Subgroups — #27353219

Order 13568.

602 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283832	C₂	1	1	yes	maximal, characteristic
2	#20283826	C₂	1	1	yes	maximal, characteristic
3	#20273504	C₂	1	2	yes	maximal, complemented
4	#20273500	C₂	1	2	yes	maximal, complemented
5	#20273495	C₂	1	2	yes	maximal, complemented
6	#20273486	C₂	1	2	yes	maximal, complemented
7	#20273467	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273467	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273463	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16810374	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2922	C₂	1	1	yes	maximal, characteristic
12	#16810754	—	2	1	no	—
13	Q₈×#88322	—	2	1	no	—
14	#16810701	—	2	1	no	—
15	#16810695	—	2	2	no	—
16	D₄×#88322	—	2	2	no	—
17	D₄×#88322	—	2	1	no	—
18	#16810680	C₂²	1	2	yes	—
19	#16810676	C₂²	1	1	yes	characteristic
20	#16810676	C₂²	1	1	yes	characteristic
21	#16810676	C₂²	1	2	yes	—
22	#16810647	C₂²	1	2	yes	—
23	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊#88322	C₂²	1	2	yes	—
26	#16810376	C₂²	1	1	yes	characteristic
27	#16810376	C₂²	1	1	yes	characteristic
28	#16810376	C₂²	1	2	yes	—
29	#16810376	C₂²	1	2	yes	—
30	#16810374	C₂²	1	2	yes	—
31	#16810374	C₂²	1	2	yes	—
32	#16810374	C₂²	1	1	yes	characteristic
33	#16810374	C₂²	1	1	yes	characteristic
34	#16810374	C₂²	1	2	yes	—
35	#16810374	C₂²	1	2	yes	—
36	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#433	C₄	1	4	yes	—
38	C₅₃×#431	C₂²	1	2	yes	—
39	C₅₃×#431	C₂²	1	2	yes	—
40	C₅₃×#424	C₄	1	2	yes	—
41	C₅₃×#408	C₄	1	2	yes	—
42	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
43	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
44	C₂×#11719917	—	2	2	no	—
45	C₂²×#88322	—	2	1	no	—
46	C₂²×#88322	—	2	2	no	—
47	C₂²×#88322	—	2	1	no	—
48	C₄⋊#88322	—	2	1	no	—
49	C₄⋊#88322	—	2	2	no	—
50	C₄⋊#88322	—	2	1	no	—
51	C₄⋊#88322	—	2	2	no	—
52	C₄⋊#88322	C₂³	1	2	yes	—
53	C₄⋊#88322	C₂³	1	1	yes	characteristic
54	C₄⋊#88322	C₂³	1	2	yes	—
55	C₄⋊#88322	C₂³	1	1	yes	characteristic
56	C₄⋊#88322	C₂³	1	1	yes	characteristic
57	C₄⋊#88322	C₂³	1	1	yes	characteristic
58	C₂²⋊#88322	—	2	1	no	—
59	C₂²⋊#88322	—	2	2	no	—
60	C₂²⋊#88322	—	2	1	no	—
61	C₂²⋊#88322	—	2	2	no	—
62	#13133128	—	2	2	no	—
63	C₄×#88322	—	2	1	no	—
64	C₄×#88322	—	2	1	no	—
65	C₄×#88322	—	2	2	no	—
66	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
68	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
69	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
70	C₄×Q₈×C₅₃	Q₈	1	1	yes	characteristic
71	C₄×D₄×C₅₃	D₄	1	2	yes	—
72	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
73	C₄×D₄×C₅₃	Q₈	1	1	yes	characteristic
74	C₄×D₄×C₅₃	Q₈	1	2	yes	—
75	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
76	#117×C₅₃	C₂×C₄	1	2	yes	—
77	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
79	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
80	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
81	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
82	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
83	#97×C₅₃	C₂×C₄	1	2	yes	—
84	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
85	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₅₃	C₂³	1	2	yes	—
88	#41×C₅₃	—	2	4	no	—
89	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
90	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
91	C₂×D₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
97	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
98	M₁₆×C₅₃	—	2	4	no	—
99	C₂×C₈×C₅₃	—	2	4	no	—
100	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
103	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
104	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
105	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
106	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
107	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
108	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
109	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
110	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
112	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂×D₄	1	2	yes	—
