Subgroups — C₅₃×#9689 #27217201

Order 13568.

596 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×#2858	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1767	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1745	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#329	C₂	1	4	yes	maximal, complemented
5	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
6	C₂×#127×C₅₃	C₄	1	1	yes	characteristic
7	C₂×#124×C₅₃	C₄	1	2	yes	—
8	C₂×#124×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
10	C₂×#97×C₅₃	C₂²	1	2	yes	—
11	C₂×#97×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#390	—	2	1	no	—
13	C₅₃×#390	—	2	1	no	—
14	C₅₃×#390	—	2	1	no	—
15	C₅₃×#390	—	2	1	no	—
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#336	—	2	1	no	—
18	C₅₃×#336	—	2	1	no	—
19	C₅₃×#336	—	2	1	no	—
20	C₅₃×#329	—	2	8	no	—
21	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×D₄×C₅₃	—	2	1	no	—
23	C₂²×D₄×C₅₃	—	2	1	no	—
24	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₅₃	—	2	2	no	—
26	C₂²×C₈×C₅₃	—	2	2	no	—
27	C₄⋊Q₈×C₅₃	—	4	2	no	—
28	#127×C₅₃	—	4	2	no	—
29	#124×C₅₃	—	2	4	no	—
30	#124×C₅₃	—	2	4	no	—
31	#124×C₅₃	—	4	4	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₅₃	—	2	4	no	—
46	#97×C₅₃	—	2	4	no	—
47	#97×C₅₃	—	2	2	no	—
48	#97×C₅₃	D₄	1	4	yes	complemented
49	C₂⁴×C₅₃	—	2	2	no	—
50	C₂×Q₈×C₅₃	—	2	4	no	—
51	C₂×Q₈×C₅₃	—	4	2	no	—
52	C₂×Q₈×C₅₃	—	4	2	no	—
53	C₂×D₄×C₅₃	—	2	4	no	—
54	C₂×D₄×C₅₃	—	4	2	no	—
55	C₂×D₄×C₅₃	—	4	2	no	—
56	C₂×D₄×C₅₃	—	4	2	no	—
57	C₂×D₄×C₅₃	—	4	2	no	—
58	C₂×D₄×C₅₃	—	4	2	no	—
59	C₂×D₄×C₅₃	—	4	2	no	—
60	C₂×D₄×C₅₃	—	4	2	no	—
61	C₂×D₄×C₅₃	—	4	2	no	—
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	—	2	4	no	—
71	C₂×C₈×C₅₃	—	2	4	no	—
72	C₂×C₈×C₅₃	—	4	2	no	—
73	C₂×C₈×C₅₃	—	4	2	no	—
74	C₂×C₈×C₅₃	—	4	2	no	—
75	C₂×C₈×C₅₃	—	4	2	no	—
76	C₄⋊C₄×C₅₃	—	4	2	no	—
77	C₄⋊C₄×C₅₃	—	4	2	no	—
78	C₂²⋊C₄×C₅₃	—	4	4	no	—
79	C₂²⋊C₄×C₅₃	—	4	4	no	—
80	C₂²⋊C₄×C₅₃	—	4	4	no	—
81	C₂²⋊C₄×C₅₃	—	4	2	no	—
82	C₂²⋊C₄×C₅₃	—	4	2	no	—
83	C₄²×C₅₃	—	2	2	no	—
84	C₄²×C₅₃	—	2	2	no	—
85	C₄²×C₅₃	—	2	2	no	—
86	C₄²×C₅₃	—	4	1	no	—
87	C₄²×C₅₃	—	4	1	no	—
88	C₄²×C₅₃	—	4	1	no	—
89	C₄²×C₅₃	C₂×D₄	1	2	yes	—
90	C₄²×C₅₃	#41	1	2	yes	—
91	C₂³×C₅₃	—	4	2	no	—
92	C₂³×C₅₃	—	4	2	no	—
93	C₂³×C₅₃	—	4	2	no	—
94	C₂³×C₅₃	—	4	2	no	—
95	C₂³×C₅₃	—	4	2	no	—
96	C₂³×C₅₃	—	4	2	no	—
97	C₂³×C₅₃	—	4	2	no	—
98	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	Q₈×C₅₃	—	8	4	no	—
100	D₄×C₅₃	—	8	4	no	—
101	D₄×C₅₃	—	8	4	no	—
102	D₄×C₅₃	—	8	4	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	1	no	—
113	C₂×C₄×C₅₃	—	2	1	no	—
114	C₂×C₄×C₅₃	—	2	1	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	2	1	no	—
118	C₂×C₄×C₅₃	—	4	2	no	—
119	C₂×C₄×C₅₃	—	4	2	no	—
120	C₂×C₄×C₅₃	—	4	2	no	—
121	C₂×C₄×C₅₃	#100	1	2	yes	—
122	C₂×C₄×C₅₃	#100	1	2	yes	—
123	C₂×C₄×C₅₃	#104	1	2	yes	—
