Subgroups — #27429215

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×#16810890	C₂	1	1	yes	maximal, characteristic
2	#20283567	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276172	C₂	1	2	yes	maximal, complemented
4	#20275987	C₂	1	2	yes	maximal, complemented
5	#20275506	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275200	C₂	1	2	yes	maximal, complemented
7	#20274839	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274824	C₂	1	2	yes	maximal, complemented
9	#20274770	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274516	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1978	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133243	C₂²	1	2	yes	—
13	C₂×#13133223	C₂²	1	1	yes	characteristic
14	#16811098	C₂²	1	2	yes	—
15	#16810890	—	2	2	no	—
16	#16810890	—	2	2	no	—
17	#16810735	—	2	2	no	—
18	D₈×Dic₅₃	—	2	2	no	—
19	#16810672	C₂²	1	2	yes	—
20	C₂×#13133153	C₂²	1	1	yes	characteristic
21	#16810661	C₂²	1	1	yes	characteristic
22	#16810639	C₂²	1	2	yes	—
23	#16810628	C₂²	1	2	yes	—
24	#16810605	C₂²	1	1	yes	characteristic
25	C₂×#13133134	C₂²	1	2	yes	—
26	#16810596	C₂²	1	1	yes	characteristic
27	#16810578	C₂²	1	1	yes	characteristic
28	#16810395	C₂²	1	2	yes	—
29	#16810395	C₂²	1	1	yes	characteristic
30	#16810394	C₂²	1	2	yes	—
31	#16810394	C₂²	1	2	yes	—
32	#16810393	C₂²	1	1	yes	characteristic
33	#16810379	C₂²	1	2	yes	—
34	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#418	C₂²	1	2	yes	—
37	C₂×#100×C₅₃	C₂²	1	2	yes	—
38	C₅₃×#405	C₂²	1	1	yes	characteristic
39	C₂×#11719945	C₂³	1	1	yes	characteristic
40	#13133243	—	2	4	no	—
41	#13133243	—	4	2	no	—
42	C₂×#11719923	—	2	2	no	—
43	C₂×#11719922	—	2	2	no	—
44	#13133223	—	2	1	no	—
45	#13133223	—	2	1	no	—
46	#13133223	—	2	1	no	—
47	#13133223	—	2	1	no	—
48	C₂×#11719917	C₂³	1	1	yes	characteristic
49	C₂×#11719917	C₂³	1	1	yes	characteristic
50	C₂×#11719917	C₂³	1	2	yes	—
51	C₂×#11719912	—	2	1	no	—
52	C₂×C₈×D₅₃	—	2	1	no	—
53	D₄×Dic₅₃	—	2	4	no	—
54	D₄×Dic₅₃	—	4	2	no	—
55	#13133176	C₂³	1	2	yes	—
56	#13133176	C₂³	1	1	yes	characteristic
57	C₂×C₂²⋊Dic₅₃	—	2	4	no	—
58	#13133153	—	2	1	no	—
59	#13133153	—	2	1	no	—
60	#13133153	—	4	2	no	—
61	#13133151	—	2	1	no	—
62	#13133149	—	2	1	no	—
63	#13133143	—	4	2	no	—
64	#13133138	—	2	4	no	—
65	#13133135	—	2	1	no	—
66	#13133134	—	2	4	no	—
67	#13133133	—	2	1	no	—
68	#13133131	—	2	2	no	—
69	#13133127	—	2	1	no	—
70	C₈×Dic₅₃	—	2	1	no	—
71	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
72	C₂×D₈×C₅₃	—	2	1	no	—
73	C₂×D₈×C₅₃	—	2	1	no	—
74	C₂×D₈×C₅₃	—	2	1	no	—
75	C₂×D₈×C₅₃	—	2	1	no	—
76	C₂×D₈×C₅₃	D₄	1	2	yes	—
77	C₂×D₈×C₅₃	D₄	1	2	yes	—
78	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
80	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
81	#117×C₅₃	C₂³	1	1	yes	characteristic
82	#108×C₅₃	—	2	1	no	—
83	#108×C₅₃	—	2	1	no	—
84	#102×C₅₃	—	4	2	no	—
85	#100×C₅₃	—	4	2	no	—
86	#97×C₅₃	—	2	1	no	—
87	C₄×C₈×C₅₃	—	2	1	no	—
88	C₂⁴×C₅₃	—	2	2	no	—
89	C₂×D₄×C₅₃	—	2	2	no	—
90	C₂×D₄×C₅₃	—	2	2	no	—
