Subgroups — C₂×#20283131 #28247488

Order 13568.

3342 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719912	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#117×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20283131	C₂	1	16	yes	maximal, direct product factor
5	C₂×#16810581	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810580	C₂	1	2	yes	maximal, complemented
7	C₂×#16810579	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16810578	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16810577	C₂	1	2	yes	maximal, complemented
10	C₂×#16810575	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×#11719912	C₂	1	2	yes	maximal, complemented
12	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
14	C₂²×#11719917	C₂²	1	1	yes	characteristic
15	C₂²×#11719912	C₄	1	16	yes	complemented
16	C₂²×#11719912	C₂²	1	2	yes	—
17	C₂²×#11719912	C₂²	1	1	yes	characteristic
18	C₂²×#11719912	C₂²	1	1	yes	characteristic
19	C₂²×#11719912	C₂²	1	1	yes	characteristic
20	C₂²×#11719912	C₂²	1	1	yes	characteristic
21	C₂²×#11719912	C₂²	1	8	yes	—
22	C₂×M₁₆×D₅₃	C₄	1	16	yes	complemented
23	C₂×M₁₆×D₅₃	C₂²	1	8	yes	—
24	C₂³×#88322	C₂²	1	1	yes	characteristic
25	C₂²×C₈×D₅₃	C₂²	1	2	yes	—
26	C₂×#13133176	C₂²	1	1	yes	characteristic
27	C₂×#13133175	C₄	1	2	yes	—
28	C₂×#13133174	C₂²	1	2	yes	—
29	C₂×#13133173	C₄	1	2	yes	—
30	#117×D₅₃	C₄	1	8	yes	—
31	#117×D₅₃	C₂²	1	8	yes	—
32	C₂×C₄⋊C₄×D₅₃	C₄	1	2	yes	—
33	C₂×C₂²⋊C₄×D₅₃	C₄	1	2	yes	—
34	C₂×C₄²×D₅₃	C₂²	1	2	yes	—
35	#16810581	C₂²	1	8	yes	complemented
36	#16810580	C₂²	1	16	yes	complemented
37	#16810579	C₂²	1	8	yes	complemented
38	#16810578	C₂²	1	8	yes	complemented
39	#16810577	C₂²	1	16	yes	complemented
40	#16810575	C₂²	1	16	yes	complemented
41	C₄×#11719912	C₂²	1	16	yes	complemented
42	C₂×#13133128	C₂²	1	2	yes	complemented
43	C₂×#13133127	C₂²	1	2	yes	complemented
44	C₂×#13133127	C₂²	1	2	yes	complemented
45	C₂×C₄×#88322	C₂²	1	2	yes	complemented
46	C₂×C₈×Dic₅₃	C₂²	1	2	yes	complemented
47	C₂×C₈×Dic₅₃	C₂²	1	2	yes	complemented
48	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
49	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
50	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
51	C₅₃×#405	C₂²	1	8	yes	complemented
52	C₂×#97×C₅₃	C₂²	1	2	yes	complemented
53	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	complemented
54	C₂⁴×D₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂³×Dic₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₄×D₅₃	C₂×C₄	1	4	yes	—
57	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₄×D₅₃	C₂×C₄	1	4	yes	—
