Subgroups — C₂×C₅₃×#3182 #27268201

Order 13568.

2234 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#532	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3182	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#468	C₂	1	4	yes	maximal, complemented
9	C₂×C₅₃×#440	C₂	1	4	yes	maximal, complemented
10	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₅₃	—	2	1	no	—
12	C₂²×SD₁₆×C₅₃	C₂²	1	4	yes	—
13	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#533	—	2	4	no	—
15	C₅₃×#533	C₂²	1	8	yes	complemented
16	C₅₃×#532	—	2	4	no	—
17	C₅₃×#532	C₂²	1	8	yes	—
18	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#125×C₅₃	—	2	1	no	—
23	C₂×#124×C₅₃	—	2	1	no	—
24	C₂×C₂²⋊Q₈×C₅₃	—	2	1	no	—
25	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
26	C₂×#121×C₅₃	—	2	1	no	—
27	C₂×#121×C₅₃	C₂²	1	2	yes	—
28	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
29	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₅₃	—	2	1	no	—
31	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
32	C₂×#117×C₅₃	—	2	1	no	—
33	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#500	C₂²	1	8	yes	complemented
35	C₅₃×#492	C₂²	1	8	yes	complemented
36	C₅₃×#487	C₂²	1	8	yes	complemented
37	C₅₃×#486	C₂²	1	8	yes	complemented
38	C₅₃×#468	C₂²	1	32	yes	complemented
39	C₅₃×#440	C₂²	1	32	yes	complemented
40	C₂×#107×C₅₃	C₂²	1	4	yes	—
41	C₂×#103×C₅₃	C₂²	1	4	yes	—
42	C₂×#102×C₅₃	C₂²	1	4	yes	—
43	C₄×M₁₆×C₅₃	C₂²	1	8	yes	complemented
44	C₂×#97×C₅₃	C₂²	1	2	yes	—
45	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
47	C₂×#41×C₅₃	—	2	8	no	—
48	C₂×#41×C₅₃	—	4	8	no	—
49	C₂²×Q₈×C₅₃	—	2	1	no	—
50	C₂²×Q₈×C₅₃	—	2	1	no	—
51	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
52	C₂²×D₄×C₅₃	—	2	2	no	—
53	C₂²×D₄×C₅₃	—	2	1	no	—
54	C₂²×D₄×C₅₃	—	2	1	no	—
55	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
56	C₂³×C₄×C₅₃	—	2	2	no	—
57	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
58	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
59	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
60	C₂×SD₁₆×C₅₃	—	2	16	no	—
61	C₂×SD₁₆×C₅₃	C₂³	1	16	yes	—
62	C₂×M₁₆×C₅₃	—	2	8	no	—
63	C₂×M₁₆×C₅₃	C₂³	1	8	yes	—
64	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
65	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
66	C₄⋊Q₈×C₅₃	—	2	2	no	—
67	C₄⋊Q₈×C₅₃	—	2	2	no	—
68	C₄⋊Q₈×C₅₃	—	2	2	no	—
69	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
70	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
71	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
72	#127×C₅₃	—	2	2	no	—
73	#127×C₅₃	C₂³	1	4	yes	—
74	#125×C₅₃	—	2	8	no	—
75	#124×C₅₃	—	2	8	no	—
76	C₂²⋊Q₈×C₅₃	—	2	8	no	—
77	C₂²⋊Q₈×C₅₃	—	2	4	no	—
78	C₂²⋊Q₈×C₅₃	C₂³	1	8	yes	—
79	#121×C₅₃	—	2	8	no	—
80	#121×C₅₃	—	2	4	no	—
81	#121×C₅₃	C₂³	1	8	yes	—
82	C₄×Q₈×C₅₃	—	2	8	no	—
83	C₄×Q₈×C₅₃	—	2	4	no	—
84	C₄×Q₈×C₅₃	C₂³	1	8	yes	—
85	C₄×D₄×C₅₃	—	2	8	no	—
86	C₄×D₄×C₅₃	—	2	4	no	—
87	C₄×D₄×C₅₃	C₂³	1	8	yes	—
88	#117×C₅₃	—	2	8	no	—
89	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
