2	1	no	—
118	C₂²×D₅₃	—	2	1	no	—
119	C₂²×D₅₃	—	2	2	no	—
120	C₂²×D₅₃	—	2	2	no	—
121	C₂²×D₅₃	C₄×C₈	1	1	yes	characteristic
122	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
123	C₂×#6157	—	2	2	no	—
124	C₂×#6157	—	2	2	no	—
125	C₂×#6157	—	2	2	no	—
126	C₂×#6157	—	2	2	no	—
127	C₂×#6157	—	2	2	no	—
128	C₂×#6157	M₃₂	1	4	yes	complemented
129	C₂×C₄×C₅₃	—	2	1	no	—
130	C₂×C₄×C₅₃	—	2	1	no	—
131	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
133	C₂×Dic₅₃	—	2	1	no	—
134	C₂×Dic₅₃	—	2	1	no	—
135	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
136	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
137	C₄×D₅₃	—	2	1	no	—
138	C₄×D₅₃	—	2	1	no	—
139	C₄×D₅₃	—	2	1	no	—
140	C₄×D₅₃	—	2	1	no	—
141	C₄×D₅₃	—	2	1	no	—
142	C₄×D₅₃	—	2	1	no	—
143	C₄×D₅₃	C₄×D₄	1	1	yes	characteristic
144	C₄×D₅₃	C₄×D₄	1	1	yes	characteristic
145	C₄×D₅₃	#131	1	1	yes	characteristic
146	C₄×D₅₃	#131	1	1	yes	characteristic
147	#88324	—	2	2	no	—
148	#88324	—	2	2	no	—
149	C₈×C₅₃	—	2	1	no	—
150	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
151	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
152	#88322	—	2	1	no	—
153	#88322	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	#88322	#117	1	1	yes	characteristic
155	#12612	—	53	1	no	maximal, Sylow
156	C₂²×C₅₃	—	2	1	no	—
157	C₂²×C₅₃	—	2	1	no	—
158	C₂²×C₅₃	—	2	1	no	—
159	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
160	C₂×D₅₃	—	2	1	no	—
161	C₂×D₅₃	—	2	1	no	—
162	C₂×D₅₃	—	2	1	no	—
163	C₂×D₅₃	—	4	2	no	—
164	C₂×D₅₃	—	4	2	no	—
165	C₂×D₅₃	C₈×D₄	1	1	yes	characteristic
166	C₂×D₅₃	C₈×D₄	1	1	yes	characteristic
167	C₂×D₅₃	C₂×M₃₂	1	2	yes	—
168	C₂×D₅₃	#504	1	2	yes	—
169	#6157	—	2	4	no	—
170	#6157	—	4	2	no	—
171	C₄×C₅₃	—	2	1	no	—
172	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
173	C₄×C₅₃	#405	1	1	yes	characteristic
174	Dic₅₃	—	2	1	no	—
175	Dic₅₃	C₈×D₄	1	1	yes	characteristic
176	Dic₅₃	C₈×D₄	1	1	yes	characteristic
177	C₂²×M₃₂	—	53	1	no	—
178	C₂×C₂²⋊C₁₆	—	53	1	no	—
179	C₂×#346	—	53	1	no	—
180	C₂×C₄×C₁₆	—	53	1	no	—
181	C₈×C₂²⋊C₄	—	53	1	no	—
182	#1374	—	53	1	no	—
183	#1374	—	53	1	no	—
184	C₂×C₅₃	—	4	1	no	—
185	C₂×C₅₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
186	C₂×C₅₃	C₄×M₃₂	1	1	yes	characteristic
187	C₂×C₅₃	#2108	1	1	yes	characteristic
188	D₅₃	—	4	1	no	—
189	D₅₃	#2168	1	2	yes	—
190	D₅₃	#2169	1	2	yes	—
191	C₂³×C₈	—	53	1	no	—
192	C₂×M₃₂	—	53	2	no	—
193	C₂×M₃₂	—	53	2	no	—
194	C₂×M₃₂	—	106	2	no	—
195	C₂×M₃₂	—	106	2	no	—
196	C₂²×C₁₆	—	53	1	no	—
197	C₂²×C₁₆	—	53	1	no	—
