Subgroups — C₅₃×#32691 #27240203

Order 13568.

610 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2584	C₂	1	2	yes	maximal
2	C₅₃×#2087	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2086	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2059	C₂	1	2	yes	maximal
5	C₅₃×#2059	C₂	1	2	yes	maximal
6	C₅₃×#1985	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1943	C₂	1	2	yes	maximal
8	C₅₃×#1921	C₂	1	2	yes	maximal
9	C₅₃×#1918	C₂	1	2	yes	maximal
10	C₂×#125×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#107×C₅₃	C₂²	1	2	yes	—
15	C₂×#106×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#398	—	2	2	no	—
19	C₅₃×#395	—	2	2	no	—
20	C₅₃×#389	—	2	2	no	—
21	C₅₃×#389	—	2	2	no	—
22	C₅₃×#389	C₂²	1	2	yes	—
23	C₅₃×#389	C₂²	1	2	yes	—
24	C₅₃×#389	C₂²	1	2	yes	—
25	C₅₃×#389	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
27	C₅₃×#340	C₂²	1	1	yes	characteristic
28	C₅₃×#340	C₂²	1	2	yes	—
29	C₅₃×#340	C₂²	1	2	yes	—
30	C₅₃×#340	C₂²	1	1	yes	characteristic
31	C₅₃×#340	C₂²	1	2	yes	—
32	C₅₃×#340	C₂²	1	2	yes	—
33	C₅₃×#336	C₂²	1	2	yes	—
34	C₅₃×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
39	#125×C₅₃	—	4	2	no	—
40	#125×C₅₃	—	4	2	no	—
41	C₄×Q₈×C₅₃	—	4	1	no	—
42	C₄×Q₈×C₅₃	—	4	1	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
47	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
48	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₅₃	—	2	2	no	—
56	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
57	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
58	#107×C₅₃	—	4	1	no	—
59	#107×C₅₃	—	4	1	no	—
60	#107×C₅₃	Q₈	1	8	yes	complemented
61	#106×C₅₃	—	4	2	no	—
62	C₄⋊C₈×C₅₃	—	4	1	no	—
63	#97×C₅₃	—	4	1	no	—
64	#95×C₅₃	—	2	2	no	—
65	#95×C₅₃	—	2	2	no	—
66	#95×C₅₃	—	2	2	no	—
67	#95×C₅₃	—	2	2	no	—
68	#95×C₅₃	—	2	2	no	—
69	C₂×Q₈×C₅₃	—	4	1	no	—
70	C₂×Q₈×C₅₃	—	4	2	no	—
71	C₂²×C₄×C₅₃	—	2	2	no	—
72	C₂²×C₄×C₅₃	—	2	2	no	—
73	C₂²×C₄×C₅₃	—	2	2	no	—
74	C₂²×C₄×C₅₃	—	2	2	no	—
75	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₅₃	—	2	2	no	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	2	1	no	—
83	C₂×C₈×C₅₃	—	2	1	no	—
84	C₂×C₈×C₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	—	2	2	no	—
86	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
87	C₄⋊C₄×C₅₃	—	2	2	no	—
88	C₄⋊C₄×C₅₃	—	2	2	no	—
89	C₄⋊C₄×C₅₃	—	2	2	no	—
90	C₄⋊C₄×C₅₃	—	2	2	no	—
91	C₄⋊C₄×C₅₃	—	2	2	no	—
92	C₄⋊C₄×C₅₃	—	2	2	no	—
93	C₄⋊C₄×C₅₃	—	4	1	no	—
94	C₄⋊C₄×C₅₃	—	4	2	no	—
95	C₄⋊C₄×C₅₃	—	4	4	no	—
96	C₄⋊C₄×C₅₃	—	4	4	no	—
97	C₄⋊C₄×C₅₃	—	4	4	no	—
98	C₄⋊C₄×C₅₃	—	4	4	no	—
99	C₄⋊C₄×C₅₃	—	8	2	no	—
100	C₄⋊C₄×C₅₃	—	8	2	no	—
101	C₄⋊C₄×C₅₃	C₂×Q₈	1	4	yes	—
102	C₄⋊C₄×C₅₃	C₂×Q₈	1	4	yes	—
103	C₄²×C₅₃	—	2	1	no	—
104	C₄²×C₅₃	—	2	1	no	—
105	C₄²×C₅₃	—	2	2	no	—
106	C₄²×C₅₃	—	2	2	no	—
107	C₄²×C₅₃	—	8	2	no	—
108	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
