Subgroups — C₅₃×#60651 #27268163

Order 13568.

3138 conjugacy classes of subgroups.

The subgroup lattice has 500 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	2	yes	maximal, complemented
2	C₂×C₅₃×#577	C₂	1	2	yes	maximal, complemented
3	C₅₃×#3500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3499	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#3156	C₂	1	6	yes	maximal, complemented
6	C₅₃×#3154	C₂	1	6	yes	maximal, complemented
7	C₅₃×#3153	C₂	1	3	yes	maximal, complemented
8	C₅₃×#3152	C₂	1	3	yes	maximal, complemented
9	C₅₃×#3150	C₂	1	3	yes	maximal, complemented
10	C₂×C₅₃×#497	C₂	1	3	yes	maximal, complemented
11	C₅₃×#2875	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	2	yes	—
13	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#578	—	2	16	no	—
16	C₅₃×#577	—	2	16	no	—
17	C₂×#137×C₅₃	C₂²	1	6	yes	—
18	C₂×#137×C₅₃	C₂²	1	6	yes	complemented
19	C₂×#137×C₅₃	C₂²	1	6	yes	complemented
20	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	6	yes	—
21	C₂×#135×C₅₃	C₂²	1	6	yes	—
22	C₂×#135×C₅₃	C₂²	1	6	yes	complemented
23	C₂²×Q₁₆×C₅₃	C₂²	1	6	yes	complemented
24	C₂²×SD₁₆×C₅₃	C₂²	1	6	yes	—
25	C₂×#131×C₅₃	C₂²	1	2	yes	—
26	D₄×Q₈×C₅₃	—	2	2	no	—
27	C₅₃×#548	—	2	2	no	—
28	C₅₃×#547	—	2	6	no	—
29	C₅₃×#546	—	2	6	no	—
30	C₅₃×#536	C₂²	1	3	yes	—
31	C₅₃×#536	C₂²	1	6	yes	complemented
32	C₅₃×#535	C₂²	1	3	yes	—
33	C₅₃×#532	C₂²	1	3	yes	—
34	C₅₃×#532	C₂²	1	3	yes	complemented
35	C₂×C₄⋊Q₈×C₅₃	C₂²	1	3	yes	complemented
36	C₂×#124×C₅₃	C₂²	1	3	yes	—
37	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#497	C₂²	1	3	yes	complemented
39	C₅₃×#497	C₂²	1	6	yes	complemented
40	C₅₃×#497	C₂²	1	12	yes	complemented
41	C₅₃×#497	C₂²	1	3	yes	complemented
42	C₅₃×#497	C₂²	1	6	yes	complemented
43	C₅₃×#496	C₂²	1	1	yes	characteristic, complemented
44	C₅₃×#496	C₂²	1	3	yes	complemented
45	C₅₃×#496	C₂²	1	6	yes	complemented
46	C₅₃×#495	C₂²	1	12	yes	complemented
47	C₅₃×#494	C₂²	1	6	yes	complemented
48	C₅₃×#492	C₂²	1	6	yes	complemented
49	C₅₃×#405	C₂²	1	6	yes	complemented
50	C₄×M₁₆×C₅₃	C₂²	1	3	yes	complemented
51	C₂×#97×C₅₃	C₂²	1	3	yes	complemented
52	2₋¹⁺⁴×C₅₃	—	2	8	no	—
53	2₋¹⁺⁴×C₅₃	—	2	8	no	—
54	2₋¹⁺⁴×C₅₃	—	2	8	no	—
55	H(5,𝔽₂)×C₅₃	—	2	8	no	—
56	C₂×#41×C₅₃	—	2	1	no	—