198	C₂²×C₁₆	—	53	1	no	—
199	C₂²×C₁₆	—	53	1	no	—
200	C₂×C₂²⋊C₈	—	53	1	no	—
201	C₂×C₄×C₈	—	53	1	no	—
202	C₂×C₄×C₈	—	53	1	no	—
203	C₄×C₂²⋊C₄	—	53	1	no	—
204	C₂²⋊C₁₆	—	53	1	no	—
205	C₂²⋊C₁₆	—	53	1	no	—
206	C₂²⋊C₁₆	—	53	1	no	—
207	C₂²⋊C₁₆	#6157	1	1	yes	characteristic, complemented
208	#346	—	106	2	no	—
209	C₄×C₁₆	—	106	2	no	—
210	#336	—	53	1	no	—
211	#336	—	53	1	no	—
212	C₅₃	#12612	1	1	yes	characteristic, complemented, Sylow
213	C₂³×C₄	—	53	1	no	—
214	C₂²×C₈	—	53	1	no	—
215	C₂²×C₈	—	53	1	no	—
216	C₂²×C₈	—	53	1	no	—
217	C₂²×C₈	—	53	1	no	—
218	C₂²×C₈	—	53	1	no	—
219	C₂²×C₈	—	53	1	no	—
220	C₂²×C₈	—	53	1	no	—
221	C₂²×C₈	—	106	1	no	—
222	C₂²×C₈	—	106	1	no	—
223	C₂²×C₈	—	106	1	no	—
224	C₂²×C₈	—	106	1	no	—
225	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
226	C₂×C₂²⋊C₄	—	53	2	no	—
227	C₂×C₄²	—	53	1	no	—
228	C₂×C₄²	—	53	1	no	—
229	M₃₂	—	106	4	no	—
230	M₃₂	—	106	4	no	—
231	C₂×C₁₆	—	53	2	no	—
232	C₂×C₁₆	—	53	2	no	—
233	C₂×C₁₆	—	53	2	no	—
234	C₂×C₁₆	—	53	2	no	—
235	C₂×C₁₆	—	53	1	no	—
236	C₂×C₁₆	—	53	1	no	—
237	C₂×C₁₆	—	106	1	no	—
238	C₂×C₁₆	—	106	1	no	—
239	C₂×C₁₆	—	106	1	no	—
240	C₂×C₁₆	—	106	1	no	—
241	C₂×C₁₆	—	106	1	no	—
242	C₂×C₁₆	—	106	1	no	—
243	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
244	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
245	C₂²⋊C₈	—	53	2	no	—
246	C₂²⋊C₈	—	53	2	no	—
247	C₄×C₈	—	106	1	no	—
248	C₄×C₈	—	106	1	no	—
249	C₄×C₈	—	106	1	no	—
250	C₄×C₈	—	106	1	no	—
251	#95	—	53	2	no	—
252	C₂⁴	—	53	1	no	—
253	C₂²×C₄	—	53	1	no	—
254	C₂²×C₄	—	53	2	no	—
255	C₂²×C₄	—	53	1	no	—
256	C₂²×C₄	—	53	1	no	—
257	C₂²×C₄	—	53	1	no	—
258	C₂²×C₄	—	53	1	no	—
259	C₂²×C₄	—	53	2	no	—
260	C₂²×C₄	—	106	1	no	—
261	C₂²×C₄	—	106	1	no	—
262	C₂²×C₄	—	106	1	no	—
263	C₂²×C₄	—	106	1	no	—
264	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	53	1	no	—
268	C₂×C₈	—	53	1	no	—
269	C₂×C₈	—	53	1	no	—
270	C₂×C₈	—	53	1	no	—
271	C₂×C₈	—	53	1	no	—
272	C₂×C₈	—	53	2	no	—
273	C₂×C₈	—	53	2	no	—
274	C₂×C₈	—	53	1	no	—
275	C₂×C₈	—	106	1	no	—
276	C₂×C₈	—	106	1	no	—
277	C₂×C₈	—	106	1	no	—
278	C₂×C₈	—	106	1	no	—
279	C₂×C₈	—	106	1	no	—
280	C₂×C₈	—	106	1	no	—
281	C₂×C₈	—	106	1	no	—
282	C₂×C₈	—	106	1	no	—
283	C₂×C₈	—	106	1	no	—
284	C₂×C₈	—	106	1	no	—
285	C₂×C₈	—	106	1	no	—
286	C₂×C₈	—	106	1	no	—
287	C₂×C₈	C₄×#6157	1	1	yes	characteristic
288	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
289	C₂²⋊C₄	—	106	4	no	—
290	C₄²	—	106	1	no	—
291	C₄²	—	106	1	no	—
292	C₄²	—	106	1	no	—
293	C₄²	—	106	1	no	—
294	C₁₆	—	2	1	no	—
