Subgroups — C₂×C₅₃×#3205 #27268286

Order 13568.

3102 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#127×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3205	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#493	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#486	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#459	C₂	1	4	yes	maximal, complemented
7	C₂×C₅₃×#447	C₂	1	4	yes	maximal, complemented
8	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₅₃	—	2	1	no	—
10	C₂³×D₄×C₅₃	—	2	1	no	—
11	C₂³×D₄×C₅₃	—	2	1	no	—
12	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
13	C₂²×D₈×C₅₃	C₂²	1	8	yes	complemented
14	C₅₃×#532	—	2	4	no	—
15	C₅₃×#532	C₂²	1	8	yes	complemented
16	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
17	C₂×#127×C₅₃	—	2	1	no	—
18	C₂×#127×C₅₃	—	2	1	no	—
19	C₂×#127×C₅₃	—	2	4	no	—
20	C₂×#127×C₅₃	C₂²	1	2	yes	—
21	C₂×#127×C₅₃	C₂²	1	2	yes	—
22	C₂×#127×C₅₃	C₂²	1	8	yes	complemented
23	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic, complemented
24	C₂×#124×C₅₃	—	2	1	no	—
25	C₂×#121×C₅₃	—	2	1	no	—
26	C₂×#121×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₅₃	—	2	1	no	—
28	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
29	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#493	C₂²	1	16	yes	complemented
31	C₅₃×#486	C₂²	1	16	yes	complemented
32	C₅₃×#459	C₂²	1	32	yes	complemented
33	C₅₃×#447	C₂²	1	32	yes	complemented
34	C₅₃×#431	C₂²	1	8	yes	complemented
35	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	complemented
36	C₂×#102×C₅₃	C₂²	1	8	yes	complemented
37	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	complemented
39	C₂⁵×C₅₃	—	2	2	no	—
40	C₂×#41×C₅₃	—	2	8	no	—
41	C₂×#41×C₅₃	—	4	8	no	—
42	C₂²×Q₈×C₅₃	—	2	1	no	—
43	C₂²×D₄×C₅₃	—	2	2	no	—
44	C₂²×D₄×C₅₃	—	2	1	no	—
45	C₂²×D₄×C₅₃	—	2	1	no	—
46	C₂²×D₄×C₅₃	—	2	4	no	—
47	C₂²×D₄×C₅₃	—	2	2	no	—
48	C₂²×D₄×C₅₃	—	2	2	no	—
49	C₂²×D₄×C₅₃	—	2	2	no	—
50	C₂²×D₄×C₅₃	—	2	2	no	—
51	C₂²×D₄×C₅₃	—	2	2	no	—
52	C₂²×D₄×C₅₃	—	2	8	no	—
53	C₂²×D₄×C₅₃	—	2	8	no	—
54	C₂²×D₄×C₅₃	—	4	8	no	—
55	C₂²×D₄×C₅₃	—	4	8	no	—
56	C₂²×D₄×C₅₃	—	4	8	no	—
57	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
58	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
59	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
60	C₂³×C₄×C₅₃	—	2	2	no	—
61	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
62	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
63	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
64	C₂×D₈×C₅₃	—	4	16	no	—
65	C₂×D₈×C₅₃	C₂³	1	32	yes	complemented
66	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
67	C₄⋊Q₈×C₅₃	—	2	2	no	—
