Subgroups — C₅₃×#60649 #27268161

Order 13568.

3178 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#578	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#577	C₂	1	2	yes	maximal, complemented
3	C₂×C₅₃×#576	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3500	C₂	1	2	yes	maximal, complemented
5	C₅₃×#3156	C₂	1	3	yes	maximal, complemented
6	C₅₃×#3155	C₂	1	3	yes	maximal, complemented
7	C₅₃×#3154	C₂	1	6	yes	maximal, complemented
8	C₅₃×#3153	C₂	1	6	yes	maximal, complemented
9	C₅₃×#3147	C₂	1	3	yes	maximal, complemented
10	C₂×C₅₃×#495	C₂	1	3	yes	maximal, complemented
11	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	2	yes	—
13	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#578	—	2	8	no	—
15	C₅₃×#577	—	2	16	no	—
16	C₅₃×#576	—	2	8	no	—
17	C₂×#137×C₅₃	C₂²	1	6	yes	complemented
18	C₂×#137×C₅₃	C₂²	1	6	yes	complemented
19	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	6	yes	—
20	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	6	yes	complemented
21	C₂×#135×C₅₃	C₂²	1	3	yes	—
22	C₂×#135×C₅₃	C₂²	1	6	yes	complemented
23	C₂×#135×C₅₃	C₂²	1	3	yes	complemented
24	C₂²×Q₁₆×C₅₃	C₂²	1	3	yes	complemented
25	C₂²×SD₁₆×C₅₃	C₂²	1	6	yes	complemented
26	C₂²×D₈×C₅₃	C₂²	1	3	yes	—
27	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#548	—	2	4	no	—
30	C₅₃×#546	—	2	12	no	—
31	C₅₃×#536	C₂²	1	6	yes	complemented
32	C₅₃×#535	C₂²	1	6	yes	complemented
33	C₅₃×#532	C₂²	1	6	yes	complemented
34	C₂×#124×C₅₃	C₂²	1	6	yes	complemented
35	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#497	C₂²	1	12	yes	complemented
37	C₅₃×#496	C₂²	1	12	yes	complemented
38	C₅₃×#495	C₂²	1	3	yes	complemented
39	C₅₃×#495	C₂²	1	6	yes	complemented
40	C₅₃×#495	C₂²	1	6	yes	complemented
41	C₅₃×#495	C₂²	1	6	yes	complemented
42	C₅₃×#495	C₂²	1	1	yes	characteristic, complemented
43	C₅₃×#495	C₂²	1	6	yes	complemented
44	C₅₃×#494	C₂²	1	3	yes	complemented
45	C₅₃×#493	C₂²	1	3	yes	complemented
46	C₅₃×#492	C₂²	1	6	yes	complemented
47	C₅₃×#405	C₂²	1	3	yes	complemented
48	C₅₃×#405	C₂²	1	3	yes	complemented
49	C₄×M₁₆×C₅₃	C₂²	1	3	yes	complemented
50	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	complemented
51	2₋¹⁺⁴×C₅₃	—	2	8	no	—
52	2₋¹⁺⁴×C₅₃	—	2	8	no	—
53	H(5,𝔽₂)×C₅₃	—	2	8	no	—
