Subgroups — #28316116

Order 13568.

762 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291392	C₂	1	2	yes	maximal
2	#20290278	C₂	1	4	yes	maximal, complemented
3	#20290274	C₂	1	2	yes	maximal, complemented
4	#20290271	C₂	1	2	yes	maximal, complemented
5	#20290264	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
7	#16811645	C₂²	1	4	yes	—
8	#16811644	—	2	2	no	—
9	#16811642	—	2	2	no	—
10	#16811639	C₂²	1	4	yes	—
11	#16811627	C₂²	1	4	yes	—
12	#16811594	—	2	2	no	—
13	C₄×#11719936	—	2	2	no	—
14	#16811487	C₂²	1	8	yes	complemented
15	#16811485	C₂²	1	8	yes	complemented
16	C₂²×C₈×D₅₃	C₄	1	2	yes	—
17	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×#88322	C₄	1	1	yes	characteristic
19	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
20	C₄×C₈×D₅₃	C₄	1	1	yes	characteristic
21	C₄×C₈×D₅₃	C₄	1	1	yes	characteristic
22	C₄×C₈×D₅₃	C₄	1	2	yes	—
23	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
24	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
25	C₄×C₈×D₅₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
27	C₂×#11719936	—	2	4	no	—
28	C₄⋊#88324	—	2	4	no	—
29	C₄⋊#88324	C₂³	1	8	yes	—
30	#13133274	—	2	4	no	—
31	C₂²⋊#88324	—	2	4	no	—
32	#13133272	—	2	4	no	—
33	C₄×#88324	—	2	4	no	—
34	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
37	C₂²×#88322	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₅₃	—	2	8	no	—
39	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
44	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
45	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
46	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
48	C₄²×D₅₃	D₄	1	1	yes	characteristic
49	C₄²×D₅₃	D₄	1	1	yes	characteristic
50	C₄²×D₅₃	Q₈	1	1	yes	characteristic
51	C₄²×D₅₃	Q₈	1	1	yes	characteristic
52	C₄²×D₅₃	C₂³	1	1	yes	characteristic
53	C₄²×D₅₃	C₂³	1	1	yes	characteristic
54	C₄×#88322	C₂×C₄	1	1	yes	characteristic
55	C₄×#88322	C₂×C₄	1	1	yes	characteristic
56	C₄×#88322	C₂×C₄	1	2	yes	—
57	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
58	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
59	C₈×Dic₅₃	C₂³	1	2	yes	—
60	C₈×Dic₅₃	C₂³	1	2	yes	—
61	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
66	C₂³×D₅₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₄×D₅₃	—	2	4	no	—
74	C₂×C₄×D₅₃	—	2	4	no	—
75	C₂×C₄×D₅₃	—	2	4	no	—
76	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
88	C₂×#88324	—	2	8	no	—
89	#11719936	—	4	8	no	—
90	C₂×C₈×C₅₃	—	2	4	no	—
91	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
93	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
94	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
95	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
98	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
99	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
100	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
103	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
104	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
105	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
106	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
107	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
108	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
109	C₂×#88322	M₁₆	1	8	yes	complemented
110	C₂×#88322	C₂²×C₄	1	2	yes	—
111	C₂×#88322	C₂²×C₄	1	2	yes	—
112	C₈×D₅₃	—	2	4	no	—
113	C₈×D₅₃	—	2	4	no	—
114	C₈×D₅₃	—	2	4	no	—
115	C₈×D₅₃	—	2	4	no	—
116	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×D₅₃	—	2	2	no	—