59	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₄×D₅₃	C₂×C₄	1	8	yes	—
61	C₂²×C₄×D₅₃	C₂×C₄	1	8	yes	—
62	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
63	C₂²×C₄×D₅₃	C₂³	1	4	yes	—
64	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
65	C₂×#11719917	C₂×C₄	1	8	yes	complemented
66	C₂×#11719917	C₂³	1	4	yes	—
67	C₂×#11719912	C₂×C₄	1	16	yes	—
68	C₂×#11719912	C₂×C₄	1	8	yes	—
69	C₂×#11719912	C₂×C₄	1	8	yes	—
70	C₂×#11719912	C₂×C₄	1	64	yes	complemented
71	C₂×#11719912	C₂×C₄	1	8	yes	complemented
72	C₂×#11719912	C₂×C₄	1	8	yes	complemented
73	C₂×#11719912	C₂³	1	4	yes	—
74	C₂×#11719912	C₂³	1	8	yes	—
75	C₂×#11719912	C₂³	1	4	yes	—
76	C₂×#11719912	C₂³	1	4	yes	—
77	C₂×#11719912	C₂³	1	4	yes	—
78	M₁₆×D₅₃	C₂×C₄	1	64	yes	complemented
79	C₂²×#88322	C₂×C₄	1	8	yes	complemented
80	C₂²×#88322	C₂³	1	1	yes	characteristic
81	C₂²×#88322	C₂³	1	2	yes	—
82	C₂²×#88322	C₂³	1	1	yes	characteristic
83	C₂²×#88322	C₂³	1	4	yes	—
84	C₂×C₈×D₅₃	C₂×C₄	1	16	yes	complemented
85	C₂×C₈×D₅₃	C₂³	1	8	yes	—
86	#13133176	C₂×C₄	1	4	yes	—
87	#13133176	C₂³	1	4	yes	—
88	#13133175	C₂×C₄	1	16	yes	—
89	#13133174	C₂×C₄	1	8	yes	—
90	#13133174	C₂³	1	8	yes	—
91	#13133173	C₂×C₄	1	16	yes	—
92	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
93	C₂×#11719919	C₂×C₄	1	4	yes	—
94	C₄⋊C₄×D₅₃	C₂×C₄	1	16	yes	—
95	C₂×C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
96	C₂×#11719921	C₂×C₄	1	4	yes	—
97	C₂²⋊C₄×D₅₃	C₂×C₄	1	16	yes	—
98	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
99	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
100	C₄²×D₅₃	C₂×C₄	1	8	yes	—
101	C₄²×D₅₃	C₂³	1	8	yes	—
102	#13133128	C₂³	1	8	yes	complemented
103	#13133127	C₂³	1	8	yes	complemented
104	#13133127	C₂³	1	8	yes	complemented
105	C₄×#88322	C₂³	1	8	yes	complemented
106	C₈×Dic₅₃	C₂³	1	8	yes	complemented
107	C₈×Dic₅₃	C₂³	1	8	yes	complemented
108	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
109	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	complemented
110	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
111	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	complemented
112	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
113	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
114	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
115	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
116	#117×C₅₃	C₂×C₄	1	4	yes	—
117	#117×C₅₃	C₂³	1	4	yes	—
118	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
119	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
120	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