57	C₂×#41×C₅₃	—	2	3	no	—
58	C₂×#41×C₅₃	—	2	6	no	—
59	C₂×#41×C₅₃	C₂³	1	3	yes	—
60	C₂×#41×C₅₃	C₂³	1	3	yes	—
61	C₂×#41×C₅₃	C₂³	1	6	yes	—
62	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
63	C₂²×Q₈×C₅₃	—	2	1	no	—
64	C₂²×Q₈×C₅₃	—	2	2	no	—
65	C₂²×Q₈×C₅₃	C₂³	1	3	yes	—
66	C₂²×Q₈×C₅₃	C₂³	1	6	yes	—
67	C₂²×D₄×C₅₃	—	2	3	no	—
68	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
69	#137×C₅₃	—	2	24	no	—
70	#137×C₅₃	—	2	24	no	—
71	#137×C₅₃	—	2	24	no	—
72	AGL(1,ℤ/8)×C₅₃	—	2	24	no	—
73	#135×C₅₃	—	2	24	no	—
74	#135×C₅₃	—	2	24	no	—
75	C₂×Q₁₆×C₅₃	—	2	24	no	—
76	C₂×Q₁₆×C₅₃	C₂³	1	6	yes	—
77	C₂×Q₁₆×C₅₃	C₂³	1	12	yes	—
78	C₂×Q₁₆×C₅₃	C₂³	1	6	yes	—
79	C₂×SD₁₆×C₅₃	—	2	24	no	—
80	C₂×SD₁₆×C₅₃	C₂³	1	6	yes	—
81	C₂×SD₁₆×C₅₃	C₂³	1	12	yes	—
82	C₂×SD₁₆×C₅₃	C₂³	1	6	yes	—
83	C₂×SD₁₆×C₅₃	C₂³	1	6	yes	—
84	C₂×SD₁₆×C₅₃	C₂³	1	2	yes	—
85	C₂×D₈×C₅₃	C₂³	1	6	yes	—
86	C₂×D₈×C₅₃	C₂³	1	2	yes	—
87	#131×C₅₃	D₄	1	16	yes	complemented
88	C₂×M₁₆×C₅₃	C₂³	1	6	yes	—
89	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
90	C₄⋊Q₈×C₅₃	—	2	6	no	—
91	C₄⋊Q₈×C₅₃	C₂³	1	6	yes	—
92	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
93	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
94	#127×C₅₃	C₂³	1	3	yes	—
95	#127×C₅₃	C₂³	1	1	yes	characteristic
96	#124×C₅₃	—	2	6	no	—
97	#124×C₅₃	C₂³	1	6	yes	—
98	#124×C₅₃	C₂³	1	6	yes	—
99	#124×C₅₃	C₂³	1	3	yes	—
100	#124×C₅₃	C₂³	1	1	yes	characteristic
101	#123×C₅₃	—	2	6	no	—
102	#123×C₅₃	—	2	12	no	—
103	C₂²⋊Q₈×C₅₃	—	2	6	no	—
104	C₂²⋊Q₈×C₅₃	—	2	12	no	—
105	#121×C₅₃	—	2	6	no	—
106	#121×C₅₃	—	2	6	no	—
107	#121×C₅₃	—	2	6	no	—
108	#120×C₅₃	—	2	6	no	—
109	C₄×Q₈×C₅₃	D₄	1	2	yes	—
110	C₄×D₄×C₅₃	—	2	6	no	—
111	C₄×D₄×C₅₃	—	2	6	no	—
112	C₄×D₄×C₅₃	D₄	1	6	yes	—
113	#117×C₅₃	C₂³	1	3	yes	—
114	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
115	C₂×C₂²⋊C₄×C₅₃	—	2	6	no	—
116	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
117	#97×C₅₃	C₂³	1	1	yes	characteristic
118	#97×C₅₃	C₂³	1	3	yes	—
119	#97×C₅₃	C₂³	1	6	yes	—
120	C₄×C₈×C₅₃	C₂³	1	6	yes	—
121	C₂⁴×C₅₃	—	2	3	no	—
122	#41×C₅₃	—	2	12	no	—
123	#41×C₅₃	—	2	12	no	—
124	#41×C₅₃	—	2	12	no	—
125	#41×C₅₃	—	2	12	no	—
126	#41×C₅₃	—	4	4	no	—
127	#41×C₅₃	—	4	12	no	—
128	#41×C₅₃	—	4	24	no	—