90	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
91	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
92	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
93	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
94	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
95	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
97	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
98	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
99	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
100	C₂×C₄²×C₅₃	—	2	2	no	—
101	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
102	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
103	C₂×C₄²×C₅₃	D₄	1	4	yes	—
104	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
107	#107×C₅₃	C₂³	1	16	yes	—
108	#103×C₅₃	C₂³	1	16	yes	—
109	#102×C₅₃	C₂³	1	16	yes	—
110	#97×C₅₃	C₂³	1	8	yes	—
111	C₄×C₈×C₅₃	C₂³	1	4	yes	—
112	C₄×C₈×C₅₃	C₂³	1	4	yes	—
113	C₂⁴×C₅₃	—	2	2	no	—
114	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
115	#41×C₅₃	—	4	32	no	—
116	C₂×Q₈×C₅₃	—	2	2	no	—
117	C₂×Q₈×C₅₃	—	2	4	no	—
118	C₂×Q₈×C₅₃	—	2	4	no	—
119	C₂×Q₈×C₅₃	—	2	8	no	—
120	C₂×Q₈×C₅₃	—	4	4	no	—
121	C₂×Q₈×C₅₃	—	4	4	no	—
122	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
123	C₂×D₄×C₅₃	—	2	8	no	—
124	C₂×D₄×C₅₃	—	2	2	no	—
125	C₂×D₄×C₅₃	—	2	4	no	—
126	C₂×D₄×C₅₃	—	2	4	no	—
127	C₂×D₄×C₅₃	—	2	8	no	—
128	C₂×D₄×C₅₃	—	4	8	no	—
129	C₂×D₄×C₅₃	—	4	4	no	—
130	C₂×D₄×C₅₃	—	4	4	no	—
131	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
132	C₂²×C₄×C₅₃	—	2	4	no	—
133	C₂²×C₄×C₅₃	—	2	4	no	—
134	C₂²×C₄×C₅₃	—	2	4	no	—
135	C₂²×C₄×C₅₃	—	2	2	no	—
136	C₂²×C₄×C₅₃	—	2	2	no	—
137	C₂²×C₄×C₅₃	—	2	2	no	—
138	C₂²×C₄×C₅₃	—	2	8	no	—
139	C₂²×C₄×C₅₃	—	4	8	no	—
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	4	yes	—
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	1	yes	characteristic
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	SD₁₆×C₅₃	—	2	32	no	—
151	M₁₆×C₅₃	—	2	16	no	—
152	C₂×C₈×C₅₃	—	2	4	no	—
153	C₂×C₈×C₅₃	#41	1	8	yes	complemented
154	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
155	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
156	C₄⋊C₄×C₅₃	—	2	4	no	—
157	C₄⋊C₄×C₅₃	—	2	4	no	—
158	C₄⋊C₄×C₅₃	—	2	4	no	—
159	C₄⋊C₄×C₅₃	—	2	4	no	—
160	C₄⋊C₄×C₅₃	—	2	8	no	—
161	C₄⋊C₄×C₅₃	—	2	8	no	—
162	C₄⋊C₄×C₅₃	—	2	2	no	—
163	C₄⋊C₄×C₅₃	—	2	2	no	—
164	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
165	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
166	C₂²⋊C₄×C₅₃	—	2	8	no	—
167	C₂²⋊C₄×C₅₃	—	2	8	no	—
168	C₂²⋊C₄×C₅₃	—	2	8	no	—
169	C₄²×C₅₃	—	2	8	no	—
170	C₄²×C₅₃	C₂×D₄	1	2	yes	—
171	C₄²×C₅₃	C₂×D₄	1	4	yes	—
172	C₄²×C₅₃	C₂×D₄	1	4	yes	—
173	C₄²×C₅₃	C₂×D₄	1	2	yes	—
174	C₄²×C₅₃	C₂⁴	1	2	yes	—
175	C₄²×C₅₃	C₂⁴	1	2	yes	—
176	C₂³×C₅₃	—	2	2	no	—
177	C₂³×C₅₃	—	2	2	no	—
178	C₂³×C₅₃	—	2	2	no	—
179	C₂³×C₅₃	—	2	4	no	—
180	C₂³×C₅₃	—	4	4	no	—
181	C₂³×C₅₃	—	4	4	no	—
182	C₂³×C₅₃	—	4	4	no	—
183	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
