Subgroups — C₂×#20272802 #27298958

Order 13568.

586 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133157	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810578	C₂	1	1	yes	maximal, characteristic
3	#20272802	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	C₄	1	1	yes	characteristic
6	C₂×#13133176	C₄	1	1	yes	characteristic
7	C₂×#13133157	—	2	1	no	—
8	C₂×#13133157	—	2	1	no	—
9	C₂×#13133157	—	2	1	no	—
10	C₂×#13133157	—	2	1	no	—
11	C₂×#13133157	C₄	1	2	yes	complemented
12	C₂×#13133157	C₂²	1	2	yes	—
13	C₂²×#11719908	C₄	1	2	yes	complemented
14	#16810578	—	2	1	no	—
15	#16810578	—	2	1	no	—
16	#16810578	C₄	1	2	yes	—
17	#16810578	C₂²	1	2	yes	—
18	C₂×#13133127	—	2	1	no	—
19	C₂×C₈×Dic₅₃	—	2	1	no	—
20	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×M₃₂×C₅₃	—	2	1	no	—
22	C₂×M₃₂×C₅₃	—	2	1	no	—
23	C₂×M₃₂×C₅₃	—	2	1	no	—
24	C₂×M₃₂×C₅₃	C₄	1	2	yes	complemented
25	C₂×M₃₂×C₅₃	C₄	1	2	yes	complemented
26	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
27	C₂²×C₁₆×C₅₃	—	2	1	no	—
28	C₂×#11719917	—	2	1	no	—
29	C₂×#11719917	—	2	1	no	—
30	C₂×#11719917	—	2	1	no	—
31	C₂×#11719917	C₈	1	2	yes	—
32	C₂×#11719917	C₈	1	2	yes	—
33	C₂×#11719917	C₂×C₄	1	2	yes	—
34	C₂²×#88322	—	2	1	no	—
35	#13133176	—	2	1	no	—
36	#13133176	—	2	1	no	—
37	#13133176	C₈	1	2	yes	—
38	#13133176	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊Dic₅₃	—	2	1	no	—
40	C₂×C₂²⋊Dic₅₃	C₈	1	2	yes	—
41	C₂×C₄×Dic₅₃	—	2	1	no	—
42	#13133157	—	2	2	no	—
43	#13133157	—	2	2	no	—
44	#13133157	—	2	2	no	—
45	#13133157	—	2	2	no	—
46	C₂×#11719908	—	2	2	no	—
47	C₂×#11719908	—	2	2	no	—
48	C₂×#11719908	C₂×C₄	1	4	yes	complemented
49	#13133127	—	2	2	no	—
50	#13133127	—	2	2	no	—
51	C₈×Dic₅₃	—	2	2	no	—
52	C₈×Dic₅₃	—	2	2	no	—
53	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
65	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
66	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
67	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	M₃₂×C₅₃	—	2	2	no	—
69	M₃₂×C₅₃	—	2	2	no	—
70	M₃₂×C₅₃	—	2	2	no	—
71	M₃₂×C₅₃	C₂×C₄	1	4	yes	complemented
72	C₂×C₁₆×C₅₃	—	2	2	no	—
73	C₂×C₁₆×C₅₃	—	2	2	no	—
74	C₂×C₁₆×C₅₃	—	2	2	no	—
75	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×Dic₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
119	C₂²⋊Dic₅₃	—	2	2	no	—
120	C₂²⋊Dic₅₃	C₂×C₈	1	4	yes	—
121	C₄⋊Dic₅₃	—	2	2	no	—
122	C₄⋊Dic₅₃	—	2	2	no	—
123	C₄×Dic₅₃	—	2	2	no	—
124	C₄×Dic₅₃	—	2	2	no	—
125	#11719917	—	2	2	no	—
126	#11719917	—	2	2	no	—
127	#11719917	—	2	2	no	—
128	#11719917	C₂×C₈	1	4	yes	—
129	C₂×#88322	—	2	2	no	—
130	C₂×#88322	—	2	2	no	—
131	C₂×#88322	—	2	2	no	—
132	C₁₆×C₅₃	—	2	4	no	—
133	#11719908	—	2	4	no	—
134	C₂³×C₅₃	—	4	2	no	—
135	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
136	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
137	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
138	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
139	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
140	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
141	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
143	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
144	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
145	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
148	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
149	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
150	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
151	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
169	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
170	C₂×Dic₅₃	—	2	4	no	—
171	C₂×Dic₅₃	—	4	4	no	—