54	H(5,𝔽₂)×C₅₃	—	2	8	no	—
55	C₂×#41×C₅₃	—	2	2	no	—
56	C₂×#41×C₅₃	—	2	6	no	—
57	C₂×#41×C₅₃	C₂³	1	6	yes	—
58	C₂×#41×C₅₃	C₂³	1	6	yes	—
59	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
60	C₂²×Q₈×C₅₃	—	2	2	no	—
61	C₂²×Q₈×C₅₃	C₂³	1	6	yes	—
62	C₂²×D₄×C₅₃	—	2	6	no	—
63	C₂²×D₄×C₅₃	C₂³	1	6	yes	—
64	#137×C₅₃	—	2	24	no	—
65	#137×C₅₃	—	2	24	no	—
66	AGL(1,ℤ/8)×C₅₃	—	2	24	no	—
67	AGL(1,ℤ/8)×C₅₃	—	2	24	no	—
68	#135×C₅₃	—	2	12	no	—
69	#135×C₅₃	—	2	24	no	—
70	#135×C₅₃	—	2	12	no	—
71	C₂×Q₁₆×C₅₃	—	2	12	no	—
72	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
73	C₂×Q₁₆×C₅₃	C₂³	1	3	yes	—
74	C₂×Q₁₆×C₅₃	C₂³	1	6	yes	—
75	C₂×Q₁₆×C₅₃	C₂³	1	6	yes	—
76	C₂×SD₁₆×C₅₃	—	2	24	no	—
77	C₂×SD₁₆×C₅₃	C₂³	1	2	yes	—
78	C₂×SD₁₆×C₅₃	C₂³	1	6	yes	—
79	C₂×SD₁₆×C₅₃	C₂³	1	6	yes	—
80	C₂×SD₁₆×C₅₃	C₂³	1	6	yes	—
81	C₂×SD₁₆×C₅₃	C₂³	1	12	yes	—
82	C₂×D₈×C₅₃	—	2	12	no	—
83	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
84	C₂×D₈×C₅₃	C₂³	1	3	yes	—
85	C₂×D₈×C₅₃	C₂³	1	6	yes	—
86	C₂×D₈×C₅₃	C₂³	1	6	yes	—
87	#131×C₅₃	D₄	1	8	yes	complemented
88	#131×C₅₃	D₄	1	8	yes	complemented
89	C₂×M₁₆×C₅₃	C₂³	1	3	yes	—
90	C₂×M₁₆×C₅₃	C₂³	1	3	yes	—
91	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
92	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
93	C₄⋊Q₈×C₅₃	C₂³	1	6	yes	—
94	#127×C₅₃	C₂³	1	6	yes	—
95	#124×C₅₃	—	2	12	no	—
96	#124×C₅₃	C₂³	1	6	yes	—
97	#124×C₅₃	C₂³	1	2	yes	—
98	#124×C₅₃	C₂³	1	12	yes	—
99	#123×C₅₃	—	2	12	no	—
100	C₂²⋊Q₈×C₅₃	—	2	12	no	—
101	#121×C₅₃	—	2	12	no	—
102	#121×C₅₃	—	2	12	no	—
103	#120×C₅₃	—	2	12	no	—
104	C₄×Q₈×C₅₃	D₄	1	2	yes	—
105	C₄×D₄×C₅₃	—	2	12	no	—
106	C₄×D₄×C₅₃	D₄	1	6	yes	—
107	#117×C₅₃	C₂³	1	3	yes	—
108	C₂×C₂²⋊C₄×C₅₃	—	2	12	no	—
109	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
110	#97×C₅₃	C₂³	1	6	yes	—
111	C₄×C₈×C₅₃	C₂³	1	3	yes	—
112	C₄×C₈×C₅₃	C₂³	1	3	yes	—
113	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
114	C₄×C₈×C₅₃	C₂³	1	3	yes	—
115	C₂⁴×C₅₃	—	2	6	no	—
116	#41×C₅₃	—	2	12	no	—
117	#41×C₅₃	—	2	12	no	—
118	#41×C₅₃	—	2	12	no	—
119	#41×C₅₃	—	2	12	no	—
120	#41×C₅₃	—	4	8	no	—
121	#41×C₅₃	—	4	24	no	—
122	#41×C₅₃	C₂×D₄	1	4	yes	—
123	#41×C₅₃	C₂×D₄	1	4	yes	—
124	C₂×Q₈×C₅₃	—	2	6	no	—
125	C₂×Q₈×C₅₃	—	2	12	no	—
126	C₂×Q₈×C₅₃	—	2	12	no	—
