Subgroups — #28310529

Order 13568.

474 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₃⋊(C₂²⋊C₁₆)	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	C₂×M₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×D₅₃⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂×D₅₃⋊C₁₆	C₂²	1	1	yes	characteristic
10	D₅₃⋊(C₂²⋊C₈)	C₄	1	1	yes	characteristic
11	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
12	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
13	C₄×C₂²⋊#6157	C₄	1	1	yes	characteristic
14	#16811518	C₄	1	2	yes	complemented
15	C₂²⋊#11719910	C₄	1	2	yes	complemented
16	#16811510	—	2	2	no	—
17	C₁₆×#6157	—	2	2	no	—
18	(C₂×#6157)⋊C₈	C₄	1	1	yes	characteristic
19	(C₂×#6157)⋊C₈	C₄	1	1	yes	characteristic
20	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
21	C₂×#13133157	C₄	1	1	yes	characteristic, complemented
22	C₂×#13133156	C₄	1	1	yes	characteristic, complemented
23	C₂×#13133156	C₄	1	1	yes	characteristic, complemented
24	M₃₂×D₅₃	—	2	2	no	—
25	M₃₂×D₅₃	—	2	2	no	—
26	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
28	C₂×M₃₂×C₅₃	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	D₅₃⋊C₁₆	—	2	1	no	—
35	D₅₃⋊C₁₆	—	2	1	no	—
36	D₅₃⋊C₁₆	—	2	1	no	—
37	D₅₃⋊C₁₆	—	2	1	no	—
38	C₂×#11719910	C₂×C₄	1	2	yes	—
39	C₂×#11719910	C₂×C₄	1	2	yes	—
40	#13133274	C₈	1	2	yes	—
41	C₂²⋊#88324	C₈	1	2	yes	—
42	C₈×#6157	—	2	1	no	—
43	C₈×#6157	—	2	1	no	—
44	C₈×#6157	—	2	1	no	—
45	C₈×#6157	—	2	1	no	—
46	(C₂×#6157)⋊C₄	C₈	1	2	yes	—
47	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₅₃	—	2	1	no	—
50	C₂×C₈×D₅₃	—	2	1	no	—
51	C₂×C₈×D₅₃	—	2	1	no	—
52	C₂×C₈×D₅₃	—	2	1	no	—
53	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
57	#13133157	—	2	2	no	—
58	#13133156	—	2	2	no	—
59	#13133156	—	2	2	no	—
60	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
61	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
62	C₁₆×D₅₃	—	2	1	no	—
63	C₁₆×D₅₃	—	2	1	no	—
64	C₁₆×D₅₃	D₄	1	2	yes	—
65	C₁₆×D₅₃	D₄	1	2	yes	—
66	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
67	M₃₂×C₅₃	—	2	2	no	—
68	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
71	C₂²×#6157	C₂×C₈	1	2	yes	—
72	C₂²×#6157	C₂×C₈	1	2	yes	—
73	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
74	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
75	C₂×C₄×D₅₃	—	2	1	no	—
76	C₂×C₄×D₅₃	—	2	1	no	—
77	C₂×C₄×D₅₃	—	2	1	no	—
78	C₂×C₄×D₅₃	—	2	1	no	—
79	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
80	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²⋊#6157	—	2	4	no	—
84	C₂×#88324	C₂×C₈	1	2	yes	—
85	C₂×#88324	C₂×C₈	1	2	yes	—
86	C₄×#6157	—	2	1	no	—
87	C₄×#6157	—	2	1	no	—
88	C₄×#6157	—	2	1	no	—
89	C₄×#6157	—	2	1	no	—