109	Q₈×C₅₃	—	4	4	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	2	1	no	—
113	C₂×C₄×C₅₃	—	2	2	no	—
114	C₂×C₄×C₅₃	—	2	4	no	—
115	C₂×C₄×C₅₃	—	2	4	no	—
116	C₂×C₄×C₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	2	2	no	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	2	1	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	4	2	no	—
124	C₂×C₄×C₅₃	—	4	2	no	—
125	C₂×C₄×C₅₃	—	4	2	no	—
126	C₂×C₄×C₅₃	—	4	2	no	—
127	C₂×C₄×C₅₃	—	4	2	no	—
128	C₂×C₄×C₅₃	—	4	2	no	—
129	C₂×C₄×C₅₃	—	4	2	no	—
130	C₂×C₄×C₅₃	—	4	2	no	—
131	C₂×C₄×C₅₃	—	4	2	no	—
132	C₂×C₄×C₅₃	—	4	2	no	—
133	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
134	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
136	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
137	C₈×C₅₃	—	2	4	no	—
138	C₈×C₅₃	—	4	1	no	—
139	C₈×C₅₃	—	4	1	no	—
140	#32691	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂²×C₅₃	#534	1	1	yes	characteristic
142	C₂²×C₅₃	D₄×Q₈	1	2	yes	—
143	C₂²×C₅₃	#575	1	2	yes	—
144	C₂²×C₅₃	#576	1	1	yes	characteristic
145	C₂²×C₅₃	#578	1	1	yes	characteristic
146	C₄×C₅₃	—	4	4	no	—
147	C₄×C₅₃	—	4	4	no	—
148	C₄×C₅₃	—	4	1	no	—
149	C₄×C₅₃	—	4	2	no	—
150	C₄×C₅₃	—	4	1	no	—
151	C₄×C₅₃	—	4	2	no	—
152	C₄×C₅₃	—	8	2	no	—
153	C₄×C₅₃	—	8	2	no	—
154	C₄×C₅₃	#544	1	2	yes	—
155	C₄×C₅₃	Q₈²	1	1	yes	characteristic
156	C₄×C₅₃	#563	1	1	yes	characteristic
157	#2584	C₂×C₅₃	1	2	yes	—
158	#2087	C₂×C₅₃	1	1	yes	characteristic
159	#2086	C₂×C₅₃	1	1	yes	characteristic
160	#2059	C₂×C₅₃	1	2	yes	—
161	#2059	C₂×C₅₃	1	2	yes	—
162	#1985	C₂×C₅₃	1	1	yes	characteristic
163	#1943	C₂×C₅₃	1	2	yes	—
164	#1921	C₂×C₅₃	1	2	yes	—
165	#1918	C₂×C₅₃	1	2	yes	—
166	C₂×C₅₃	#2734	1	1	yes	characteristic
167	C₂×C₅₃	#3197	1	1	yes	characteristic
168	C₂×C₅₃	#3203	1	1	yes	characteristic
169	C₂×C₅₃	#3389	1	2	yes	—
170	C₂×C₅₃	#3390	1	2	yes	—
171	C₂×#125	C₂²×C₅₃	1	2	yes	—
172	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
173	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
174	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
175	C₂×#107	C₂²×C₅₃	1	2	yes	—
176	C₂×#106	C₂²×C₅₃	1	2	yes	—
177	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
178	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
179	#398	—	2	2	no	—
180	#395	—	2	2	no	—
181	#389	—	2	2	no	—
182	#389	—	2	2	no	—
183	#389	C₂²×C₅₃	1	2	yes	—
184	#389	C₂²×C₅₃	1	2	yes	—
185	#389	C₂²×C₅₃	1	2	yes	—
186	#389	C₂²×C₅₃	1	2	yes	—
187	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
188	#340	C₂²×C₅₃	1	2	yes	—
189	#340	C₂²×C₅₃	1	2	yes	—
190	#340	C₂²×C₅₃	1	2	yes	—
191	#340	C₂²×C₅₃	1	2	yes	—
192	#340	C₂²×C₅₃	1	1	yes	characteristic
193	#340	C₂²×C₅₃	1	1	yes	characteristic
194	#336	C₂²×C₅₃	1	2	yes	—
195	#336	C₂²×C₅₃	1	2	yes	—
196	C₅₃	#32691	1	1	yes	characteristic, direct product factor, Sylow
