Subgroups — #27429217

Order 13568.

656 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810893	C₂	1	1	yes	maximal, characteristic
2	#20283568	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276178	C₂	1	2	yes	maximal, complemented
4	#20275988	C₂	1	2	yes	maximal, complemented
5	#20275625	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275505	C₂	1	1	yes	maximal, characteristic, complemented
7	#20275205	C₂	1	2	yes	maximal, complemented
8	#20274839	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274825	C₂	1	2	yes	maximal, complemented
10	#20274769	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1757	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133242	C₂²	1	2	yes	—
13	C₂²×D₄₂₄	C₂²	1	1	yes	characteristic
14	#16811097	C₂²	1	2	yes	—
15	#16810893	—	2	2	no	—
16	#16810893	—	2	2	no	—
17	#16810736	—	2	2	no	—
18	C₄×D₄₂₄	—	2	2	no	—
19	#16810671	C₂²	1	2	yes	—
20	C₂×#13133153	C₂²	1	1	yes	characteristic
21	C₂×#13133153	C₂²	1	1	yes	characteristic
22	#16810661	C₂²	1	1	yes	characteristic
23	#16810640	C₂²	1	2	yes	—
24	#16810629	C₂²	1	2	yes	—
25	#16810627	C₂²	1	2	yes	—
26	C₂×#13133133	C₂²	1	2	yes	—
27	C₂×#13133133	C₂²	1	2	yes	—
28	#16810395	C₂²	1	2	yes	—
29	#16810394	C₂²	1	2	yes	—
30	#16810393	C₂²	1	2	yes	—
31	#16810379	C₂²	1	2	yes	—
32	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#413	C₂²	1	1	yes	characteristic
34	C₅₃×#410	C₂²	1	1	yes	characteristic
35	C₅₃×#405	C₂²	1	1	yes	characteristic
36	C₅₃×#405	C₂²	1	1	yes	characteristic
37	C₅₃×#344	C₂²	1	1	yes	characteristic
38	C₅₃×#344	C₂²	1	1	yes	characteristic
39	C₂²×D₂₁₂	C₂³	1	2	yes	—
40	#13133242	—	2	4	no	—
41	#13133242	—	4	2	no	—
42	C₂×#11719924	—	2	2	no	—
43	C₂×D₄₂₄	—	2	1	no	—
44	C₂×D₄₂₄	—	2	1	no	—
45	C₂×D₄₂₄	—	2	1	no	—
46	C₂×D₄₂₄	—	2	1	no	—
47	C₂×D₄₂₄	D₄	1	2	yes	—
48	C₂×D₄₂₄	D₄	1	2	yes	—
49	C₂×#11719922	—	2	2	no	—
50	C₂×#11719917	C₂³	1	2	yes	—
51	C₂×#11719917	C₂³	1	2	yes	—
52	C₄×D₂₁₂	—	2	4	no	—
53	#13133179	—	4	2	no	—
54	#13133176	C₂³	1	2	yes	—
55	C₂²⋊C₄×D₅₃	—	2	4	no	—
56	#13133153	—	2	1	no	—
57	#13133153	—	2	1	no	—
58	#13133153	—	2	1	no	—
59	#13133153	—	2	1	no	—
60	#13133152	—	4	2	no	—
61	#13133151	—	2	1	no	—
62	#13133149	—	2	1	no	—
63	#13133144	—	4	2	no	—
64	#13133139	—	2	4	no	—
65	#13133137	—	4	2	no	—
66	#13133133	—	2	4	no	—
67	#13133133	—	4	2	no	—
68	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
69	#131×C₅₃	—	2	1	no	—
70	#131×C₅₃	—	2	1	no	—
71	#131×C₅₃	—	2	1	no	—
72	#131×C₅₃	—	2	1	no	—
73	C₂×M₁₆×C₅₃	—	2	1	no	—
74	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