91	C₂×D₄×C₅₃	—	2	2	no	—
92	C₂×D₄×C₅₃	—	2	2	no	—
93	C₂×D₄×C₅₃	—	2	2	no	—
94	C₂×D₄×C₅₃	C₂×D₄	1	4	yes	—
95	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
96	C₂×C₂²⋊D₅₃	—	2	1	no	—
97	C₂²×Dic₅₃	—	4	4	no	—
98	#11719945	—	2	1	no	—
99	#11719945	—	2	1	no	—
100	#11719945	—	2	1	no	—
101	#11719945	—	2	1	no	—
102	C₂×D₂₁₂	#41	1	1	yes	characteristic
103	C₂×C₄×D₅₃	—	2	1	no	—
104	C₂×Dic₁₀₆	#41	1	1	yes	characteristic
105	D₈×C₅₃	—	2	2	no	—
106	D₈×C₅₃	—	2	2	no	—
107	D₈×C₅₃	—	2	2	no	—
108	D₈×C₅₃	—	2	2	no	—
109	M₁₆×C₅₃	—	2	2	no	—
110	M₁₆×C₅₃	—	2	1	no	—
111	M₁₆×C₅₃	—	2	1	no	—
112	M₁₆×C₅₃	—	2	2	no	—
113	C₂×C₈×C₅₃	—	2	1	no	—
114	C₂×C₈×C₅₃	—	2	2	no	—
115	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
118	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
119	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
120	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
121	C₄⋊C₄×C₅₃	—	2	1	no	—
122	C₂²⋊C₄×C₅₃	—	2	1	no	—
123	C₄²×C₅₃	—	2	1	no	—
124	C₂²⋊Dic₅₃	—	2	2	no	—
125	C₂²⋊Dic₅₃	—	4	4	no	—
126	C₂²⋊Dic₅₃	#41	1	2	yes	—
127	#11719923	—	4	4	no	—
128	#11719922	—	4	4	no	—
129	C₄⋊Dic₅₃	—	2	1	no	—
130	C₄⋊Dic₅₃	—	2	2	no	—
131	C₄×Dic₅₃	—	2	1	no	—
132	C₄×Dic₅₃	—	2	2	no	—
133	#11719917	—	2	1	no	—
134	#11719917	—	2	2	no	—
135	#11719917	—	2	1	no	—
136	#11719917	—	2	1	no	—
137	#11719917	—	2	1	no	—
138	#11719917	—	2	2	no	—
139	C₂×#88322	—	2	1	no	—
140	C₂×#88322	—	2	1	no	—
141	C₂×#88322	—	2	2	no	—
142	#11719912	—	4	1	no	—
143	#11719912	—	4	1	no	—
144	C₈×D₅₃	—	4	1	no	—
145	C₈×D₅₃	—	4	1	no	—
146	C₂³×C₅₃	—	2	4	no	—
147	C₂³×C₅₃	—	4	2	no	—
148	C₂³×C₅₃	—	4	2	no	—
149	C₂³×C₅₃	—	16	2	no	—
150	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
151	C₂²×D₅₃	—	2	1	no	—
152	D₄×C₅₃	—	2	4	no	—
153	D₄×C₅₃	—	2	4	no	—
154	D₄×C₅₃	—	4	2	no	—
155	D₄×C₅₃	—	4	2	no	—
156	C₂×C₄×C₅₃	—	4	1	no	—
157	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
161	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
163	C₂²⋊D₅₃	—	4	1	no	—
164	C₂²⋊D₅₃	—	4	1	no	—
165	C₂×Dic₅₃	—	2	1	no	—
166	C₂×Dic₅₃	—	2	2	no	—
167	C₂×Dic₅₃	—	4	2	no	—
168	C₂×Dic₅₃	—	16	4	no	—
169	D₂₁₂	—	2	1	no	—
170	D₂₁₂	—	2	1	no	—
171	C₄×D₅₃	—	4	1	no	—
172	C₄×D₅₃	—	4	1	no	—
173	Dic₁₀₆	—	2	1	no	—
174	Dic₁₀₆	—	2	1	no	—
175	C₈×C₅₃	—	4	2	no	—
176	C₈×C₅₃	—	4	1	no	—
177	C₈×C₅₃	#123	1	1	yes	characteristic
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	#88322	—	4	1	no	—
182	#88322	—	4	1	no	—
183	#88322	—	4	2	no	—
184	#24988	—	53	1	no	maximal, Sylow
185	C₂²×C₅₃	—	4	1	no	—
186	C₂²×C₅₃	—	4	4	no	—
187	C₂²×C₅₃	—	8	4	no	—
188	C₂²×C₅₃	—	16	2	no	—
189	C₂²×C₅₃	—	16	2	no	—
190	C₂²×C₅₃	#529	1	1	yes	characteristic
191	C₂²×C₅₃	#546	1	1	yes	characteristic
192	C₂²×C₅₃	#555	1	1	yes	characteristic
193	C₂×D₅₃	—	4	1	no	—
194	C₂×D₅₃	—	8	1	no	—
195	C₄×C₅₃	—	8	1	no	—
196	C₄×C₅₃	C₂×#123	1	1	yes	characteristic
197	C₄×C₅₃	#540	1	1	yes	characteristic
198	C₄×C₅₃	#546	1	1	yes	characteristic
199	C₄×C₅₃	#553	1	1	yes	characteristic
200	Dic₅₃	—	4	2	no	—
201	Dic₅₃	—	4	1	no	—