127	C₂×Q₈×C₅₃	—	4	8	no	—
128	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Q₈×C₅₃	C₂⁴	1	6	yes	—
130	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
131	C₂×D₄×C₅₃	—	2	6	no	—
132	C₂×D₄×C₅₃	—	2	6	no	—
133	C₂×D₄×C₅₃	—	2	12	no	—
134	C₂×D₄×C₅₃	—	2	12	no	—
135	C₂×D₄×C₅₃	—	2	6	no	—
136	C₂×D₄×C₅₃	—	2	12	no	—
137	C₂×D₄×C₅₃	—	4	24	no	—
138	C₂×D₄×C₅₃	C₂×D₄	1	3	yes	—
139	C₂×D₄×C₅₃	C₂⁴	1	6	yes	—
140	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
141	C₂²×C₄×C₅₃	—	2	6	no	—
142	C₂²×C₄×C₅₃	—	2	6	no	—
143	C₂²×C₄×C₅₃	—	2	6	no	—
144	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
145	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
146	Q₁₆×C₅₃	—	2	6	no	—
147	Q₁₆×C₅₃	—	2	12	no	—
148	Q₁₆×C₅₃	—	2	2	no	—
149	Q₁₆×C₅₃	—	2	12	no	—
150	SD₁₆×C₅₃	—	2	12	no	—
151	SD₁₆×C₅₃	—	2	24	no	—
152	SD₁₆×C₅₃	—	2	4	no	—
153	SD₁₆×C₅₃	—	2	12	no	—
154	SD₁₆×C₅₃	—	2	12	no	—
155	D₈×C₅₃	—	2	6	no	—
156	D₈×C₅₃	—	2	12	no	—
157	D₈×C₅₃	—	2	2	no	—
158	D₈×C₅₃	—	2	12	no	—
159	M₁₆×C₅₃	C₂×D₄	1	12	yes	—
160	M₁₆×C₅₃	C₂×D₄	1	12	yes	—
161	C₂×C₈×C₅₃	C₂×D₄	1	12	yes	—
162	C₂×C₈×C₅₃	C₂×D₄	1	12	yes	—
163	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
164	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
165	C₂×C₈×C₅₃	C₂⁴	1	3	yes	—
166	C₂×C₈×C₅₃	C₂⁴	1	3	yes	—
167	C₄⋊C₄×C₅₃	—	2	12	no	—
168	C₄⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
169	C₂²⋊C₄×C₅₃	—	2	12	no	—
170	C₂²⋊C₄×C₅₃	—	2	24	no	—
171	C₂²⋊C₄×C₅₃	—	2	4	no	—
172	C₂²⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
173	C₄²×C₅₃	C₂×D₄	1	6	yes	—
174	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
175	C₄²×C₅₃	C₂⁴	1	3	yes	—
176	C₂³×C₅₃	—	2	2	no	—
177	C₂³×C₅₃	—	2	6	no	—
178	C₂³×C₅₃	—	4	12	no	—
179	C₂³×C₅₃	—	8	12	no	—
180	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
181	Q₈×C₅₃	—	2	6	no	—
182	Q₈×C₅₃	—	2	6	no	—
183	Q₈×C₅₃	—	2	2	no	—
184	Q₈×C₅₃	—	2	2	no	—
185	Q₈×C₅₃	—	4	12	no	—
186	Q₈×C₅₃	#127	1	4	yes	—
187	D₄×C₅₃	—	2	6	no	—
188	D₄×C₅₃	—	2	6	no	—
189	D₄×C₅₃	—	2	2	no	—
190	D₄×C₅₃	—	2	2	no	—
191	D₄×C₅₃	—	4	12	no	—
192	D₄×C₅₃	—	4	12	no	—
193	D₄×C₅₃	—	4	12	no	—
194	D₄×C₅₃	—	8	12	no	—
195	D₄×C₅₃	#127	1	12	yes	—
196	C₂×C₄×C₅₃	—	2	2	no	—
197	C₂×C₄×C₅₃	—	2	6	no	—
198	C₂×C₄×C₅₃	—	4	6	no	—
199	C₂×C₄×C₅₃	—	4	12	no	—