114	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
115	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
116	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
117	#11719917	—	8	2	no	—
118	C₂×#88322	—	2	1	no	—
119	C₂×#88322	—	2	1	no	—
120	C₂×#88322	—	2	1	no	—
121	C₂×#88322	—	2	2	no	—
122	C₂×#88322	—	2	1	no	—
123	C₂×#88322	—	2	2	no	—
124	C₂×#88322	—	8	1	no	—
125	C₂×#88322	—	8	2	no	—
126	C₂×#88322	—	8	1	no	—
127	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	Q₈×C₅₃	—	2	2	no	—
130	D₄×C₅₃	—	2	4	no	—
131	D₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	4	no	—
133	C₂×C₄×C₅₃	—	2	1	no	—
134	C₂×C₄×C₅₃	—	2	1	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	4	1	no	—
138	C₂×C₄×C₅₃	—	4	2	no	—
139	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
144	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	#131	1	4	yes	—
146	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
149	C₈×C₅₃	—	2	2	no	—
150	C₈×C₅₃	—	2	2	no	—
151	C₈×C₅₃	—	2	4	no	—
152	#88322	—	4	1	no	—
153	#88322	—	4	1	no	—
154	#88322	—	4	1	no	—
155	#88322	—	4	2	no	—
156	#88322	—	4	2	no	—
157	#88322	—	4	1	no	—
158	#17811	—	53	1	no	maximal, Sylow
159	C₂²×C₅₃	—	2	1	no	—
160	C₂²×C₅₃	—	2	1	no	—
161	C₂²×C₅₃	—	2	2	no	—
162	C₂²×C₅₃	—	2	2	no	—
163	C₂²×C₅₃	—	2	2	no	—
164	C₂²×C₅₃	#424	1	2	yes	—
165	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
166	C₄×C₅₃	—	2	2	no	—
167	C₄×C₅₃	—	2	1	no	—
168	C₄×C₅₃	—	2	2	no	—
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	—	2	1	no	—
171	C₄×C₅₃	—	2	2	no	—
172	C₄×C₅₃	#424	1	2	yes	—
173	C₄×C₅₃	C₂×#131	1	2	yes	—
174	C₄×C₅₃	#577	1	1	yes	characteristic
175	C₄×C₅₃	#577	1	1	yes	characteristic
176	#2989	—	53	1	no	—
177	#2988	—	53	1	no	—
178	#2922	D₅₃	1	1	yes	characteristic
179	#1574	—	53	2	no	—
180	#1573	—	53	2	no	—
181	#1572	—	53	2	no	—
182	#1570	—	53	2	no	—
183	#1565	—	53	1	no	—
184	#1565	—	53	1	no	—
185	#1564	—	53	1	no	—
186	C₂×#334	—	53	1	no	—
187	C₂×C₅₃	—	2	2	no	—
188	C₂×C₅₃	—	4	2	no	—
189	C₂×C₅₃	C₂×#424	1	1	yes	characteristic
190	C₂×C₅₃	#2912	1	1	yes	characteristic
191	C₂×C₅₃	#2916	1	1	yes	characteristic
192	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
193	#446	—	106	1	no	—
194	C₈×Q₈	—	106	1	no	—
195	#436	—	106	1	no	—
196	#435	—	106	2	no	—
197	C₈×D₄	—	106	2	no	—
198	C₈×D₄	—	106	1	no	—
199	#433	Dic₅₃	1	4	yes	—
200	#432	—	53	2	no	—
201	#431	—	53	1	no	—
202	#431	—	53	1	no	—
203	#431	—	53	2	no	—
204	#431	C₂×D₅₃	1	2	yes	—
205	#431	C₂×D₅₃	1	2	yes	—
206	#424	Dic₅₃	1	2	yes	—
207	#423	—	53	2	no	—
208	C₂×C₄⋊C₈	—	53	1	no	—
209	C₂×C₄⋊C₈	—	53	1	no	—
210	C₂×C₄⋊C₈	—	53	2	no	—
211	#408	Dic₅₃	1	2	yes	—
212	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
213	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
214	#335	—	53	1	no	—
215	#335	—	53	1	no	—
216	#335	—	53	2	no	—
217	#335	—	53	2	no	—
218	#334	—	53	2	no	—
219	#334	—	53	2	no	—
220	#334	—	53	1	no	—
221	#334	—	53	1	no	—
222	#334	—	53	2	no	—
223	#334	—	53	2	no	—
224	C₅₃	#17811	1	1	yes	characteristic, complemented, Sylow
225	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
226	C₂×M₁₆	—	106	2	no	—
227	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
228	C₂²×C₈	—	106	1	no	—
229	C₂²×C₈	—	106	2	no	—
230	C₂²×C₈	—	106	1	no	—
231	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
232	C₄×Q₈	Dic₁₀₆	1	1	yes	characteristic
233	C₄×Q₈	D₂₁₂	1	1	yes	characteristic
234	C₄×D₄	Dic₁₀₆	1	2	yes	—
235	C₄×D₄	Dic₁₀₆	1	1	yes	characteristic
236	C₄×D₄	D₂₁₂	1	2	yes	—
237	C₄×D₄	D₂₁₂	1	1	yes	characteristic
238	#117	C₂×Dic₅₃	1	1	yes	characteristic
239	#117	C₂×Dic₅₃	1	2	yes	—
240	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
241	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
242	C₄⋊C₈	—	53	2	no	—
243	C₄⋊C₈	—	53	1	no	—
244	C₄⋊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	—	106	1	no	—
249	C₄⋊C₈	—	106	2	no	—
250	C₄⋊C₈	—	106	1	no	—
251	C₄⋊C₈	—	106	2	no	—
252	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
253	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
254	C₂²⋊C₈	—	106	1	no	—
255	C₂²⋊C₈	—	106	2	no	—
256	C₂²⋊C₈	—	106	1	no	—
257	C₂²⋊C₈	—	106	2	no	—
258	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
