Subgroups — #28328332

Order 13568.

734 conjugacy classes of subgroups.

The subgroup lattice has 356 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	2	yes	maximal, complemented
2	#20291101	C₂	1	2	yes	maximal, complemented
3	#20291098	C₂	1	4	yes	maximal, complemented
4	#20291094	C₂	1	2	yes	maximal, complemented
5	#20291047	C₂	1	1	yes	maximal, characteristic, complemented
6	#20290944	C₂	1	2	yes	maximal, complemented
7	C₄×#13133291	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16810688	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133292	C₂²	1	4	yes	complemented
10	D₅₃⋊M₃₂	—	2	8	no	—
11	C₂×#13133291	C₂²	1	2	yes	—
12	C₂×#13133291	C₂²	1	1	yes	characteristic
13	C₂×#13133291	C₂²	1	1	yes	characteristic
14	C₂×D₅₃⋊C₁₆	C₂²	1	4	yes	complemented
15	C₄⋊#11719910	C₂²	1	2	yes	complemented
16	C₄⋊#11719910	C₂²	1	2	yes	complemented
17	#16811518	C₂²	1	4	yes	complemented
18	#16811518	C₂²	1	4	yes	complemented
19	#16811511	C₂²	1	2	yes	complemented
20	C₄×#11719910	C₂²	1	2	yes	complemented
21	C₂²×C₈×D₅₃	C₂²	1	2	yes	—
22	C₂×C₄×D₂₁₂	C₄	1	1	yes	characteristic
23	#16810688	C₄	1	1	yes	characteristic
24	#16810688	C₄	1	2	yes	—
25	#16810688	C₄	1	1	yes	characteristic
26	#16810688	C₂²	1	1	yes	characteristic
27	#16810688	C₂²	1	2	yes	—
28	#16810688	C₂²	1	1	yes	characteristic
29	C₂×#13133129	C₄	1	2	yes	—
30	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
32	#13133292	—	2	8	no	—
33	#13133291	—	2	4	no	—
34	#13133291	—	2	2	no	—
35	#13133291	D₄	1	4	yes	complemented
36	D₅₃⋊C₁₆	—	2	8	no	—
37	C₂×#11719910	C₂³	1	4	yes	—
38	C₂×#11719910	C₂³	1	2	yes	—
39	C₂×#11719910	C₂³	1	2	yes	—
40	C₂²×D₂₁₂	C₈	1	1	yes	characteristic
41	C₂²×C₄×D₅₃	C₂×C₄	1	2	yes	—
42	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
43	C₂²×#88322	C₂³	1	1	yes	characteristic
44	C₂×C₈×D₅₃	—	2	4	no	—
45	C₂×C₈×D₅₃	—	2	4	no	—
46	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₅₃	C₂³	1	2	yes	—
49	C₂×C₈×D₅₃	C₂³	1	2	yes	—
50	C₄×D₂₁₂	C₈	1	1	yes	characteristic
51	C₄×D₂₁₂	C₈	1	2	yes	—
52	C₄×D₂₁₂	C₈	1	1	yes	characteristic
53	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
54	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
55	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
57	C₂×#11719921	C₈	1	2	yes	—
58	#13133129	C₂×C₄	1	2	yes	—
59	#13133129	C₂×C₄	1	2	yes	—
60	#13133129	C₂×C₄	1	2	yes	—
61	#13133129	C₂×C₄	1	2	yes	—
62	C₄×#88322	C₂×C₄	1	1	yes	characteristic
63	C₄×#88322	C₂×C₄	1	1	yes	characteristic
64	C₄×#88322	C₂³	1	1	yes	characteristic
65	C₄×#88322	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
67	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
70	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂³×D₅₃	C₂×C₈	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
76	C₂×D₂₁₂	—	2	4	no	—
77	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
78	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
