Subgroups — C₅₃×#60646 #27268158

Order 13568.

3250 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#577	C₂	1	4	yes	maximal, complemented
2	C₅₃×#3499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3498	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3154	C₂	1	12	yes	maximal, complemented
5	C₅₃×#3152	C₂	1	3	yes	maximal, complemented
6	C₅₃×#3151	C₂	1	3	yes	maximal, complemented
7	C₅₃×#3147	C₂	1	3	yes	maximal, complemented
8	C₂×C₅₃×#492	C₂	1	3	yes	maximal, complemented
9	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	2	yes	—
11	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#577	—	2	32	no	—
13	C₂×#137×C₅₃	C₂²	1	12	yes	complemented
14	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	12	yes	complemented
15	C₂×#135×C₅₃	C₂²	1	12	yes	complemented
16	C₂²×SD₁₆×C₅₃	C₂²	1	12	yes	complemented
17	C₂×#131×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#550	—	2	2	no	—
19	D₄×Q₈×C₅₃	—	2	2	no	—
20	C₅₃×#547	—	2	6	no	—
21	D₄²×C₅₃	—	2	6	no	—
22	C₅₃×#536	C₂²	1	6	yes	complemented
23	C₅₃×#535	C₂²	1	6	yes	—
24	C₅₃×#532	C₂²	1	3	yes	—
25	C₅₃×#532	C₂²	1	3	yes	complemented
26	C₂×C₄⋊Q₈×C₅₃	C₂²	1	3	yes	complemented
27	C₂×#127×C₅₃	C₂²	1	3	yes	—
28	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#497	C₂²	1	12	yes	complemented
30	C₅₃×#496	C₂²	1	12	yes	complemented
31	C₅₃×#495	C₂²	1	12	yes	complemented
32	C₅₃×#494	C₂²	1	3	yes	complemented
33	C₅₃×#493	C₂²	1	3	yes	complemented
34	C₅₃×#492	C₂²	1	3	yes	complemented
35	C₅₃×#492	C₂²	1	6	yes	complemented
36	C₅₃×#492	C₂²	1	12	yes	complemented
37	C₅₃×#492	C₂²	1	1	yes	characteristic, complemented
38	C₅₃×#405	C₂²	1	6	yes	complemented
39	C₄×M₁₆×C₅₃	C₂²	1	3	yes	complemented
40	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	complemented
41	2₋¹⁺⁴×C₅₃	—	2	16	no	—
42	H(5,𝔽₂)×C₅₃	—	2	16	no	—
43	C₂×#41×C₅₃	—	2	2	no	—
44	C₂×#41×C₅₃	—	2	6	no	—
45	C₂×#41×C₅₃	C₂³	1	6	yes	—
46	C₂×#41×C₅₃	C₂³	1	6	yes	—
47	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×Q₈×C₅₃	—	2	2	no	—
49	C₂²×Q₈×C₅₃	C₂³	1	6	yes	—
50	C₂²×D₄×C₅₃	—	2	6	no	—
51	C₂²×D₄×C₅₃	—	2	6	no	—
52	C₂²×D₄×C₅₃	C₂³	1	6	yes	—
53	#137×C₅₃	—	2	48	no	—
54	AGL(1,ℤ/8)×C₅₃	—	2	48	no	—
55	#135×C₅₃	—	2	48	no	—
56	C₂×Q₁₆×C₅₃	C₂³	1	12	yes	—
57	C₂×SD₁₆×C₅₃	—	2	48	no	—
58	C₂×SD₁₆×C₅₃	C₂³	1	12	yes	—
59	C₂×SD₁₆×C₅₃	C₂³	1	24	yes	—
60	C₂×SD₁₆×C₅₃	C₂³	1	4	yes	—
61	C₂×D₈×C₅₃	C₂³	1	12	yes	—
62	#131×C₅₃	D₄	1	16	yes	complemented
63	C₂×M₁₆×C₅₃	C₂³	1	6	yes	—
64	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
