Subgroups — #27402614

Order 13568.

812 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283632	C₂	1	2	yes	maximal
2	#20283326	C₂	1	2	yes	maximal, complemented
3	#20275506	C₂	1	2	yes	maximal, complemented
4	#20274957	C₂	1	4	yes	maximal, complemented
5	#20274887	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274769	C₂	1	2	yes	maximal, complemented
7	#20274594	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#443	C₂	1	1	yes	maximal, characteristic
9	#16811390	C₂²	1	4	yes	—
10	#16810745	—	2	4	no	—
11	#16810740	—	2	4	no	—
12	#16810672	—	2	4	no	—
13	#16810672	C₂²	1	4	yes	—
14	C₂×#13133153	—	2	4	no	—
15	C₂×#13133153	C₂²	1	2	yes	—
16	C₂×#13133153	C₂²	1	2	yes	—
17	C₂×#13133153	C₂²	1	1	yes	characteristic
18	#16810661	C₂²	1	1	yes	characteristic
19	#16810647	C₂²	1	2	yes	—
20	#16810640	C₂²	1	4	yes	—
21	#16810578	C₂²	1	2	yes	—
22	#16810395	C₂²	1	4	yes	—
23	#16810383	C₂²	1	2	yes	—
24	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#131×C₅₃	C₂²	1	2	yes	—
26	C₅₃×#443	C₄	1	2	yes	complemented
27	C₅₃×#443	C₄	1	2	yes	complemented
28	C₅₃×#443	C₄	1	2	yes	complemented
29	C₅₃×#443	C₄	1	2	yes	complemented
30	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×#104×C₅₃	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
33	#13133224	—	2	8	no	—
34	#13133224	—	4	4	no	—
35	C₂×#11719917	—	2	8	no	—
36	C₂×#11719917	C₂³	1	4	yes	—
37	C₂×#11719917	C₂³	1	2	yes	—
38	#13133176	C₂³	1	2	yes	—
39	#13133153	—	2	2	no	—
40	#13133153	—	2	2	no	—
41	#13133153	—	2	2	no	—
42	#13133153	—	2	2	no	—
43	#13133153	—	2	8	no	—
44	#13133153	D₄	1	2	yes	—
45	#13133153	D₄	1	2	yes	—
46	#13133151	—	2	1	no	—
47	#13133149	—	2	1	no	—
48	C₄⋊#88322	—	4	2	no	—
49	#13133144	—	2	8	no	—
50	#13133127	—	2	2	no	—
51	C₈×Dic₅₃	—	2	2	no	—
52	C₂×#41×C₅₃	C₂³	1	2	yes	—
53	#135×C₅₃	C₂×C₄	1	2	yes	—
54	#135×C₅₃	C₂×C₄	1	2	yes	—
55	#135×C₅₃	D₄	1	2	yes	—
56	#135×C₅₃	D₄	1	2	yes	—
57	C₂×Q₁₆×C₅₃	Q₈	1	1	yes	characteristic
58	C₂×SD₁₆×C₅₃	Q₈	1	2	yes	—
59	C₂×D₈×C₅₃	Q₈	1	1	yes	characteristic
60	#131×C₅₃	—	2	2	no	—
61	#131×C₅₃	—	2	2	no	—
62	#131×C₅₃	C₂×C₄	1	4	yes	—
63	#131×C₅₃	C₂×C₄	1	4	yes	—
64	C₂×M₁₆×C₅₃	—	2	2	no	—
65	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
66	C₂²×C₈×C₅₃	—	2	2	no	—
67	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
70	#108×C₅₃	C₂×C₄	1	2	yes	—
71	#108×C₅₃	C₂×C₄	1	2	yes	—
72	#104×C₅₃	C₂×C₄	1	4	yes	—
73	#104×C₅₃	C₂×C₄	1	4	yes	—
74	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
75	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
78	#41×C₅₃	—	2	2	no	—
79	#41×C₅₃	—	2	2	no	—
80	#41×C₅₃	C₂²×C₄	1	4	yes	—
81	#41×C₅₃	C₂×D₄	1	4	yes	—
82	C₂×Q₈×C₅₃	C₂×Q₈	1	2	yes	—
83	C₂×D₄×C₅₃	—	2	2	no	—
84	C₂×D₄×C₅₃	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₅₃	—	2	2	no	—
86	C₂²×C₄×C₅₃	—	2	1	no	—
87	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
88	Q₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
89	Q₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