200	C₂×C₄×C₅₃	—	4	12	no	—
201	C₂×C₄×C₅₃	—	4	12	no	—
202	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
203	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
204	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
205	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
206	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
207	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
208	C₈×C₅₃	C₂²×D₄	1	2	yes	—
209	C₈×C₅₃	C₂²×D₄	1	2	yes	—
210	C₈×C₅₃	C₂²×D₄	1	6	yes	—
211	C₈×C₅₃	C₂²×D₄	1	6	yes	—
212	#60649	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂²×C₅₃	—	2	6	no	—
214	C₂²×C₅₃	—	4	4	no	—
215	C₂²×C₅₃	—	4	12	no	—
216	C₂²×C₅₃	—	4	2	no	—
217	C₂²×C₅₃	—	4	6	no	—
218	C₂²×C₅₃	—	16	12	no	—
219	C₂²×C₅₃	C₂×#127	1	6	yes	—
220	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
221	C₄×C₅₃	—	2	2	no	—
222	C₄×C₅₃	—	2	6	no	—
223	C₄×C₅₃	—	4	2	no	—
224	C₄×C₅₃	—	4	6	no	—
225	C₄×C₅₃	C₂×#127	1	6	yes	—
226	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
227	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
228	C₂×#578	C₂×C₅₃	1	1	yes	characteristic, complemented
229	C₂×#577	C₂×C₅₃	1	2	yes	complemented
230	C₂×#576	C₂×C₅₃	1	1	yes	characteristic, complemented
231	#3500	C₂×C₅₃	1	2	yes	complemented
232	#3156	C₂×C₅₃	1	3	yes	complemented
233	#3155	C₂×C₅₃	1	3	yes	complemented
234	#3154	C₂×C₅₃	1	6	yes	complemented
235	#3153	C₂×C₅₃	1	6	yes	complemented
236	#3147	C₂×C₅₃	1	3	yes	complemented
237	C₂×#495	C₂×C₅₃	1	3	yes	complemented
238	C₄×#131	C₂×C₅₃	1	1	yes	characteristic, complemented
239	C₂×C₅₃	—	2	6	no	—
240	C₂×C₅₃	—	8	2	no	—
241	C₂×C₅₃	—	8	6	no	—
242	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
243	C₂×C₅₃	#3584	1	2	yes	—
244	C₂×2₋¹⁺⁴	C₂²×C₅₃	1	2	yes	—
245	C₂×H(5,𝔽₂)	C₂²×C₅₃	1	2	yes	—
246	#578	—	2	8	no	—
247	#577	—	2	16	no	—
248	#576	—	2	8	no	—
249	C₂×#137	C₂²×C₅₃	1	6	yes	complemented
250	C₂×#137	C₂²×C₅₃	1	6	yes	complemented
251	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	6	yes	—
252	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	6	yes	complemented
253	C₂×#135	C₂²×C₅₃	1	3	yes	—
254	C₂×#135	C₂²×C₅₃	1	6	yes	complemented
255	C₂×#135	C₂²×C₅₃	1	3	yes	complemented
256	C₂²×Q₁₆	C₂²×C₅₃	1	3	yes	complemented