121	#97×C₅₃	C₂³	1	8	yes	complemented
122	C₄×C₈×C₅₃	C₂³	1	8	yes	complemented
123	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₂³×D₅₃	—	4	4	no	—
125	C₂³×D₅₃	C₄²	1	8	yes	—
126	C₂³×D₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₂³×D₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₂³×D₅₃	C₂²×C₄	1	4	yes	—
129	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
130	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
132	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
133	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
134	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
135	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
136	C₂²×Dic₅₃	C₄²	1	8	yes	—
137	C₂²×Dic₅₃	C₂²×C₄	1	4	yes	—
138	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
139	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
140	C₂²×Dic₅₃	C₂²×C₄	1	4	yes	—
141	C₂×C₄×D₅₃	—	4	8	no	—
142	C₂×C₄×D₅₃	C₄²	1	8	yes	—
143	C₂×C₄×D₅₃	C₄²	1	8	yes	—
144	C₂×C₄×D₅₃	C₄²	1	32	yes	—
145	C₂×C₄×D₅₃	C₂²×C₄	1	16	yes	—
146	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
147	C₂×C₄×D₅₃	C₂²×C₄	1	4	yes	—
148	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
149	C₂×C₄×D₅₃	C₂²×C₄	1	4	yes	—
150	C₂×C₄×D₅₃	C₂²×C₄	1	4	yes	—
151	C₂×C₄×D₅₃	C₂²×C₄	1	16	yes	—
152	C₂×C₄×D₅₃	C₂²×C₄	1	4	yes	—
153	C₂×C₄×D₅₃	C₂²×C₄	1	4	yes	—
154	C₂×C₄×D₅₃	C₂²×C₄	1	16	yes	—
155	C₂×C₄×D₅₃	C₂²×C₄	1	4	yes	—
156	C₂×C₄×D₅₃	C₂⁴	1	2	yes	—
157	C₂×C₄×D₅₃	C₂⁴	1	2	yes	—
158	M₁₆×C₅₃	C₂²×C₄	1	16	yes	complemented
159	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
160	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
161	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
162	C₂×C₈×C₅₃	C₂²×C₄	1	16	yes	complemented
163	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
164	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
165	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
166	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
167	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
168	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
169	C₄²×C₅₃	C₂⁴	1	4	yes	—
170	C₂²⋊Dic₅₃	C₂²×C₄	1	8	yes	—
171	#11719921	C₂²×C₄	1	16	yes	—
172	C₄⋊Dic₅₃	C₂²×C₄	1	8	yes	—
173	#11719919	C₂²×C₄	1	16	yes	—
174	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
175	C₄×Dic₅₃	C₂²×C₄	1	16	yes	—
176	C₄×Dic₅₃	C₂⁴	1	4	yes	—
177	#11719917	C₂²×C₄	1	16	yes	complemented
178	C₂×#88322	C₂²×C₄	1	8	yes	—
179	C₂×#88322	C₂²×C₄	1	4	yes	—
180	C₂×#88322	C₂²×C₄	1	4	yes	—
181	C₂×#88322	C₂²×C₄	1	16	yes	complemented
182	C₂×#88322	C₂⁴	1	2	yes	—