68	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	complemented
69	#127×C₅₃	—	2	4	no	—
70	#127×C₅₃	—	2	4	no	—
71	#127×C₅₃	—	2	8	no	—
72	#127×C₅₃	—	2	8	no	—
73	#127×C₅₃	—	2	2	no	—
74	#127×C₅₃	C₂³	1	8	yes	—
75	#127×C₅₃	C₂³	1	8	yes	—
76	#127×C₅₃	C₂³	1	4	yes	complemented
77	#124×C₅₃	—	2	8	no	—
78	#121×C₅₃	—	2	8	no	—
79	#121×C₅₃	—	2	4	no	—
80	#121×C₅₃	C₂³	1	8	yes	—
81	C₄×D₄×C₅₃	—	2	8	no	—
82	C₄×D₄×C₅₃	—	2	4	no	—
83	C₄×D₄×C₅₃	C₂³	1	8	yes	complemented
84	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
85	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
86	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
88	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
89	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
90	C₂×C₄²×C₅₃	D₄	1	4	yes	—
91	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
92	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
93	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
94	C₄⋊C₈×C₅₃	C₂³	1	8	yes	complemented
95	#102×C₅₃	C₂³	1	32	yes	complemented
96	C₂²⋊C₈×C₅₃	C₂³	1	8	yes	—
97	C₄×C₈×C₅₃	C₂³	1	8	yes	complemented
98	C₂⁴×C₅₃	—	2	8	no	—
99	C₂⁴×C₅₃	—	2	2	no	—
100	C₂⁴×C₅₃	—	2	4	no	—
101	C₂⁴×C₅₃	—	4	8	no	—
102	C₂⁴×C₅₃	—	4	8	no	—
103	C₂⁴×C₅₃	—	4	8	no	—
104	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
105	#41×C₅₃	—	4	32	no	—
106	C₂×Q₈×C₅₃	—	2	4	no	—
107	C₂×Q₈×C₅₃	—	4	4	no	—
108	C₂×D₄×C₅₃	—	2	8	no	—
109	C₂×D₄×C₅₃	—	2	2	no	—
110	C₂×D₄×C₅₃	—	2	4	no	—
111	C₂×D₄×C₅₃	—	2	4	no	—
112	C₂×D₄×C₅₃	—	2	8	no	—
113	C₂×D₄×C₅₃	—	2	4	no	—
114	C₂×D₄×C₅₃	—	2	8	no	—
115	C₂×D₄×C₅₃	—	2	8	no	—
116	C₂×D₄×C₅₃	—	2	8	no	—
117	C₂×D₄×C₅₃	—	2	8	no	—
118	C₂×D₄×C₅₃	—	2	8	no	—
119	C₂×D₄×C₅₃	—	2	16	no	—
120	C₂×D₄×C₅₃	—	4	16	no	—
121	C₂×D₄×C₅₃	—	4	8	no	—
122	C₂×D₄×C₅₃	—	4	8	no	—
123	C₂×D₄×C₅₃	—	4	16	no	—
124	C₂×D₄×C₅₃	—	4	32	no	—
125	C₂×D₄×C₅₃	—	4	8	no	—
126	C₂×D₄×C₅₃	—	4	8	no	—
127	C₂×D₄×C₅₃	—	4	8	no	—
128	C₂×D₄×C₅₃	—	4	4	no	—
129	C₂×D₄×C₅₃	—	4	32	no	—
130	C₂×D₄×C₅₃	—	4	4	no	—
131	C₂×D₄×C₅₃	—	4	8	no	—
132	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
133	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
134	C₂²×C₄×C₅₃	—	2	4	no	—
135	C₂²×C₄×C₅₃	—	2	4	no	—
136	C₂²×C₄×C₅₃	—	2	2	no	—
137	C₂²×C₄×C₅₃	—	2	8	no	—
138	C₂²×C₄×C₅₃	—	4	8	no	—
139	C₂²×C₄×C₅₃	D₈	1	8	yes	complemented
140	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
141	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
142	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
147	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
149	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
150	D₈×C₅₃	—	4	32	no	—
151	C₂×C₈×C₅₃	—	2	8	no	—
152	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
