Subgroups — C₅₃×#18715 #27226227

Order 13568.

1056 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3549	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#3227	C₂	1	2	yes	maximal, complemented
3	C₅₃×#3152	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1634	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1629	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1543	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1484	C₂	1	2	yes	maximal, complemented
8	C₂×#137×C₅₃	—	2	2	no	—
9	C₅₃×H(3,𝔽₄)	—	2	2	no	—
10	C₅₃×#555	—	2	2	no	—
11	C₅₃×#555	C₂²	1	2	yes	—
12	D₄²×C₅₃	—	2	2	no	—
13	D₄²×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#535	—	2	2	no	—
15	C₅₃×#535	—	2	4	no	—
16	C₅₃×#532	C₂²	1	1	yes	characteristic
17	C₅₃×#532	C₂²	1	2	yes	—
18	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
19	C₅₃×#498	—	2	2	no	—
20	C₅₃×#497	—	2	2	no	—
21	C₅₃×#494	C₂²	1	2	yes	complemented
22	C₅₃×#492	C₂²	1	2	yes	—
23	C₅₃×#486	—	2	2	no	—
24	C₅₃×#481	—	2	4	no	—
25	C₅₃×#476	C₂²	1	8	yes	complemented
26	C₅₃×#431	C₂²	1	2	yes	complemented
27	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
28	C₅₃×#333	C₂²	1	2	yes	complemented
29	C₅₃×#331	C₂²	1	2	yes	—
30	C₅₃×D₄⋊C₈	C₂²	1	8	yes	complemented
31	C₂×#41×C₅₃	—	2	2	no	—
32	C₂×#41×C₅₃	D₄	1	2	yes	—
33	C₂²×Q₈×C₅₃	—	2	1	no	—
34	C₂²×D₄×C₅₃	—	2	2	no	—
35	C₂²×D₄×C₅₃	—	2	4	no	—
36	#137×C₅₃	—	4	8	no	—
37	C₂×Q₁₆×C₅₃	—	2	4	no	—
38	C₂×SD₁₆×C₅₃	—	2	4	no	—
39	C₂×M₁₆×C₅₃	—	2	1	no	—
40	C₄⋊Q₈×C₅₃	—	2	1	no	—
41	C₄⋊Q₈×C₅₃	—	2	2	no	—
42	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
43	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
45	#127×C₅₃	—	2	4	no	—
46	#127×C₅₃	D₄	1	2	yes	—
47	#127×C₅₃	C₂³	1	1	yes	characteristic
48	#124×C₅₃	—	2	8	no	—
49	#124×C₅₃	—	2	2	no	—
50	#124×C₅₃	—	2	4	no	—
51	#124×C₅₃	D₄	1	2	yes	—
52	#121×C₅₃	—	2	4	no	—
53	#121×C₅₃	—	2	4	no	—
54	#121×C₅₃	—	2	4	no	—
55	#121×C₅₃	—	2	2	no	—
56	#121×C₅₃	—	2	2	no	—
57	#121×C₅₃	D₄	1	4	yes	—
58	#120×C₅₃	—	2	4	no	—
59	#120×C₅₃	—	2	8	no	—
60	#120×C₅₃	—	2	4	no	—
61	C₄×D₄×C₅₃	—	2	2	no	—
62	C₄×D₄×C₅₃	—	2	2	no	—
63	C₄×D₄×C₅₃	C₂³	1	4	yes	—
64	#117×C₅₃	—	2	2	no	—
65	#117×C₅₃	—	2	4	no	—
66	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	#106×C₅₃	—	2	8	no	—
69	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
70	#102×C₅₃	—	2	8	no	—
71	C₂²⋊C₈×C₅₃	—	2	2	no	—
72	#97×C₅₃	—	2	1	no	—
73	C₄×C₈×C₅₃	—	2	2	no	—
74	C₄×C₈×C₅₃	C₂³	1	2	yes	—
75	C₂⁴×C₅₃	—	2	4	no	—
76	#41×C₅₃	—	4	4	no	—
77	#41×C₅₃	—	8	4	no	—
78	C₂×Q₈×C₅₃	—	2	2	no	—
79	C₂×Q₈×C₅₃	—	2	2	no	—
80	C₂×Q₈×C₅₃	—	4	4	no	—
81	C₂×Q₈×C₅₃	—	4	1	no	—
82	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
83	C₂×D₄×C₅₃	—	2	8	no	—
84	C₂×D₄×C₅₃	—	2	4	no	—