79	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄×D₅₃	—	2	4	no	—
81	C₂×C₄×D₅₃	—	2	4	no	—
82	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
83	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
84	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₅₃	—	2	4	no	—
87	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
90	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
91	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	#11719921	C₂×C₈	1	2	yes	—
94	#11719921	C₂×C₈	1	2	yes	—
95	#11719921	C₂×C₈	1	2	yes	—
96	#11719921	C₂×C₈	1	2	yes	—
97	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
98	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
99	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
100	C₂×#88322	—	2	2	no	—
101	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
102	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
103	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
104	C₂×#88322	C₂×D₄	1	2	yes	—
105	C₂×#88322	#41	1	2	yes	—
106	C₂×#88322	C₂⁴	1	1	yes	characteristic
107	C₈×D₅₃	—	2	4	no	—
108	C₈×D₅₃	—	2	4	no	—
109	C₈×D₅₃	—	2	2	no	—
110	C₈×D₅₃	—	2	2	no	—
111	C₈×D₅₃	—	2	2	no	—
112	C₈×D₅₃	—	2	2	no	—
113	#11719910	—	2	4	no	—
114	#11719910	—	2	2	no	—
115	#11719910	C₂×D₄	1	4	yes	complemented
116	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
117	C₂²×D₅₃	—	2	4	no	—
118	C₂²×D₅₃	—	2	4	no	—
119	C₂²×D₅₃	—	4	2	no	—
120	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
121	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	4	1	no	—
124	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₅₃	#131	1	2	yes	—
131	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×Dic₅₃	—	2	4	no	—
133	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
134	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
135	D₂₁₂	—	2	4	no	—
136	D₂₁₂	—	2	2	no	—
137	D₂₁₂	M₃₂	1	4	yes	complemented
138	C₄×D₅₃	—	2	2	no	—
139	C₄×D₅₃	—	2	2	no	—
140	C₄×D₅₃	—	2	2	no	—
141	C₄×D₅₃	—	2	2	no	—
142	C₄×D₅₃	—	2	4	no	—
143	C₄×D₅₃	—	2	4	no	—
144	C₈×C₅₃	—	2	2	no	—
145	C₈×C₅₃	—	2	2	no	—
146	#88322	—	2	1	no	—
147	#88322	—	2	1	no	—
148	#88322	C₄×D₄	1	2	yes	—
149	#88322	C₂²×D₄	1	1	yes	characteristic
150	#88322	C₂×#41	1	1	yes	characteristic
151	D₄×M₃₂	—	53	1	no	maximal, Sylow
152	C₂²×C₅₃	—	2	1	no	—
153	C₂²×C₅₃	—	2	1	no	—
154	C₂²×C₅₃	C₈×D₄	1	2	yes	—
155	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
156	C₂×D₅₃	—	2	2	no	—
157	C₂×D₅₃	—	2	2	no	—
158	C₂×D₅₃	—	2	2	no	—
159	C₂×D₅₃	—	2	2	no	—
160	C₂×D₅₃	—	2	2	no	—
161	C₂×D₅₃	—	4	8	no	—
162	C₂×D₅₃	C₂×M₃₂	1	4	yes	complemented
163	C₄×C₅₃	—	2	1	no	—
164	C₄×C₅₃	—	2	1	no	—
165	C₄×C₅₃	—	2	1	no	—
166	C₄×C₅₃	C₈×D₄	1	2	yes	—
167	C₄×C₅₃	C₂×M₃₂	1	2	yes	complemented
168	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
169	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
170	Dic₅₃	—	2	2	no	—
171	Dic₅₃	—	2	2	no	—
172	C₂²×M₃₂	—	53	2	no	—
173	C₂×C₈×D₄	—	53	1	no	—
174	#2169	—	53	2	no	—
175	#2168	—	53	4	no	—
