Subgroups — #20283378

Order 6784.

392 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811091	C₂	1	1	yes	maximal, characteristic
2	#16810684	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810683	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810683	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810678	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810677	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810674	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810673	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810645	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#16810588	C₂	1	1	yes	maximal, characteristic
12	#16810588	C₂	1	1	yes	maximal, characteristic
13	#16810580	C₂	1	1	yes	maximal, characteristic, complemented
14	#16810577	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#433	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11719945	C₂²	1	1	yes	characteristic
17	C₂×#11719912	C₂²	1	1	yes	characteristic
18	C₂×#11719912	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
21	#13133182	C₄	1	1	yes	characteristic
22	#13133182	C₄	1	1	yes	characteristic
23	#13133179	C₄	1	1	yes	characteristic
24	#13133179	C₄	1	1	yes	characteristic
25	#13133177	C₄	1	1	yes	characteristic
26	#13133177	C₄	1	1	yes	characteristic
27	#13133177	C₄	1	1	yes	characteristic
28	#13133177	C₄	1	1	yes	characteristic
29	#13133174	C₂²	1	1	yes	characteristic
30	#13133174	C₂²	1	1	yes	characteristic
31	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
32	C₄²×D₅₃	C₂²	1	1	yes	characteristic
33	C₄²×D₅₃	C₂²	1	1	yes	characteristic
34	C₄⋊#88322	C₂²	1	1	yes	characteristic
35	C₄⋊#88322	C₂²	1	1	yes	characteristic
36	#13133146	C₂²	1	1	yes	characteristic
37	#13133146	C₂²	1	1	yes	characteristic
38	#13133146	C₂²	1	1	yes	characteristic
39	#13133146	C₂²	1	1	yes	characteristic
40	C₂²⋊#88322	C₂²	1	1	yes	characteristic
41	C₂²⋊#88322	C₂²	1	1	yes	characteristic
42	#13133129	C₂²	1	1	yes	characteristic
43	#13133129	C₂²	1	1	yes	characteristic
44	#13133129	C₂²	1	1	yes	characteristic
45	#13133129	C₂²	1	1	yes	characteristic
46	#13133128	C₂²	1	1	yes	characteristic
47	#13133127	C₂²	1	1	yes	characteristic
48	#13133127	C₂²	1	1	yes	characteristic
49	C₄×#88322	C₂²	1	1	yes	characteristic
50	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
51	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
52	#117×C₅₃	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
55	C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
57	#97×C₅₃	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
64	#11719945	—	2	2	no	—
65	#11719945	—	2	2	no	—
66	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
73	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
80	C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
81	C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₄²×C₅₃	C₂³	1	1	yes	characteristic
83	C₄²×C₅₃	C₂³	1	1	yes	characteristic
84	#11719921	C₂×C₄	1	1	yes	characteristic
85	#11719921	C₂×C₄	1	1	yes	characteristic
86	#11719921	C₂×C₄	1	1	yes	characteristic
87	#11719921	C₂×C₄	1	1	yes	characteristic
88	#11719919	C₂×C₄	1	1	yes	characteristic
89	#11719919	C₂×C₄	1	1	yes	characteristic
90	#11719919	C₂×C₄	1	1	yes	characteristic
91	#11719919	C₂×C₄	1	1	yes	characteristic
92	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
96	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
97	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