75	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
76	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₅₃	—	2	1	no	—
78	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	#117×C₅₃	C₂³	1	1	yes	characteristic
80	#117×C₅₃	C₂³	1	1	yes	characteristic
81	#101×C₅₃	—	2	1	no	—
82	#100×C₅₃	—	2	1	no	—
83	#99×C₅₃	—	2	2	no	—
84	#97×C₅₃	—	2	1	no	—
85	#97×C₅₃	—	2	1	no	—
86	C₄×C₈×C₅₃	—	2	1	no	—
87	C₄×C₈×C₅₃	—	2	1	no	—
88	C₂³×D₅₃	—	2	2	no	—
89	#41×C₅₃	—	2	1	no	—
90	#41×C₅₃	—	2	1	no	—
91	#41×C₅₃	—	2	1	no	—
92	#41×C₅₃	—	2	1	no	—
93	C₂×Q₈×C₅₃	#41	1	1	yes	characteristic
94	C₂×D₄×C₅₃	—	2	1	no	—
95	C₂×D₄×C₅₃	#41	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	—	2	1	no	—
97	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
98	C₂×D₂₁₂	—	2	2	no	—
99	C₂×D₂₁₂	—	2	2	no	—
100	C₂×D₂₁₂	—	2	2	no	—
101	C₂×D₂₁₂	—	2	2	no	—
102	C₂×D₂₁₂	—	2	2	no	—
103	C₂×D₂₁₂	C₂×D₄	1	4	yes	—
104	C₂×C₄×D₅₃	—	4	4	no	—
105	M₁₆×C₅₃	—	2	1	no	—
106	M₁₆×C₅₃	—	2	1	no	—
107	M₁₆×C₅₃	—	2	1	no	—
108	M₁₆×C₅₃	—	2	1	no	—
109	M₁₆×C₅₃	—	2	1	no	—
110	M₁₆×C₅₃	—	2	1	no	—
111	M₁₆×C₅₃	—	4	1	no	—
112	M₁₆×C₅₃	—	4	1	no	—
113	C₂×C₈×C₅₃	—	2	1	no	—
114	C₂×C₈×C₅₃	—	2	1	no	—
115	C₂×C₈×C₅₃	—	2	1	no	—
116	C₂×C₈×C₅₃	—	4	1	no	—
117	C₂×C₈×C₅₃	—	4	1	no	—
118	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
121	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
122	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
123	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
124	C₄⋊C₄×C₅₃	—	2	1	no	—
125	C₄⋊C₄×C₅₃	—	2	1	no	—
126	C₂²⋊C₄×C₅₃	—	2	1	no	—
127	C₂²⋊C₄×C₅₃	#41	1	2	yes	—
128	C₄²×C₅₃	—	2	1	no	—
129	C₄²×C₅₃	—	2	1	no	—
130	C₂²⋊Dic₅₃	—	2	2	no	—
131	#11719924	—	4	4	no	—
132	#11719922	—	4	4	no	—
133	#11719921	—	4	4	no	—
134	C₄⋊Dic₅₃	—	2	2	no	—
135	C₄×Dic₅₃	—	2	2	no	—
136	#11719917	—	2	2	no	—
137	#11719917	—	2	2	no	—
138	#11719917	—	2	2	no	—
139	#11719917	—	2	2	no	—
140	C₂×#88322	—	2	2	no	—
141	C₂×#88322	—	2	2	no	—
142	D₄₂₄	—	2	2	no	—
143	D₄₂₄	—	2	2	no	—
144	D₄₂₄	—	2	2	no	—
145	D₄₂₄	—	2	2	no	—
146	C₂³×C₅₃	—	2	1	no	—
147	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
148	C₂²×D₅₃	—	2	4	no	—
149	C₂²×D₅₃	—	4	2	no	—
150	C₂²×D₅₃	—	4	2	no	—
151	C₂²×D₅₃	—	16	2	no	—
152	Q₈×C₅₃	—	2	1	no	—
153	Q₈×C₅₃	—	2	1	no	—
154	D₄×C₅₃	—	2	1	no	—
155	D₄×C₅₃	—	2	1	no	—
156	D₄×C₅₃	—	4	1	no	—
157	D₄×C₅₃	—	4	1	no	—
158	C₂×C₄×C₅₃	—	2	1	no	—
159	C₂×C₄×C₅₃	—	2	2	no	—
160	C₂×C₄×C₅₃	—	4	1	no	—
161	C₂×C₄×C₅₃	—	4	1	no	—
162	C₂×C₄×C₅₃	—	4	1	no	—
163	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
167	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