184	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
185	Q₈×C₅₃	—	2	8	no	—
186	Q₈×C₅₃	—	4	4	no	—
187	Q₈×C₅₃	—	4	8	no	—
188	Q₈×C₅₃	—	4	8	no	—
189	D₄×C₅₃	—	2	8	no	—
190	D₄×C₅₃	—	4	16	no	—
191	D₄×C₅₃	—	4	4	no	—
192	D₄×C₅₃	—	4	8	no	—
193	D₄×C₅₃	—	4	8	no	—
194	C₂×C₄×C₅₃	—	2	2	no	—
195	C₂×C₄×C₅₃	—	2	2	no	—
196	C₂×C₄×C₅₃	—	2	2	no	—
197	C₂×C₄×C₅₃	—	2	4	no	—
198	C₂×C₄×C₅₃	—	2	8	no	—
199	C₂×C₄×C₅₃	—	2	8	no	—
200	C₂×C₄×C₅₃	—	2	4	no	—
201	C₂×C₄×C₅₃	—	2	4	no	—
202	C₂×C₄×C₅₃	—	2	4	no	—
203	C₂×C₄×C₅₃	—	2	4	no	—
204	C₂×C₄×C₅₃	—	4	16	no	—
205	C₂×C₄×C₅₃	—	4	4	no	—
206	C₂×C₄×C₅₃	—	4	4	no	—
207	C₂×C₄×C₅₃	—	4	4	no	—
208	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
209	C₂×C₄×C₅₃	#137	1	4	yes	—
210	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
211	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
212	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
213	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
215	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
216	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
217	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
218	C₈×C₅₃	—	2	4	no	—
219	C₈×C₅₃	C₂×#41	1	8	yes	complemented
220	C₂×#3182	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
221	C₂²×C₅₃	—	2	2	no	—
222	C₂²×C₅₃	—	2	2	no	—
223	C₂²×C₅₃	—	2	2	no	—
224	C₂²×C₅₃	—	4	4	no	—
225	C₂²×C₅₃	—	4	4	no	—
226	C₂²×C₅₃	—	4	4	no	—
227	C₂²×C₅₃	—	4	4	no	—
228	C₂²×C₅₃	—	8	8	no	—
229	C₂²×C₅₃	#532	1	2	yes	—
230	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
231	C₂²×C₅₃	C₂×#137	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	—	2	2	no	—
236	C₄×C₅₃	—	4	4	no	—
237	C₄×C₅₃	—	4	4	no	—
238	C₄×C₅₃	—	4	4	no	—
239	C₄×C₅₃	#532	1	2	yes	—
240	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
241	C₄×C₅₃	C₂×#137	1	4	yes	—
242	C₄×C₅₃	C₂²×#41	1	2	yes	—
243	C₂×#533	C₂×C₅₃	1	1	yes	characteristic, complemented
244	C₂×#532	C₂×C₅₃	1	1	yes	characteristic
245	#3182	C₂×C₅₃	1	16	yes	direct product factor
246	C₂×#500	C₂×C₅₃	1	1	yes	characteristic, complemented
247	C₂×#492	C₂×C₅₃	1	1	yes	characteristic, complemented
248	C₂×#487	C₂×C₅₃	1	1	yes	characteristic, complemented
249	C₂×#486	C₂×C₅₃	1	1	yes	characteristic, complemented
250	C₂×#468	C₂×C₅₃	1	4	yes	complemented
251	C₂×#440	C₂×C₅₃	1	4	yes	complemented
252	C₂×C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
253	C₂×C₅₃	—	4	2	no	—
254	C₂×C₅₃	—	8	4	no	—
255	C₂×C₅₃	#3182	1	4	yes	direct product factor
256	C₂×C₅₃	C₂×#532	1	1	yes	characteristic
257	C₂×C₅₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
258	C₂×C₅₃	C₂²×#137	1	1	yes	characteristic
259	C₂²×#41	—	2	1	no	—
260	C₂²×SD₁₆	C₂²×C₅₃	1	4	yes	—
261	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
262	#533	—	2	4	no	—
263	#533	C₂²×C₅₃	1	8	yes	complemented
264	#532	—	2	4	no	—
265	#532	C₂²×C₅₃	1	8	yes	—
266	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
267	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
268	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