129	#41×C₅₃	C₂×D₄	1	8	yes	—
130	C₂×Q₈×C₅₃	—	2	3	no	—
131	C₂×Q₈×C₅₃	—	2	6	no	—
132	C₂×Q₈×C₅₃	—	2	12	no	—
133	C₂×Q₈×C₅₃	—	2	12	no	—
134	C₂×Q₈×C₅₃	—	2	12	no	—
135	C₂×Q₈×C₅₃	—	4	4	no	—
136	C₂×Q₈×C₅₃	—	4	8	no	—
137	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
138	C₂×Q₈×C₅₃	C₂⁴	1	3	yes	—
139	C₂×Q₈×C₅₃	C₂⁴	1	6	yes	—
140	C₂×Q₈×C₅₃	C₂⁴	1	1	yes	characteristic
141	C₂×D₄×C₅₃	—	2	3	no	—
142	C₂×D₄×C₅₃	—	2	3	no	—
143	C₂×D₄×C₅₃	—	2	6	no	—
144	C₂×D₄×C₅₃	—	2	12	no	—
145	C₂×D₄×C₅₃	—	2	2	no	—
146	C₂×D₄×C₅₃	—	2	3	no	—
147	C₂×D₄×C₅₃	—	2	6	no	—
148	C₂×D₄×C₅₃	—	4	12	no	—
149	C₂×D₄×C₅₃	C₂×D₄	1	3	yes	—
150	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
151	C₂×D₄×C₅₃	C₂⁴	1	3	yes	—
152	C₂×D₄×C₅₃	C₂⁴	1	1	yes	characteristic
153	C₂²×C₄×C₅₃	—	2	3	no	—
154	C₂²×C₄×C₅₃	—	2	6	no	—
155	C₂²×C₄×C₅₃	—	2	3	no	—
156	C₂²×C₄×C₅₃	—	2	3	no	—
157	C₂²×C₄×C₅₃	—	2	6	no	—
158	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
159	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
160	Q₁₆×C₅₃	—	2	12	no	—
161	Q₁₆×C₅₃	—	2	24	no	—
162	Q₁₆×C₅₃	—	2	12	no	—
163	SD₁₆×C₅₃	—	2	12	no	—
164	SD₁₆×C₅₃	—	2	24	no	—
165	SD₁₆×C₅₃	—	2	4	no	—
166	SD₁₆×C₅₃	—	2	12	no	—
167	SD₁₆×C₅₃	—	2	12	no	—
168	D₈×C₅₃	—	2	4	no	—
169	D₈×C₅₃	—	2	12	no	—
170	M₁₆×C₅₃	C₂×D₄	1	24	yes	—
171	C₂×C₈×C₅₃	C₂×D₄	1	24	yes	—
172	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
173	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
174	C₄⋊C₄×C₅₃	—	2	6	no	—
175	C₄⋊C₄×C₅₃	—	2	12	no	—
176	C₄⋊C₄×C₅₃	—	2	6	no	—
177	C₄⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
178	C₂²⋊C₄×C₅₃	—	2	6	no	—
179	C₂²⋊C₄×C₅₃	—	2	12	no	—
180	C₂²⋊C₄×C₅₃	—	2	2	no	—
181	C₂²⋊C₄×C₅₃	—	2	12	no	—
182	C₂²⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
183	C₄²×C₅₃	C₂×D₄	1	6	yes	—
184	C₄²×C₅₃	C₂⁴	1	3	yes	—
185	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
186	C₂³×C₅₃	—	2	1	no	—
187	C₂³×C₅₃	—	2	2	no	—
188	C₂³×C₅₃	—	2	3	no	—
189	C₂³×C₅₃	—	4	6	no	—
190	C₂³×C₅₃	—	8	6	no	—
191	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
192	Q₈×C₅₃	—	2	6	no	—
193	Q₈×C₅₃	—	2	6	no	—
194	Q₈×C₅₃	—	2	6	no	—
195	Q₈×C₅₃	—	2	2	no	—
196	Q₈×C₅₃	—	4	6	no	—
197	Q₈×C₅₃	—	4	12	no	—
198	Q₈×C₅₃	#127	1	4	yes	—
199	D₄×C₅₃	—	2	6	no	—
200	D₄×C₅₃	—	2	2	no	—