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	D₅₃	1	1	yes	characteristic, complemented
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	—	53	1	no	—
214	C₂×C₅₃	—	2	1	no	—
215	C₂×C₅₃	—	2	1	no	—
216	C₂×C₅₃	—	2	1	no	—
217	C₂×C₅₃	—	8	4	no	—
218	C₂×C₅₃	#2420	1	1	yes	characteristic
219	D₅₃	—	8	1	no	—
220	C₂×AGL(1,ℤ/8)	—	53	2	no	—
221	C₂²×D₈	C₂×D₅₃	1	1	yes	characteristic
222	C₂×#131	—	53	1	no	—
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	C₂×D₅₃	1	1	yes	characteristic
231	#427	—	53	1	no	—
232	#421	—	53	2	no	—
233	#418	—	53	2	no	—
234	#418	C₂×D₅₃	1	2	yes	—
235	#413	—	53	1	no	—
236	C₂×#100	—	53	2	no	—
237	C₂×#100	C₂×D₅₃	1	2	yes	—
238	#410	—	53	1	no	—
239	#405	—	53	1	no	—
240	#405	C₂×D₅₃	1	1	yes	characteristic
241	#344	—	53	2	no	—
242	#344	—	53	2	no	—
243	#344	—	53	1	no	—
244	#344	—	53	2	no	—
245	#344	—	53	1	no	—
246	H(3,ℤ/4)	—	53	2	no	—
247	C₅₃	#24988	1	1	yes	characteristic, complemented, Sylow
248	C₂×#41	—	53	1	no	—
249	C₂²×D₄	C₂²×D₅₃	1	2	yes	—
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	—	2	1	no	—
254	C₂×D₈	—	2	1	no	—
255	C₂×D₈	—	2	1	no	—
256	C₂×D₈	—	2	1	no	—
257	C₂×D₈	—	106	2	no	—
258	C₂×D₈	D₂₁₂	1	2	yes	—
259	C₂×D₈	D₂₁₂	1	2	yes	—
260	#131	—	106	1	no	—
261	#131	—	106	1	no	—
262	#131	—	106	1	no	—
263	#131	—	106	1	no	—
264	C₂×M₁₆	—	53	1	no	—
265	C₂×M₁₆	—	53	2	no	—
266	C₂×M₁₆	—	53	1	no	—
267	C₂×M₁₆	—	106	1	no	—
268	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
269	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
270	C₂²×C₈	—	106	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	1	no	—
275	#117	—	53	2	no	—
276	#117	C₂²×D₅₃	1	1	yes	characteristic
277	C₂×C₂²⋊C₄	—	106	4	no	—
278	#108	—	2	1	no	—
279	#108	—	2	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	—	4	2	no	—
287	#102	—	106	4	no	—
288	#101	—	106	1	no	—
289	#100	—	4	2	no	—
290	#100	—	106	1	no	—
291	#100	—	106	4	no	—
292	#99	—	106	2	no	—
293	#97	—	2	1	no	—
294	#97	—	106	1	no	—
295	C₄×C₈	—	2	1	no	—
296	C₄×C₈	—	106	1	no	—
297	C₂⁴	—	2	2	no	—
298	#41	—	106	1	no	—
299	#41	—	106	1	no	—
300	#41	—	106	1	no	—
301	#41	—	106	1	no	—
302	C₂×Q₈	—	53	1	no	—
303	C₂×D₄	—	2	2	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	2	2	no	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	—	53	1	no	—
309	C₂×D₄	—	106	1	no	—
310	C₂×D₄	C₂×D₂₁₂	1	4	yes	—
311	C₂²×C₄	—	106	1	no	—
312	C₂²×C₄	—	212	4	no	—
313	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
314	SD₁₆	—	212	4	no	—
315	D₈	—	2	2	no	—
316	D₈	—	2	2	no	—
317	D₈	—	2	2	no	—
318	D₈	—	2	2	no	—
319	D₈	—	212	4	no	—
320	M₁₆	—	2	1	no	—
321	M₁₆	—	2	2	no	—
322	M₁₆	—	2	1	no	—
323	M₁₆	—	2	2	no	—
324	M₁₆	—	106	1	no	—
325	M₁₆	—	106	2	no	—
326	M₁₆	—	106	1	no	—
327	M₁₆	—	106	1	no	—
328	M₁₆	—	106	1	no	—
329	M₁₆	—	106	2	no	—
330	M₁₆	—	212	1	no	—
331	M₁₆	—	212	1	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	106	1	no	—
335	C₂×C₈	—	106	2	no	—
336	C₂×C₈	—	106	1	no	—
337	C₂×C₈	—	212	1	no	—
338	C₂×C₈	—	212	1	no	—
339	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
