Subgroups — C₅₃×#23834 #27231346

Order 13568.

608 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2961	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#2955	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2103	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2102	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2080	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2053	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#2050	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#2031	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1765	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1757	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1747	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#575	C₂²	1	1	yes	characteristic
13	C₂×#131×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#533	C₂²	1	1	yes	characteristic
15	C₅₃×#446	—	2	2	no	—
16	C₅₃×#446	—	2	2	no	—
17	C₅₃×#444	—	2	2	no	—
18	C₅₃×#443	—	2	2	no	—
19	C₅₃×#430	C₂²	1	1	yes	characteristic
20	C₅₃×#424	C₂²	1	1	yes	characteristic
21	C₅₃×#424	C₂²	1	1	yes	characteristic
22	C₅₃×#423	C₂²	1	1	yes	characteristic
23	C₅₃×#421	C₂²	1	2	yes	—
24	C₂×#104×C₅₃	C₂²	1	2	yes	—
25	C₅₃×#416	C₂²	1	2	yes	—
26	C₅₃×#413	C₂²	1	2	yes	—
27	C₅₃×#410	C₂²	1	2	yes	—
28	C₅₃×#405	C₂²	1	1	yes	characteristic
29	C₅₃×#405	C₂²	1	2	yes	—
30	C₅₃×#405	C₂²	1	1	yes	characteristic
31	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#344	C₂²	1	2	yes	—
33	C₅₃×#344	C₂²	1	2	yes	—
34	C₅₃×#341	C₂²	1	2	yes	—
35	C₅₃×#339	C₂²	1	2	yes	—
36	C₅₃×#338	C₂²	1	1	yes	characteristic
37	C₅₃×H(3,ℤ/4)	C₂²	1	2	yes	—
38	C₅₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
39	C₂×#41×C₅₃	C₂³	1	2	yes	—
40	#137×C₅₃	—	2	2	no	—
41	AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
42	#135×C₅₃	—	2	2	no	—
43	#131×C₅₃	—	2	2	no	—
44	#131×C₅₃	—	2	2	no	—
45	#131×C₅₃	—	2	2	no	—
46	#131×C₅₃	—	2	2	no	—
47	C₂×M₁₆×C₅₃	—	2	2	no	—
48	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
49	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
50	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	—	2	2	no	—
53	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈×C₅₃	—	2	1	no	—
56	#125×C₅₃	—	2	1	no	—
57	C₂²⋊Q₈×C₅₃	—	2	2	no	—
58	C₄×Q₈×C₅₃	—	2	2	no	—
59	#117×C₅₃	C₂³	1	2	yes	—
60	#117×C₅₃	C₂³	1	1	yes	characteristic
61	#117×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
63	#108×C₅₃	—	2	2	no	—
64	C₄⋊C₈×C₅₃	—	2	2	no	—
65	C₄⋊C₈×C₅₃	—	2	2	no	—
66	C₄⋊C₈×C₅₃	—	2	2	no	—
67	#104×C₅₃	—	2	4	no	—
68	#104×C₅₃	—	2	2	no	—
69	#104×C₅₃	—	2	2	no	—
70	#103×C₅₃	—	2	2	no	—
71	#102×C₅₃	—	2	2	no	—
72	#101×C₅₃	—	2	2	no	—
73	#100×C₅₃	—	2	2	no	—
74	#99×C₅₃	—	2	4	no	—
75	#97×C₅₃	—	2	1	no	—
76	#97×C₅₃	—	2	1	no	—
77	#97×C₅₃	—	2	1	no	—
78	#97×C₅₃	—	2	2	no	—
79	#97×C₅₃	D₄	1	2	yes	—
80	C₄×C₈×C₅₃	—	2	1	no	—
81	C₄×C₈×C₅₃	—	2	2	no	—
82	C₄×C₈×C₅₃	D₄	1	2	yes	—