201	D₄×C₅₃	—	2	2	no	—
202	D₄×C₅₃	—	2	2	no	—
203	D₄×C₅₃	—	4	4	no	—
204	D₄×C₅₃	—	4	6	no	—
205	D₄×C₅₃	—	4	6	no	—
206	D₄×C₅₃	—	4	12	no	—
207	D₄×C₅₃	—	4	6	no	—
208	D₄×C₅₃	—	8	6	no	—
209	D₄×C₅₃	#127	1	12	yes	—
210	C₂×C₄×C₅₃	—	2	1	no	—
211	C₂×C₄×C₅₃	—	2	3	no	—
212	C₂×C₄×C₅₃	—	2	6	no	—
213	C₂×C₄×C₅₃	—	4	6	no	—
214	C₂×C₄×C₅₃	—	4	6	no	—
215	C₂×C₄×C₅₃	—	4	12	no	—
216	C₂×C₄×C₅₃	—	4	6	no	—
217	C₂×C₄×C₅₃	—	4	6	no	—
218	C₂×C₄×C₅₃	—	4	12	no	—
219	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
220	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
221	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
222	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
223	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
224	C₈×C₅₃	C₂²×D₄	1	4	yes	—
225	C₈×C₅₃	C₂²×D₄	1	12	yes	—
226	#60651	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
227	C₂²×C₅₃	—	2	6	no	—
228	C₂²×C₅₃	—	4	4	no	—
229	C₂²×C₅₃	—	4	2	no	—
230	C₂²×C₅₃	—	4	6	no	—
231	C₂²×C₅₃	—	4	2	no	—
232	C₂²×C₅₃	—	4	1	no	—
233	C₂²×C₅₃	—	4	3	no	—
234	C₂²×C₅₃	—	16	6	no	—
235	C₂²×C₅₃	C₂×#127	1	6	yes	—
236	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
237	C₄×C₅₃	—	2	2	no	—
238	C₄×C₅₃	—	2	6	no	—
239	C₄×C₅₃	—	4	1	no	—
240	C₄×C₅₃	—	4	3	no	—
241	C₄×C₅₃	—	4	6	no	—
242	C₄×C₅₃	C₂×#127	1	6	yes	—
243	C₄×C₅₃	C₂³×D₄	1	2	yes	—
244	C₂×#578	C₂×C₅₃	1	2	yes	complemented
245	C₂×#577	C₂×C₅₃	1	2	yes	complemented
246	#3500	C₂×C₅₃	1	1	yes	characteristic, complemented
247	#3499	C₂×C₅₃	1	1	yes	characteristic, complemented
248	#3156	C₂×C₅₃	1	6	yes	complemented
249	#3154	C₂×C₅₃	1	6	yes	complemented
250	#3153	C₂×C₅₃	1	3	yes	complemented
251	#3152	C₂×C₅₃	1	3	yes	complemented
252	#3150	C₂×C₅₃	1	3	yes	complemented
253	C₂×#497	C₂×C₅₃	1	3	yes	complemented
254	#2875	C₂×C₅₃	1	1	yes	characteristic, complemented
255	C₂×C₅₃	—	2	6	no	—
256	C₂×C₅₃	—	8	2	no	—
257	C₂×C₅₃	—	8	1	no	—
258	C₂×C₅₃	—	8	3	no	—
259	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
260	C₂×C₅₃	#3586	1	2	yes	—
261	C₂×2₋¹⁺⁴	C₂²×C₅₃	1	2	yes	—
262	C₂×2₋¹⁺⁴	C₂²×C₅₃	1	1	yes	characteristic
263	C₂×H(5,𝔽₂)	C₂²×C₅₃	1	1	yes	characteristic
264	#578	—	2	16	no	—
265	#577	—	2	16	no	—
266	C₂×#137	C₂²×C₅₃	1	6	yes	—
267	C₂×#137	C₂²×C₅₃	1	6	yes	complemented