118	C₂²×D₅₃	—	2	2	no	—
119	C₂²×D₅₃	—	2	2	no	—
120	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
130	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
131	C₂×Dic₅₃	—	2	2	no	—
132	C₂×Dic₅₃	—	2	2	no	—
133	C₂×Dic₅₃	—	2	2	no	—
134	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
140	C₄×D₅₃	—	2	2	no	—
141	C₄×D₅₃	—	2	2	no	—
142	C₄×D₅₃	—	2	2	no	—
143	C₄×D₅₃	—	2	2	no	—
144	C₄×D₅₃	—	2	2	no	—
145	C₄×D₅₃	—	2	2	no	—
146	C₄×D₅₃	—	2	2	no	—
147	C₄×D₅₃	—	2	2	no	—
148	C₄×D₅₃	—	2	2	no	—
149	C₄×D₅₃	—	2	2	no	—
150	C₄×D₅₃	#108	1	4	yes	—
151	C₄×D₅₃	#135	1	4	yes	—
152	#88324	—	4	8	no	—
153	C₈×C₅₃	—	2	2	no	—
154	C₈×C₅₃	—	2	2	no	—
155	#88322	C₂×M₁₆	1	4	yes	—
156	#88322	C₂×M₁₆	1	4	yes	—
157	#17793	—	53	1	no	maximal, Sylow
158	C₂²×C₅₃	—	2	1	no	—
159	C₂²×C₅₃	—	2	1	no	—
160	C₂²×C₅₃	#423	1	2	yes	—
161	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
162	C₂×D₅₃	—	2	8	no	—
163	C₂×D₅₃	—	2	1	no	—
164	C₂×D₅₃	—	2	1	no	—
165	C₂×D₅₃	—	2	1	no	—
166	C₂×D₅₃	—	2	1	no	—
167	C₂×D₅₃	#427	1	2	yes	—
168	C₂×D₅₃	C₂×#108	1	2	yes	—
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	—	2	1	no	—
171	C₄×C₅₃	#423	1	2	yes	—
172	C₄×C₅₃	#427	1	2	yes	—
173	C₄×C₅₃	C₂×#108	1	2	yes	—
174	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
175	Dic₅₃	—	2	1	no	—
176	Dic₅₃	—	2	1	no	—
177	Dic₅₃	—	2	1	no	—
178	Dic₅₃	—	2	1	no	—
179	Dic₅₃	C₂×#108	1	2	yes	—
180	Dic₅₃	C₂×#135	1	2	yes	—
181	#2920	—	53	2	no	—
182	C₂²×C₄×C₈	—	53	1	no	—
183	#1568	—	53	4	no	—
184	#1567	—	53	2	no	—
185	#1566	—	53	2	no	—
186	#1564	—	53	4	no	—
187	C₂×C₅₃	—	2	2	no	—
188	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
189	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
190	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
191	D₅₃	—	2	2	no	—
192	D₅₃	—	2	2	no	—
193	C₂³×C₈	—	53	2	no	—
194	C₂²×C₄²	—	53	1	no	—
195	#432	—	53	4	no	—
196	#432	—	106	2	no	—
197	#431	—	53	4	no	—
198	#431	—	106	2	no	—
199	#423	—	53	4	no	—
200	#407	—	106	2	no	—
201	C₄×M₁₆	—	106	2	no	—
202	C₂×C₄×C₈	—	53	2	no	—
203	C₂×C₄×C₈	—	53	1	no	—
204	C₂×C₄×C₈	—	53	1	no	—
205	C₂×C₄×C₈	—	53	1	no	—
206	C₂×C₄×C₈	—	53	2	no	—
207	C₂×C₄×C₈	—	53	1	no	—
208	C₂×C₄×C₈	—	53	1	no	—
209	C₂×C₄×C₈	—	53	2	no	—
210	C₂×C₄×C₈	#6157	1	1	yes	characteristic
211	#335	—	53	8	no	—
212	#334	—	53	8	no	—
213	C₅₃	#17793	1	1	yes	characteristic, complemented, Sylow
214	C₂³×C₄	—	53	1	no	—
215	C₂³×C₄	—	53	1	no	—
216	C₂³×C₄	—	53	1	no	—
217	C₂×M₁₆	—	106	4	no	—
218	C₂²×C₈	—	53	2	no	—
219	C₂²×C₈	—	53	2	no	—
220	C₂²×C₈	—	53	2	no	—
221	C₂²×C₈	—	53	2	no	—
222	C₂²×C₈	—	53	2	no	—
223	C₂²×C₈	—	106	8	no	—
224	C₂²×C₈	C₂×#6157	1	2	yes	—
225	C₂×C₄²	—	53	1	no	—
226	C₂×C₄²	—	53	1	no	—
227	C₂×C₄²	—	53	1	no	—
228	C₂×C₄²	—	53	1	no	—
229	C₂×C₄²	—	53	1	no	—
230	C₂×C₄²	—	53	1	no	—
231	C₂×C₄²	—	53	1	no	—
232	C₂×C₄²	—	53	1	no	—
233	C₂×C₄²	—	53	1	no	—
234	C₂×C₄²	—	53	1	no	—
235	C₂×C₄²	—	53	1	no	—
236	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
237	C₄⋊C₈	—	53	8	no	—
238	C₄⋊C₈	—	106	4	no	—
239	C₂²⋊C₈	—	106	4	no	—
240	C₂²⋊C₈	—	106	4	no	—
241	#97	—	106	4	no	—
242	C₄×C₈	—	53	2	no	—
243	C₄×C₈	—	53	2	no	—
244	C₄×C₈	—	53	2	no	—
245	C₄×C₈	—	53	2	no	—
246	C₄×C₈	—	53	1	no	—
247	C₄×C₈	—	53	1	no	—
248	C₄×C₈	—	53	2	no	—
249	C₄×C₈	—	106	4	no	—
250	C₄×C₈	C₂×#6157	1	1	yes	characteristic
251	C₄×C₈	C₂×#6157	1	1	yes	characteristic
252	C₄×C₈	C₂×#6157	1	2	yes	—
253	C₂⁴	—	53	1	no	—
254	C₂²×C₄	—	53	1	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	1	no	—
260	C₂²×C₄	—	53	1	no	—
261	C₂²×C₄	—	53	1	no	—
262	C₂²×C₄	—	53	1	no	—