183	C₂×#88322	C₂⁴	1	4	yes	—
184	C₂×#88322	C₂⁴	1	2	yes	—
185	#11719912	C₂²×C₄	1	32	yes	—
186	#11719912	C₂²×C₄	1	16	yes	—
187	#11719912	C₂²×C₄	1	16	yes	—
188	#11719912	C₂²×C₄	1	16	yes	complemented
189	#11719912	C₂²×C₄	1	16	yes	complemented
190	C₈×D₅₃	C₂²×C₄	1	32	yes	complemented
191	C₂³×C₅₃	—	2	4	no	—
192	C₂³×C₅₃	C₂×C₄²	1	4	yes	—
193	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
194	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
195	C₂²×D₅₃	—	2	4	no	—
196	C₂²×D₅₃	—	2	4	no	—
197	C₂²×D₅₃	—	4	16	no	—
198	C₂²×D₅₃	—	4	8	no	—
199	C₂²×D₅₃	C₂×C₄²	1	4	yes	—
200	C₂²×D₅₃	C₂×C₄²	1	4	yes	—
201	C₂²×D₅₃	C₂×C₄²	1	16	yes	—
202	C₂²×D₅₃	C₂³×C₄	1	2	yes	—
203	C₂²×D₅₃	C₂³×C₄	1	2	yes	—
204	C₂×C₄×C₅₃	—	2	8	no	—
205	C₂×C₄×C₅₃	C₂×C₄²	1	8	yes	—
206	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
207	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
208	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
209	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
210	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
211	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
212	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
214	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
215	C₂×Dic₅₃	—	2	8	no	—
216	C₂×Dic₅₃	C₂×C₄²	1	16	yes	—
217	C₂×Dic₅₃	C₂×C₄²	1	4	yes	—
218	C₂×Dic₅₃	C₂×C₄²	1	4	yes	—
219	C₂×Dic₅₃	C₂³×C₄	1	8	yes	—
220	C₂×Dic₅₃	C₂³×C₄	1	2	yes	—
221	C₂×Dic₅₃	C₂³×C₄	1	2	yes	—
222	C₄×D₅₃	—	4	16	no	—
223	C₄×D₅₃	C₂×C₄²	1	8	yes	—
224	C₄×D₅₃	C₂×C₄²	1	16	yes	—
225	C₄×D₅₃	C₂×C₄²	1	8	yes	—
226	C₄×D₅₃	C₂×C₄²	1	16	yes	—
227	C₄×D₅₃	C₂³×C₄	1	4	yes	—
228	C₄×D₅₃	C₂³×C₄	1	4	yes	—
229	C₄×D₅₃	C₂³×C₄	1	4	yes	—
230	C₄×D₅₃	C₂³×C₄	1	4	yes	—
231	C₈×C₅₃	C₂³×C₄	1	8	yes	—
232	C₈×C₅₃	C₂³×C₄	1	4	yes	—
233	C₈×C₅₃	C₂³×C₄	1	4	yes	—
234	#88322	C₂³×C₄	1	8	yes	—
235	#88322	C₂³×C₄	1	4	yes	—
236	#88322	C₂³×C₄	1	4	yes	—
237	C₂×#2873	—	53	1	no	maximal, Sylow
238	C₂²×C₅₃	—	2	8	no	—
239	C₂²×C₅₃	—	2	4	no	—
240	C₂²×C₅₃	C₂²×C₄²	1	4	yes	—
241	C₂²×C₅₃	C₂²×C₄²	1	2	yes	—
242	C₂²×C₅₃	#568	1	4	yes	—
243	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
244	C₂×D₅₃	—	2	8	no	—
245	C₂×D₅₃	—	2	8	no	—
246	C₂×D₅₃	—	2	4	no	—
247	C₂×D₅₃	—	2	4	no	—
248	C₂×D₅₃	—	4	16	no	—
249	C₂×D₅₃	C₂²×C₄²	1	4	yes	—
250	C₂×D₅₃	C₂²×C₄²	1	4	yes	—
251	C₄×C₅₃	—	2	8	no	—
252	C₄×C₅₃	C₂²×C₄²	1	2	yes	—
253	C₄×C₅₃	C₂²×C₄²	1	4	yes	—
254	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
255	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
256	Dic₅₃	—	2	8	no	—
257	Dic₅₃	C₂²×C₄²	1	4	yes	—
258	Dic₅₃	C₂²×C₄²	1	4	yes	—