153	C₄⋊C₄×C₅₃	—	2	4	no	—
154	C₄⋊C₄×C₅₃	—	2	2	no	—
155	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
156	C₂²⋊C₄×C₅₃	—	2	8	no	—
157	C₂²⋊C₄×C₅₃	—	2	8	no	—
158	C₄²×C₅₃	C₂×D₄	1	2	yes	—
159	C₄²×C₅₃	C₂×D₄	1	4	yes	—
160	C₄²×C₅₃	C₂×D₄	1	4	yes	—
161	C₄²×C₅₃	C₂×D₄	1	2	yes	—
162	C₄²×C₅₃	C₂⁴	1	2	yes	—
163	C₄²×C₅₃	C₂⁴	1	2	yes	—
164	C₂³×C₅₃	—	2	2	no	—
165	C₂³×C₅₃	—	2	2	no	—
166	C₂³×C₅₃	—	2	4	no	—
167	C₂³×C₅₃	—	2	8	no	—
168	C₂³×C₅₃	—	4	16	no	—
169	C₂³×C₅₃	—	4	16	no	—
170	C₂³×C₅₃	—	4	8	no	—
171	C₂³×C₅₃	—	4	4	no	—
172	C₂³×C₅₃	—	4	4	no	—
173	C₂³×C₅₃	—	4	16	no	—
174	C₂³×C₅₃	—	4	8	no	—
175	C₂³×C₅₃	—	4	8	no	—
176	C₂³×C₅₃	—	4	4	no	—
177	C₂³×C₅₃	—	8	32	no	—
178	C₂³×C₅₃	C₂×D₈	1	4	yes	—
179	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
180	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
181	Q₈×C₅₃	—	4	8	no	—
182	D₄×C₅₃	—	2	16	no	—
183	D₄×C₅₃	—	2	8	no	—
184	D₄×C₅₃	—	4	16	no	—
185	D₄×C₅₃	—	4	16	no	—
186	D₄×C₅₃	—	4	16	no	—
187	D₄×C₅₃	—	4	4	no	—
188	D₄×C₅₃	—	4	8	no	—
189	D₄×C₅₃	—	4	16	no	—
190	D₄×C₅₃	—	4	16	no	—
191	D₄×C₅₃	—	4	8	no	—
192	D₄×C₅₃	—	4	8	no	—
193	D₄×C₅₃	—	4	16	no	—
194	C₂×C₄×C₅₃	—	2	2	no	—
195	C₂×C₄×C₅₃	—	2	2	no	—
196	C₂×C₄×C₅₃	—	2	4	no	—
197	C₂×C₄×C₅₃	—	2	8	no	—
198	C₂×C₄×C₅₃	—	2	4	no	—
199	C₂×C₄×C₅₃	—	2	4	no	—
200	C₂×C₄×C₅₃	—	4	4	no	—
201	C₂×C₄×C₅₃	—	4	16	no	—
202	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
203	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
204	C₂×C₄×C₅₃	C₂×D₈	1	16	yes	complemented
205	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
206	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
207	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
208	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
210	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
211	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
212	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
213	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
214	C₈×C₅₃	—	2	8	no	—
215	C₂×#3205	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
216	C₂²×C₅₃	—	2	2	no	—
217	C₂²×C₅₃	—	2	8	no	—
218	C₂²×C₅₃	—	2	2	no	—
219	C₂²×C₅₃	—	4	8	no	—
220	C₂²×C₅₃	—	4	4	no	—
221	C₂²×C₅₃	—	4	4	no	—
222	C₂²×C₅₃	—	4	8	no	—
223	C₂²×C₅₃	—	4	8	no	—
224	C₂²×C₅₃	—	4	4	no	—
225	C₂²×C₅₃	—	8	32	no	—
226	C₂²×C₅₃	—	8	8	no	—
227	C₂²×C₅₃	—	8	16	no	—
228	C₂²×C₅₃	#535	1	2	yes	—
229	C₂²×C₅₃	C₂²×D₈	1	4	yes	—
230	C₂²×C₅₃	C₂²×D₈	1	2	yes	—
231	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
232	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
233	C₄×C₅₃	—	2	2	no	—
234	C₄×C₅₃	—	2	2	no	—
235	C₄×C₅₃	—	4	4	no	—
236	C₄×C₅₃	#535	1	2	yes	—
237	C₄×C₅₃	C₂²×D₈	1	4	yes	—