124	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₅₃	#121	1	2	yes	—
126	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
127	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
128	C₈×C₅₃	—	4	4	no	—
129	C₈×C₅₃	—	4	4	no	—
130	#9689	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₅₃	—	8	2	no	—
132	C₂²×C₅₃	—	8	2	no	—
133	C₂²×C₅₃	—	8	2	no	—
134	C₂²×C₅₃	—	8	2	no	—
135	C₂²×C₅₃	—	8	2	no	—
136	C₂²×C₅₃	—	8	2	no	—
137	C₂²×C₅₃	—	8	2	no	—
138	C₂²×C₅₃	#390	1	1	yes	characteristic
139	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
140	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
141	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
142	C₂²×C₅₃	#421	1	1	yes	characteristic
143	C₂²×C₅₃	#421	1	1	yes	characteristic
144	C₂²×C₅₃	#421	1	1	yes	characteristic
145	C₄×C₅₃	—	2	2	no	—
146	C₄×C₅₃	—	2	2	no	—
147	C₄×C₅₃	—	2	1	no	—
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	2	2	no	—
150	C₄×C₅₃	—	2	2	no	—
151	C₄×C₅₃	—	2	2	no	—
152	C₄×C₅₃	—	4	1	no	—
153	C₄×C₅₃	—	8	2	no	—
154	#2858	C₂×C₅₃	1	1	yes	characteristic
155	#1767	C₂×C₅₃	1	1	yes	characteristic, complemented
156	#1745	C₂×C₅₃	1	1	yes	characteristic, complemented
157	C₂×#329	C₂×C₅₃	1	4	yes	complemented
158	C₂×C₅₃	—	16	2	no	—
159	C₂×C₅₃	#1527	1	1	yes	characteristic
160	C₂×C₅₃	#1530	1	1	yes	characteristic
161	C₂×C₅₃	#1533	1	1	yes	characteristic
162	C₂×C₅₃	#1536	1	1	yes	characteristic
163	C₂×C₅₃	#1963	1	1	yes	characteristic
164	C₂×C₅₃	#1965	1	1	yes	characteristic
165	C₂×C₅₃	#1980	1	1	yes	characteristic
166	C₂×C₄⋊Q₈	C₄×C₅₃	1	1	yes	characteristic
167	C₂×#127	C₄×C₅₃	1	1	yes	characteristic
168	C₂×#124	C₄×C₅₃	1	2	yes	—
169	C₂×#124	C₂²×C₅₃	1	2	yes	—
170	C₂×C₄⋊C₈	—	2	1	no	—
171	C₂×#97	C₂²×C₅₃	1	2	yes	—
172	C₂×#97	C₂²×C₅₃	1	2	yes	—
173	#390	—	2	1	no	—
174	#390	—	2	1	no	—
175	#390	—	2	1	no	—
176	#390	—	2	1	no	—
177	C₄³	C₂²×C₅₃	1	1	yes	characteristic
178	#336	—	2	1	no	—
179	#336	—	2	1	no	—
180	#336	—	2	1	no	—
181	#329	—	2	8	no	—
182	C₅₃	#9689	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
184	C₂²×D₄	—	2	1	no	—
185	C₂²×D₄	—	2	1	no	—
186	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₄⋊Q₈	—	4	2	no	—
190	#127	—	4	2	no	—
191	#124	—	2	4	no	—
192	#124	—	2	4	no	—
193	#124	—	4	4	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₂²⋊C₄	—	2	2	no	—
197	C₂×C₂²⋊C₄	—	2	2	no	—
198	C₂×C₂²⋊C₄	—	2	2	no	—
199	C₂×C₂²⋊C₄	—	2	2	no	—
200	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
201	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
202	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
203	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
204	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
205	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
206	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
207	C₄⋊C₈	—	2	4	no	—
208	#97	—	2	4	no	—
209	#97	—	2	2	no	—
210	#97	D₄×C₅₃	1	4	yes	complemented