259	C₂³×M₁₆	—	53	1	no	—
260	C₂³×M₁₆	—	53	1	no	—
261	C₂²×#117	—	53	1	no	—
262	#2873	—	53	16	no	—
263	C₂×#405	—	53	2	no	—
264	C₂×#405	—	53	2	no	—
265	C₂×#405	—	53	1	no	—
266	C₂×#405	—	53	1	no	—
267	C₂×#405	—	53	1	no	—
268	C₂×#405	D₅₃	1	1	yes	characteristic, complemented
269	C₂×C₄×M₁₆	—	53	2	no	—
270	C₂×C₄×M₁₆	—	53	2	no	—
271	C₂×C₅₃	—	2	4	no	—
272	C₂×C₅₃	#2873	1	4	yes	complemented
273	C₂×C₅₃	C₂³×C₄²	1	1	yes	characteristic
274	C₂×C₅₃	C₂×#568	1	2	yes	—
275	D₅₃	—	2	4	no	—
276	D₅₃	—	2	4	no	—
277	C₂⁴×C₄	—	53	1	no	—
278	C₂²×M₁₆	—	53	1	no	—
279	C₂²×M₁₆	—	53	16	no	—
280	C₂²×M₁₆	—	53	8	no	—
281	C₂²×M₁₆	—	53	16	no	—
282	C₂²×M₁₆	—	53	2	no	—
283	C₂²×M₁₆	—	53	1	no	—
284	C₂²×M₁₆	—	53	1	no	—
285	C₂²×M₁₆	—	53	1	no	—
286	C₂²×M₁₆	—	53	1	no	—
287	C₂²×M₁₆	—	53	8	no	—
288	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
289	C₂³×C₈	—	53	1	no	—
290	C₂³×C₈	—	53	2	no	—
291	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
292	C₂×#117	—	53	1	no	—
293	C₂×#117	—	53	8	no	—
294	C₂×#117	—	53	8	no	—
295	C₂×#117	—	53	2	no	—
296	C₂×#117	—	53	2	no	—
297	C₂×#117	—	53	2	no	—
298	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
299	C₂²×C₄⋊C₄	—	53	2	no	—
300	C₂²×C₂²⋊C₄	—	53	2	no	—
301	C₂²×C₄²	—	53	2	no	—
302	#405	—	53	8	no	—
303	#405	—	53	8	no	—
304	#405	—	53	8	no	—
305	#405	—	53	16	no	—
306	#405	—	53	16	no	—
307	#405	C₂×D₅₃	1	8	yes	complemented
308	C₄×M₁₆	—	53	16	no	—
309	C₄×M₁₆	—	53	16	no	—
310	C₂×#97	—	53	2	no	—
311	C₂×#97	—	53	2	no	—
312	C₂×#97	—	53	2	no	—
313	C₂×#97	C₂×D₅₃	1	2	yes	complemented
314	C₂×C₄×C₈	—	53	2	no	—
315	C₂×C₄×C₈	—	53	2	no	—
316	C₂×C₄×C₈	—	53	2	no	—
317	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	complemented
318	C₅₃	C₂×#2873	1	1	yes	characteristic, complemented, Sylow
319	C₂⁵	—	53	1	no	—
320	C₂³×C₄	—	53	1	no	—
321	C₂³×C₄	—	53	1	no	—
322	C₂³×C₄	—	53	1	no	—
323	C₂³×C₄	—	53	4	no	—
324	C₂³×C₄	—	53	8	no	—
325	C₂³×C₄	—	53	8	no	—
326	C₂³×C₄	—	53	4	no	—
327	C₂³×C₄	—	53	1	no	—
328	C₂³×C₄	—	53	1	no	—
329	C₂³×C₄	—	53	4	no	—
330	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
331	C₂×M₁₆	—	53	8	no	—
332	C₂×M₁₆	—	53	4	no	—
333	C₂×M₁₆	—	53	64	no	—
334	C₂×M₁₆	—	53	16	no	—
335	C₂×M₁₆	—	53	8	no	—
336	C₂×M₁₆	—	53	8	no	—
337	C₂×M₁₆	—	53	8	no	—
338	C₂×M₁₆	—	53	8	no	—
339	C₂×M₁₆	—	53	64	no	—
340	C₂×M₁₆	—	53	4	no	—
341	C₂×M₁₆	—	53	8	no	—
342	C₂×M₁₆	—	53	4	no	—
343	C₂×M₁₆	—	53	4	no	—
344	C₂×M₁₆	—	53	4	no	—
345	C₂×M₁₆	C₄×D₅₃	1	8	yes	complemented
346	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
347	C₂²×C₈	—	53	8	no	—
348	C₂²×C₈	—	53	4	no	—
349	C₂²×C₈	—	53	1	no	—
350	C₂²×C₈	—	53	1	no	—