65	C₄⋊Q₈×C₅₃	—	2	6	no	—
66	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
67	C₄⋊Q₈×C₅₃	C₂³	1	6	yes	—
68	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
69	#127×C₅₃	—	2	6	no	—
70	#127×C₅₃	C₂³	1	3	yes	—
71	#127×C₅₃	C₂³	1	6	yes	—
72	#127×C₅₃	C₂³	1	1	yes	characteristic
73	#124×C₅₃	C₂³	1	6	yes	—
74	#124×C₅₃	C₂³	1	6	yes	—
75	#123×C₅₃	—	2	12	no	—
76	C₂²⋊Q₈×C₅₃	—	2	12	no	—
77	#121×C₅₃	—	2	12	no	—
78	#121×C₅₃	—	2	6	no	—
79	#121×C₅₃	—	2	6	no	—
80	#120×C₅₃	—	2	12	no	—
81	C₄×Q₈×C₅₃	D₄	1	2	yes	—
82	C₄×D₄×C₅₃	—	2	6	no	—
83	C₄×D₄×C₅₃	—	2	6	no	—
84	C₄×D₄×C₅₃	D₄	1	6	yes	—
85	#117×C₅₃	C₂³	1	3	yes	—
86	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
87	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
88	#97×C₅₃	C₂³	1	6	yes	—
89	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₅₃	C₂³	1	3	yes	—
91	C₄×C₈×C₅₃	C₂³	1	6	yes	—
92	C₂⁴×C₅₃	—	2	6	no	—
93	#41×C₅₃	—	2	24	no	—
94	#41×C₅₃	—	2	24	no	—
95	#41×C₅₃	—	4	8	no	—
96	#41×C₅₃	—	4	24	no	—
97	#41×C₅₃	C₂×D₄	1	8	yes	—
98	C₂×Q₈×C₅₃	—	2	6	no	—
99	C₂×Q₈×C₅₃	—	2	24	no	—
100	C₂×Q₈×C₅₃	—	4	8	no	—
101	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
103	C₂×Q₈×C₅₃	C₂⁴	1	6	yes	—
104	C₂×D₄×C₅₃	—	2	6	no	—
105	C₂×D₄×C₅₃	—	2	6	no	—
106	C₂×D₄×C₅₃	—	2	24	no	—
107	C₂×D₄×C₅₃	—	2	2	no	—
108	C₂×D₄×C₅₃	—	2	6	no	—
109	C₂×D₄×C₅₃	—	2	6	no	—
110	C₂×D₄×C₅₃	—	2	12	no	—
111	C₂×D₄×C₅₃	—	4	24	no	—
112	C₂×D₄×C₅₃	—	8	12	no	—
113	C₂×D₄×C₅₃	C₂×D₄	1	3	yes	—
114	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
115	C₂×D₄×C₅₃	C₂⁴	1	6	yes	—
116	C₂²×C₄×C₅₃	—	2	6	no	—
117	C₂²×C₄×C₅₃	—	2	6	no	—
118	C₂²×C₄×C₅₃	—	2	6	no	—
119	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
120	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
121	Q₁₆×C₅₃	—	2	24	no	—
122	SD₁₆×C₅₃	—	2	24	no	—
123	SD₁₆×C₅₃	—	2	48	no	—
124	SD₁₆×C₅₃	—	2	8	no	—
125	D₈×C₅₃	—	2	24	no	—
126	M₁₆×C₅₃	C₂×D₄	1	24	yes	complemented
127	C₂×C₈×C₅₃	C₂×D₄	1	24	yes	complemented
128	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
129	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
130	C₄⋊C₄×C₅₃	—	2	6	no	—
131	C₄⋊C₄×C₅₃	—	2	12	no	—
132	C₄⋊C₄×C₅₃	—	2	2	no	—
133	C₄⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
134	C₂²⋊C₄×C₅₃	—	2	12	no	—
135	C₂²⋊C₄×C₅₃	—	2	12	no	—
136	C₂²⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
137	C₄²×C₅₃	C₂×D₄	1	6	yes	—
138	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
139	C₄²×C₅₃	C₂⁴	1	3	yes	—
140	C₂³×C₅₃	—	2	2	no	—