98	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
99	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
100	C₂×#88322	C₂³	1	1	yes	characteristic
101	C₂×#88322	C₂³	1	1	yes	characteristic
102	C₂×#88322	C₂³	1	1	yes	characteristic
103	C₂×#88322	C₂³	1	1	yes	characteristic
104	#11719912	—	2	2	no	—
105	#11719912	—	2	2	no	—
106	C₈×D₅₃	—	2	2	no	—
107	C₈×D₅₃	—	2	2	no	—
108	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	1	no	—
113	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₅₃	—	2	2	no	—
120	C₂²⋊D₅₃	—	2	2	no	—
121	C₂×Dic₅₃	—	2	2	no	—
122	C₂×Dic₅₃	—	4	1	no	—
123	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
125	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
129	D₂₁₂	—	2	2	no	—
130	C₄×D₅₃	—	2	1	no	—
131	C₄×D₅₃	—	2	1	no	—
132	C₄×D₅₃	—	2	2	no	—
133	C₄×D₅₃	—	2	2	no	—
134	C₄×D₅₃	—	4	1	no	—
135	C₄×D₅₃	—	4	1	no	—
136	C₄×D₅₃	#41	1	2	yes	—
137	C₄×D₅₃	#41	1	2	yes	—
138	Dic₁₀₆	—	2	2	no	—
139	C₈×C₅₃	—	2	1	no	—
140	C₈×C₅₃	—	2	1	no	—
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	—	2	1	no	—
143	#88322	—	2	1	no	—
144	#88322	—	2	1	no	—
145	#88322	—	2	1	no	—
146	#88322	—	2	1	no	—
147	C₂²×C₅₃	—	2	1	no	—
148	C₂²×C₅₃	—	2	1	no	—
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
151	C₂×D₅₃	—	2	1	no	—
152	C₂×D₅₃	—	2	1	no	—
153	C₂×D₅₃	—	2	1	no	—
154	C₂×D₅₃	—	2	1	no	—
155	C₂×D₅₃	—	2	1	no	—
156	C₂×D₅₃	#117	1	2	yes	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	—	2	1	no	—
161	C₄×C₅₃	—	2	1	no	—
162	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
163	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
164	Dic₅₃	—	2	1	no	—
165	Dic₅₃	—	2	1	no	—
166	Dic₅₃	—	2	1	no	—
167	Dic₅₃	—	2	1	no	—
168	Dic₅₃	#117	1	2	yes	—
169	Dic₅₃	#131	1	2	yes	—
170	#2925	—	53	1	no	maximal, Sylow
171	C₂×C₅₃	—	4	1	no	—
172	C₂×C₅₃	C₂×#117	1	1	yes	characteristic
173	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
174	C₂×C₅₃	#568	1	1	yes	characteristic
175	D₅₃	—	2	2	no	—
176	D₅₃	—	4	1	no	—
177	C₄×#41	—	53	1	no	—
178	#433	—	53	1	no	—
179	#433	—	53	1	no	—
180	#433	—	53	1	no	—
181	#433	D₅₃	1	1	yes	characteristic, complemented
182	#432	—	53	1	no	—
183	#432	—	53	1	no	—
184	#431	—	53	1	no	—
185	#431	—	53	1	no	—
186	#423	—	53	1	no	—
187	C₂×C₄⋊C₈	—	53	1	no	—
188	#408	—	53	1	no	—
189	#408	—	53	1	no	—
190	#405	—	53	1	no	—
191	#405	—	53	1	no	—
192	C₅₃	#2925	1	1	yes	characteristic, complemented, Sylow
193	C₂×#41	—	53	1	no	—
194	C₂×M₁₆	—	53	1	no	—
195	C₂×M₁₆	—	53	1	no	—
196	C₂²×C₈	—	53	1	no	—
197	C₂²×C₈	—	53	1	no	—
198	C₄×Q₈	—	53	1	no	—
199	C₄×Q₈	—	53	1	no	—
200	C₄×D₄	—	53	1	no	—
201	C₄×D₄	—	53	1	no	—
202	C₄×D₄	—	53	1	no	—
203	C₄×D₄	—	53	1	no	—
204	C₄×D₄	—	53	1	no	—
205	C₄×D₄	—	53	1	no	—
206	#117	—	53	1	no	—
207	#117	—	53	1	no	—
208	#117	C₂×D₅₃	1	1	yes	characteristic
209	C₂×C₄²	—	53	1	no	—
210	C₂×C₄²	—	53	1	no	—
211	C₂×C₄²	—	53	1	no	—
212	C₄⋊C₈	—	53	1	no	—
213	C₄⋊C₈	—	53	1	no	—
214	C₄⋊C₈	—	53	1	no	—
215	C₄⋊C₈	—	53	1	no	—
216	C₄⋊C₈	—	53	1	no	—
217	C₄⋊C₈	—	53	1	no	—
218	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
219	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
220	C₂²⋊C₈	—	53	1	no	—
221	C₂²⋊C₈	—	53	1	no	—
222	C₂²⋊C₈	—	53	1	no	—
223	C₂²⋊C₈	—	53	1	no	—
224	C₂²⋊C₈	—	53	1	no	—
225	C₂²⋊C₈	—	53	1	no	—
226	C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
227	C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
228	#97	—	53	1	no	—
229	#97	—	53	1	no	—
230	#97	—	53	1	no	—
231	#97	C₂×D₅₃	1	1	yes	characteristic