268	C₂×#137	C₂²×C₅₃	1	6	yes	complemented
269	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	6	yes	—
270	C₂×#135	C₂²×C₅₃	1	6	yes	—
271	C₂×#135	C₂²×C₅₃	1	6	yes	complemented
272	C₂²×Q₁₆	C₂²×C₅₃	1	6	yes	complemented
273	C₂²×SD₁₆	C₂²×C₅₃	1	6	yes	—
274	C₂×#131	C₂²×C₅₃	1	2	yes	—
275	D₄×Q₈	—	2	2	no	—
276	#548	—	2	2	no	—
277	#547	—	2	6	no	—
278	#546	—	2	6	no	—
279	#536	C₂²×C₅₃	1	3	yes	—
280	#536	C₂²×C₅₃	1	6	yes	complemented
281	#535	C₂²×C₅₃	1	3	yes	—
282	#532	C₂²×C₅₃	1	3	yes	—
283	#532	C₂²×C₅₃	1	3	yes	complemented
284	C₂×C₄⋊Q₈	C₂²×C₅₃	1	3	yes	complemented
285	C₂×#124	C₂²×C₅₃	1	3	yes	—
286	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
287	#497	C₂²×C₅₃	1	3	yes	complemented
288	#497	C₂²×C₅₃	1	6	yes	complemented
289	#497	C₂²×C₅₃	1	12	yes	complemented
290	#497	C₂²×C₅₃	1	3	yes	complemented
291	#497	C₂²×C₅₃	1	6	yes	complemented
292	#496	C₂²×C₅₃	1	1	yes	characteristic, complemented
293	#496	C₂²×C₅₃	1	3	yes	complemented
294	#496	C₂²×C₅₃	1	6	yes	complemented
295	#495	C₂²×C₅₃	1	12	yes	complemented
296	#494	C₂²×C₅₃	1	6	yes	complemented
297	#492	C₂²×C₅₃	1	6	yes	complemented
298	#405	C₂²×C₅₃	1	6	yes	complemented
299	C₄×M₁₆	C₂²×C₅₃	1	3	yes	complemented
300	C₂×#97	C₂²×C₅₃	1	3	yes	complemented
301	C₅₃	#60651	1	1	yes	characteristic, direct product factor, Sylow
302	2₋¹⁺⁴	—	2	8	no	—
303	2₋¹⁺⁴	—	2	8	no	—
304	2₋¹⁺⁴	—	2	8	no	—
305	H(5,𝔽₂)	—	2	8	no	—
306	C₂×#41	—	2	1	no	—
307	C₂×#41	—	2	3	no	—
308	C₂×#41	—	2	6	no	—
309	C₂×#41	C₂³×C₅₃	1	3	yes	—
310	C₂×#41	C₂³×C₅₃	1	3	yes	—
311	C₂×#41	C₂³×C₅₃	1	6	yes	—
312	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
313	C₂²×Q₈	—	2	1	no	—
314	C₂²×Q₈	—	2	2	no	—
315	C₂²×Q₈	C₂³×C₅₃	1	3	yes	—
316	C₂²×Q₈	C₂³×C₅₃	1	6	yes	—
317	C₂²×D₄	—	2	3	no	—
318	C₂²×D₄	C₂³×C₅₃	1	3	yes	—
319	#137	—	2	24	no	—
320	#137	—	2	24	no	—
321	#137	—	2	24	no	—
322	AGL(1,ℤ/8)	—	2	24	no	—
323	#135	—	2	24	no	—
324	#135	—	2	24	no	—
325	C₂×Q₁₆	—	2	24	no	—
326	C₂×Q₁₆	C₂³×C₅₃	1	6	yes	—
327	C₂×Q₁₆	C₂³×C₅₃	1	12	yes	—
328	C₂×Q₁₆	C₂³×C₅₃	1	6	yes	—
329	C₂×SD₁₆	—	2	24	no	—
330	C₂×SD₁₆	C₂³×C₅₃	1	6	yes	—
331	C₂×SD₁₆	C₂³×C₅₃	1	12	yes	—
332	C₂×SD₁₆	C₂³×C₅₃	1	2	yes	—
333	C₂×SD₁₆	C₂³×C₅₃	1	6	yes	—
334	C₂×SD₁₆	C₂³×C₅₃	1	6	yes	—
335	C₂×D₈	C₂³×C₅₃	1	6	yes	—