259	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
260	#97	—	106	2	no	—
261	#97	C₂×Dic₅₃	1	2	yes	—
262	C₄×C₈	—	106	1	no	—
263	C₄×C₈	—	106	2	no	—
264	C₄×C₈	—	106	1	no	—
265	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
266	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
268	C₄×C₈	C₂²×D₅₃	1	2	yes	—
269	#41	—	2	4	no	—
270	C₂×Q₈	C₄⋊Dic₅₃	1	1	yes	characteristic
271	C₂×D₄	C₄⋊Dic₅₃	1	2	yes	—
272	C₂×D₄	C₄⋊Dic₅₃	1	1	yes	characteristic
273	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
274	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
277	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
278	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
279	M₁₆	—	2	4	no	—
280	M₁₆	—	424	2	no	—
281	C₂×C₈	—	2	4	no	—
282	C₂×C₈	—	106	1	no	—
283	C₂×C₈	—	106	1	no	—
284	C₂×C₈	—	106	1	no	—
285	C₂×C₈	—	106	2	no	—
286	C₂×C₈	—	106	1	no	—
287	C₂×C₈	—	106	2	no	—
288	C₂×C₈	—	424	1	no	—
289	C₂×C₈	—	424	2	no	—
290	C₂×C₈	—	424	1	no	—
291	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
292	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
293	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
294	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
295	C₄⋊C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
296	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
297	C₄⋊C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
298	C₂²⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
299	C₂²⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
300	C₂²⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
301	C₄²	C₂×Dic₁₀₆	1	2	yes	—
302	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
303	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
304	C₄²	C₂×D₂₁₂	1	2	yes	—
305	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
306	C₄²	C₂²×Dic₅₃	1	2	yes	—
307	C₄²	C₂³×D₅₃	1	1	yes	characteristic
308	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
309	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
310	Q₈	—	2	2	no	—
311	D₄	—	2	4	no	—
312	D₄	—	2	2	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
321	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
322	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
323	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
324	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
325	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
326	C₂×C₄	#13133224	1	4	yes	—
327	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
328	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
329	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
330	C₈	—	2	2	no	—
331	C₈	—	2	2	no	—
332	C₈	—	2	4	no	—
333	C₈	—	212	1	no	—
334	C₈	—	212	1	no	—
335	C₈	—	212	1	no	—
336	C₈	—	212	2	no	—
337	C₈	—	212	2	no	—
338	C₈	—	212	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	#16810650	1	2	yes	—
345	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
346	C₄	—	2	2	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	2	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	2	no	—
352	C₄	#16810650	1	2	yes	—
353	C₄	C₂×#13133224	1	2	yes	—
354	C₄	#16811435	1	1	yes	characteristic
355	C₄	#16811435	1	1	yes	characteristic
356	C₂	—	2	2	no	—
357	C₂	—	4	2	no	—
358	C₂	C₂×#16810650	1	1	yes	characteristic
359	C₂	#20283310	1	1	yes	characteristic
360	C₂	#20283326	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#2922	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#2922	D₅₃
hypocenter	C₅₃	#17811
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ #2922

Complements

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

N	H ≅ G/N	type
3. #20273504	C₂	semidirect
4. #20273500	C₂	semidirect
5. #20273495	C₂	semidirect
6. #20273486	C₂	semidirect
7. #20273467	C₂	semidirect
8. #20273467	C₂	semidirect
9. #20273463	C₂	semidirect
10. C₂×#16810374	C₂	semidirect
224. C₅₃	#17811	semidirect

Maximal subgroups

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

subgroup	index
#20283832	2
#20283826	2
#20273504	2
#20273500	2
#20273495	2
#20273486	2
#20273467	2
#20273463	2
C₂×#16810374	2
C₅₃×#2922	2
#17811	53