257	C₂²×SD₁₆	C₂²×C₅₃	1	6	yes	complemented
258	C₂²×D₈	C₂²×C₅₃	1	3	yes	—
259	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
260	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
261	#548	—	2	4	no	—
262	#546	—	2	12	no	—
263	#536	C₂²×C₅₃	1	6	yes	complemented
264	#535	C₂²×C₅₃	1	6	yes	complemented
265	#532	C₂²×C₅₃	1	6	yes	complemented
266	C₂×#124	C₂²×C₅₃	1	6	yes	complemented
267	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
268	#497	C₂²×C₅₃	1	12	yes	complemented
269	#496	C₂²×C₅₃	1	12	yes	complemented
270	#495	C₂²×C₅₃	1	3	yes	complemented
271	#495	C₂²×C₅₃	1	6	yes	complemented
272	#495	C₂²×C₅₃	1	6	yes	complemented
273	#495	C₂²×C₅₃	1	6	yes	complemented
274	#495	C₂²×C₅₃	1	1	yes	characteristic, complemented
275	#495	C₂²×C₅₃	1	6	yes	complemented
276	#494	C₂²×C₅₃	1	3	yes	complemented
277	#493	C₂²×C₅₃	1	3	yes	complemented
278	#492	C₂²×C₅₃	1	6	yes	complemented
279	#405	C₂²×C₅₃	1	3	yes	complemented
280	#405	C₂²×C₅₃	1	3	yes	complemented
281	C₄×M₁₆	C₂²×C₅₃	1	3	yes	complemented
282	C₂×C₄×C₈	C₂²×C₅₃	1	3	yes	complemented
283	C₅₃	#60649	1	1	yes	characteristic, direct product factor, Sylow
284	2₋¹⁺⁴	—	2	8	no	—
285	2₋¹⁺⁴	—	2	8	no	—
286	H(5,𝔽₂)	—	2	8	no	—
287	H(5,𝔽₂)	—	2	8	no	—
288	C₂×#41	—	2	2	no	—
289	C₂×#41	—	2	6	no	—
290	C₂×#41	C₂³×C₅₃	1	6	yes	—
291	C₂×#41	C₂³×C₅₃	1	6	yes	—
292	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
293	C₂²×Q₈	—	2	2	no	—
294	C₂²×Q₈	C₂³×C₅₃	1	6	yes	—
295	C₂²×D₄	—	2	6	no	—
296	C₂²×D₄	C₂³×C₅₃	1	6	yes	—
297	#137	—	2	24	no	—
298	#137	—	2	24	no	—
299	AGL(1,ℤ/8)	—	2	24	no	—
300	AGL(1,ℤ/8)	—	2	24	no	—
301	#135	—	2	12	no	—
302	#135	—	2	24	no	—
303	#135	—	2	12	no	—
304	C₂×Q₁₆	—	2	12	no	—
305	C₂×Q₁₆	C₂³×C₅₃	1	3	yes	—
306	C₂×Q₁₆	C₂³×C₅₃	1	6	yes	—
307	C₂×Q₁₆	C₂³×C₅₃	1	1	yes	characteristic
308	C₂×Q₁₆	C₂³×C₅₃	1	6	yes	—
309	C₂×SD₁₆	—	2	24	no	—
310	C₂×SD₁₆	C₂³×C₅₃	1	6	yes	—
311	C₂×SD₁₆	C₂³×C₅₃	1	12	yes	—
312	C₂×SD₁₆	C₂³×C₅₃	1	6	yes	—
313	C₂×SD₁₆	C₂³×C₅₃	1	2	yes	—
314	C₂×SD₁₆	C₂³×C₅₃	1	6	yes	—
315	C₂×D₈	—	2	12	no	—
316	C₂×D₈	C₂³×C₅₃	1	3	yes	—
317	C₂×D₈	C₂³×C₅₃	1	6	yes	—
318	C₂×D₈	C₂³×C₅₃	1	1	yes	characteristic
319	C₂×D₈	C₂³×C₅₃	1	6	yes	—
320	#131	D₄×C₅₃	1	8	yes	complemented
321	#131	D₄×C₅₃	1	8	yes	complemented
322	C₂×M₁₆	C₂³×C₅₃	1	3	yes	—