90	D₅₃⋊C₈	—	2	1	no	—
91	D₅₃⋊C₈	—	2	1	no	—
92	D₅₃⋊C₈	—	2	1	no	—
93	D₅₃⋊C₈	—	2	1	no	—
94	C₂×C₈×C₅₃	—	2	1	no	—
95	C₂×C₈×C₅₃	—	2	1	no	—
96	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×#88322	—	2	1	no	—
99	C₂×#88322	—	2	1	no	—
100	C₂×#88322	C₄²	1	1	yes	characteristic
101	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
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	—	2	1	no	—
107	C₈×D₅₃	—	2	1	no	—
108	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
109	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
110	C₈×D₅₃	#41	1	1	yes	characteristic
111	C₈×D₅₃	#41	1	1	yes	characteristic
112	#11719910	—	2	2	no	—
113	#11719910	—	2	2	no	—
114	C₁₆×C₅₃	—	2	1	no	—
115	C₁₆×C₅₃	C₂²⋊C₄	1	2	yes	complemented
116	#11719908	—	2	1	no	—
117	#11719908	C₂²⋊C₄	1	2	yes	complemented
118	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
119	C₂²×D₅₃	—	2	1	no	—
120	C₂²×D₅₃	—	2	1	no	—
121	C₂²×D₅₃	—	2	2	no	—
122	C₂²×D₅₃	—	2	2	no	—
123	C₂²×D₅₃	C₄×C₈	1	1	yes	characteristic
124	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
125	C₂×#6157	—	2	2	no	—
126	C₂×#6157	—	2	2	no	—
127	C₂×#6157	—	2	2	no	—
128	C₂×#6157	—	2	2	no	—
129	C₂×#6157	—	2	2	no	—
130	C₂×#6157	M₃₂	1	4	yes	complemented
131	C₂×C₄×C₅₃	—	2	1	no	—
132	C₂×C₄×C₅₃	—	2	1	no	—
133	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
135	C₂×Dic₅₃	—	2	1	no	—
136	C₂×Dic₅₃	—	2	1	no	—
137	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
138	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
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	—	2	1	no	—
144	C₄×D₅₃	—	2	1	no	—
145	C₄×D₅₃	C₄×D₄	1	1	yes	characteristic
146	C₄×D₅₃	C₄×D₄	1	1	yes	characteristic
147	C₄×D₅₃	#131	1	1	yes	characteristic
148	C₄×D₅₃	#131	1	1	yes	characteristic
149	#88324	—	2	2	no	—
150	#88324	—	2	2	no	—
151	C₈×C₅₃	—	2	1	no	—
152	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₈×C₅₃	#117	1	1	yes	characteristic
154	#88322	—	2	1	no	—
155	#88322	C₄×D₄	1	1	yes	characteristic
156	#88322	C₄×D₄	1	1	yes	characteristic
157	#12612	—	53	1	no	maximal, Sylow
158	C₂²×C₅₃	—	2	1	no	—
159	C₂²×C₅₃	—	2	1	no	—
160	C₂²×C₅₃	—	2	1	no	—
161	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
162	C₂×D₅₃	—	2	1	no	—
163	C₂×D₅₃	—	2	1	no	—
164	C₂×D₅₃	—	2	1	no	—
165	C₂×D₅₃	—	4	2	no	—
166	C₂×D₅₃	—	4	2	no	—
167	C₂×D₅₃	C₈×D₄	1	1	yes	characteristic
168	C₂×D₅₃	C₈×D₄	1	1	yes	characteristic
169	C₂×D₅₃	C₂×M₃₂	1	2	yes	—
170	C₂×D₅₃	#504	1	2	yes	—
171	#6157	—	2	4	no	—
172	#6157	—	4	2	no	—
173	C₄×C₅₃	—	2	1	no	—
174	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
175	C₄×C₅₃	#405	1	1	yes	characteristic
176	Dic₅₃	—	2	1	no	—
177	Dic₅₃	C₈×D₄	1	1	yes	characteristic
178	Dic₅₃	C₈×D₄	1	1	yes	characteristic