83	#41×C₅₃	—	2	2	no	—
84	#41×C₅₃	—	2	2	no	—
85	#41×C₅₃	—	2	2	no	—
86	#41×C₅₃	—	2	2	no	—
87	C₂×Q₈×C₅₃	—	2	2	no	—
88	C₂×Q₈×C₅₃	#41	1	2	yes	—
89	C₂×D₄×C₅₃	—	2	2	no	—
90	C₂×D₄×C₅₃	#41	1	2	yes	—
91	C₂²×C₄×C₅₃	—	2	1	no	—
92	C₂²×C₄×C₅₃	—	2	2	no	—
93	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
94	Q₁₆×C₅₃	—	2	2	no	—
95	SD₁₆×C₅₃	—	2	4	no	—
96	D₈×C₅₃	—	2	2	no	—
97	M₁₆×C₅₃	—	2	2	no	—
98	M₁₆×C₅₃	—	2	2	no	—
99	M₁₆×C₅₃	—	2	2	no	—
100	M₁₆×C₅₃	—	2	2	no	—
101	M₁₆×C₅₃	—	2	2	no	—
102	M₁₆×C₅₃	—	2	2	no	—
103	M₁₆×C₅₃	—	4	2	no	—
104	M₁₆×C₅₃	—	4	2	no	—
105	C₂×C₈×C₅₃	—	2	1	no	—
106	C₂×C₈×C₅₃	—	2	1	no	—
107	C₂×C₈×C₅₃	—	2	2	no	—
108	C₂×C₈×C₅₃	—	2	2	no	—
109	C₂×C₈×C₅₃	—	2	2	no	—
110	C₂×C₈×C₅₃	—	4	2	no	—
111	C₂×C₈×C₅₃	—	4	2	no	—
112	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₅₃	#41	1	2	yes	—
116	C₂×C₈×C₅₃	#41	1	2	yes	—
117	C₄⋊C₄×C₅₃	—	2	4	no	—
118	C₄⋊C₄×C₅₃	—	2	2	no	—
119	C₄⋊C₄×C₅₃	—	2	1	no	—
120	C₄⋊C₄×C₅₃	—	2	1	no	—
121	C₄⋊C₄×C₅₃	—	2	2	no	—
122	C₂²⋊C₄×C₅₃	—	2	1	no	—
123	C₂²⋊C₄×C₅₃	—	2	2	no	—
124	C₂²⋊C₄×C₅₃	#41	1	2	yes	—
125	C₄²×C₅₃	—	2	1	no	—
126	C₄²×C₅₃	—	2	1	no	—
127	C₄²×C₅₃	—	2	2	no	—
128	C₄²×C₅₃	C₂×D₄	1	2	yes	—
129	C₄²×C₅₃	#41	1	1	yes	characteristic
130	C₄²×C₅₃	#41	1	1	yes	characteristic
131	C₂³×C₅₃	—	2	2	no	—
132	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
133	Q₈×C₅₃	—	2	2	no	—
134	Q₈×C₅₃	—	2	2	no	—
135	Q₈×C₅₃	—	8	2	no	—
136	D₄×C₅₃	—	2	2	no	—
137	D₄×C₅₃	—	2	2	no	—
138	D₄×C₅₃	—	4	2	no	—
139	D₄×C₅₃	—	4	2	no	—
140	C₂×C₄×C₅₃	—	2	1	no	—
141	C₂×C₄×C₅₃	—	2	1	no	—
142	C₂×C₄×C₅₃	—	2	2	no	—
143	C₂×C₄×C₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	—	4	2	no	—
146	C₂×C₄×C₅₃	—	4	2	no	—
147	C₂×C₄×C₅₃	—	4	2	no	—
148	C₂×C₄×C₅₃	—	8	1	no	—
149	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
151	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
152	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
153	C₈×C₅₃	—	2	2	no	—
154	C₈×C₅₃	—	4	1	no	—
155	C₈×C₅₃	—	4	1	no	—
156	C₈×C₅₃	—	4	2	no	—
157	C₈×C₅₃	—	4	2	no	—
158	C₈×C₅₃	#124	1	1	yes	characteristic
159	C₈×C₅₃	#124	1	2	yes	—
160	C₈×C₅₃	#124	1	1	yes	characteristic
161	#23834	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂²×C₅₃	—	4	1	no	—
163	C₂²×C₅₃	—	4	2	no	—
164	C₂²×C₅₃	—	8	2	no	—
165	C₂²×C₅₃	#525	1	1	yes	characteristic
166	C₂²×C₅₃	#529	1	2	yes	—
167	C₄×C₅₃	—	4	1	no	—
168	C₄×C₅₃	—	4	1	no	—
169	C₄×C₅₃	—	4	2	no	—
170	C₄×C₅₃	—	4	2	no	—
171	C₄×C₅₃	—	4	2	no	—
172	C₄×C₅₃	—	8	2	no	—
173	C₄×C₅₃	#525	1	1	yes	characteristic
174	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
175	C₄×C₅₃	#529	1	2	yes	—
176	#2961	C₂×C₅₃	1	1	yes	characteristic, complemented
177	#2955	C₂×C₅₃	1	1	yes	characteristic
178	#2103	C₂×C₅₃	1	1	yes	characteristic, complemented
179	#2102	C₂×C₅₃	1	1	yes	characteristic, complemented
180	#2080	C₂×C₅₃	1	2	yes	complemented
181	#2053	C₂×C₅₃	1	1	yes	characteristic
182	#2050	C₂×C₅₃	1	1	yes	characteristic