336	C₂×D₈	C₂³×C₅₃	1	2	yes	—
337	#131	D₄×C₅₃	1	16	yes	complemented
338	C₂×M₁₆	C₂³×C₅₃	1	6	yes	—
339	C₂²×C₈	C₂³×C₅₃	1	6	yes	—
340	C₄⋊Q₈	—	2	6	no	—
341	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
342	C₄⋊Q₈	C₂³×C₅₃	1	6	yes	—
343	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
344	#127	C₂³×C₅₃	1	3	yes	—
345	#127	C₂³×C₅₃	1	1	yes	characteristic
346	#124	—	2	6	no	—
347	#124	C₂³×C₅₃	1	3	yes	—
348	#124	C₂³×C₅₃	1	6	yes	—
349	#124	C₂³×C₅₃	1	1	yes	characteristic
350	#124	C₂³×C₅₃	1	6	yes	—
351	#123	—	2	6	no	—
352	#123	—	2	12	no	—
353	C₂²⋊Q₈	—	2	6	no	—
354	C₂²⋊Q₈	—	2	12	no	—
355	#121	—	2	6	no	—
356	#121	—	2	6	no	—
357	#121	—	2	6	no	—
358	#120	—	2	6	no	—
359	C₄×Q₈	D₄×C₅₃	1	2	yes	—
360	C₄×D₄	—	2	6	no	—
361	C₄×D₄	—	2	6	no	—
362	C₄×D₄	D₄×C₅₃	1	6	yes	—
363	#117	C₂³×C₅₃	1	3	yes	—
364	C₂×C₄⋊C₄	—	2	6	no	—
365	C₂×C₂²⋊C₄	—	2	6	no	—
366	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
367	#97	C₂³×C₅₃	1	1	yes	characteristic
368	#97	C₂³×C₅₃	1	3	yes	—
369	#97	C₂³×C₅₃	1	6	yes	—
370	C₄×C₈	C₂³×C₅₃	1	6	yes	—
371	C₂⁴	—	2	3	no	—
372	#41	—	2	12	no	—
373	#41	—	2	12	no	—
374	#41	—	2	12	no	—
375	#41	—	2	12	no	—
376	#41	—	4	4	no	—
377	#41	—	4	12	no	—
378	#41	—	4	24	no	—
379	#41	C₂×D₄×C₅₃	1	8	yes	—
380	C₂×Q₈	—	2	3	no	—
381	C₂×Q₈	—	2	6	no	—
382	C₂×Q₈	—	2	12	no	—
383	C₂×Q₈	—	2	12	no	—
384	C₂×Q₈	—	2	12	no	—
385	C₂×Q₈	—	4	4	no	—
386	C₂×Q₈	—	4	8	no	—
387	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
388	C₂×Q₈	C₂⁴×C₅₃	1	3	yes	—
389	C₂×Q₈	C₂⁴×C₅₃	1	6	yes	—
390	C₂×Q₈	C₂⁴×C₅₃	1	1	yes	characteristic
391	C₂×D₄	—	2	3	no	—
392	C₂×D₄	—	2	3	no	—
393	C₂×D₄	—	2	6	no	—
394	C₂×D₄	—	2	12	no	—
395	C₂×D₄	—	2	2	no	—
396	C₂×D₄	—	2	3	no	—
397	C₂×D₄	—	2	6	no	—
398	C₂×D₄	—	4	12	no	—
399	C₂×D₄	C₂×D₄×C₅₃	1	3	yes	—
400	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
401	C₂×D₄	C₂⁴×C₅₃	1	3	yes	—
402	C₂×D₄	C₂⁴×C₅₃	1	1	yes	characteristic
403	C₂²×C₄	—	2	3	no	—
404	C₂²×C₄	—	2	6	no	—
405	C₂²×C₄	—	2	3	no	—
406	C₂²×C₄	—	2	3	no	—
407	C₂²×C₄	—	2	6	no	—
408	C₂²×C₄	C₂×D₄×C₅₃	1	6	yes	—
409	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
410	Q₁₆	—	2	12	no	—
411	Q₁₆	—	2	24	no	—
412	Q₁₆	—	2	12	no	—
413	SD₁₆	—	2	12	no	—
414	SD₁₆	—	2	24	no	—
415	SD₁₆	—	2	4	no	—
416	SD₁₆	—	2	12	no	—