141	C₂³×C₅₃	—	2	6	no	—
142	C₂³×C₅₃	—	4	12	no	—
143	C₂³×C₅₃	—	8	12	no	—
144	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
145	Q₈×C₅₃	—	2	12	no	—
146	Q₈×C₅₃	—	2	4	no	—
147	Q₈×C₅₃	—	4	12	no	—
148	Q₈×C₅₃	#127	1	4	yes	—
149	D₄×C₅₃	—	2	12	no	—
150	D₄×C₅₃	—	2	4	no	—
151	D₄×C₅₃	—	4	12	no	—
152	D₄×C₅₃	—	4	12	no	—
153	D₄×C₅₃	—	4	12	no	—
154	D₄×C₅₃	—	4	4	no	—
155	D₄×C₅₃	—	4	12	no	—
156	D₄×C₅₃	—	8	12	no	—
157	D₄×C₅₃	#127	1	12	yes	—
158	C₂×C₄×C₅₃	—	2	12	no	—
159	C₂×C₄×C₅₃	—	2	2	no	—
160	C₂×C₄×C₅₃	—	2	6	no	—
161	C₂×C₄×C₅₃	—	4	12	no	—
162	C₂×C₄×C₅₃	—	4	12	no	—
163	C₂×C₄×C₅₃	—	4	12	no	—
164	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
165	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
166	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
167	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
168	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
169	C₈×C₅₃	C₂²×D₄	1	4	yes	—
170	C₈×C₅₃	C₂²×D₄	1	12	yes	—
171	#60646	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂²×C₅₃	—	2	6	no	—
173	C₂²×C₅₃	—	4	2	no	—
174	C₂²×C₅₃	—	4	6	no	—
175	C₂²×C₅₃	—	4	12	no	—
176	C₂²×C₅₃	—	4	4	no	—
177	C₂²×C₅₃	—	16	12	no	—
178	C₂²×C₅₃	C₂×#127	1	6	yes	—
179	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
180	C₄×C₅₃	—	2	6	no	—
181	C₄×C₅₃	—	4	2	no	—
182	C₄×C₅₃	—	4	6	no	—
183	C₄×C₅₃	C₂×#127	1	6	yes	—
184	C₄×C₅₃	#577	1	4	yes	complemented
185	C₄×C₅₃	C₂³×D₄	1	2	yes	—
186	C₂×#577	C₂×C₅₃	1	4	yes	complemented
187	#3499	C₂×C₅₃	1	1	yes	characteristic, complemented
188	#3498	C₂×C₅₃	1	1	yes	characteristic, complemented
189	#3154	C₂×C₅₃	1	12	yes	complemented
190	#3152	C₂×C₅₃	1	3	yes	complemented
191	#3151	C₂×C₅₃	1	3	yes	complemented
192	#3147	C₂×C₅₃	1	3	yes	complemented
193	C₂×#492	C₂×C₅₃	1	3	yes	complemented
194	C₄×#131	C₂×C₅₃	1	1	yes	characteristic, complemented
195	C₂×C₅₃	—	2	6	no	—
196	C₂×C₅₃	—	8	2	no	—
197	C₂×C₅₃	—	8	6	no	—
198	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
199	C₂×C₅₃	C₂×#577	1	2	yes	—
200	C₂×2₋¹⁺⁴	C₂²×C₅₃	1	2	yes	—
201	C₂×H(5,𝔽₂)	C₂²×C₅₃	1	2	yes	—
202	#577	—	2	32	no	—
203	C₂×#137	C₂²×C₅₃	1	12	yes	complemented
204	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	12	yes	complemented
205	C₂×#135	C₂²×C₅₃	1	12	yes	complemented
206	C₂²×SD₁₆	C₂²×C₅₃	1	12	yes	complemented
207	C₂×#131	C₂²×C₅₃	1	2	yes	—
208	#550	—	2	2	no	—
209	D₄×Q₈	—	2	2	no	—
210	#547	—	2	6	no	—
211	D₄²	—	2	6	no	—
212	#536	C₂²×C₅₃	1	6	yes	complemented
213	#535	C₂²×C₅₃	1	6	yes	—