238	C₄×C₅₃	C₂²×D₈	1	4	yes	—
239	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
240	C₄×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
241	C₂×#532	C₂×C₅₃	1	1	yes	characteristic, complemented
242	C₂²×#127	C₂×C₅₃	1	1	yes	characteristic, complemented
243	#3205	C₂×C₅₃	1	16	yes	direct product factor
244	C₂×#493	C₂×C₅₃	1	2	yes	complemented
245	C₂×#486	C₂×C₅₃	1	2	yes	complemented
246	C₂×#459	C₂×C₅₃	1	4	yes	complemented
247	C₂×#447	C₂×C₅₃	1	4	yes	complemented
248	C₂×#431	C₂×C₅₃	1	1	yes	characteristic, complemented
249	C₂×C₅₃	—	2	4	no	—
250	C₂×C₅₃	—	8	4	no	—
251	C₂×C₅₃	—	8	8	no	—
252	C₂×C₅₃	#3205	1	4	yes	direct product factor
253	C₂×C₅₃	C₂×#535	1	1	yes	characteristic
254	C₂×C₅₃	C₂³×D₈	1	1	yes	characteristic
255	C₂×C₅₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
256	C₂²×#41	—	2	1	no	—
257	C₂³×D₄	—	2	1	no	—
258	C₂³×D₄	—	2	1	no	—
259	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
260	C₂²×D₈	C₂²×C₅₃	1	8	yes	complemented
261	#532	—	2	4	no	—
262	#532	C₂²×C₅₃	1	8	yes	complemented
263	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
264	C₂×#127	—	2	1	no	—
265	C₂×#127	—	2	1	no	—
266	C₂×#127	—	2	4	no	—
267	C₂×#127	C₂²×C₅₃	1	2	yes	—
268	C₂×#127	C₂²×C₅₃	1	2	yes	—
269	C₂×#127	C₂²×C₅₃	1	8	yes	complemented
270	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic, complemented
271	C₂×#124	—	2	1	no	—
272	C₂×#121	—	2	1	no	—
273	C₂×#121	C₂²×C₅₃	1	2	yes	—
274	C₂×C₄×D₄	—	2	1	no	—
275	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	complemented
276	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
277	#493	C₂²×C₅₃	1	16	yes	complemented
278	#486	C₂²×C₅₃	1	16	yes	complemented
279	#459	C₂²×C₅₃	1	32	yes	complemented
280	#447	C₂²×C₅₃	1	32	yes	complemented
281	#431	C₂²×C₅₃	1	8	yes	complemented
282	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
283	C₂×#102	C₂²×C₅₃	1	8	yes	complemented
284	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
285	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	complemented
286	C₅₃	C₂×#3205	1	1	yes	characteristic, direct product factor, Sylow
287	C₂⁵	—	2	2	no	—
288	C₂×#41	—	2	8	no	—
289	C₂×#41	—	4	8	no	—
290	C₂²×Q₈	—	2	1	no	—
291	C₂²×D₄	—	2	2	no	—
292	C₂²×D₄	—	2	1	no	—
293	C₂²×D₄	—	2	1	no	—
294	C₂²×D₄	—	2	2	no	—
295	C₂²×D₄	—	2	4	no	—
296	C₂²×D₄	—	2	2	no	—
297	C₂²×D₄	—	2	2	no	—
298	C₂²×D₄	—	2	2	no	—
299	C₂²×D₄	—	2	2	no	—
300	C₂²×D₄	—	2	8	no	—
301	C₂²×D₄	—	2	8	no	—
302	C₂²×D₄	—	4	8	no	—
303	C₂²×D₄	—	4	8	no	—
304	C₂²×D₄	—	4	8	no	—
305	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
306	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
307	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