183	#2031	C₂×C₅₃	1	2	yes	complemented
184	#1765	C₂×C₅₃	1	2	yes	complemented
185	#1757	C₂×C₅₃	1	2	yes	complemented
186	#1747	C₂×C₅₃	1	1	yes	characteristic, complemented
187	C₂×C₅₃	—	2	1	no	—
188	C₂×C₅₃	—	2	2	no	—
189	C₂×C₅₃	—	8	2	no	—
190	C₂×C₅₃	#2398	1	1	yes	characteristic
191	#575	C₂²×C₅₃	1	1	yes	characteristic
192	C₂×#131	C₂²×C₅₃	1	2	yes	—
193	#533	C₂²×C₅₃	1	1	yes	characteristic
194	#446	—	2	2	no	—
195	#446	—	2	2	no	—
196	#444	—	2	2	no	—
197	#443	—	2	2	no	—
198	#430	C₂²×C₅₃	1	1	yes	characteristic
199	#424	C₂²×C₅₃	1	1	yes	characteristic
200	#424	C₂²×C₅₃	1	1	yes	characteristic
201	#423	C₂²×C₅₃	1	1	yes	characteristic
202	#421	C₂²×C₅₃	1	2	yes	—
203	C₂×#104	C₂²×C₅₃	1	2	yes	—
204	#416	C₂²×C₅₃	1	2	yes	—
205	#413	C₂²×C₅₃	1	2	yes	—
206	#410	C₂²×C₅₃	1	2	yes	—
207	#405	C₂²×C₅₃	1	1	yes	characteristic
208	#405	C₂²×C₅₃	1	2	yes	—
209	#405	C₂²×C₅₃	1	1	yes	characteristic
210	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
211	#344	C₂²×C₅₃	1	2	yes	—
212	#344	C₂²×C₅₃	1	2	yes	—
213	#341	C₂²×C₅₃	1	2	yes	—
214	#339	C₂²×C₅₃	1	2	yes	—
215	#338	C₂²×C₅₃	1	1	yes	characteristic
216	H(3,ℤ/4)	C₂²×C₅₃	1	1	yes	characteristic
217	H(3,ℤ/4)	C₂²×C₅₃	1	2	yes	—
218	C₅₃	#23834	1	1	yes	characteristic, direct product factor, Sylow
219	C₂×#41	C₂³×C₅₃	1	2	yes	—
220	#137	—	2	2	no	—
221	AGL(1,ℤ/8)	—	2	2	no	—
222	#135	—	2	2	no	—
223	#131	—	2	2	no	—
224	#131	—	2	2	no	—
225	#131	—	2	2	no	—
226	#131	—	2	2	no	—
227	C₂×M₁₆	—	2	2	no	—
228	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
229	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
230	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
231	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
232	C₂²×C₈	—	2	2	no	—
233	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
234	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
235	C₄⋊Q₈	—	2	1	no	—
236	#125	—	2	1	no	—
237	C₂²⋊Q₈	—	2	2	no	—
238	C₄×Q₈	—	2	2	no	—
239	#117	C₂³×C₅₃	1	2	yes	—
240	#117	C₂³×C₅₃	1	1	yes	characteristic
241	#117	C₂³×C₅₃	1	1	yes	characteristic
242	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
243	#108	—	2	2	no	—
244	C₄⋊C₈	—	2	2	no	—
245	C₄⋊C₈	—	2	2	no	—
246	C₄⋊C₈	—	2	2	no	—
247	#104	—	2	4	no	—
248	#104	—	2	2	no	—
249	#104	—	2	2	no	—
250	#103	—	2	2	no	—
251	#102	—	2	2	no	—
252	#101	—	2	2	no	—
253	#100	—	2	2	no	—
254	#99	—	2	4	no	—
255	#97	—	2	1	no	—
256	#97	—	2	1	no	—
257	#97	—	2	1	no	—
258	#97	—	2	2	no	—
259	#97	D₄×C₅₃	1	2	yes	—
260	C₄×C₈	—	2	1	no	—
261	C₄×C₈	—	2	2	no	—
262	C₄×C₈	D₄×C₅₃	1	2	yes	—
263	#41	—	2	2	no	—
264	#41	—	2	2	no	—
265	#41	—	2	2	no	—
266	#41	—	2	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	#41×C₅₃	1	2	yes	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	#41×C₅₃	1	2	yes	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
274	Q₁₆	—	2	2	no	—
275	SD₁₆	—	2	4	no	—
276	D₈	—	2	2	no	—