179	C₂²×M₃₂	—	53	1	no	—
180	C₂×C₂²⋊C₁₆	—	53	1	no	—
181	C₂×#346	—	53	1	no	—
182	C₂×C₄×C₁₆	—	53	1	no	—
183	C₈×C₂²⋊C₄	—	53	1	no	—
184	#1374	—	53	1	no	—
185	#1374	—	53	1	no	—
186	C₂×C₅₃	—	4	1	no	—
187	C₂×C₅₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
188	C₂×C₅₃	C₄×M₃₂	1	1	yes	characteristic
189	C₂×C₅₃	#2108	1	1	yes	characteristic
190	D₅₃	—	4	1	no	—
191	D₅₃	#2168	1	2	yes	—
192	D₅₃	#2169	1	2	yes	—
193	C₂³×C₈	—	53	1	no	—
194	C₂×M₃₂	—	53	1	no	—
195	C₂×M₃₂	—	53	1	no	—
196	C₂×M₃₂	—	53	1	no	—
197	C₂×M₃₂	—	106	2	no	—
198	C₂×M₃₂	—	106	2	no	—
199	C₂×M₃₂	#6157	1	1	yes	characteristic, complemented
200	C₂²×C₁₆	—	53	1	no	—
201	C₂²×C₁₆	—	53	1	no	—
202	C₂²×C₁₆	—	53	1	no	—
203	C₂²×C₁₆	—	53	1	no	—
204	C₂×C₂²⋊C₈	—	53	1	no	—
205	C₂×C₄×C₈	—	53	1	no	—
206	C₂×C₄×C₈	—	53	1	no	—
207	C₄×C₂²⋊C₄	—	53	1	no	—
208	C₂²⋊C₁₆	—	53	2	no	—
209	C₂²⋊C₁₆	—	53	2	no	—
210	#346	—	106	2	no	—
211	C₄×C₁₆	—	106	2	no	—
212	#336	—	53	1	no	—
213	#336	—	53	1	no	—
214	C₅₃	#12612	1	1	yes	characteristic, complemented, Sylow
215	C₂³×C₄	—	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	—	53	1	no	—
222	C₂²×C₈	—	53	1	no	—
223	C₂²×C₈	—	106	1	no	—
224	C₂²×C₈	—	106	1	no	—
225	C₂²×C₈	—	106	1	no	—
226	C₂²×C₈	—	106	1	no	—
227	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
228	C₂×C₂²⋊C₄	—	53	2	no	—
229	C₂×C₄²	—	53	1	no	—
230	C₂×C₄²	—	53	1	no	—
231	M₃₂	—	2	2	no	—
232	M₃₂	—	106	2	no	—
233	M₃₂	—	106	2	no	—
234	M₃₂	—	106	2	no	—
235	C₂×C₁₆	—	53	1	no	—
236	C₂×C₁₆	—	53	2	no	—
237	C₂×C₁₆	—	53	2	no	—
238	C₂×C₁₆	—	53	2	no	—
239	C₂×C₁₆	—	53	2	no	—
240	C₂×C₁₆	—	53	1	no	—
241	C₂×C₁₆	—	106	1	no	—
242	C₂×C₁₆	—	106	1	no	—
243	C₂×C₁₆	—	106	1	no	—
244	C₂×C₁₆	—	106	1	no	—
245	C₂×C₁₆	—	106	1	no	—
246	C₂×C₁₆	—	106	1	no	—
247	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
248	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
249	C₂²⋊C₈	—	53	2	no	—
250	C₂²⋊C₈	—	53	2	no	—
251	C₄×C₈	—	106	1	no	—
252	C₄×C₈	—	106	1	no	—
253	C₄×C₈	—	106	1	no	—
254	C₄×C₈	—	106	1	no	—
255	#95	—	53	2	no	—
256	C₂⁴	—	53	1	no	—
257	C₂²×C₄	—	53	1	no	—
258	C₂²×C₄	—	53	2	no	—
259	C₂²×C₄	—	53	1	no	—
260	C₂²×C₄	—	53	1	no	—
261	C₂²×C₄	—	53	1	no	—
262	C₂²×C₄	—	53	1	no	—
263	C₂²×C₄	—	53	2	no	—
264	C₂²×C₄	—	106	1	no	—
265	C₂²×C₄	—	106	1	no	—
266	C₂²×C₄	—	106	1	no	—
267	C₂²×C₄	—	106	1	no	—
268	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	53	1	no	—
272	C₂×C₈	—	53	1	no	—
273	C₂×C₈	—	53	1	no	—
274	C₂×C₈	—	53	1	no	—
275	C₂×C₈	—	53	1	no	—
276	C₂×C₈	—	53	2	no	—
277	C₂×C₈	—	53	2	no	—
278	C₂×C₈	—	53	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	—	106	1	no	—
288	C₂×C₈	—	106	1	no	—
289	C₂×C₈	—	106	1	no	—
290	C₂×C₈	—	106	1	no	—