295	C₁₆	—	2	1	no	—
296	C₁₆	—	53	4	no	—
297	C₁₆	—	106	2	no	—
298	C₁₆	—	106	1	no	—
299	C₁₆	—	106	1	no	—
300	C₂³	—	53	1	no	—
301	C₂³	—	53	1	no	—
302	C₂³	—	106	2	no	—
303	C₂³	—	106	1	no	—
304	C₂³	—	106	1	no	—
305	C₂³	—	106	2	no	—
306	C₂³	C₈×#6157	1	1	yes	characteristic
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	53	1	no	—
310	C₂×C₄	—	53	1	no	—
311	C₂×C₄	—	53	1	no	—
312	C₂×C₄	—	53	1	no	—
313	C₂×C₄	—	53	1	no	—
314	C₂×C₄	—	53	4	no	—
315	C₂×C₄	—	53	1	no	—
316	C₂×C₄	—	106	1	no	—
317	C₂×C₄	—	106	1	no	—
318	C₂×C₄	—	106	1	no	—
319	C₂×C₄	—	106	1	no	—
320	C₂×C₄	—	106	1	no	—
321	C₂×C₄	—	106	1	no	—
322	C₂×C₄	—	106	2	no	—
323	C₂×C₄	—	106	2	no	—
324	C₂×C₄	—	106	1	no	—
325	C₂×C₄	—	106	1	no	—
326	C₂×C₄	—	106	2	no	—
327	C₂×C₄	—	106	2	no	—
328	C₂×C₄	—	106	2	no	—
329	C₂×C₄	C₈×#6157	1	1	yes	characteristic
330	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	53	1	no	—
333	C₈	—	53	1	no	—
334	C₈	—	106	2	no	—
335	C₈	—	106	2	no	—
336	C₈	—	106	1	no	—
337	C₈	D₄×#6157	1	1	yes	characteristic
338	C₈	D₄×#6157	1	1	yes	characteristic
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	53	2	no	—
343	C₂²	—	53	2	no	—
344	C₂²	—	53	1	no	—
345	C₂²	—	53	1	no	—
346	C₂²	—	106	1	no	—
347	C₂²	—	106	1	no	—
348	C₂²	—	106	1	no	—
349	C₂²	—	212	2	no	—
350	C₂²	—	212	2	no	—
351	C₂²	C₂×C₈×#6157	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	53	1	no	—
354	C₄	—	53	1	no	—
355	C₄	—	106	4	no	—
356	C₄	—	106	1	no	—
357	C₄	—	212	2	no	—
358	C₄	C₂²⋊C₄×#6157	1	1	yes	characteristic
359	C₄	#16811589	1	1	yes	characteristic
360	C₂	—	4	1	no	—
361	C₂	—	53	2	no	—
362	C₂	—	53	2	no	—
363	C₂	—	212	1	no	—
364	C₂	#20290450	1	1	yes	characteristic
365	C₂	M₃₂×#6157	1	1	yes	characteristic
366	C₂	#20290923	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₅₃×C₂²⋊C₁₆	C₄
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	C₂²⋊C₁₆	#6157
hypocenter	C₅₃	#12612
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#16811510	C₂	semidirect
3. C₂×C₁₆×#6157	C₂	semidirect
5. (C₂×#6157)⋊C₁₆	C₂	semidirect
6. (C₂×#6157)⋊C₁₆	C₂	semidirect
8. C₂×#13133292	C₄	semidirect
11. C₂×#13133291	C₄	semidirect
25. C₂²⋊#11719908	C₄	semidirect
26. #16810402	C₄	semidirect
27. #16810402	C₄	semidirect
28. C₅₃×C₂²⋊C₁₆	C₄	semidirect
111. #11719910	C₂²⋊C₄	semidirect
128. C₂×#6157	M₃₂	semidirect
207. C₂²⋊C₁₆	#6157	semidirect
212. C₅₃	#12612	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅₃⋊M₃₂	2
C₂×#16811510	2
C₂×C₁₆×#6157	2
C₈×C₂²⋊#6157	2
(C₂×#6157)⋊C₁₆	2
C₂²⋊C₁₆×D₅₃	2
#12612	53