232	C₄×C₈	—	53	1	no	—
233	C₄×C₈	—	53	1	no	—
234	C₄×C₈	—	53	1	no	—
235	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
236	#41	—	106	2	no	—
237	#41	—	106	2	no	—
238	C₂×Q₈	—	53	1	no	—
239	C₂×D₄	—	53	1	no	—
240	C₂×D₄	—	53	1	no	—
241	C₂×D₄	—	53	1	no	—
242	C₂²×C₄	—	53	1	no	—
243	C₂²×C₄	—	53	1	no	—
244	C₂²×C₄	—	53	1	no	—
245	C₂²×C₄	—	53	1	no	—
246	C₂²×C₄	—	53	1	no	—
247	C₂²×C₄	—	53	1	no	—
248	C₂²×C₄	—	53	1	no	—
249	C₂²×C₄	—	53	1	no	—
250	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
251	M₁₆	—	106	2	no	—
252	M₁₆	—	106	2	no	—
253	C₂×C₈	—	53	1	no	—
254	C₂×C₈	—	53	1	no	—
255	C₂×C₈	—	53	1	no	—
256	C₂×C₈	—	53	1	no	—
257	C₂×C₈	—	106	2	no	—
258	C₂×C₈	—	106	2	no	—
259	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
260	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
263	C₄⋊C₄	—	53	1	no	—
264	C₄⋊C₄	—	53	1	no	—
265	C₄⋊C₄	—	53	1	no	—
266	C₄⋊C₄	—	53	1	no	—
267	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
268	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
269	C₂²⋊C₄	—	53	1	no	—
270	C₂²⋊C₄	—	53	1	no	—
271	C₂²⋊C₄	—	53	1	no	—
272	C₂²⋊C₄	—	53	1	no	—
273	C₂²⋊C₄	C₄×D₅₃	1	1	yes	characteristic
274	C₂²⋊C₄	C₄×D₅₃	1	1	yes	characteristic
275	C₄²	—	53	1	no	—
276	C₄²	—	53	1	no	—
277	C₄²	—	53	1	no	—
278	C₄²	—	53	1	no	—
279	C₄²	—	53	1	no	—
280	C₄²	—	53	1	no	—
281	C₄²	—	53	1	no	—
282	C₄²	—	53	1	no	—
283	C₄²	C₂²×D₅₃	1	1	yes	characteristic
284	C₄²	C₂²×D₅₃	1	1	yes	characteristic
285	C₂³	—	53	1	no	—
286	C₂³	—	53	1	no	—
287	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
288	Q₈	—	106	2	no	—
289	D₄	—	106	2	no	—
290	D₄	—	106	2	no	—
291	D₄	—	106	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	53	1	no	—
295	C₂×C₄	—	53	1	no	—
296	C₂×C₄	—	53	2	no	—
297	C₂×C₄	—	53	2	no	—
298	C₂×C₄	—	53	1	no	—
299	C₂×C₄	—	53	1	no	—
300	C₂×C₄	—	53	1	no	—
301	C₂×C₄	—	53	1	no	—
302	C₂×C₄	—	106	2	no	—
303	C₂×C₄	—	106	2	no	—
304	C₂×C₄	—	106	2	no	—
305	C₂×C₄	—	106	1	no	—
306	C₂×C₄	—	106	1	no	—
307	C₂×C₄	—	212	1	no	—
308	C₂×C₄	—	212	1	no	—
309	C₂×C₄	—	212	1	no	—
310	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
311	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
312	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
313	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	106	1	no	—
321	C₈	—	106	1	no	—
322	C₈	—	106	1	no	—
323	C₈	—	106	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	53	2	no	—
328	C₂²	—	106	1	no	—
329	C₂²	—	106	1	no	—
330	C₂²	—	106	1	no	—
331	C₂²	—	106	1	no	—
332	C₂²	—	106	1	no	—
333	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	53	2	no	—
340	C₄	—	53	2	no	—
341	C₄	—	106	1	no	—
342	C₄	—	106	1	no	—
343	C₄	—	106	1	no	—
344	C₄	—	106	1	no	—
345	C₄	#41×D₅₃	1	1	yes	characteristic
346	C₄	#41×D₅₃	1	1	yes	characteristic
347	C₂	—	4	1	no	—
348	C₂	—	106	2	no	—
349	C₂	—	212	1	no	—
350	C₂	#117×D₅₃	1	1	yes	characteristic
351	C₂	#131×D₅₃	1	1	yes	characteristic
352	C₂	#16811388	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#433	C₂
hypercenter	#433	D₅₃
hypocenter	C₅₃	#2925
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. #16810684	C₂	semidirect
3. #16810683	C₂	semidirect
4. #16810683	C₂	semidirect
5. #16810678	C₂	semidirect
6. #16810677	C₂	semidirect
7. #16810674	C₂	semidirect
8. #16810673	C₂	semidirect
9. #16810645	C₂	semidirect
10. C₄⋊C₈×D₅₃	C₂	semidirect
13. #16810580	C₂	semidirect
14. #16810577	C₂	semidirect
15. C₅₃×#433	C₂	semidirect
181. #433	D₅₃	semidirect
192. C₅₃	#2925	semidirect