351	C₂²×C₈	—	53	2	no	—
352	C₂²×C₈	—	53	16	no	—
353	C₂²×C₈	—	53	8	no	—
354	C₂²×C₈	C₄×D₅₃	1	8	yes	complemented
355	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
356	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
357	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
358	C₂²×C₈	C₂²×D₅₃	1	4	yes	—
359	#117	—	53	4	no	—
360	#117	—	53	4	no	—
361	#117	—	53	16	no	—
362	#117	—	53	16	no	—
363	#117	—	53	8	no	—
364	#117	—	53	8	no	—
365	#117	C₄×D₅₃	1	4	yes	—
366	#117	C₂²×D₅₃	1	4	yes	—
367	C₂×C₄⋊C₄	—	53	4	no	—
368	C₂×C₄⋊C₄	—	53	2	no	—
369	C₂×C₄⋊C₄	—	53	16	no	—
370	C₂×C₄⋊C₄	C₄×D₅₃	1	2	yes	—
371	C₂×C₂²⋊C₄	—	53	2	no	—
372	C₂×C₂²⋊C₄	—	53	4	no	—
373	C₂×C₂²⋊C₄	—	53	16	no	—
374	C₂×C₂²⋊C₄	C₄×D₅₃	1	2	yes	—
375	C₂×C₄²	—	53	4	no	—
376	C₂×C₄²	—	53	2	no	—
377	C₂×C₄²	—	53	8	no	—
378	C₂×C₄²	—	53	8	no	—
379	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
380	#97	—	53	8	no	—
381	#97	—	53	8	no	—
382	#97	—	53	8	no	—
383	#97	C₂²×D₅₃	1	8	yes	complemented
384	C₄×C₈	—	53	8	no	—
385	C₄×C₈	—	53	8	no	—
386	C₄×C₈	—	53	8	no	—
387	C₄×C₈	C₂²×D₅₃	1	8	yes	complemented
388	C₂⁴	—	53	1	no	—
389	C₂⁴	—	53	1	no	—
390	C₂⁴	—	53	8	no	—
391	C₂⁴	—	53	4	no	—
392	C₂⁴	—	212	4	no	—
393	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
394	C₂²×C₄	—	53	8	no	—
395	C₂²×C₄	—	53	1	no	—
396	C₂²×C₄	—	53	1	no	—
397	C₂²×C₄	—	53	4	no	—
398	C₂²×C₄	—	53	8	no	—
399	C₂²×C₄	—	53	8	no	—
400	C₂²×C₄	—	53	32	no	—
401	C₂²×C₄	—	53	2	no	—
402	C₂²×C₄	—	53	4	no	—
403	C₂²×C₄	—	53	4	no	—
404	C₂²×C₄	—	53	2	no	—
405	C₂²×C₄	—	53	4	no	—
406	C₂²×C₄	—	53	16	no	—
407	C₂²×C₄	—	53	4	no	—
408	C₂²×C₄	—	53	4	no	—
409	C₂²×C₄	—	53	4	no	—
410	C₂²×C₄	—	53	16	no	—
411	C₂²×C₄	—	53	2	no	—
412	C₂²×C₄	—	53	2	no	—
413	C₂²×C₄	—	53	4	no	—
414	C₂²×C₄	—	53	16	no	—
415	C₂²×C₄	—	212	8	no	—
416	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
417	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
418	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
419	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
420	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
421	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
422	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
423	M₁₆	—	53	16	no	—
424	M₁₆	—	53	32	no	—
425	M₁₆	—	53	16	no	—
426	M₁₆	—	53	16	no	—
427	M₁₆	—	53	16	no	—
428	M₁₆	—	53	16	no	—
429	M₁₆	C₂×C₄×D₅₃	1	16	yes	complemented
430	C₂×C₈	—	53	4	no	—
431	C₂×C₈	—	53	4	no	—
432	C₂×C₈	—	53	8	no	—
433	C₂×C₈	—	53	16	no	—
434	C₂×C₈	—	53	2	no	—
435	C₂×C₈	—	53	4	no	—
436	C₂×C₈	—	53	2	no	—
437	C₂×C₈	—	53	32	no	—
438	C₂×C₈	C₂×C₄×D₅₃	1	8	yes	—