291	C₂×C₈	C₄×#6157	1	1	yes	characteristic
292	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
293	C₂²⋊C₄	—	106	4	no	—
294	C₄²	—	106	1	no	—
295	C₄²	—	106	1	no	—
296	C₄²	—	106	1	no	—
297	C₄²	—	106	1	no	—
298	C₁₆	—	2	1	no	—
299	C₁₆	—	53	2	no	—
300	C₁₆	—	106	2	no	—
301	C₁₆	—	106	2	no	—
302	C₁₆	—	106	1	no	—
303	C₁₆	C₂²⋊#6157	1	2	yes	complemented
304	C₂³	—	53	1	no	—
305	C₂³	—	53	1	no	—
306	C₂³	—	106	2	no	—
307	C₂³	—	106	1	no	—
308	C₂³	—	106	1	no	—
309	C₂³	—	106	2	no	—
310	C₂³	C₈×#6157	1	1	yes	characteristic
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	53	1	no	—
314	C₂×C₄	—	53	1	no	—
315	C₂×C₄	—	53	1	no	—
316	C₂×C₄	—	53	1	no	—
317	C₂×C₄	—	53	1	no	—
318	C₂×C₄	—	53	4	no	—
319	C₂×C₄	—	53	1	no	—
320	C₂×C₄	—	106	1	no	—
321	C₂×C₄	—	106	1	no	—
322	C₂×C₄	—	106	1	no	—
323	C₂×C₄	—	106	1	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	1	no	—
329	C₂×C₄	—	106	1	no	—
330	C₂×C₄	—	106	2	no	—
331	C₂×C₄	—	106	2	no	—
332	C₂×C₄	—	106	2	no	—
333	C₂×C₄	C₈×#6157	1	1	yes	characteristic
334	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	53	1	no	—
337	C₈	—	53	1	no	—
338	C₈	—	106	2	no	—
339	C₈	—	106	2	no	—
340	C₈	—	106	1	no	—
341	C₈	C₂×C₂²⋊#6157	1	1	yes	characteristic
342	C₈	#13133296	1	1	yes	characteristic
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	53	2	no	—
347	C₂²	—	53	2	no	—
348	C₂²	—	53	1	no	—
349	C₂²	—	53	1	no	—
350	C₂²	—	106	1	no	—
351	C₂²	—	106	1	no	—
352	C₂²	—	106	1	no	—
353	C₂²	—	212	2	no	—
354	C₂²	—	212	2	no	—
355	C₂²	C₂×C₈×#6157	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	53	1	no	—
358	C₄	—	53	1	no	—
359	C₄	—	106	4	no	—
360	C₄	—	106	1	no	—
361	C₄	—	212	2	no	—
362	C₄	C₄×C₂²⋊#6157	1	1	yes	characteristic
363	C₄	#16811586	1	1	yes	characteristic
364	C₂	—	4	1	no	—
365	C₂	—	53	2	no	—
366	C₂	—	53	2	no	—
367	C₂	—	212	1	no	—
368	C₂	C₈×C₂²⋊#6157	1	1	yes	characteristic
369	C₂	M₃₂×#6157	1	1	yes	characteristic
370	C₂	#20290927	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂×M₃₂×C₅₃	C₄
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	C₂×M₃₂	#6157
hypocenter	C₅₃	#12612
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

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
14. #16811518	C₄	semidirect
15. C₂²⋊#11719910	C₄	semidirect
21. C₂×#13133157	C₄	semidirect
22. C₂×#13133156	C₄	semidirect
23. C₂×#13133156	C₄	semidirect
28. C₂×M₃₂×C₅₃	C₄	semidirect
115. C₁₆×C₅₃	C₂²⋊C₄	semidirect
117. #11719908	C₂²⋊C₄	semidirect
130. C₂×#6157	M₃₂	semidirect
199. C₂×M₃₂	#6157	semidirect
214. C₅₃	#12612	semidirect
303. C₁₆	C₂²⋊#6157	semidirect

Maximal subgroups

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

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