197	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
199	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
200	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
201	#125	—	4	2	no	—
202	#125	—	4	2	no	—
203	C₄×Q₈	—	4	1	no	—
204	C₄×Q₈	—	4	1	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
212	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
213	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
214	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
215	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
216	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
219	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
220	#107	—	4	1	no	—
221	#107	—	4	1	no	—
222	#107	Q₈×C₅₃	1	8	yes	complemented
223	#106	—	4	2	no	—
224	C₄⋊C₈	—	4	1	no	—
225	#97	—	4	1	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	2	no	—
230	#95	—	2	2	no	—
231	C₂×Q₈	—	4	1	no	—
232	C₂×Q₈	—	4	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
239	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
241	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₄⋊C₄	—	4	4	no	—
259	C₄⋊C₄	—	4	4	no	—
260	C₄⋊C₄	—	4	4	no	—
261	C₄⋊C₄	—	8	2	no	—
262	C₄⋊C₄	—	8	2	no	—
263	C₄⋊C₄	C₂×Q₈×C₅₃	1	4	yes	—
264	C₄⋊C₄	C₂×Q₈×C₅₃	1	4	yes	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	8	2	no	—
270	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
271	Q₈	—	4	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
296	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
297	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
298	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
299	C₈	—	2	4	no	—
300	C₈	—	4	1	no	—
301	C₈	—	4	1	no	—
302	C₂²	C₅₃×#534	1	1	yes	characteristic
303	C₂²	D₄×Q₈×C₅₃	1	2	yes	—
304	C₂²	C₅₃×#575	1	2	yes	—
305	C₂²	C₅₃×#576	1	1	yes	characteristic
306	C₂²	C₅₃×#578	1	1	yes	characteristic
307	C₄	—	4	2	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	2	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	4	no	—
312	C₄	—	4	4	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	2	no	—
315	C₄	C₅₃×#544	1	2	yes	—
316	C₄	Q₈²×C₅₃	1	1	yes	characteristic
317	C₄	C₅₃×#563	1	1	yes	characteristic
318	C₂	C₅₃×#2734	1	1	yes	characteristic
319	C₂	C₅₃×#3197	1	1	yes	characteristic
320	C₂	C₅₃×#3203	1	1	yes	characteristic
321	C₂	C₅₃×#3389	1	2	yes	—
322	C₂	C₅₃×#3390	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄×Q₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
60. #107×C₅₃	Q₈	semidirect
140. #32691	C₅₃	direct (N × H)
196. C₅₃	#32691	direct (N × H)
222. #107	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2584	2
C₅₃×#2087	2
C₅₃×#2086	2
C₅₃×#2059	2
C₅₃×#1985	2
C₅₃×#1943	2
C₅₃×#1921	2
C₅₃×#1918	2
#32691	53

Subgroups — C53×#32691 #27240203

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#32691 #27240203