277	M₁₆	—	2	2	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	2	2	no	—
280	M₁₆	—	2	2	no	—
281	M₁₆	—	2	2	no	—
282	M₁₆	—	2	2	no	—
283	M₁₆	—	4	2	no	—
284	M₁₆	—	4	2	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	4	2	no	—
291	C₂×C₈	—	4	2	no	—
292	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
293	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
294	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
295	C₂×C₈	#41×C₅₃	1	2	yes	—
296	C₂×C₈	#41×C₅₃	1	2	yes	—
297	C₄⋊C₄	—	2	4	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₂²⋊C₄	—	2	1	no	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	#41×C₅₃	1	2	yes	—
305	C₄²	—	2	1	no	—
306	C₄²	—	2	1	no	—
307	C₄²	—	2	2	no	—
308	C₄²	C₂×D₄×C₅₃	1	2	yes	—
309	C₄²	#41×C₅₃	1	1	yes	characteristic
310	C₄²	#41×C₅₃	1	1	yes	characteristic
311	C₂³	—	2	2	no	—
312	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
313	Q₈	—	2	2	no	—
314	Q₈	—	2	2	no	—
315	Q₈	—	8	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	2	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	8	1	no	—
329	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
330	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
331	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
332	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	—	4	1	no	—
335	C₈	—	4	1	no	—
336	C₈	—	4	2	no	—
337	C₈	—	4	2	no	—
338	C₈	#124×C₅₃	1	1	yes	characteristic
339	C₈	#124×C₅₃	1	2	yes	—
340	C₈	#124×C₅₃	1	1	yes	characteristic
341	C₂²	—	4	1	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	C₅₃×#525	1	1	yes	characteristic
345	C₂²	C₅₃×#529	1	2	yes	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	2	no	—
349	C₄	—	4	2	no	—
350	C₄	—	4	2	no	—
351	C₄	—	8	2	no	—
352	C₄	C₅₃×#525	1	1	yes	characteristic
353	C₄	C₂×#124×C₅₃	1	1	yes	characteristic
354	C₄	C₅₃×#529	1	2	yes	—
355	C₂	—	2	1	no	—
356	C₂	—	2	2	no	—
357	C₂	—	8	2	no	—
358	C₂	C₅₃×#2398	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	C₂×#124
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂×C₅₃	#2398
prime radical	C₅₃×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#2961	C₂	semidirect
3. C₅₃×#2103	C₂	semidirect
4. C₅₃×#2102	C₂	semidirect
5. C₅₃×#2080	C₂	semidirect
8. C₅₃×#2031	C₂	semidirect
9. C₅₃×#1765	C₂	semidirect
10. C₅₃×#1757	C₂	semidirect
11. C₅₃×#1747	C₂	semidirect
161. #23834	C₅₃	direct (N × H)
176. #2961	C₂×C₅₃	semidirect
178. #2103	C₂×C₅₃	semidirect
179. #2102	C₂×C₅₃	semidirect
180. #2080	C₂×C₅₃	semidirect
183. #2031	C₂×C₅₃	semidirect
184. #1765	C₂×C₅₃	semidirect
185. #1757	C₂×C₅₃	semidirect
186. #1747	C₂×C₅₃	semidirect
218. C₅₃	#23834	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#2961	2
C₅₃×#2955	2
C₅₃×#2103	2
C₅₃×#2102	2
C₅₃×#2080	2
C₅₃×#2053	2
C₅₃×#2050	2
C₅₃×#2031	2
C₅₃×#1765	2
C₅₃×#1757	2
C₅₃×#1747	2
#23834	53

Subgroups — C53×#23834 #27231346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#23834 #27231346