417	SD₁₆	—	2	12	no	—
418	D₈	—	2	4	no	—
419	D₈	—	2	12	no	—
420	M₁₆	C₂×D₄×C₅₃	1	24	yes	—
421	C₂×C₈	C₂×D₄×C₅₃	1	24	yes	—
422	C₂×C₈	C₂⁴×C₅₃	1	6	yes	—
423	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
424	C₄⋊C₄	—	2	6	no	—
425	C₄⋊C₄	—	2	12	no	—
426	C₄⋊C₄	—	2	6	no	—
427	C₄⋊C₄	C₂×D₄×C₅₃	1	6	yes	—
428	C₂²⋊C₄	—	2	6	no	—
429	C₂²⋊C₄	—	2	12	no	—
430	C₂²⋊C₄	—	2	2	no	—
431	C₂²⋊C₄	—	2	12	no	—
432	C₂²⋊C₄	C₂×D₄×C₅₃	1	6	yes	—
433	C₄²	C₂×D₄×C₅₃	1	6	yes	—
434	C₄²	C₂⁴×C₅₃	1	3	yes	—
435	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
436	C₂³	—	2	2	no	—
437	C₂³	—	2	3	no	—
438	C₂³	—	2	1	no	—
439	C₂³	—	4	6	no	—
440	C₂³	—	8	6	no	—
441	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
442	Q₈	—	2	6	no	—
443	Q₈	—	2	6	no	—
444	Q₈	—	2	6	no	—
445	Q₈	—	2	2	no	—
446	Q₈	—	4	6	no	—
447	Q₈	—	4	12	no	—
448	Q₈	#127×C₅₃	1	4	yes	—
449	D₄	—	2	2	no	—
450	D₄	—	2	6	no	—
451	D₄	—	2	2	no	—
452	D₄	—	2	2	no	—
453	D₄	—	4	4	no	—
454	D₄	—	4	6	no	—
455	D₄	—	4	6	no	—
456	D₄	—	4	12	no	—
457	D₄	—	4	6	no	—
458	D₄	—	8	6	no	—
459	D₄	#127×C₅₃	1	12	yes	—
460	C₂×C₄	—	2	3	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	6	no	—
463	C₂×C₄	—	4	6	no	—
464	C₂×C₄	—	4	6	no	—
465	C₂×C₄	—	4	12	no	—
466	C₂×C₄	—	4	6	no	—
467	C₂×C₄	—	4	6	no	—
468	C₂×C₄	—	4	12	no	—
469	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
470	C₂×C₄	C₂²×D₄×C₅₃	1	6	yes	—
471	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
472	C₂×C₄	C₂²×D₄×C₅₃	1	12	yes	—
473	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
474	C₈	C₂²×D₄×C₅₃	1	4	yes	—
475	C₈	C₂²×D₄×C₅₃	1	12	yes	—
476	C₂²	—	2	6	no	—
477	C₂²	—	4	4	no	—
478	C₂²	—	4	2	no	—
479	C₂²	—	4	6	no	—
480	C₂²	—	4	2	no	—
481	C₂²	—	4	1	no	—
482	C₂²	—	4	3	no	—
483	C₂²	—	16	6	no	—
484	C₂²	C₂×#127×C₅₃	1	6	yes	—
485	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
486	C₄	—	2	2	no	—
487	C₄	—	2	6	no	—
488	C₄	—	4	1	no	—
489	C₄	—	4	3	no	—
490	C₄	—	4	6	no	—
491	C₄	C₂×#127×C₅₃	1	6	yes	—
492	C₄	C₂³×D₄×C₅₃	1	2	yes	—
493	C₂	—	2	6	no	—
494	C₂	—	8	3	no	—
495	C₂	—	8	2	no	—
496	C₂	—	8	1	no	—
497	C₂	C₂²×#127×C₅₃	1	1	yes	characteristic
498	C₂	C₅₃×#3586	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. C₅₃×#3500	C₂	semidirect
4. C₅₃×#3499	C₂	semidirect