439	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
440	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
441	C₂×C₈	C₂×C₄×D₅₃	1	16	yes	complemented
442	C₂×C₈	C₂³×D₅₃	1	2	yes	—
443	C₂×C₈	C₂³×D₅₃	1	4	yes	—
444	C₂×C₈	C₂³×D₅₃	1	2	yes	—
445	C₄⋊C₄	—	53	16	no	—
446	C₄⋊C₄	—	53	8	no	—
447	C₄⋊C₄	C₂×C₄×D₅₃	1	8	yes	—
448	C₂²⋊C₄	—	53	8	no	—
449	C₂²⋊C₄	—	53	16	no	—
450	C₂²⋊C₄	C₂×C₄×D₅₃	1	8	yes	—
451	C₄²	—	53	16	no	—
452	C₄²	—	53	4	no	—
453	C₄²	—	53	4	no	—
454	C₄²	C₂×C₄×D₅₃	1	4	yes	—
455	C₄²	C₂³×D₅₃	1	4	yes	—
456	C₂³	—	2	4	no	—
457	C₂³	—	53	4	no	—
458	C₂³	—	53	4	no	—
459	C₂³	—	53	2	no	—
460	C₂³	—	53	2	no	—
461	C₂³	—	53	16	no	—
462	C₂³	—	106	4	no	—
463	C₂³	—	106	4	no	—
464	C₂³	—	212	8	no	—
465	C₂³	—	212	16	no	—
466	C₂³	C₄²×D₅₃	1	4	yes	—
467	C₂³	C₂²×C₄×D₅₃	1	2	yes	—
468	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
469	C₂×C₄	—	2	8	no	—
470	C₂×C₄	—	53	16	no	—
471	C₂×C₄	—	53	4	no	—
472	C₂×C₄	—	53	4	no	—
473	C₂×C₄	—	53	2	no	—
474	C₂×C₄	—	53	2	no	—
475	C₂×C₄	—	53	8	no	—
476	C₂×C₄	—	53	16	no	—
477	C₂×C₄	—	53	8	no	—
478	C₂×C₄	—	53	16	no	—
479	C₂×C₄	—	53	4	no	—
480	C₂×C₄	—	53	4	no	—
481	C₂×C₄	—	53	4	no	—
482	C₂×C₄	—	53	4	no	—
483	C₂×C₄	—	53	8	no	—
484	C₂×C₄	—	106	8	no	—
485	C₂×C₄	—	212	16	no	—
486	C₂×C₄	C₄²×D₅₃	1	8	yes	—
487	C₂×C₄	C₄²×D₅₃	1	4	yes	—
488	C₂×C₄	C₂²×C₄×D₅₃	1	8	yes	—
489	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
490	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
491	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
492	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
493	C₂×C₄	C₂²×C₄×D₅₃	1	4	yes	—
494	C₂×C₄	C₂²×C₄×D₅₃	1	4	yes	—
495	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
496	C₈	—	53	8	no	—
497	C₈	—	53	4	no	—
498	C₈	—	53	4	no	—
499	C₈	C₂²×C₄×D₅₃	1	8	yes	—
500	C₈	C₂²×C₄×D₅₃	1	4	yes	—
501	C₈	C₂²×C₄×D₅₃	1	4	yes	—
502	C₂²	—	2	4	no	—
503	C₂²	—	2	8	no	—
504	C₂²	—	53	4	no	—
505	C₂²	—	53	4	no	—
506	C₂²	—	106	4	no	—
507	C₂²	—	106	4	no	—
508	C₂²	—	106	8	no	—
509	C₂²	—	106	8	no	—
510	C₂²	—	212	16	no	—
511	C₂²	C₂×C₄²×D₅₃	1	4	yes	—
512	C₂²	C₂×C₄²×D₅₃	1	2	yes	—
513	C₂²	#16811388	1	4	yes	—
514	C₂²	C₂³×C₄×D₅₃	1	1	yes	characteristic
515	C₄	—	2	8	no	—
516	C₄	—	53	4	no	—
517	C₄	—	53	4	no	—
518	C₄	—	106	8	no	—
519	C₄	C₂×C₄²×D₅₃	1	2	yes	—
520	C₄	C₂×C₄²×D₅₃	1	4	yes	—
521	C₄	C₂³×C₄×D₅₃	1	1	yes	characteristic
522	C₄	C₂³×C₄×D₅₃	1	1	yes	characteristic
523	C₂	—	2	4	no	—
524	C₂	—	106	4	no	—
525	C₂	—	106	4	no	—
526	C₂	#20283131	1	4	yes	direct product factor
527	C₂	C₂²×C₄²×D₅₃	1	1	yes	characteristic
528	C₂	C₂×#16811388	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₅₃