90	SD₁₆×C₅₃	C₄⋊C₄	1	4	yes	—
91	SD₁₆×C₅₃	C₄⋊C₄	1	4	yes	—
92	D₈×C₅₃	C₄⋊C₄	1	2	yes	—
93	D₈×C₅₃	C₄⋊C₄	1	2	yes	—
94	M₁₆×C₅₃	—	2	2	no	—
95	M₁₆×C₅₃	—	2	4	no	—
96	M₁₆×C₅₃	—	2	4	no	—
97	M₁₆×C₅₃	C₂²×C₄	1	4	yes	—
98	C₂×C₈×C₅₃	—	2	2	no	—
99	C₂×C₈×C₅₃	—	2	4	no	—
100	C₂×C₈×C₅₃	—	2	4	no	—
101	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
110	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
111	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
112	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
113	C₄²×C₅₃	—	2	1	no	—
114	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂²⋊Dic₅₃	—	2	2	no	—
117	C₄⋊Dic₅₃	—	2	2	no	—
118	C₄×Dic₅₃	—	2	2	no	—
119	#11719917	—	2	2	no	—
120	#11719917	—	2	4	no	—
121	#11719917	—	2	4	no	—
122	#11719917	—	4	8	no	—
123	#11719917	C₂×D₄	1	4	yes	—
124	C₂×#88322	—	2	2	no	—
125	C₂×#88322	—	2	4	no	—
126	C₂×#88322	—	4	8	no	—
127	C₂³×C₅₃	—	2	2	no	—
128	C₂³×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
129	Q₈×C₅₃	—	2	2	no	—
130	Q₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
131	D₄×C₅₃	—	2	4	no	—
132	D₄×C₅₃	—	2	4	no	—
133	D₄×C₅₃	—	2	2	no	—
134	D₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
135	C₂×C₄×C₅₃	—	2	1	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	4	no	—
138	C₂×C₄×C₅₃	—	2	4	no	—
139	C₂×C₄×C₅₃	—	2	1	no	—
140	C₂×C₄×C₅₃	—	4	2	no	—
141	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×Dic₅₃	—	4	2	no	—
148	C₈×C₅₃	—	2	4	no	—
149	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
154	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
155	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
156	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
157	#88322	—	2	4	no	—
158	#88322	—	4	4	no	—
159	#22694	—	53	1	no	maximal, Sylow
160	C₂²×C₅₃	—	2	4	no	—
161	C₂²×C₅₃	—	4	1	no	—
162	C₂²×C₅₃	—	8	2	no	—
163	C₂²×C₅₃	#520	1	1	yes	characteristic
164	C₂²×C₅₃	#547	1	1	yes	characteristic
165	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
166	C₄×C₅₃	—	2	2	no	—
167	C₄×C₅₃	—	2	4	no	—
168	C₄×C₅₃	—	4	1	no	—
169	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
170	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
171	C₄×C₅₃	D₄²	1	1	yes	characteristic
172	C₄×C₅₃	#554	1	1	yes	characteristic
173	Dic₅₃	—	8	2	no	—
174	#2957	—	53	2	no	—
175	C₂×#443	D₅₃	1	1	yes	characteristic
176	#2916	—	53	2	no	—
177	#1953	—	53	2	no	—
178	#1861	—	53	4	no	—
179	#1848	—	53	1	no	—
180	#1814	—	53	2	no	—
181	#1772	—	53	1	no	—
182	C₂×C₅₃	—	2	1	no	—
183	C₂×C₅₃	—	2	1	no	—
184	C₂×C₅₃	—	2	1	no	—
185	C₂×C₅₃	—	4	4	no	—
186	C₂×C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic
187	C₂×#135	C₂×D₅₃	1	1	yes	characteristic
188	#568	—	53	4	no	—
189	C₂×#131	C₂×D₅₃	1	2	yes	—
190	#444	—	106	4	no	—
191	#443	—	106	4	no	—
192	#443	Dic₅₃	1	2	yes	complemented
193	#443	Dic₅₃	1	2	yes	complemented
194	#443	Dic₅₃	1	2	yes	complemented
195	#443	Dic₅₃	1	2	yes	complemented
196	#430	—	53	4	no	—
197	#430	—	106	4	no	—