323	C₂×M₁₆	C₂³×C₅₃	1	3	yes	—
324	C₂²×C₈	C₂³×C₅₃	1	3	yes	—
325	C₂²×C₈	C₂³×C₅₃	1	3	yes	—
326	C₄⋊Q₈	C₂³×C₅₃	1	6	yes	—
327	#127	C₂³×C₅₃	1	6	yes	—
328	#124	—	2	12	no	—
329	#124	C₂³×C₅₃	1	6	yes	—
330	#124	C₂³×C₅₃	1	12	yes	—
331	#124	C₂³×C₅₃	1	2	yes	—
332	#123	—	2	12	no	—
333	C₂²⋊Q₈	—	2	12	no	—
334	#121	—	2	12	no	—
335	#121	—	2	12	no	—
336	#120	—	2	12	no	—
337	C₄×Q₈	D₄×C₅₃	1	2	yes	—
338	C₄×D₄	—	2	12	no	—
339	C₄×D₄	D₄×C₅₃	1	6	yes	—
340	#117	C₂³×C₅₃	1	3	yes	—
341	C₂×C₂²⋊C₄	—	2	12	no	—
342	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
343	#97	C₂³×C₅₃	1	6	yes	—
344	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
345	C₄×C₈	C₂³×C₅₃	1	3	yes	—
346	C₄×C₈	C₂³×C₅₃	1	3	yes	—
347	C₄×C₈	C₂³×C₅₃	1	3	yes	—
348	C₂⁴	—	2	6	no	—
349	#41	—	2	12	no	—
350	#41	—	2	12	no	—
351	#41	—	2	12	no	—
352	#41	—	2	12	no	—
353	#41	—	4	8	no	—
354	#41	—	4	24	no	—
355	#41	C₂×D₄×C₅₃	1	4	yes	—
356	#41	C₂×D₄×C₅₃	1	4	yes	—
357	C₂×Q₈	—	2	6	no	—
358	C₂×Q₈	—	2	12	no	—
359	C₂×Q₈	—	2	12	no	—
360	C₂×Q₈	—	4	8	no	—
361	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
362	C₂×Q₈	C₂⁴×C₅₃	1	6	yes	—
363	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
364	C₂×D₄	—	2	6	no	—
365	C₂×D₄	—	2	6	no	—
366	C₂×D₄	—	2	12	no	—
367	C₂×D₄	—	2	12	no	—
368	C₂×D₄	—	2	6	no	—
369	C₂×D₄	—	2	12	no	—
370	C₂×D₄	—	4	24	no	—
371	C₂×D₄	C₂×D₄×C₅₃	1	3	yes	—
372	C₂×D₄	C₂⁴×C₅₃	1	6	yes	—
373	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
374	C₂²×C₄	—	2	6	no	—
375	C₂²×C₄	—	2	6	no	—
376	C₂²×C₄	—	2	6	no	—
377	C₂²×C₄	C₂×D₄×C₅₃	1	6	yes	—
378	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
379	Q₁₆	—	2	6	no	—
380	Q₁₆	—	2	12	no	—
381	Q₁₆	—	2	2	no	—
382	Q₁₆	—	2	12	no	—
383	SD₁₆	—	2	12	no	—
384	SD₁₆	—	2	24	no	—
385	SD₁₆	—	2	4	no	—
386	SD₁₆	—	2	12	no	—
387	SD₁₆	—	2	12	no	—
388	D₈	—	2	6	no	—
389	D₈	—	2	12	no	—
390	D₈	—	2	2	no	—
391	D₈	—	2	12	no	—
392	M₁₆	C₂×D₄×C₅₃	1	12	yes	—
393	M₁₆	C₂×D₄×C₅₃	1	12	yes	—
394	C₂×C₈	C₂×D₄×C₅₃	1	12	yes	—
395	C₂×C₈	C₂×D₄×C₅₃	1	12	yes	—
396	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
397	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
398	C₂×C₈	C₂⁴×C₅₃	1	3	yes	—
399	C₂×C₈	C₂⁴×C₅₃	1	3	yes	—
400	C₄⋊C₄	—	2	12	no	—
401	C₄⋊C₄	C₂×D₄×C₅₃	1	6	yes	—
402	C₂²⋊C₄	—	2	12	no	—