85	C₂×D₄×C₅₃	—	2	2	no	—
86	C₂×D₄×C₅₃	—	2	4	no	—
87	C₂×D₄×C₅₃	—	2	2	no	—
88	C₂×D₄×C₅₃	—	2	2	no	—
89	C₂×D₄×C₅₃	—	2	2	no	—
90	C₂×D₄×C₅₃	—	8	8	no	—
91	C₂×D₄×C₅₃	—	8	4	no	—
92	C₂×D₄×C₅₃	C₂×D₄	1	4	yes	—
93	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₅₃	—	2	2	no	—
95	C₂²×C₄×C₅₃	—	2	4	no	—
96	C₂²×C₄×C₅₃	—	4	1	no	—
97	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
99	Q₁₆×C₅₃	—	4	8	no	—
100	SD₁₆×C₅₃	—	4	8	no	—
101	M₁₆×C₅₃	—	2	4	no	—
102	C₂×C₈×C₅₃	—	2	2	no	—
103	C₂×C₈×C₅₃	—	2	4	no	—
104	C₄⋊C₄×C₅₃	—	2	2	no	—
105	C₄⋊C₄×C₅₃	—	2	2	no	—
106	C₄⋊C₄×C₅₃	—	2	4	no	—
107	C₄⋊C₄×C₅₃	—	4	2	no	—
108	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
109	C₂²⋊C₄×C₅₃	—	2	4	no	—
110	C₂²⋊C₄×C₅₃	—	2	8	no	—
111	C₂²⋊C₄×C₅₃	—	2	4	no	—
112	C₂²⋊C₄×C₅₃	—	2	4	no	—
113	C₂²⋊C₄×C₅₃	—	2	4	no	—
114	C₄²×C₅₃	—	2	4	no	—
115	C₄²×C₅₃	C₂×D₄	1	2	yes	—
116	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
118	C₂³×C₅₃	—	2	4	no	—
119	C₂³×C₅₃	—	2	2	no	—
120	C₂³×C₅₃	—	8	8	no	—
121	C₂³×C₅₃	—	8	4	no	—
122	C₂³×C₅₃	#120	1	1	yes	characteristic
123	Q₈×C₅₃	—	4	2	no	—
124	Q₈×C₅₃	—	4	4	no	—
125	Q₈×C₅₃	—	8	2	no	—
126	Q₈×C₅₃	—	8	2	no	—
127	D₄×C₅₃	—	2	8	no	—
128	D₄×C₅₃	—	4	2	no	—
129	D₄×C₅₃	—	4	4	no	—
130	D₄×C₅₃	—	4	4	no	—
131	D₄×C₅₃	—	8	2	no	—
132	D₄×C₅₃	—	8	2	no	—
133	D₄×C₅₃	—	8	4	no	—
134	D₄×C₅₃	—	8	8	no	—
135	D₄×C₅₃	—	8	4	no	—
136	C₂×C₄×C₅₃	—	2	4	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	4	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	4	8	no	—
141	C₂×C₄×C₅₃	—	4	4	no	—
142	C₂×C₄×C₅₃	—	4	2	no	—
143	C₂×C₄×C₅₃	—	8	2	no	—
144	C₂×C₄×C₅₃	#120	1	2	yes	—
145	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
148	C₈×C₅₃	—	2	4	no	—
149	C₈×C₅₃	—	4	4	no	—
150	#18715	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂²×C₅₃	—	2	2	no	—
152	C₂²×C₅₃	—	2	1	no	—
153	C₂²×C₅₃	—	4	4	no	—
154	C₂²×C₅₃	—	4	4	no	—
155	C₂²×C₅₃	—	4	4	no	—
156	C₂²×C₅₃	—	4	2	no	—
157	C₂²×C₅₃	—	4	4	no	—
158	C₂²×C₅₃	—	16	8	no	—
159	C₂²×C₅₃	C₂×#120	1	1	yes	characteristic
160	C₄×C₅₃	—	2	2	no	—
161	C₄×C₅₃	—	2	1	no	—
162	C₄×C₅₃	—	4	4	no	—
163	C₄×C₅₃	—	4	2	no	—
164	C₄×C₅₃	—	8	2	no	—
165	C₄×C₅₃	#453	1	2	yes	—
166	C₄×C₅₃	#577	1	4	yes	—
167	#3549	C₂×C₅₃	1	1	yes	characteristic
168	#3227	C₂×C₅₃	1	2	yes	complemented
169	#3152	C₂×C₅₃	1	1	yes	characteristic, complemented
170	#1634	C₂×C₅₃	1	4	yes	complemented
171	#1629	C₂×C₅₃	1	4	yes	complemented
172	#1543	C₂×C₅₃	1	1	yes	characteristic, complemented
173	#1484	C₂×C₅₃	1	2	yes	complemented
174	C₂×C₅₃	—	4	1	no	—
175	C₂×C₅₃	—	8	4	no	—
176	C₂×C₅₃	—	8	2	no	—
177	C₂×C₅₃	#3009	1	2	yes	—
178	C₂×C₅₃	C₂×#453	1	1	yes	characteristic
179	C₂×#137	—	2	2	no	—
180	H(3,𝔽₄)	—	2	2	no	—
181	#555	—	2	2	no	—
182	#555	C₂²×C₅₃	1	2	yes	—