168	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂×Dic₅₃	—	4	2	no	—
170	D₂₁₂	—	2	4	no	—
171	D₂₁₂	—	2	4	no	—
172	D₂₁₂	—	4	2	no	—
173	D₂₁₂	—	4	2	no	—
174	C₄×D₅₃	—	16	4	no	—
175	C₈×C₅₃	—	4	1	no	—
176	C₈×C₅₃	—	4	1	no	—
177	C₈×C₅₃	—	4	1	no	—
178	C₈×C₅₃	#123	1	1	yes	characteristic
179	C₈×C₅₃	#123	1	1	yes	characteristic
180	C₈×C₅₃	#123	1	1	yes	characteristic
181	C₈×C₅₃	#123	1	1	yes	characteristic
182	#88322	—	4	2	no	—
183	#88322	—	4	2	no	—
184	#24988	—	53	1	no	maximal, Sylow
185	C₂²×C₅₃	—	4	1	no	—
186	C₂²×C₅₃	—	4	1	no	—
187	C₂²×C₅₃	—	8	1	no	—
188	C₂²×C₅₃	#529	1	1	yes	characteristic
189	C₂²×C₅₃	#546	1	1	yes	characteristic
190	C₂²×C₅₃	#555	1	1	yes	characteristic
191	C₂×D₅₃	—	4	4	no	—
192	C₂×D₅₃	—	8	4	no	—
193	C₂×D₅₃	—	16	2	no	—
194	C₂×D₅₃	—	16	2	no	—
195	C₄×C₅₃	—	4	2	no	—
196	C₄×C₅₃	—	4	1	no	—
197	C₄×C₅₃	—	8	1	no	—
198	C₄×C₅₃	C₂×#123	1	1	yes	characteristic
199	C₄×C₅₃	#540	1	1	yes	characteristic
200	C₄×C₅₃	#546	1	1	yes	characteristic
201	C₄×C₅₃	#553	1	1	yes	characteristic
202	Dic₅₃	—	8	2	no	—
203	C₂×#472	—	53	1	no	—
204	#2948	—	53	1	no	—
205	#2081	—	53	2	no	—
206	#2048	—	53	2	no	—
207	#1978	—	53	1	no	—
208	#1953	—	53	1	no	—
209	#1904	—	53	2	no	—
210	#1833	—	53	1	no	—
211	#1829	—	53	2	no	—
212	#1814	—	53	1	no	—
213	#1757	D₅₃	1	1	yes	characteristic, complemented
214	C₂×C₅₃	—	2	1	no	—
215	C₂×C₅₃	—	2	1	no	—
216	C₂×C₅₃	—	2	1	no	—
217	C₂×C₅₃	—	8	1	no	—
218	C₂×C₅₃	#2420	1	1	yes	characteristic
219	D₅₃	—	8	4	no	—
220	C₂×AGL(1,ℤ/8)	—	53	2	no	—
221	C₂²×D₈	—	53	1	no	—
222	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
223	#518	—	53	2	no	—
224	#472	—	106	2	no	—
225	#472	—	106	2	no	—
226	#442	—	106	2	no	—
227	C₄×D₈	—	106	2	no	—
228	#430	—	53	2	no	—
229	C₂×#108	—	53	1	no	—
230	C₂×#108	—	53	1	no	—
231	#427	—	53	1	no	—
232	#421	—	53	2	no	—
233	#418	—	53	2	no	—
234	#418	—	53	2	no	—
235	#413	C₂×D₅₃	1	1	yes	characteristic
236	C₂×#100	—	53	2	no	—
237	C₂×#100	—	53	2	no	—
238	#410	C₂×D₅₃	1	1	yes	characteristic
239	#405	C₂×D₅₃	1	1	yes	characteristic
240	#405	C₂×D₅₃	1	1	yes	characteristic
241	#344	—	53	2	no	—
242	#344	—	53	2	no	—
243	#344	—	53	2	no	—
244	#344	C₂×D₅₃	1	1	yes	characteristic
245	#344	C₂×D₅₃	1	1	yes	characteristic
246	H(3,ℤ/4)	—	53	2	no	—
247	C₅₃	#24988	1	1	yes	characteristic, complemented, Sylow
248	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
249	C₂²×D₄	—	53	2	no	—
250	AGL(1,ℤ/8)	—	106	4	no	—
251	AGL(1,ℤ/8)	—	212	2	no	—
252	C₂×SD₁₆	—	106	2	no	—
253	C₂×D₈	—	53	2	no	—
254	C₂×D₈	—	53	2	no	—
255	C₂×D₈	—	106	2	no	—
256	C₂×D₈	—	106	1	no	—