211	C₂⁴	—	2	2	no	—
212	C₂×Q₈	—	2	4	no	—
213	C₂×Q₈	—	4	2	no	—
214	C₂×Q₈	—	4	2	no	—
215	C₂×D₄	—	2	4	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂×D₄	—	4	2	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	2	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
226	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
227	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
228	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
229	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	4	no	—
241	C₂²⋊C₄	—	4	4	no	—
242	C₂²⋊C₄	—	4	4	no	—
243	C₂²⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	4	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	4	1	no	—
249	C₄²	—	4	1	no	—
250	C₄²	—	4	1	no	—
251	C₄²	C₂×D₄×C₅₃	1	2	yes	—
252	C₄²	#41×C₅₃	1	2	yes	—
253	C₂³	—	4	2	no	—
254	C₂³	—	4	2	no	—
255	C₂³	—	4	2	no	—
256	C₂³	—	4	2	no	—
257	C₂³	—	4	2	no	—
258	C₂³	—	4	2	no	—
259	C₂³	—	4	2	no	—
260	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
261	Q₈	—	8	4	no	—
262	D₄	—	8	4	no	—
263	D₄	—	8	4	no	—
264	D₄	—	8	4	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	#100×C₅₃	1	2	yes	—
284	C₂×C₄	#100×C₅₃	1	2	yes	—
285	C₂×C₄	#104×C₅₃	1	2	yes	—
286	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
287	C₂×C₄	#121×C₅₃	1	2	yes	—
288	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
289	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
290	C₈	—	4	4	no	—
291	C₈	—	4	4	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	—	8	2	no	—
295	C₂²	—	8	2	no	—
296	C₂²	—	8	2	no	—
297	C₂²	—	8	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	C₅₃×#390	1	1	yes	characteristic
300	C₂²	C₂×#100×C₅₃	1	1	yes	characteristic
301	C₂²	C₂×#100×C₅₃	1	1	yes	characteristic
302	C₂²	C₂×#104×C₅₃	1	1	yes	characteristic
303	C₂²	C₅₃×#421	1	1	yes	characteristic
304	C₂²	C₅₃×#421	1	1	yes	characteristic
305	C₂²	C₅₃×#421	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	4	1	no	—
314	C₄	—	8	2	no	—
315	C₂	—	16	2	no	—
316	C₂	C₅₃×#1527	1	1	yes	characteristic
317	C₂	C₅₃×#1530	1	1	yes	characteristic
318	C₂	C₅₃×#1533	1	1	yes	characteristic
319	C₂	C₅₃×#1536	1	1	yes	characteristic
320	C₂	C₅₃×#1963	1	1	yes	characteristic
321	C₂	C₅₃×#1965	1	1	yes	characteristic
322	C₂	C₅₃×#1980	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂×#127×C₅₃	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1767	C₂	semidirect
3. C₅₃×#1745	C₂	semidirect
4. C₂×C₅₃×#329	C₂	semidirect
48. #97×C₅₃	D₄	semidirect
130. #9689	C₅₃	direct (N × H)
155. #1767	C₂×C₅₃	semidirect
156. #1745	C₂×C₅₃	semidirect
157. C₂×#329	C₂×C₅₃	semidirect
182. C₅₃	#9689	direct (N × H)
210. #97	D₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2858	2
C₅₃×#1767	2
C₅₃×#1745	2
C₂×C₅₃×#329	2
#9689	53

Subgroups — C53×#9689 #27217201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9689 #27217201