176	D₄×C₁₆	—	53	2	no	—
177	#2150	—	53	1	no	—
178	#2112	—	53	2	no	—
179	C₄×M₃₂	—	53	1	no	—
180	C₂×C₅₃	—	2	2	no	—
181	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
182	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
183	C₂×C₅₃	#3407	1	1	yes	characteristic
184	D₅₃	—	2	4	no	—
185	D₅₃	—	4	2	no	—
186	C₂³×C₈	—	53	2	no	—
187	C₂×C₄×D₄	—	53	1	no	—
188	C₂×M₃₂	—	53	1	no	—
189	C₂×M₃₂	—	53	1	no	—
190	C₂×M₃₂	—	53	2	no	—
191	C₂×M₃₂	—	53	4	no	—
192	C₂×M₃₂	—	106	8	no	—
193	C₂²×C₁₆	—	53	4	no	—
194	C₈×D₄	—	53	1	no	—
195	C₈×D₄	—	53	2	no	—
196	C₈×D₄	—	53	1	no	—
197	C₈×D₄	—	53	2	no	—
198	C₈×D₄	—	53	1	no	—
199	C₈×D₄	—	53	1	no	—
200	C₂×C₄⋊C₈	#6157	1	1	yes	characteristic
201	C₂×C₂²⋊C₈	—	53	2	no	—
202	C₂×C₄×C₈	—	53	1	no	—
203	C₄⋊C₁₆	—	53	2	no	—
204	C₄⋊C₁₆	—	53	2	no	—
205	C₂²⋊C₁₆	—	53	4	no	—
206	C₂²⋊C₁₆	—	53	4	no	—
207	#346	—	53	2	no	—
208	C₄×C₁₆	—	53	2	no	—
209	C₅₃	D₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
210	C₂²×D₄	—	53	1	no	—
211	C₂³×C₄	—	53	2	no	—
212	C₂²×C₈	—	53	1	no	—
213	C₂²×C₈	—	53	1	no	—
214	C₂²×C₈	—	53	2	no	—
215	C₂²×C₈	—	53	2	no	—
216	C₂²×C₈	—	53	2	no	—
217	C₂²×C₈	—	53	2	no	—
218	C₂²×C₈	—	106	4	no	—
219	C₂²×C₈	—	106	4	no	—
220	C₂²×C₈	C₂×#6157	1	2	yes	—
221	C₄×D₄	—	53	1	no	—
222	C₄×D₄	—	53	2	no	—
223	C₄×D₄	—	53	1	no	—
224	C₄×D₄	—	53	1	no	—
225	C₄×D₄	—	53	2	no	—
226	C₄×D₄	—	53	1	no	—
227	C₂×C₄⋊C₄	—	53	1	no	—
228	C₂×C₂²⋊C₄	—	53	2	no	—
229	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
230	M₃₂	—	53	4	no	—
231	M₃₂	—	106	8	no	—
232	M₃₂	—	106	2	no	—
233	M₃₂	—	106	4	no	—
234	C₂×C₁₆	—	53	2	no	—
235	C₂×C₁₆	—	53	2	no	—
236	C₂×C₁₆	—	53	4	no	—
237	C₂×C₁₆	—	106	8	no	—
238	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic
239	C₄⋊C₈	C₂×#6157	1	2	yes	—
240	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic
241	C₂²⋊C₈	—	53	2	no	—
242	C₂²⋊C₈	—	53	2	no	—
243	C₂²⋊C₈	—	53	2	no	—
244	C₂²⋊C₈	—	53	2	no	—
245	C₄×C₈	—	53	1	no	—
246	C₄×C₈	—	53	1	no	—
247	C₄×C₈	—	53	1	no	—
248	C₄×C₈	—	53	1	no	—
249	C₂⁴	—	53	2	no	—
250	C₂×D₄	—	53	1	no	—
251	C₂×D₄	—	53	2	no	—
252	C₂×D₄	—	53	1	no	—
253	C₂×D₄	—	106	4	no	—
254	C₂²×C₄	—	53	2	no	—
255	C₂²×C₄	—	53	2	no	—
256	C₂²×C₄	—	53	2	no	—
257	C₂²×C₄	—	53	2	no	—
258	C₂²×C₄	—	53	2	no	—
259	C₂²×C₄	—	106	4	no	—
260	C₂²×C₄	—	106	4	no	—
261	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
262	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
263	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	53	2	no	—
266	C₂×C₈	—	53	2	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	—	106	2	no	—
272	C₂×C₈	—	106	2	no	—
273	C₂×C₈	—	106	4	no	—
274	C₂×C₈	—	106	4	no	—
275	C₂×C₈	—	106	2	no	—
276	C₂×C₈	—	106	2	no	—
277	C₂×C₈	—	106	2	no	—
278	C₂×C₈	C₂²×#6157	1	2	yes	—
279	C₂×C₈	C₂²×#6157	1	2	yes	—
280	C₄⋊C₄	—	53	2	no	—
281	C₄⋊C₄	—	53	1	no	—
282	C₄⋊C₄	—	53	1	no	—
283	C₂²⋊C₄	—	53	2	no	—
284	C₂²⋊C₄	—	53	2	no	—
285	C₂²⋊C₄	—	53	2	no	—
286	C₂²⋊C₄	—	53	2	no	—