183	D₄²	—	2	2	no	—
184	D₄²	C₂²×C₅₃	1	2	yes	—
185	#535	—	2	2	no	—
186	#535	—	2	4	no	—
187	#532	C₂²×C₅₃	1	1	yes	characteristic
188	#532	C₂²×C₅₃	1	2	yes	—
189	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
190	#498	—	2	2	no	—
191	#497	—	2	2	no	—
192	#494	C₂²×C₅₃	1	2	yes	complemented
193	#492	C₂²×C₅₃	1	2	yes	—
194	#486	—	2	2	no	—
195	#481	—	2	4	no	—
196	#476	C₂²×C₅₃	1	8	yes	complemented
197	#431	C₂²×C₅₃	1	2	yes	complemented
198	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic, complemented
199	#333	C₂²×C₅₃	1	2	yes	complemented
200	#331	C₂²×C₅₃	1	2	yes	—
201	D₄⋊C₈	C₂²×C₅₃	1	8	yes	complemented
202	C₅₃	#18715	1	1	yes	characteristic, direct product factor, Sylow
203	C₂×#41	—	2	2	no	—
204	C₂×#41	D₄×C₅₃	1	2	yes	—
205	C₂²×Q₈	—	2	1	no	—
206	C₂²×D₄	—	2	2	no	—
207	C₂²×D₄	—	2	4	no	—
208	#137	—	4	8	no	—
209	C₂×Q₁₆	—	2	4	no	—
210	C₂×SD₁₆	—	2	4	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₄⋊Q₈	—	2	2	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
215	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
216	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
217	#127	—	2	4	no	—
218	#127	D₄×C₅₃	1	2	yes	—
219	#127	C₂³×C₅₃	1	1	yes	characteristic
220	#124	—	2	8	no	—
221	#124	—	2	4	no	—
222	#124	—	2	2	no	—
223	#124	D₄×C₅₃	1	2	yes	—
224	#121	—	2	4	no	—
225	#121	—	2	4	no	—
226	#121	—	2	2	no	—
227	#121	—	2	2	no	—
228	#121	—	2	4	no	—
229	#121	D₄×C₅₃	1	4	yes	—
230	#120	—	2	4	no	—
231	#120	—	2	8	no	—
232	#120	—	2	4	no	—
233	C₄×D₄	—	2	2	no	—
234	C₄×D₄	—	2	2	no	—
235	C₄×D₄	C₂³×C₅₃	1	4	yes	—
236	#117	—	2	2	no	—
237	#117	—	2	4	no	—
238	C₂×C₄⋊C₄	—	2	2	no	—
239	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
240	#106	—	2	8	no	—
241	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
242	#102	—	2	8	no	—
243	C₂²⋊C₈	—	2	2	no	—
244	#97	—	2	1	no	—
245	C₄×C₈	—	2	2	no	—
246	C₄×C₈	C₂³×C₅₃	1	2	yes	—
247	C₂⁴	—	2	4	no	—
248	#41	—	4	4	no	—
249	#41	—	8	4	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	4	4	no	—
253	C₂×Q₈	—	4	1	no	—
254	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
255	C₂×D₄	—	2	4	no	—
256	C₂×D₄	—	2	8	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	4	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	8	8	no	—
263	C₂×D₄	—	8	4	no	—
264	C₂×D₄	C₂×D₄×C₅₃	1	4	yes	—
265	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	4	no	—
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
270	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
271	Q₁₆	—	4	8	no	—
272	SD₁₆	—	4	8	no	—
273	M₁₆	—	2	4	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	4	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	4	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