308	C₂³×C₄	—	2	2	no	—
309	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
310	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
311	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
312	C₂×D₈	—	4	16	no	—
313	C₂×D₈	C₂³×C₅₃	1	32	yes	complemented
314	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
315	C₄⋊Q₈	—	2	2	no	—
316	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	complemented
317	#127	—	2	4	no	—
318	#127	—	2	4	no	—
319	#127	—	2	8	no	—
320	#127	—	2	8	no	—
321	#127	—	2	2	no	—
322	#127	C₂³×C₅₃	1	8	yes	—
323	#127	C₂³×C₅₃	1	8	yes	—
324	#127	C₂³×C₅₃	1	4	yes	complemented
325	#124	—	2	8	no	—
326	#121	—	2	8	no	—
327	#121	—	2	4	no	—
328	#121	C₂³×C₅₃	1	8	yes	—
329	C₄×D₄	—	2	8	no	—
330	C₄×D₄	—	2	4	no	—
331	C₄×D₄	C₂³×C₅₃	1	8	yes	complemented
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
334	C₂×C₂²⋊C₄	—	2	2	no	—
335	C₂×C₂²⋊C₄	—	2	2	no	—
336	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
337	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
338	C₂×C₄²	D₄×C₅₃	1	4	yes	—
339	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
340	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
341	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
342	C₄⋊C₈	C₂³×C₅₃	1	8	yes	complemented
343	#102	C₂³×C₅₃	1	32	yes	complemented
344	C₂²⋊C₈	C₂³×C₅₃	1	8	yes	—
345	C₄×C₈	C₂³×C₅₃	1	8	yes	complemented
346	C₂⁴	—	2	2	no	—
347	C₂⁴	—	2	8	no	—
348	C₂⁴	—	2	4	no	—
349	C₂⁴	—	4	8	no	—
350	C₂⁴	—	4	8	no	—
351	C₂⁴	—	4	8	no	—
352	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
353	#41	—	4	32	no	—
354	C₂×Q₈	—	2	4	no	—
355	C₂×Q₈	—	4	4	no	—
356	C₂×D₄	—	2	8	no	—
357	C₂×D₄	—	2	2	no	—
358	C₂×D₄	—	2	4	no	—
359	C₂×D₄	—	2	4	no	—
360	C₂×D₄	—	2	8	no	—
361	C₂×D₄	—	2	4	no	—
362	C₂×D₄	—	2	8	no	—
363	C₂×D₄	—	2	8	no	—
364	C₂×D₄	—	2	8	no	—
365	C₂×D₄	—	2	8	no	—
366	C₂×D₄	—	2	8	no	—
367	C₂×D₄	—	2	16	no	—
368	C₂×D₄	—	4	16	no	—
369	C₂×D₄	—	4	8	no	—
370	C₂×D₄	—	4	8	no	—
371	C₂×D₄	—	4	16	no	—
372	C₂×D₄	—	4	32	no	—
373	C₂×D₄	—	4	8	no	—
374	C₂×D₄	—	4	8	no	—
375	C₂×D₄	—	4	8	no	—
376	C₂×D₄	—	4	4	no	—
377	C₂×D₄	—	4	32	no	—
378	C₂×D₄	—	4	4	no	—
379	C₂×D₄	—	4	8	no	—
380	C₂×D₄	C₂⁴×C₅₃	1	8	yes	—
381	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	8	no	—
386	C₂²×C₄	—	4	8	no	—
387	C₂²×C₄	D₈×C₅₃	1	8	yes	complemented
388	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
389	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
390	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
391	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
392	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
393	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