403	C₂²⋊C₄	—	2	24	no	—
404	C₂²⋊C₄	—	2	4	no	—
405	C₂²⋊C₄	C₂×D₄×C₅₃	1	6	yes	—
406	C₄²	C₂×D₄×C₅₃	1	6	yes	—
407	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
408	C₄²	C₂⁴×C₅₃	1	3	yes	—
409	C₂³	—	2	6	no	—
410	C₂³	—	2	2	no	—
411	C₂³	—	4	12	no	—
412	C₂³	—	8	12	no	—
413	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
414	Q₈	—	2	6	no	—
415	Q₈	—	2	2	no	—
416	Q₈	—	2	6	no	—
417	Q₈	—	2	2	no	—
418	Q₈	—	4	12	no	—
419	Q₈	#127×C₅₃	1	4	yes	—
420	D₄	—	2	6	no	—
421	D₄	—	2	2	no	—
422	D₄	—	2	6	no	—
423	D₄	—	2	2	no	—
424	D₄	—	4	12	no	—
425	D₄	—	4	12	no	—
426	D₄	—	4	12	no	—
427	D₄	—	8	12	no	—
428	D₄	#127×C₅₃	1	12	yes	—
429	C₂×C₄	—	2	6	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	4	6	no	—
432	C₂×C₄	—	4	12	no	—
433	C₂×C₄	—	4	12	no	—
434	C₂×C₄	—	4	12	no	—
435	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
436	C₂×C₄	C₂²×D₄×C₅₃	1	6	yes	—
437	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
438	C₂×C₄	C₂²×D₄×C₅₃	1	6	yes	—
439	C₂×C₄	C₂²×D₄×C₅₃	1	6	yes	—
440	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
441	C₈	C₂²×D₄×C₅₃	1	2	yes	—
442	C₈	C₂²×D₄×C₅₃	1	2	yes	—
443	C₈	C₂²×D₄×C₅₃	1	6	yes	—
444	C₈	C₂²×D₄×C₅₃	1	6	yes	—
445	C₂²	—	2	6	no	—
446	C₂²	—	4	4	no	—
447	C₂²	—	4	12	no	—
448	C₂²	—	4	2	no	—
449	C₂²	—	4	6	no	—
450	C₂²	—	16	12	no	—
451	C₂²	C₂×#127×C₅₃	1	6	yes	—
452	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
453	C₄	—	2	2	no	—
454	C₄	—	2	6	no	—
455	C₄	—	4	2	no	—
456	C₄	—	4	6	no	—
457	C₄	C₂×#127×C₅₃	1	6	yes	—
458	C₄	C₂³×D₄×C₅₃	1	1	yes	characteristic
459	C₄	C₂³×D₄×C₅₃	1	1	yes	characteristic
460	C₂	—	2	6	no	—
461	C₂	—	8	6	no	—
462	C₂	—	8	2	no	—
463	C₂	C₂²×#127×C₅₃	1	1	yes	characteristic
464	C₂	C₅₃×#3584	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×#578	C₂	semidirect
2. C₂×C₅₃×#577	C₂	semidirect
3. C₂×C₅₃×#576	C₂	semidirect
4. C₅₃×#3500	C₂	semidirect
5. C₅₃×#3156	C₂	semidirect
6. C₅₃×#3155	C₂	semidirect
7. C₅₃×#3154	C₂	semidirect
8. C₅₃×#3153	C₂	semidirect
9. C₅₃×#3147	C₂	semidirect
10. C₂×C₅₃×#495	C₂	semidirect
11. C₄×#131×C₅₃	C₂	semidirect
17. C₂×#137×C₅₃	C₂²	semidirect
18. C₂×#137×C₅₃	C₂²	semidirect
20. C₂×AGL(1,ℤ/8)×C₅₃	C₂²	semidirect
22. C₂×#135×C₅₃	C₂²	semidirect
23. C₂×#135×C₅₃	C₂²	semidirect
24. C₂²×Q₁₆×C₅₃	C₂²	semidirect