198	C₂×#108	—	53	1	no	—
199	C₂×#108	—	53	2	no	—
200	C₂×#108	—	53	2	no	—
201	C₂×#108	—	106	4	no	—
202	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
203	#427	—	53	1	no	—
204	#423	—	53	2	no	—
205	#421	—	53	4	no	—
206	C₂×#104	C₂×D₅₃	1	2	yes	—
207	#405	—	53	2	no	—
208	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
209	#344	—	53	4	no	—
210	#339	—	53	2	no	—
211	C₅₃	#22694	1	1	yes	characteristic, complemented, Sylow
212	C₂×#41	C₂²×D₅₃	1	2	yes	—
213	#135	D₂₁₂	1	2	yes	—
214	#135	D₂₁₂	1	2	yes	—
215	#135	C₂×Dic₅₃	1	2	yes	—
216	#135	C₂×Dic₅₃	1	2	yes	—
217	C₂×Q₁₆	Dic₁₀₆	1	1	yes	characteristic
218	C₂×SD₁₆	Dic₁₀₆	1	2	yes	—
219	C₂×D₈	Dic₁₀₆	1	1	yes	characteristic
220	#131	—	2	2	no	—
221	#131	—	2	2	no	—
222	#131	—	106	8	no	—
223	#131	—	212	4	no	—
224	#131	C₂×Dic₅₃	1	4	yes	—
225	#131	C₂×Dic₅₃	1	4	yes	—
226	C₂×M₁₆	—	2	2	no	—
227	C₂×M₁₆	—	53	2	no	—
228	C₂×M₁₆	—	53	4	no	—
229	C₂×M₁₆	—	106	8	no	—
230	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
231	C₂²×C₈	—	2	2	no	—
232	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
234	#117	—	53	2	no	—
235	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
236	#108	—	53	2	no	—
237	#108	—	53	2	no	—
238	#108	—	106	8	no	—
239	#108	—	106	2	no	—
240	#108	—	106	2	no	—
241	#108	—	106	2	no	—
242	#108	—	106	2	no	—
243	#108	C₂×Dic₅₃	1	2	yes	—
244	#108	C₂×Dic₅₃	1	2	yes	—
245	#107	—	106	1	no	—
246	#106	—	106	1	no	—
247	C₄⋊C₈	—	212	2	no	—
248	#104	—	106	8	no	—
249	#104	C₂×Dic₅₃	1	4	yes	—
250	#104	C₂×Dic₅₃	1	4	yes	—
251	#97	—	106	2	no	—
252	C₄×C₈	—	106	2	no	—
253	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
254	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
255	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
256	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
257	#41	—	2	2	no	—
258	#41	—	2	2	no	—
259	#41	C₂×D₂₁₂	1	4	yes	—
260	#41	C₂²×Dic₅₃	1	4	yes	—
261	C₂×Q₈	C₂×Dic₁₀₆	1	2	yes	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	C₂×Dic₁₀₆	1	2	yes	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	1	no	—
266	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
267	Q₁₆	C₄⋊Dic₅₃	1	2	yes	—
268	Q₁₆	C₄⋊Dic₅₃	1	2	yes	—
269	SD₁₆	C₄⋊Dic₅₃	1	4	yes	—
270	SD₁₆	C₄⋊Dic₅₃	1	4	yes	—
271	D₈	C₄⋊Dic₅₃	1	2	yes	—
272	D₈	C₄⋊Dic₅₃	1	2	yes	—
273	M₁₆	—	2	2	no	—
274	M₁₆	—	2	4	no	—
275	M₁₆	—	2	4	no	—
276	M₁₆	—	53	4	no	—
277	M₁₆	—	106	2	no	—
278	M₁₆	—	106	4	no	—
279	M₁₆	—	106	4	no	—
280	M₁₆	—	212	8	no	—
281	M₁₆	C₂²×Dic₅₃	1	4	yes	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	—	2	4	no	—
285	C₂×C₈	—	106	2	no	—
286	C₂×C₈	—	106	4	no	—
287	C₂×C₈	—	212	8	no	—
288	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
289	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
290	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
291	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
292	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
293	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