340	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
341	C₂×C₈	#11719947	1	1	yes	characteristic
342	C₂×C₈	#11719947	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	—	106	1	no	—
347	C₄⋊C₄	—	106	2	no	—
348	C₂²⋊C₄	—	2	1	no	—
349	C₂²⋊C₄	—	53	2	no	—
350	C₂²⋊C₄	—	106	2	no	—
351	C₂²⋊C₄	—	212	4	no	—
352	C₄²	—	2	1	no	—
353	C₄²	—	106	1	no	—
354	C₄²	—	106	2	no	—
355	C₂³	—	2	4	no	—
356	C₂³	—	4	2	no	—
357	C₂³	—	4	2	no	—
358	C₂³	—	16	2	no	—
359	C₂³	—	106	1	no	—
360	C₂³	#41×D₅₃	1	1	yes	characteristic
361	Q₈	—	106	1	no	—
362	Q₈	—	106	1	no	—
363	D₄	—	2	4	no	—
364	D₄	—	2	4	no	—
365	D₄	—	4	2	no	—
366	D₄	—	4	2	no	—
367	D₄	—	106	1	no	—
368	D₄	—	106	1	no	—
369	D₄	—	212	1	no	—
370	D₄	—	212	1	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	848	4	no	—
373	C₂×C₄	—	106	2	no	—
374	C₂×C₄	—	106	1	no	—
375	C₂×C₄	—	212	2	no	—
376	C₂×C₄	—	212	1	no	—
377	C₂×C₄	—	212	1	no	—
378	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
379	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
380	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
381	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
382	C₂×C₄	#13133262	1	1	yes	characteristic
383	C₂×C₄	#13133262	1	1	yes	characteristic
384	C₈	—	4	1	no	—
385	C₈	—	4	2	no	—
386	C₈	—	212	1	no	—
387	C₈	—	212	2	no	—
388	C₈	—	212	1	no	—
389	C₈	#13133201	1	1	yes	characteristic
390	C₈	#13133201	1	1	yes	characteristic
391	C₈	#13133201	1	1	yes	characteristic
392	C₈	#13133201	1	1	yes	characteristic
393	C₂²	—	4	1	no	—
394	C₂²	—	4	4	no	—
395	C₂²	—	8	4	no	—
396	C₂²	—	16	2	no	—
397	C₂²	—	16	2	no	—
398	C₂²	—	424	1	no	—
399	C₂²	—	212	1	no	—
400	C₂²	#16811152	1	1	yes	characteristic
401	C₂²	#16811268	1	1	yes	characteristic
402	C₂²	#16811337	1	1	yes	characteristic
403	C₄	—	8	1	no	—
404	C₄	—	424	2	no	—
405	C₄	—	212	2	no	—
406	C₄	—	212	1	no	—
407	C₄	C₂×#13133201	1	1	yes	characteristic
408	C₄	#16811227	1	1	yes	characteristic
409	C₄	#16811268	1	1	yes	characteristic
410	C₄	#16811320	1	1	yes	characteristic
411	C₂	—	2	1	no	—
412	C₂	—	2	1	no	—
413	C₂	—	2	1	no	—
414	C₂	—	8	4	no	—
415	C₂	—	424	1	no	—
416	C₂	#20278205	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20278205
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1978	C₂
socle	C₂×C₅₃	#2420
prime radical	C₂×#16810890	C₂
hypercenter	#1978	D₅₃
hypocenter	C₅₃	#24988
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1978

Complements

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

N	H ≅ G/N	type
2. #20283567	C₂	semidirect
3. #20276172	C₂	semidirect
4. #20275987	C₂	semidirect
5. #20275506	C₂	semidirect
6. #20275200	C₂	semidirect
7. #20274839	C₂	semidirect
8. #20274824	C₂	semidirect
9. #20274770	C₂	semidirect
10. #20274516	C₂	semidirect
11. C₅₃×#1978	C₂	semidirect
207. #1978	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×#16810890	2
#20283567	2
#20276172	2
#20275987	2
#20275506	2
#20275200	2
#20274839	2
#20274824	2
#20274770	2
#20274516	2
C₅₃×#1978	2
#24988	53