214	#532	C₂²×C₅₃	1	3	yes	—
215	#532	C₂²×C₅₃	1	3	yes	complemented
216	C₂×C₄⋊Q₈	C₂²×C₅₃	1	3	yes	complemented
217	C₂×#127	C₂²×C₅₃	1	3	yes	—
218	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
219	#497	C₂²×C₅₃	1	12	yes	complemented
220	#496	C₂²×C₅₃	1	12	yes	complemented
221	#495	C₂²×C₅₃	1	12	yes	complemented
222	#494	C₂²×C₅₃	1	3	yes	complemented
223	#493	C₂²×C₅₃	1	3	yes	complemented
224	#492	C₂²×C₅₃	1	3	yes	complemented
225	#492	C₂²×C₅₃	1	6	yes	complemented
226	#492	C₂²×C₅₃	1	12	yes	complemented
227	#492	C₂²×C₅₃	1	1	yes	characteristic, complemented
228	#405	C₂²×C₅₃	1	6	yes	complemented
229	C₄×M₁₆	C₂²×C₅₃	1	3	yes	complemented
230	C₂×C₄×C₈	C₂²×C₅₃	1	3	yes	complemented
231	C₅₃	#60646	1	1	yes	characteristic, direct product factor, Sylow
232	2₋¹⁺⁴	—	2	16	no	—
233	H(5,𝔽₂)	—	2	16	no	—
234	C₂×#41	—	2	2	no	—
235	C₂×#41	—	2	6	no	—
236	C₂×#41	C₂³×C₅₃	1	6	yes	—
237	C₂×#41	C₂³×C₅₃	1	6	yes	—
238	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
239	C₂²×Q₈	—	2	2	no	—
240	C₂²×Q₈	C₂³×C₅₃	1	6	yes	—
241	C₂²×D₄	—	2	6	no	—
242	C₂²×D₄	—	2	6	no	—
243	C₂²×D₄	C₂³×C₅₃	1	6	yes	—
244	#137	—	2	48	no	—
245	AGL(1,ℤ/8)	—	2	48	no	—
246	#135	—	2	48	no	—
247	C₂×Q₁₆	C₂³×C₅₃	1	12	yes	—
248	C₂×SD₁₆	—	2	48	no	—
249	C₂×SD₁₆	C₂³×C₅₃	1	12	yes	—
250	C₂×SD₁₆	C₂³×C₅₃	1	24	yes	—
251	C₂×SD₁₆	C₂³×C₅₃	1	4	yes	—
252	C₂×D₈	C₂³×C₅₃	1	12	yes	—
253	#131	D₄×C₅₃	1	16	yes	complemented
254	C₂×M₁₆	C₂³×C₅₃	1	6	yes	—
255	C₂²×C₈	C₂³×C₅₃	1	6	yes	—
256	C₄⋊Q₈	—	2	6	no	—
257	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
258	C₄⋊Q₈	C₂³×C₅₃	1	6	yes	—
259	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
260	#127	—	2	6	no	—
261	#127	C₂³×C₅₃	1	3	yes	—
262	#127	C₂³×C₅₃	1	6	yes	—
263	#127	C₂³×C₅₃	1	1	yes	characteristic
264	#124	C₂³×C₅₃	1	6	yes	—
265	#124	C₂³×C₅₃	1	6	yes	—
266	#123	—	2	12	no	—
267	C₂²⋊Q₈	—	2	12	no	—
268	#121	—	2	6	no	—
269	#121	—	2	6	no	—
270	#121	—	2	12	no	—
271	#120	—	2	12	no	—
272	C₄×Q₈	D₄×C₅₃	1	2	yes	—
273	C₄×D₄	—	2	6	no	—
274	C₄×D₄	—	2	6	no	—
275	C₄×D₄	D₄×C₅₃	1	6	yes	—
276	#117	C₂³×C₅₃	1	3	yes	—
277	C₂×C₄⋊C₄	—	2	6	no	—
278	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
279	#97	C₂³×C₅₃	1	6	yes	—
280	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
281	C₄×C₈	C₂³×C₅₃	1	3	yes	—
282	C₄×C₈	C₂³×C₅₃	1	6	yes	—
283	C₂⁴	—	2	6	no	—
284	#41	—	2	24	no	—
285	#41	—	2	24	no	—
286	#41	—	4	8	no	—
287	#41	—	4	24	no	—
288	#41	C₂×D₄×C₅₃	1	8	yes	—
289	C₂×Q₈	—	2	6	no	—
290	C₂×Q₈	—	2	24	no	—
291	C₂×Q₈	—	4	8	no	—