294	C₂×C₈	#11719947	1	1	yes	characteristic
295	C₂×C₈	#11719947	1	1	yes	characteristic
296	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
297	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
298	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
299	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
300	C₄⋊C₄	—	106	2	no	—
301	C₂²⋊C₄	—	106	2	no	—
302	C₄²	—	2	1	no	—
303	C₄²	—	106	2	no	—
304	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
305	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
306	C₂³	—	2	2	no	—
307	C₂³	#13133264	1	1	yes	characteristic
308	Q₈	—	2	2	no	—
309	Q₈	C₂×C₄⋊Dic₅₃	1	4	yes	—
310	D₄	—	2	4	no	—
311	D₄	—	2	4	no	—
312	D₄	—	2	2	no	—
313	D₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	212	2	no	—
321	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
322	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
324	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
325	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
326	C₂×C₄	#13133263	1	1	yes	characteristic
327	C₈	—	2	4	no	—
328	C₈	—	106	4	no	—
329	C₈	—	212	4	no	—
330	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
331	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
332	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
333	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
334	C₈	D₄×Dic₅₃	1	1	yes	characteristic
335	C₈	D₄×Dic₅₃	1	1	yes	characteristic
336	C₈	D₄×Dic₅₃	1	1	yes	characteristic
337	C₈	D₄×Dic₅₃	1	1	yes	characteristic
338	C₂²	—	2	4	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	8	2	no	—
341	C₂²	#16811106	1	1	yes	characteristic
342	C₂²	#16811277	1	1	yes	characteristic
343	C₂²	D₄×Dic₁₀₆	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	2	4	no	—
346	C₄	—	4	1	no	—
347	C₄	—	424	2	no	—
348	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
349	C₄	C₂×D₄×Dic₅₃	1	1	yes	characteristic
350	C₄	D₄×D₂₁₂	1	1	yes	characteristic
351	C₄	#16811330	1	1	yes	characteristic
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	—	4	4	no	—
356	C₂	D₄×C₄⋊Dic₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#443	C₂
socle	C₂×C₅₃	D₄×C₄⋊C₄
prime radical	C₂×#135×C₅₃	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#443	D₅₃
hypocenter	C₅₃	#22694
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
2. #20283326	C₂	semidirect
3. #20275506	C₂	semidirect
4. #20274957	C₂	semidirect
5. #20274887	C₂	semidirect
6. #20274769	C₂	semidirect
7. #20274594	C₂	semidirect
26. C₅₃×#443	C₄	semidirect
27. C₅₃×#443	C₄	semidirect
28. C₅₃×#443	C₄	semidirect
29. C₅₃×#443	C₄	semidirect
192. #443	Dic₅₃	semidirect
193. #443	Dic₅₃	semidirect
194. #443	Dic₅₃	semidirect
195. #443	Dic₅₃	semidirect
211. C₅₃	#22694	semidirect

Maximal subgroups

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

subgroup	index
#20283632	2
#20283326	2
#20275506	2
#20274957	2
#20274887	2
#20274769	2
#20274594	2
C₂×C₅₃×#443	2
#22694	53