269	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
270	C₂×#125	—	2	1	no	—
271	C₂×#124	—	2	1	no	—
272	C₂×C₂²⋊Q₈	—	2	1	no	—
273	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
274	C₂×#121	—	2	1	no	—
275	C₂×#121	C₂²×C₅₃	1	2	yes	—
276	C₂×C₄×Q₈	—	2	1	no	—
277	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
278	C₂×C₄×D₄	—	2	1	no	—
279	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
280	C₂×#117	—	2	1	no	—
281	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
282	#500	C₂²×C₅₃	1	8	yes	complemented
283	#492	C₂²×C₅₃	1	8	yes	complemented
284	#487	C₂²×C₅₃	1	8	yes	complemented
285	#486	C₂²×C₅₃	1	8	yes	complemented
286	#468	C₂²×C₅₃	1	32	yes	complemented
287	#440	C₂²×C₅₃	1	32	yes	complemented
288	C₂×#107	C₂²×C₅₃	1	4	yes	—
289	C₂×#103	C₂²×C₅₃	1	4	yes	—
290	C₂×#102	C₂²×C₅₃	1	4	yes	—
291	C₄×M₁₆	C₂²×C₅₃	1	8	yes	complemented
292	C₂×#97	C₂²×C₅₃	1	2	yes	—
293	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
294	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
295	C₅₃	C₂×#3182	1	1	yes	characteristic, direct product factor, Sylow
296	C₂×#41	—	2	8	no	—
297	C₂×#41	—	4	8	no	—
298	C₂²×Q₈	—	2	1	no	—
299	C₂²×Q₈	—	2	1	no	—
300	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
301	C₂²×D₄	—	2	2	no	—
302	C₂²×D₄	—	2	1	no	—
303	C₂²×D₄	—	2	1	no	—
304	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
305	C₂³×C₄	—	2	2	no	—
306	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
307	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
308	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
309	C₂×SD₁₆	—	2	16	no	—
310	C₂×SD₁₆	C₂³×C₅₃	1	16	yes	—
311	C₂×M₁₆	—	2	8	no	—
312	C₂×M₁₆	C₂³×C₅₃	1	8	yes	—
313	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
314	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
315	C₄⋊Q₈	—	2	2	no	—
316	C₄⋊Q₈	—	2	2	no	—
317	C₄⋊Q₈	—	2	2	no	—
318	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
319	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
320	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
321	#127	—	2	2	no	—
322	#127	C₂³×C₅₃	1	4	yes	—
323	#125	—	2	8	no	—
324	#124	—	2	8	no	—
325	C₂²⋊Q₈	—	2	8	no	—
326	C₂²⋊Q₈	—	2	4	no	—
327	C₂²⋊Q₈	C₂³×C₅₃	1	8	yes	—
328	#121	—	2	8	no	—
329	#121	—	2	4	no	—
330	#121	C₂³×C₅₃	1	8	yes	—
331	C₄×Q₈	—	2	8	no	—
332	C₄×Q₈	—	2	4	no	—
333	C₄×Q₈	C₂³×C₅₃	1	8	yes	—
334	C₄×D₄	—	2	8	no	—
335	C₄×D₄	—	2	4	no	—
336	C₄×D₄	C₂³×C₅₃	1	8	yes	—
337	#117	—	2	8	no	—
338	C₂×C₄⋊C₄	—	2	1	no	—
339	C₂×C₄⋊C₄	—	2	1	no	—
340	C₂×C₄⋊C₄	—	2	1	no	—
341	C₂×C₄⋊C₄	—	2	1	no	—
342	C₂×C₄⋊C₄	—	2	2	no	—
343	C₂×C₄⋊C₄	—	2	2	no	—
344	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
345	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
346	C₂×C₂²⋊C₄	—	2	2	no	—
347	C₂×C₂²⋊C₄	—	2	2	no	—
348	C₂×C₂²⋊C₄	—	2	2	no	—
349	C₂×C₄²	—	2	2	no	—
350	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
351	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
352	C₂×C₄²	D₄×C₅₃	1	4	yes	—
353	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