292	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
293	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
294	C₂×Q₈	C₂⁴×C₅₃	1	6	yes	—
295	C₂×D₄	—	2	6	no	—
296	C₂×D₄	—	2	6	no	—
297	C₂×D₄	—	2	24	no	—
298	C₂×D₄	—	2	2	no	—
299	C₂×D₄	—	2	6	no	—
300	C₂×D₄	—	2	6	no	—
301	C₂×D₄	—	2	12	no	—
302	C₂×D₄	—	4	24	no	—
303	C₂×D₄	—	8	12	no	—
304	C₂×D₄	C₂×D₄×C₅₃	1	3	yes	—
305	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
306	C₂×D₄	C₂⁴×C₅₃	1	6	yes	—
307	C₂²×C₄	—	2	6	no	—
308	C₂²×C₄	—	2	6	no	—
309	C₂²×C₄	—	2	6	no	—
310	C₂²×C₄	C₂×D₄×C₅₃	1	6	yes	—
311	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
312	Q₁₆	—	2	24	no	—
313	SD₁₆	—	2	24	no	—
314	SD₁₆	—	2	48	no	—
315	SD₁₆	—	2	8	no	—
316	D₈	—	2	24	no	—
317	M₁₆	C₂×D₄×C₅₃	1	24	yes	complemented
318	C₂×C₈	C₂×D₄×C₅₃	1	24	yes	complemented
319	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
320	C₂×C₈	C₂⁴×C₅₃	1	6	yes	—
321	C₄⋊C₄	—	2	6	no	—
322	C₄⋊C₄	—	2	12	no	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	C₂×D₄×C₅₃	1	6	yes	—
325	C₂²⋊C₄	—	2	12	no	—
326	C₂²⋊C₄	—	2	12	no	—
327	C₂²⋊C₄	C₂×D₄×C₅₃	1	6	yes	—
328	C₄²	C₂×D₄×C₅₃	1	6	yes	—
329	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
330	C₄²	C₂⁴×C₅₃	1	3	yes	—
331	C₂³	—	2	2	no	—
332	C₂³	—	2	6	no	—
333	C₂³	—	4	12	no	—
334	C₂³	—	8	12	no	—
335	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
336	Q₈	—	2	12	no	—
337	Q₈	—	2	4	no	—
338	Q₈	—	4	12	no	—
339	Q₈	#127×C₅₃	1	4	yes	—
340	D₄	—	2	4	no	—
341	D₄	—	2	12	no	—
342	D₄	—	4	12	no	—
343	D₄	—	4	12	no	—
344	D₄	—	4	12	no	—
345	D₄	—	4	4	no	—
346	D₄	—	4	12	no	—
347	D₄	—	8	12	no	—
348	D₄	#127×C₅₃	1	12	yes	—
349	C₂×C₄	—	2	12	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	6	no	—
352	C₂×C₄	—	4	12	no	—
353	C₂×C₄	—	4	12	no	—
354	C₂×C₄	—	4	12	no	—
355	C₂×C₄	C₂²×D₄×C₅₃	1	6	yes	—
356	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
357	C₂×C₄	C₂²×D₄×C₅₃	1	12	yes	—
358	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
359	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
360	C₈	C₂²×D₄×C₅₃	1	4	yes	—
361	C₈	C₂²×D₄×C₅₃	1	12	yes	—
362	C₂²	—	2	6	no	—
363	C₂²	—	4	12	no	—
364	C₂²	—	4	4	no	—
365	C₂²	—	4	2	no	—
366	C₂²	—	4	6	no	—
367	C₂²	—	16	12	no	—
368	C₂²	C₂×#127×C₅₃	1	6	yes	—
369	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
370	C₄	—	2	6	no	—
371	C₄	—	4	2	no	—
372	C₄	—	4	6	no	—
373	C₄	C₂×#127×C₅₃	1	6	yes	—
374	C₄	C₅₃×#577	1	4	yes	complemented
375	C₄	C₂³×D₄×C₅₃	1	2	yes	—
376	C₂	—	2	6	no	—
377	C₂	—	8	6	no	—