394	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
395	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
396	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
397	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
398	D₈	—	4	32	no	—
399	C₂×C₈	—	2	8	no	—
400	C₂×C₈	C₂⁴×C₅₃	1	8	yes	—
401	C₄⋊C₄	—	2	4	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
404	C₂²⋊C₄	—	2	8	no	—
405	C₂²⋊C₄	—	2	8	no	—
406	C₄²	C₂×D₄×C₅₃	1	4	yes	—
407	C₄²	C₂×D₄×C₅₃	1	2	yes	—
408	C₄²	C₂×D₄×C₅₃	1	4	yes	—
409	C₄²	C₂×D₄×C₅₃	1	2	yes	—
410	C₄²	C₂⁴×C₅₃	1	2	yes	—
411	C₄²	C₂⁴×C₅₃	1	2	yes	—
412	C₂³	—	2	2	no	—
413	C₂³	—	2	2	no	—
414	C₂³	—	2	4	no	—
415	C₂³	—	2	8	no	—
416	C₂³	—	4	16	no	—
417	C₂³	—	4	16	no	—
418	C₂³	—	4	8	no	—
419	C₂³	—	4	4	no	—
420	C₂³	—	4	4	no	—
421	C₂³	—	4	16	no	—
422	C₂³	—	4	8	no	—
423	C₂³	—	4	8	no	—
424	C₂³	—	4	4	no	—
425	C₂³	—	8	32	no	—
426	C₂³	C₂×D₈×C₅₃	1	4	yes	—
427	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
428	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
429	Q₈	—	4	8	no	—
430	D₄	—	2	8	no	—
431	D₄	—	2	16	no	—
432	D₄	—	4	16	no	—
433	D₄	—	4	16	no	—
434	D₄	—	4	16	no	—
435	D₄	—	4	4	no	—
436	D₄	—	4	8	no	—
437	D₄	—	4	16	no	—
438	D₄	—	4	16	no	—
439	D₄	—	4	8	no	—
440	D₄	—	4	8	no	—
441	D₄	—	4	16	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	8	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	4	4	no	—
449	C₂×C₄	—	4	16	no	—
450	C₂×C₄	C₂×D₈×C₅₃	1	4	yes	—
451	C₂×C₄	C₂×D₈×C₅₃	1	4	yes	—
452	C₂×C₄	C₂×D₈×C₅₃	1	16	yes	complemented
453	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
454	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
455	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
456	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
457	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
458	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
459	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
460	C₂×C₄	H(5,𝔽₂)×C₅₃	1	4	yes	—
461	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
462	C₈	—	2	8	no	—
463	C₂²	—	2	8	no	—
464	C₂²	—	2	2	no	—
465	C₂²	—	2	2	no	—
466	C₂²	—	4	4	no	—
467	C₂²	—	4	8	no	—
468	C₂²	—	4	8	no	—
469	C₂²	—	4	4	no	—
470	C₂²	—	4	8	no	—
471	C₂²	—	4	4	no	—
472	C₂²	—	8	32	no	—
473	C₂²	—	8	16	no	—
474	C₂²	—	8	8	no	—
475	C₂²	C₅₃×#535	1	2	yes	—
476	C₂²	C₂²×D₈×C₅₃	1	4	yes	—
477	C₂²	C₂²×D₈×C₅₃	1	2	yes	—
478	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
479	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
480	C₄	—	2	2	no	—
481	C₄	—	2	2	no	—
482	C₄	—	4	4	no	—
483	C₄	C₅₃×#535	1	2	yes	—
484	C₄	C₂²×D₈×C₅₃	1	4	yes	—
485	C₄	C₂²×D₈×C₅₃	1	4	yes	—
486	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	4	yes	—