287	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
288	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
289	C₄²	C₂²×#6157	1	1	yes	characteristic
290	C₄²	C₂²×#6157	1	1	yes	characteristic
291	C₁₆	—	53	4	no	—
292	C₁₆	—	106	4	no	—
293	C₁₆	—	106	2	no	—
294	C₂³	—	53	2	no	—
295	C₂³	—	53	2	no	—
296	C₂³	—	106	4	no	—
297	C₂³	—	106	4	no	—
298	C₂³	—	212	2	no	—
299	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
300	D₄	—	53	4	no	—
301	D₄	—	106	4	no	—
302	D₄	—	106	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	53	2	no	—
306	C₂×C₄	—	53	2	no	—
307	C₂×C₄	—	106	4	no	—
308	C₂×C₄	—	106	4	no	—
309	C₂×C₄	—	106	4	no	—
310	C₂×C₄	—	106	2	no	—
311	C₂×C₄	—	106	2	no	—
312	C₂×C₄	—	106	2	no	—
313	C₂×C₄	—	106	2	no	—
314	C₂×C₄	D₄×#6157	1	2	yes	—
315	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
316	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
317	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
318	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
319	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
320	C₂×C₄	#13133305	1	2	yes	—
321	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
322	C₈	—	2	2	no	—
323	C₈	—	2	2	no	—
324	C₈	—	53	2	no	—
325	C₈	—	53	1	no	—
326	C₈	—	53	1	no	—
327	C₈	—	106	1	no	—
328	C₈	—	106	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	53	4	no	—
332	C₂²	—	106	2	no	—
333	C₂²	—	106	2	no	—
334	C₂²	—	106	2	no	—
335	C₂²	—	106	2	no	—
336	C₂²	—	106	2	no	—
337	C₂²	—	212	8	no	—
338	C₂²	#16811651	1	2	yes	—
339	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	106	2	no	—
344	C₄	—	106	2	no	—
345	C₄	#16811651	1	2	yes	—
346	C₄	D₅₃⋊M₃₂	1	2	yes	complemented
347	C₄	C₂×D₄×#6157	1	1	yes	characteristic
348	C₄	C₂×#13133305	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	106	4	no	—
351	C₂	—	212	2	no	—
352	C₂	C₂×#16811651	1	1	yes	characteristic
353	C₂	C₂×D₅₃⋊M₃₂	1	1	yes	characteristic
354	C₂	#20291684	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₂×C₄⋊C₈×C₅₃	C₄
prime radical	C₂²×D₂₁₂	C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂×C₄⋊C₈	#6157
hypocenter	C₅₃	D₄×M₃₂
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×D₅₃⋊M₃₂	C₂	semidirect
2. #20291101	C₂	semidirect
3. #20291098	C₂	semidirect
4. #20291094	C₂	semidirect
5. #20291047	C₂	semidirect
6. #20290944	C₂	semidirect
7. C₄×#13133291	C₂	semidirect
9. C₂×#13133292	C₂²	semidirect
14. C₂×D₅₃⋊C₁₆	C₂²	semidirect
15. C₄⋊#11719910	C₂²	semidirect
16. C₄⋊#11719910	C₂²	semidirect
17. #16811518	C₂²	semidirect
18. #16811518	C₂²	semidirect
19. #16811511	C₂²	semidirect
20. C₄×#11719910	C₂²	semidirect
35. #13133291	D₄	semidirect
115. #11719910	C₂×D₄	semidirect
137. D₂₁₂	M₃₂	semidirect
162. C₂×D₅₃	C₂×M₃₂	semidirect
167. C₄×C₅₃	C₂×M₃₂	semidirect
209. C₅₃	D₄×M₃₂	semidirect
346. C₄	D₅₃⋊M₃₂	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
#20291101	2
#20291098	2
#20291094	2
#20291047	2
#20290944	2
C₄×#13133291	2
C₂×#16810688	2
D₄×M₃₂	53