354	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
355	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
356	#107	C₂³×C₅₃	1	16	yes	—
357	#103	C₂³×C₅₃	1	16	yes	—
358	#102	C₂³×C₅₃	1	16	yes	—
359	#97	C₂³×C₅₃	1	8	yes	—
360	C₄×C₈	C₂³×C₅₃	1	4	yes	—
361	C₄×C₈	C₂³×C₅₃	1	4	yes	—
362	C₂⁴	—	2	2	no	—
363	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
364	#41	—	4	32	no	—
365	C₂×Q₈	—	2	2	no	—
366	C₂×Q₈	—	2	4	no	—
367	C₂×Q₈	—	2	4	no	—
368	C₂×Q₈	—	2	8	no	—
369	C₂×Q₈	—	4	4	no	—
370	C₂×Q₈	—	4	4	no	—
371	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
372	C₂×D₄	—	2	8	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	4	no	—
375	C₂×D₄	—	2	4	no	—
376	C₂×D₄	—	2	8	no	—
377	C₂×D₄	—	4	8	no	—
378	C₂×D₄	—	4	4	no	—
379	C₂×D₄	—	4	4	no	—
380	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
381	C₂²×C₄	—	2	4	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	8	no	—
388	C₂²×C₄	—	4	8	no	—
389	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
390	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
391	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
392	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
393	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
394	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
395	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
396	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
397	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
398	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
399	SD₁₆	—	2	32	no	—
400	M₁₆	—	2	16	no	—
401	C₂×C₈	—	2	4	no	—
402	C₂×C₈	#41×C₅₃	1	8	yes	complemented
403	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
404	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
405	C₄⋊C₄	—	2	4	no	—
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	—	2	4	no	—
409	C₄⋊C₄	—	2	8	no	—
410	C₄⋊C₄	—	2	8	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
414	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
415	C₂²⋊C₄	—	2	8	no	—
416	C₂²⋊C₄	—	2	8	no	—
417	C₂²⋊C₄	—	2	8	no	—
418	C₄²	—	2	8	no	—
419	C₄²	C₂×D₄×C₅₃	1	2	yes	—
420	C₄²	C₂×D₄×C₅₃	1	4	yes	—
421	C₄²	C₂×D₄×C₅₃	1	4	yes	—
422	C₄²	C₂×D₄×C₅₃	1	2	yes	—
423	C₄²	C₂⁴×C₅₃	1	2	yes	—
424	C₄²	C₂⁴×C₅₃	1	2	yes	—
425	C₂³	—	2	2	no	—
426	C₂³	—	2	2	no	—
427	C₂³	—	2	2	no	—
428	C₂³	—	2	4	no	—
429	C₂³	—	4	4	no	—
430	C₂³	—	4	4	no	—
431	C₂³	—	4	4	no	—
432	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
433	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
434	Q₈	—	2	8	no	—
435	Q₈	—	4	4	no	—
436	Q₈	—	4	8	no	—
437	Q₈	—	4	8	no	—
438	D₄	—	2	8	no	—
439	D₄	—	4	16	no	—
440	D₄	—	4	4	no	—
441	D₄	—	4	8	no	—
442	D₄	—	4	8	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	8	no	—
448	C₂×C₄	—	2	8	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	4	16	no	—
454	C₂×C₄	—	4	4	no	—
455	C₂×C₄	—	4	4	no	—
456	C₂×C₄	—	4	4	no	—
457	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
458	C₂×C₄	#137×C₅₃	1	4	yes	—
459	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