487	C₄	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
488	C₂	—	2	4	no	—
489	C₂	—	8	8	no	—
490	C₂	—	8	4	no	—
491	C₂	C₅₃×#3205	1	4	yes	direct product factor
492	C₂	C₂×C₅₃×#535	1	1	yes	characteristic
493	C₂	C₂³×D₈×C₅₃	1	1	yes	characteristic
494	C₂	C₂²×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#532	C₂	semidirect
2. C₂²×#127×C₅₃	C₂	semidirect
3. C₅₃×#3205	C₂	direct (N × H)
4. C₂×C₅₃×#493	C₂	semidirect
5. C₂×C₅₃×#486	C₂	semidirect
6. C₂×C₅₃×#459	C₂	semidirect
7. C₂×C₅₃×#447	C₂	semidirect
8. C₂×C₅₃×#431	C₂	semidirect
13. C₂²×D₈×C₅₃	C₂²	semidirect
15. C₅₃×#532	C₂²	semidirect
16. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
22. C₂×#127×C₅₃	C₂²	semidirect
23. C₂×#127×C₅₃	C₂²	semidirect
28. C₂×C₄×D₄×C₅₃	C₂²	semidirect
30. C₅₃×#493	C₂²	semidirect
31. C₅₃×#486	C₂²	semidirect
32. C₅₃×#459	C₂²	semidirect
33. C₅₃×#447	C₂²	semidirect
34. C₅₃×#431	C₂²	semidirect
35. C₂×C₄⋊C₈×C₅₃	C₂²	semidirect
36. C₂×#102×C₅₃	C₂²	semidirect
38. C₂×C₄×C₈×C₅₃	C₂²	semidirect
65. C₂×D₈×C₅₃	C₂³	semidirect
68. C₄⋊Q₈×C₅₃	C₂³	semidirect
76. #127×C₅₃	C₂³	semidirect
83. C₄×D₄×C₅₃	C₂³	semidirect
94. C₄⋊C₈×C₅₃	C₂³	semidirect
95. #102×C₅₃	C₂³	semidirect
97. C₄×C₈×C₅₃	C₂³	semidirect
139. C₂²×C₄×C₅₃	D₈	semidirect
204. C₂×C₄×C₅₃	C₂×D₈	semidirect
215. C₂×#3205	C₅₃	direct (N × H)
241. C₂×#532	C₂×C₅₃	semidirect
242. C₂²×#127	C₂×C₅₃	semidirect
243. #3205	C₂×C₅₃	direct (N × H)
244. C₂×#493	C₂×C₅₃	semidirect
245. C₂×#486	C₂×C₅₃	semidirect
246. C₂×#459	C₂×C₅₃	semidirect
247. C₂×#447	C₂×C₅₃	semidirect
248. C₂×#431	C₂×C₅₃	semidirect
252. C₂×C₅₃	#3205	direct (N × H)
260. C₂²×D₈	C₂²×C₅₃	semidirect
262. #532	C₂²×C₅₃	semidirect
263. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
269. C₂×#127	C₂²×C₅₃	semidirect
270. C₂×#127	C₂²×C₅₃	semidirect
275. C₂×C₄×D₄	C₂²×C₅₃	semidirect
277. #493	C₂²×C₅₃	semidirect
278. #486	C₂²×C₅₃	semidirect
279. #459	C₂²×C₅₃	semidirect
280. #447	C₂²×C₅₃	semidirect
281. #431	C₂²×C₅₃	semidirect
282. C₂×C₄⋊C₈	C₂²×C₅₃	semidirect
283. C₂×#102	C₂²×C₅₃	semidirect
285. C₂×C₄×C₈	C₂²×C₅₃	semidirect
286. C₅₃	C₂×#3205	direct (N × H)
313. C₂×D₈	C₂³×C₅₃	semidirect
316. C₄⋊Q₈	C₂³×C₅₃	semidirect
324. #127	C₂³×C₅₃	semidirect
331. C₄×D₄	C₂³×C₅₃	semidirect
342. C₄⋊C₈	C₂³×C₅₃	semidirect
343. #102	C₂³×C₅₃	semidirect
345. C₄×C₈	C₂³×C₅₃	semidirect
387. C₂²×C₄	D₈×C₅₃	semidirect
452. C₂×C₄	C₂×D₈×C₅₃	semidirect
491. C₂	C₅₃×#3205	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#532	2
C₂²×#127×C₅₃	2
C₅₃×#3205	2
C₂×C₅₃×#493	2
C₂×C₅₃×#486	2
C₂×C₅₃×#459	2
C₂×C₅₃×#447	2
C₂×C₅₃×#431	2
C₂×#3205	53

Subgroups — C2×C53×#3205 #27268286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#3205 #27268286