commutator	C₂×C₅₃	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂⁴×D₅₃	C₂×C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	C₂×#2873
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
4. #20283131	C₂	direct (N × H)
5. C₂×#16810581	C₂	semidirect
6. C₂×#16810580	C₂	semidirect
7. C₂×#16810579	C₂	semidirect
8. C₂×#16810578	C₂	semidirect
9. C₂×#16810577	C₂	semidirect
10. C₂×#16810575	C₂	semidirect
11. C₂×C₄×#11719912	C₂	semidirect
12. C₂×C₅₃×#405	C₂	semidirect
15. C₂²×#11719912	C₄	semidirect
22. C₂×M₁₆×D₅₃	C₄	semidirect
35. #16810581	C₂²	semidirect
36. #16810580	C₂²	semidirect
37. #16810579	C₂²	semidirect
38. #16810578	C₂²	semidirect
39. #16810577	C₂²	semidirect
40. #16810575	C₂²	semidirect
41. C₄×#11719912	C₂²	semidirect
42. C₂×#13133128	C₂²	semidirect
43. C₂×#13133127	C₂²	semidirect
44. C₂×#13133127	C₂²	semidirect
45. C₂×C₄×#88322	C₂²	semidirect
46. C₂×C₈×Dic₅₃	C₂²	semidirect
47. C₂×C₈×Dic₅₃	C₂²	semidirect
51. C₅₃×#405	C₂²	semidirect
52. C₂×#97×C₅₃	C₂²	semidirect
53. C₂×C₄×C₈×C₅₃	C₂²	semidirect
65. C₂×#11719917	C₂×C₄	semidirect
70. C₂×#11719912	C₂×C₄	semidirect
71. C₂×#11719912	C₂×C₄	semidirect
72. C₂×#11719912	C₂×C₄	semidirect
78. M₁₆×D₅₃	C₂×C₄	semidirect
79. C₂²×#88322	C₂×C₄	semidirect
84. C₂×C₈×D₅₃	C₂×C₄	semidirect
102. #13133128	C₂³	semidirect
103. #13133127	C₂³	semidirect
104. #13133127	C₂³	semidirect
105. C₄×#88322	C₂³	semidirect
106. C₈×Dic₅₃	C₂³	semidirect
107. C₈×Dic₅₃	C₂³	semidirect
109. C₂×M₁₆×C₅₃	C₂×C₄	semidirect
111. C₂²×C₈×C₅₃	C₂×C₄	semidirect
121. #97×C₅₃	C₂³	semidirect
122. C₄×C₈×C₅₃	C₂³	semidirect
158. M₁₆×C₅₃	C₂²×C₄	semidirect
162. C₂×C₈×C₅₃	C₂²×C₄	semidirect
177. #11719917	C₂²×C₄	semidirect
181. C₂×#88322	C₂²×C₄	semidirect
188. #11719912	C₂²×C₄	semidirect
189. #11719912	C₂²×C₄	semidirect
190. C₈×D₅₃	C₂²×C₄	semidirect
268. C₂×#405	D₅₃	semidirect
272. C₂×C₅₃	#2873	semidirect
307. #405	C₂×D₅₃	semidirect
313. C₂×#97	C₂×D₅₃	semidirect
317. C₂×C₄×C₈	C₂×D₅₃	semidirect
318. C₅₃	C₂×#2873	semidirect
345. C₂×M₁₆	C₄×D₅₃	semidirect
354. C₂²×C₈	C₄×D₅₃	semidirect
383. #97	C₂²×D₅₃	semidirect
387. C₄×C₈	C₂²×D₅₃	semidirect
429. M₁₆	C₂×C₄×D₅₃	semidirect
441. C₂×C₈	C₂×C₄×D₅₃	semidirect
526. C₂	#20283131	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11719912	2
C₂²×M₁₆×D₅₃	2
C₂×#117×D₅₃	2
#20283131	2
C₂×#16810581	2
C₂×#16810580	2
C₂×#16810579	2
C₂×#16810578	2
C₂×#16810577	2
C₂×#16810575	2
C₂×C₄×#11719912	2
C₂×C₅₃×#405	2
C₂×#2873	53

Subgroups — C2×#20283131 #28247488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20283131 #28247488