25. C₂²×SD₁₆×C₅₃	C₂²	semidirect
31. C₅₃×#536	C₂²	semidirect
32. C₅₃×#535	C₂²	semidirect
33. C₅₃×#532	C₂²	semidirect
34. C₂×#124×C₅₃	C₂²	semidirect
36. C₅₃×#497	C₂²	semidirect
37. C₅₃×#496	C₂²	semidirect
38. C₅₃×#495	C₂²	semidirect
39. C₅₃×#495	C₂²	semidirect
40. C₅₃×#495	C₂²	semidirect
41. C₅₃×#495	C₂²	semidirect
42. C₅₃×#495	C₂²	semidirect
43. C₅₃×#495	C₂²	semidirect
44. C₅₃×#494	C₂²	semidirect
45. C₅₃×#493	C₂²	semidirect
46. C₅₃×#492	C₂²	semidirect
47. C₅₃×#405	C₂²	semidirect
48. C₅₃×#405	C₂²	semidirect
49. C₄×M₁₆×C₅₃	C₂²	semidirect
50. C₂×C₄×C₈×C₅₃	C₂²	semidirect
87. #131×C₅₃	D₄	semidirect
88. #131×C₅₃	D₄	semidirect
212. #60649	C₅₃	direct (N × H)
228. C₂×#578	C₂×C₅₃	semidirect
229. C₂×#577	C₂×C₅₃	semidirect
230. C₂×#576	C₂×C₅₃	semidirect
231. #3500	C₂×C₅₃	semidirect
232. #3156	C₂×C₅₃	semidirect
233. #3155	C₂×C₅₃	semidirect
234. #3154	C₂×C₅₃	semidirect
235. #3153	C₂×C₅₃	semidirect
236. #3147	C₂×C₅₃	semidirect
237. C₂×#495	C₂×C₅₃	semidirect
238. C₄×#131	C₂×C₅₃	semidirect
249. C₂×#137	C₂²×C₅₃	semidirect
250. C₂×#137	C₂²×C₅₃	semidirect
252. C₂×AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
254. C₂×#135	C₂²×C₅₃	semidirect
255. C₂×#135	C₂²×C₅₃	semidirect
256. C₂²×Q₁₆	C₂²×C₅₃	semidirect
257. C₂²×SD₁₆	C₂²×C₅₃	semidirect
263. #536	C₂²×C₅₃	semidirect
264. #535	C₂²×C₅₃	semidirect
265. #532	C₂²×C₅₃	semidirect
266. C₂×#124	C₂²×C₅₃	semidirect
268. #497	C₂²×C₅₃	semidirect
269. #496	C₂²×C₅₃	semidirect
270. #495	C₂²×C₅₃	semidirect
271. #495	C₂²×C₅₃	semidirect
272. #495	C₂²×C₅₃	semidirect
273. #495	C₂²×C₅₃	semidirect
274. #495	C₂²×C₅₃	semidirect
275. #495	C₂²×C₅₃	semidirect
276. #494	C₂²×C₅₃	semidirect
277. #493	C₂²×C₅₃	semidirect
278. #492	C₂²×C₅₃	semidirect
279. #405	C₂²×C₅₃	semidirect
280. #405	C₂²×C₅₃	semidirect
281. C₄×M₁₆	C₂²×C₅₃	semidirect
282. C₂×C₄×C₈	C₂²×C₅₃	semidirect
283. C₅₃	#60649	direct (N × H)
320. #131	D₄×C₅₃	semidirect
321. #131	D₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#578	2
C₂×C₅₃×#577	2
C₂×C₅₃×#576	2
C₅₃×#3500	2
C₅₃×#3156	2
C₅₃×#3155	2
C₅₃×#3154	2
C₅₃×#3153	2
C₅₃×#3147	2
C₂×C₅₃×#495	2
C₄×#131×C₅₃	2
#60649	53

Subgroups — C53×#60649 #27268161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60649 #27268161