257	C₂×D₈	—	106	1	no	—
258	C₂×D₈	—	106	1	no	—
259	C₂×D₈	—	106	1	no	—
260	#131	—	2	1	no	—
261	#131	—	2	1	no	—
262	#131	—	2	1	no	—
263	#131	—	2	1	no	—
264	C₂×M₁₆	—	2	1	no	—
265	C₂×M₁₆	—	53	2	no	—
266	C₂×M₁₆	—	53	2	no	—
267	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
268	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
269	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
270	C₂²×C₈	—	2	1	no	—
271	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
272	C₄×D₄	—	106	4	no	—
273	C₄×D₄	—	212	2	no	—
274	#117	—	53	2	no	—
275	#117	C₂²×D₅₃	1	1	yes	characteristic
276	#117	C₂²×D₅₃	1	1	yes	characteristic
277	C₂×C₂²⋊C₄	—	106	4	no	—
278	#108	—	106	1	no	—
279	#108	—	106	1	no	—
280	#108	—	106	1	no	—
281	#108	—	106	1	no	—
282	#108	—	212	2	no	—
283	#107	—	106	1	no	—
284	#106	—	106	1	no	—
285	#104	—	212	2	no	—
286	#102	—	106	4	no	—
287	#102	—	212	2	no	—
288	#101	—	2	1	no	—
289	#100	—	2	1	no	—
290	#100	—	106	4	no	—
291	#100	—	212	2	no	—
292	#99	—	2	2	no	—
293	#97	—	2	1	no	—
294	#97	—	2	1	no	—
295	C₄×C₈	—	2	1	no	—
296	C₄×C₈	—	2	1	no	—
297	C₂⁴	—	106	2	no	—
298	#41	—	2	1	no	—
299	#41	—	2	1	no	—
300	#41	—	2	1	no	—
301	#41	—	2	1	no	—
302	C₂×Q₈	#11719945	1	1	yes	characteristic
303	C₂×D₄	—	2	1	no	—
304	C₂×D₄	—	53	4	no	—
305	C₂×D₄	—	106	2	no	—
306	C₂×D₄	—	106	2	no	—
307	C₂×D₄	—	106	2	no	—
308	C₂×D₄	—	106	2	no	—
309	C₂×D₄	—	106	2	no	—
310	C₂×D₄	#11719945	1	1	yes	characteristic
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	212	4	no	—
313	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
314	SD₁₆	—	212	4	no	—
315	D₈	—	106	2	no	—
316	D₈	—	106	2	no	—
317	D₈	—	106	2	no	—
318	D₈	—	106	2	no	—
319	D₈	—	212	4	no	—
320	M₁₆	—	2	1	no	—
321	M₁₆	—	2	1	no	—
322	M₁₆	—	2	1	no	—
323	M₁₆	—	2	1	no	—
324	M₁₆	—	2	1	no	—
325	M₁₆	—	2	1	no	—
326	M₁₆	—	4	1	no	—
327	M₁₆	—	4	1	no	—
328	M₁₆	—	106	2	no	—
329	M₁₆	—	106	2	no	—
330	M₁₆	—	106	2	no	—
331	M₁₆	—	106	2	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	4	1	no	—
336	C₂×C₈	—	4	1	no	—
337	C₂×C₈	—	106	2	no	—
338	C₂×C₈	—	106	2	no	—
339	C₂×C₈	#11719945	1	1	yes	characteristic
340	C₂×C₈	#11719945	1	1	yes	characteristic
341	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
342	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
343	C₂×C₈	#11719947	1	1	yes	characteristic
344	C₂×C₈	#11719947	1	1	yes	characteristic
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	106	2	no	—
348	C₂²⋊C₄	—	2	1	no	—
349	C₂²⋊C₄	—	106	2	no	—
350	C₂²⋊C₄	—	212	4	no	—
351	C₂²⋊C₄	#11719945	1	2	yes	—
352	C₄²	—	2	1	no	—
353	C₄²	—	2	1	no	—