263	C₂²×C₄	—	53	1	no	—
264	C₂²×C₄	—	53	1	no	—
265	C₂²×C₄	—	53	1	no	—
266	C₂²×C₄	—	53	1	no	—
267	C₂²×C₄	—	53	1	no	—
268	C₂²×C₄	—	53	1	no	—
269	C₂²×C₄	—	106	4	no	—
270	C₂²×C₄	—	106	4	no	—
271	C₂²×C₄	—	106	4	no	—
272	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
273	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
274	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
275	M₁₆	—	212	8	no	—
276	C₂×C₈	—	2	4	no	—
277	C₂×C₈	—	53	8	no	—
278	C₂×C₈	—	53	2	no	—
279	C₂×C₈	—	53	2	no	—
280	C₂×C₈	—	106	4	no	—
281	C₂×C₈	—	106	4	no	—
282	C₂×C₈	—	106	4	no	—
283	C₂×C₈	—	106	4	no	—
284	C₂×C₈	—	106	8	no	—
285	C₂×C₈	C₂²×#6157	1	2	yes	—
286	C₂×C₈	C₂²×#6157	1	2	yes	—
287	C₄²	—	53	1	no	—
288	C₄²	—	53	1	no	—
289	C₄²	—	53	1	no	—
290	C₄²	—	53	1	no	—
291	C₄²	—	53	1	no	—
292	C₄²	—	53	1	no	—
293	C₄²	—	53	1	no	—
294	C₄²	—	53	1	no	—
295	C₄²	—	53	1	no	—
296	C₄²	—	53	1	no	—
297	C₄²	—	53	1	no	—
298	C₄²	—	53	1	no	—
299	C₄²	C₄⋊#6157	1	1	yes	characteristic
300	C₄²	C₄⋊#6157	1	1	yes	characteristic
301	C₄²	C₂²×#6157	1	1	yes	characteristic
302	C₄²	C₂²×#6157	1	1	yes	characteristic
303	C₂³	—	53	1	no	—
304	C₂³	—	53	1	no	—
305	C₂³	—	106	2	no	—
306	C₂³	—	106	2	no	—
307	C₂³	—	106	2	no	—
308	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	53	1	no	—
312	C₂×C₄	—	53	1	no	—
313	C₂×C₄	—	53	4	no	—
314	C₂×C₄	—	53	4	no	—
315	C₂×C₄	—	53	1	no	—
316	C₂×C₄	—	53	1	no	—
317	C₂×C₄	—	53	1	no	—
318	C₂×C₄	—	53	1	no	—
319	C₂×C₄	—	106	2	no	—
320	C₂×C₄	—	106	2	no	—
321	C₂×C₄	—	106	2	no	—
322	C₂×C₄	—	106	2	no	—
323	C₂×C₄	—	106	2	no	—
324	C₂×C₄	—	106	2	no	—
325	C₂×C₄	—	106	2	no	—
326	C₂×C₄	—	106	2	no	—
327	C₂×C₄	—	106	2	no	—
328	C₂×C₄	—	106	2	no	—
329	C₂×C₄	—	106	2	no	—
330	C₂×C₄	—	106	2	no	—
331	C₂×C₄	—	106	2	no	—
332	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
333	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
334	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
335	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
336	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
337	C₂×C₄	D₅₃⋊M₁₆	1	4	yes	—
338	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
339	C₈	—	2	2	no	—
340	C₈	—	2	2	no	—
341	C₈	—	53	4	no	—
342	C₈	—	53	4	no	—
343	C₈	—	212	8	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	53	2	no	—
347	C₂²	—	53	2	no	—
348	C₂²	—	106	8	no	—
349	C₂²	—	106	1	no	—
350	C₂²	—	106	1	no	—
351	C₂²	—	106	1	no	—
352	C₂²	—	106	1	no	—
353	C₂²	#16811628	1	2	yes	—
354	C₂²	C₂²×C₄⋊#6157	1	1	yes	characteristic
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	53	2	no	—
358	C₄	—	53	2	no	—
359	C₄	—	106	1	no	—
360	C₄	—	106	1	no	—
361	C₄	—	106	1	no	—
362	C₄	—	106	1	no	—
363	C₄	#16811628	1	2	yes	—
364	C₄	#16811633	1	2	yes	—
365	C₄	D₅₃⋊#108	1	2	yes	—
366	C₄	C₂×D₅₃⋊M₁₆	1	2	yes	—
367	C₂	—	2	2	no	—
368	C₂	—	106	2	no	—
369	C₂	—	106	2	no	—
370	C₂	C₂×#16811628	1	1	yes	characteristic
371	C₂	C₂×#16811633	1	1	yes	characteristic
372	C₂	C₂×D₅₃⋊#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₄⋊C₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#17793
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20290278	C₂	semidirect
3. #20290274	C₂	semidirect
4. #20290271	C₂	semidirect
5. #20290264	C₂	semidirect
14. #16811487	C₂²	semidirect
15. #16811485	C₂²	semidirect
109. C₂×#88322	M₁₆	semidirect
213. C₅₃	#17793	semidirect

Maximal subgroups

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

subgroup	index
#20291392	2
#20290278	2
#20290274	2
#20290271	2
#20290264	2
C₂×C₄×C₈×D₅₃	2
#17793	53