378	C₂	—	8	2	no	—
379	C₂	C₂²×#127×C₅₃	1	1	yes	characteristic
380	C₂	C₂×C₅₃×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#577	C₂	semidirect
2. C₅₃×#3499	C₂	semidirect
3. C₅₃×#3498	C₂	semidirect
4. C₅₃×#3154	C₂	semidirect
5. C₅₃×#3152	C₂	semidirect
6. C₅₃×#3151	C₂	semidirect
7. C₅₃×#3147	C₂	semidirect
8. C₂×C₅₃×#492	C₂	semidirect
9. C₄×#131×C₅₃	C₂	semidirect
13. C₂×#137×C₅₃	C₂²	semidirect
14. C₂×AGL(1,ℤ/8)×C₅₃	C₂²	semidirect
15. C₂×#135×C₅₃	C₂²	semidirect
16. C₂²×SD₁₆×C₅₃	C₂²	semidirect
22. C₅₃×#536	C₂²	semidirect
25. C₅₃×#532	C₂²	semidirect
26. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
29. C₅₃×#497	C₂²	semidirect
30. C₅₃×#496	C₂²	semidirect
31. C₅₃×#495	C₂²	semidirect
32. C₅₃×#494	C₂²	semidirect
33. C₅₃×#493	C₂²	semidirect
34. C₅₃×#492	C₂²	semidirect
35. C₅₃×#492	C₂²	semidirect
36. C₅₃×#492	C₂²	semidirect
37. C₅₃×#492	C₂²	semidirect
38. C₅₃×#405	C₂²	semidirect
39. C₄×M₁₆×C₅₃	C₂²	semidirect
40. C₂×C₄×C₈×C₅₃	C₂²	semidirect
62. #131×C₅₃	D₄	semidirect
126. M₁₆×C₅₃	C₂×D₄	semidirect
127. C₂×C₈×C₅₃	C₂×D₄	semidirect
171. #60646	C₅₃	direct (N × H)
184. C₄×C₅₃	#577	semidirect
186. C₂×#577	C₂×C₅₃	semidirect
187. #3499	C₂×C₅₃	semidirect
188. #3498	C₂×C₅₃	semidirect
189. #3154	C₂×C₅₃	semidirect
190. #3152	C₂×C₅₃	semidirect
191. #3151	C₂×C₅₃	semidirect
192. #3147	C₂×C₅₃	semidirect
193. C₂×#492	C₂×C₅₃	semidirect
194. C₄×#131	C₂×C₅₃	semidirect
203. C₂×#137	C₂²×C₅₃	semidirect
204. C₂×AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
205. C₂×#135	C₂²×C₅₃	semidirect
206. C₂²×SD₁₆	C₂²×C₅₃	semidirect
212. #536	C₂²×C₅₃	semidirect
215. #532	C₂²×C₅₃	semidirect
216. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
219. #497	C₂²×C₅₃	semidirect
220. #496	C₂²×C₅₃	semidirect
221. #495	C₂²×C₅₃	semidirect
222. #494	C₂²×C₅₃	semidirect
223. #493	C₂²×C₅₃	semidirect
224. #492	C₂²×C₅₃	semidirect
225. #492	C₂²×C₅₃	semidirect
226. #492	C₂²×C₅₃	semidirect
227. #492	C₂²×C₅₃	semidirect
228. #405	C₂²×C₅₃	semidirect
229. C₄×M₁₆	C₂²×C₅₃	semidirect
230. C₂×C₄×C₈	C₂²×C₅₃	semidirect
231. C₅₃	#60646	direct (N × H)
253. #131	D₄×C₅₃	semidirect
317. M₁₆	C₂×D₄×C₅₃	semidirect
318. C₂×C₈	C₂×D₄×C₅₃	semidirect
374. C₄	C₅₃×#577	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#577	2
C₅₃×#3499	2
C₅₃×#3498	2
C₅₃×#3154	2
C₅₃×#3152	2
C₅₃×#3151	2
C₅₃×#3147	2
C₂×C₅₃×#492	2
C₄×#131×C₅₃	2
#60646	53

Subgroups — C53×#60646 #27268158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60646 #27268158