5. C₅₃×#3156	C₂	semidirect
6. C₅₃×#3154	C₂	semidirect
7. C₅₃×#3153	C₂	semidirect
8. C₅₃×#3152	C₂	semidirect
9. C₅₃×#3150	C₂	semidirect
10. C₂×C₅₃×#497	C₂	semidirect
11. C₅₃×#2875	C₂	semidirect
18. C₂×#137×C₅₃	C₂²	semidirect
19. C₂×#137×C₅₃	C₂²	semidirect
22. C₂×#135×C₅₃	C₂²	semidirect
23. C₂²×Q₁₆×C₅₃	C₂²	semidirect
31. C₅₃×#536	C₂²	semidirect
34. C₅₃×#532	C₂²	semidirect
35. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
38. C₅₃×#497	C₂²	semidirect
39. C₅₃×#497	C₂²	semidirect
40. C₅₃×#497	C₂²	semidirect
41. C₅₃×#497	C₂²	semidirect
42. C₅₃×#497	C₂²	semidirect
43. C₅₃×#496	C₂²	semidirect
44. C₅₃×#496	C₂²	semidirect
45. C₅₃×#496	C₂²	semidirect
46. C₅₃×#495	C₂²	semidirect
47. C₅₃×#494	C₂²	semidirect
48. C₅₃×#492	C₂²	semidirect
49. C₅₃×#405	C₂²	semidirect
50. C₄×M₁₆×C₅₃	C₂²	semidirect
51. C₂×#97×C₅₃	C₂²	semidirect
87. #131×C₅₃	D₄	semidirect
226. #60651	C₅₃	direct (N × H)
244. C₂×#578	C₂×C₅₃	semidirect
245. C₂×#577	C₂×C₅₃	semidirect
246. #3500	C₂×C₅₃	semidirect
247. #3499	C₂×C₅₃	semidirect
248. #3156	C₂×C₅₃	semidirect
249. #3154	C₂×C₅₃	semidirect
250. #3153	C₂×C₅₃	semidirect
251. #3152	C₂×C₅₃	semidirect
252. #3150	C₂×C₅₃	semidirect
253. C₂×#497	C₂×C₅₃	semidirect
254. #2875	C₂×C₅₃	semidirect
267. C₂×#137	C₂²×C₅₃	semidirect
268. C₂×#137	C₂²×C₅₃	semidirect
271. C₂×#135	C₂²×C₅₃	semidirect
272. C₂²×Q₁₆	C₂²×C₅₃	semidirect
280. #536	C₂²×C₅₃	semidirect
283. #532	C₂²×C₅₃	semidirect
284. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
287. #497	C₂²×C₅₃	semidirect
288. #497	C₂²×C₅₃	semidirect
289. #497	C₂²×C₅₃	semidirect
290. #497	C₂²×C₅₃	semidirect
291. #497	C₂²×C₅₃	semidirect
292. #496	C₂²×C₅₃	semidirect
293. #496	C₂²×C₅₃	semidirect
294. #496	C₂²×C₅₃	semidirect
295. #495	C₂²×C₅₃	semidirect
296. #494	C₂²×C₅₃	semidirect
297. #492	C₂²×C₅₃	semidirect
298. #405	C₂²×C₅₃	semidirect
299. C₄×M₁₆	C₂²×C₅₃	semidirect
300. C₂×#97	C₂²×C₅₃	semidirect
301. C₅₃	#60651	direct (N × H)
337. #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
C₅₃×#3500	2
C₅₃×#3499	2
C₅₃×#3156	2
C₅₃×#3154	2
C₅₃×#3153	2
C₅₃×#3152	2
C₅₃×#3150	2
C₂×C₅₃×#497	2
C₅₃×#2875	2
#60651	53

Subgroups — C53×#60651 #27268163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60651 #27268163