354	C₄²	—	106	2	no	—
355	C₂³	—	2	1	no	—
356	C₂³	—	848	2	no	—
357	C₂³	—	106	4	no	—
358	C₂³	—	212	2	no	—
359	C₂³	—	212	2	no	—
360	C₂³	C₂×#11719945	1	1	yes	characteristic
361	Q₈	—	2	1	no	—
362	Q₈	—	2	1	no	—
363	D₄	—	2	1	no	—
364	D₄	—	2	1	no	—
365	D₄	—	4	1	no	—
366	D₄	—	4	1	no	—
367	D₄	—	106	4	no	—
368	D₄	—	106	4	no	—
369	D₄	—	212	2	no	—
370	D₄	—	212	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	—	848	4	no	—
377	C₂×C₄	—	212	2	no	—
378	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
379	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
380	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
381	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
382	C₂×C₄	#13133263	1	1	yes	characteristic
383	C₂×C₄	#13133263	1	1	yes	characteristic
384	C₈	—	4	1	no	—
385	C₈	—	4	1	no	—
386	C₈	—	4	1	no	—
387	C₈	—	212	2	no	—
388	C₈	—	212	2	no	—
389	C₈	#13133199	1	1	yes	characteristic
390	C₈	#13133199	1	1	yes	characteristic
391	C₈	#13133199	1	1	yes	characteristic
392	C₈	#13133199	1	1	yes	characteristic
393	C₂²	—	4	1	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	8	1	no	—
396	C₂²	—	848	2	no	—
397	C₂²	—	848	2	no	—
398	C₂²	—	424	4	no	—
399	C₂²	—	212	4	no	—
400	C₂²	#16811157	1	1	yes	characteristic
401	C₂²	#16811269	1	1	yes	characteristic
402	C₂²	#16811338	1	1	yes	characteristic
403	C₄	—	4	2	no	—
404	C₄	—	4	1	no	—
405	C₄	—	8	1	no	—
406	C₄	—	424	2	no	—
407	C₄	C₂×#13133199	1	1	yes	characteristic
408	C₄	#16811228	1	1	yes	characteristic
409	C₄	#16811269	1	1	yes	characteristic
410	C₄	#16811319	1	1	yes	characteristic
411	C₂	—	2	1	no	—
412	C₂	—	2	1	no	—
413	C₂	—	2	1	no	—
414	C₂	—	8	1	no	—
415	C₂	—	424	4	no	—
416	C₂	#20278207	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20278207
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1757	C₂
socle	C₂×C₅₃	#2420
prime radical	C₂×#16810893	C₂
hypercenter	#1757	D₅₃
hypocenter	C₅₃	#24988
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757

Complements

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

N	H ≅ G/N	type
2. #20283568	C₂	semidirect
3. #20276178	C₂	semidirect
4. #20275988	C₂	semidirect
5. #20275625	C₂	semidirect
6. #20275505	C₂	semidirect
7. #20275205	C₂	semidirect
8. #20274839	C₂	semidirect
9. #20274825	C₂	semidirect
10. #20274769	C₂	semidirect
11. C₅₃×#1757	C₂	semidirect
213. #1757	D₅₃	semidirect
247. C₅₃	#24988	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810893	2
#20283568	2
#20276178	2
#20275988	2
#20275625	2
#20275505	2
#20275205	2
#20274839	2
#20274825	2
#20274769	2
C₅₃×#1757	2
#24988	53