460	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
461	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
462	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
463	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
464	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
465	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
466	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
467	C₈	—	2	4	no	—
468	C₈	C₂×#41×C₅₃	1	8	yes	complemented
469	C₂²	—	2	2	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	2	2	no	—
472	C₂²	—	4	4	no	—
473	C₂²	—	4	4	no	—
474	C₂²	—	4	4	no	—
475	C₂²	—	4	4	no	—
476	C₂²	—	8	8	no	—
477	C₂²	C₅₃×#532	1	2	yes	—
478	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
479	C₂²	C₂×#137×C₅₃	1	2	yes	—
480	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
481	C₄	—	2	2	no	—
482	C₄	—	2	2	no	—
483	C₄	—	2	2	no	—
484	C₄	—	4	4	no	—
485	C₄	—	4	4	no	—
486	C₄	—	4	4	no	—
487	C₄	C₅₃×#532	1	2	yes	—
488	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	4	yes	—
489	C₄	C₂×#137×C₅₃	1	4	yes	—
490	C₄	C₂²×#41×C₅₃	1	2	yes	—
491	C₂	—	4	2	no	—
492	C₂	—	8	4	no	—
493	C₂	C₅₃×#3182	1	4	yes	direct product factor
494	C₂	C₂×C₅₃×#532	1	1	yes	characteristic
495	C₂	C₂²×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
496	C₂	C₂²×#137×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₂×C₅₃×#532	C₂
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×#533	C₂	semidirect
3. C₅₃×#3182	C₂	direct (N × H)
4. C₂×C₅₃×#500	C₂	semidirect
5. C₂×C₅₃×#492	C₂	semidirect
6. C₂×C₅₃×#487	C₂	semidirect
7. C₂×C₅₃×#486	C₂	semidirect
8. C₂×C₅₃×#468	C₂	semidirect
9. C₂×C₅₃×#440	C₂	semidirect
10. C₂×C₄×M₁₆×C₅₃	C₂	semidirect
15. C₅₃×#533	C₂²	semidirect
34. C₅₃×#500	C₂²	semidirect
35. C₅₃×#492	C₂²	semidirect
36. C₅₃×#487	C₂²	semidirect
37. C₅₃×#486	C₂²	semidirect
38. C₅₃×#468	C₂²	semidirect
39. C₅₃×#440	C₂²	semidirect
43. C₄×M₁₆×C₅₃	C₂²	semidirect
153. C₂×C₈×C₅₃	#41	semidirect
219. C₈×C₅₃	C₂×#41	semidirect
220. C₂×#3182	C₅₃	direct (N × H)
243. C₂×#533	C₂×C₅₃	semidirect
245. #3182	C₂×C₅₃	direct (N × H)
246. C₂×#500	C₂×C₅₃	semidirect
247. C₂×#492	C₂×C₅₃	semidirect
248. C₂×#487	C₂×C₅₃	semidirect
249. C₂×#486	C₂×C₅₃	semidirect
250. C₂×#468	C₂×C₅₃	semidirect
251. C₂×#440	C₂×C₅₃	semidirect
252. C₂×C₄×M₁₆	C₂×C₅₃	semidirect
255. C₂×C₅₃	#3182	direct (N × H)
263. #533	C₂²×C₅₃	semidirect
282. #500	C₂²×C₅₃	semidirect
283. #492	C₂²×C₅₃	semidirect
284. #487	C₂²×C₅₃	semidirect
285. #486	C₂²×C₅₃	semidirect
286. #468	C₂²×C₅₃	semidirect
287. #440	C₂²×C₅₃	semidirect
291. C₄×M₁₆	C₂²×C₅₃	semidirect
295. C₅₃	C₂×#3182	direct (N × H)
402. C₂×C₈	#41×C₅₃	semidirect
468. C₈	C₂×#41×C₅₃	semidirect
493. C₂	C₅₃×#3182	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×#533	2
C₂×C₅₃×#532	2
C₅₃×#3182	2
C₂×C₅₃×#500	2
C₂×C₅₃×#492	2
C₂×C₅₃×#487	2
C₂×C₅₃×#486	2
C₂×C₅₃×#468	2
C₂×C₅₃×#440	2
C₂×C₄×M₁₆×C₅₃	2
C₂×#3182	53

Subgroups — C2×C53×#3182 #27268201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#3182 #27268201