281	C₂²⋊C₄	—	2	8	no	—
282	C₂²⋊C₄	—	2	4	no	—
283	C₂²⋊C₄	—	2	4	no	—
284	C₂²⋊C₄	—	2	4	no	—
285	C₂²⋊C₄	—	2	4	no	—
286	C₄²	—	2	4	no	—
287	C₄²	C₂×D₄×C₅₃	1	2	yes	—
288	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
289	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
290	C₂³	—	2	4	no	—
291	C₂³	—	2	2	no	—
292	C₂³	—	8	8	no	—
293	C₂³	—	8	4	no	—
294	C₂³	#120×C₅₃	1	1	yes	characteristic
295	Q₈	—	4	2	no	—
296	Q₈	—	4	4	no	—
297	Q₈	—	8	2	no	—
298	Q₈	—	8	2	no	—
299	D₄	—	2	8	no	—
300	D₄	—	4	2	no	—
301	D₄	—	4	4	no	—
302	D₄	—	4	4	no	—
303	D₄	—	8	2	no	—
304	D₄	—	8	2	no	—
305	D₄	—	8	4	no	—
306	D₄	—	8	8	no	—
307	D₄	—	8	4	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	4	8	no	—
313	C₂×C₄	—	4	4	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	8	2	no	—
316	C₂×C₄	#120×C₅₃	1	2	yes	—
317	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
318	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
319	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
320	C₈	—	2	4	no	—
321	C₈	—	4	4	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	4	no	—
325	C₂²	—	4	4	no	—
326	C₂²	—	4	4	no	—
327	C₂²	—	4	2	no	—
328	C₂²	—	4	4	no	—
329	C₂²	—	16	8	no	—
330	C₂²	C₂×#120×C₅₃	1	1	yes	characteristic
331	C₄	—	2	2	no	—
332	C₄	—	2	1	no	—
333	C₄	—	4	4	no	—
334	C₄	—	4	2	no	—
335	C₄	—	8	2	no	—
336	C₄	C₅₃×#453	1	2	yes	—
337	C₄	C₅₃×#577	1	4	yes	—
338	C₂	—	4	1	no	—
339	C₂	—	8	4	no	—
340	C₂	—	8	2	no	—
341	C₂	C₅₃×#3009	1	2	yes	—
342	C₂	C₂×C₅₃×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#3549	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#3227	C₂	semidirect
3. C₅₃×#3152	C₂	semidirect
4. C₅₃×#1634	C₂	semidirect
5. C₅₃×#1629	C₂	semidirect
6. C₅₃×#1543	C₂	semidirect
7. C₅₃×#1484	C₂	semidirect
18. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
21. C₅₃×#494	C₂²	semidirect
25. C₅₃×#476	C₂²	semidirect
26. C₅₃×#431	C₂²	semidirect
27. C₄×M₁₆×C₅₃	C₂²	semidirect
28. C₅₃×#333	C₂²	semidirect
30. C₅₃×D₄⋊C₈	C₂²	semidirect
150. #18715	C₅₃	direct (N × H)
168. #3227	C₂×C₅₃	semidirect
169. #3152	C₂×C₅₃	semidirect
170. #1634	C₂×C₅₃	semidirect
171. #1629	C₂×C₅₃	semidirect
172. #1543	C₂×C₅₃	semidirect
173. #1484	C₂×C₅₃	semidirect
189. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
192. #494	C₂²×C₅₃	semidirect
196. #476	C₂²×C₅₃	semidirect
197. #431	C₂²×C₅₃	semidirect
198. C₄×M₁₆	C₂²×C₅₃	semidirect
199. #333	C₂²×C₅₃	semidirect
201. D₄⋊C₈	C₂²×C₅₃	semidirect
202. C₅₃	#18715	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×#3549	2
C₅₃×#3227	2
C₅₃×#3152	2
C₅₃×#1634	2
C₅₃×#1629	2
C₅₃×#1543	2
C₅₃×#1484	2
#18715	53

Subgroups — C53×#18715 #27226227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18715 #27226227