172	C₈×C₅₃	#95	1	1	yes	characteristic
173	C₈×C₅₃	#95	1	1	yes	characteristic
174	C₈×C₅₃	#95	1	1	yes	characteristic
175	C₈×C₅₃	#95	1	1	yes	characteristic
176	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
177	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
178	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
180	#88322	—	2	4	no	—
181	C₂×#1378	—	53	1	no	maximal, Sylow
182	C₂²×C₅₃	—	2	2	no	—
183	C₂²×C₅₃	—	2	2	no	—
184	C₂²×C₅₃	—	2	2	no	—
185	C₂²×C₅₃	—	4	1	no	—
186	C₂²×C₅₃	—	4	1	no	—
187	C₂²×C₅₃	—	4	1	no	—
188	C₂²×C₅₃	—	4	1	no	—
189	C₂²×C₅₃	#336	1	1	yes	characteristic
190	C₂²×C₅₃	#336	1	1	yes	characteristic
191	C₂²×C₅₃	#336	1	1	yes	characteristic
192	C₂²×C₅₃	#336	1	1	yes	characteristic
193	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
194	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
195	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
196	C₄×C₅₃	#336	1	1	yes	characteristic
197	C₄×C₅₃	#336	1	1	yes	characteristic
198	C₄×C₅₃	#336	1	1	yes	characteristic
199	C₄×C₅₃	#336	1	1	yes	characteristic
200	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
201	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
202	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
203	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
204	Dic₅₃	—	4	4	no	—
205	C₂²×M₃₂	—	53	1	no	—
206	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
207	C₂×#405	—	53	1	no	—
208	#1378	—	53	4	no	—
209	C₂×C₅₃	—	2	1	no	—
210	C₂×C₅₃	—	2	1	no	—
211	C₂×C₅₃	—	2	1	no	—
212	C₂×C₅₃	—	2	1	no	—
213	C₂×C₅₃	—	2	1	no	—
214	C₂×C₅₃	—	2	1	no	—
215	C₂×C₅₃	#1378	1	2	yes	complemented
216	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
217	C₂²×M₁₆	—	53	1	no	—
218	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
219	C₂×#117	—	53	1	no	—
220	C₂×M₃₂	—	2	1	no	—
221	C₂×M₃₂	—	2	1	no	—
222	C₂×M₃₂	—	2	1	no	—
223	C₂×M₃₂	—	53	2	no	—
224	C₂×M₃₂	—	53	2	no	—
225	C₂×M₃₂	—	106	1	no	—
226	C₂×M₃₂	—	106	1	no	—
227	C₂×M₃₂	—	106	1	no	—
228	C₂×M₃₂	—	106	1	no	—
229	C₂×M₃₂	Dic₅₃	1	2	yes	complemented
230	C₂×M₃₂	Dic₅₃	1	2	yes	complemented
231	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
232	C₂²×C₁₆	—	2	1	no	—
233	C₂²×C₁₆	—	53	2	no	—
234	#405	—	53	2	no	—
235	#405	—	53	2	no	—
236	#405	—	106	1	no	—
237	#405	—	106	1	no	—
238	C₂×#97	—	106	1	no	—
239	C₂×C₄×C₈	—	106	1	no	—
240	C₅₃	C₂×#1378	1	1	yes	characteristic, complemented, Sylow
241	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
242	C₂×M₁₆	—	53	2	no	—
243	C₂×M₁₆	—	53	2	no	—
244	C₂×M₁₆	—	53	2	no	—
245	C₂×M₁₆	—	106	1	no	—
246	C₂×M₁₆	—	106	1	no	—
247	C₂×M₁₆	—	106	1	no	—
248	C₂²×C₈	—	106	1	no	—
249	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
250	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
251	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
252	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
253	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
254	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
255	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
256	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
257	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
258	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
259	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
260	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
261	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
262	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
263	#117	—	53	2	no	—
264	#117	—	53	2	no	—
265	#117	—	106	1	no	—
266	#117	—	106	1	no	—
267	C₂×C₄⋊C₄	—	106	1	no	—
268	C₂×C₂²⋊C₄	—	53	2	no	—
269	C₂×C₄²	—	106	1	no	—
270	M₃₂	—	2	2	no	—
271	M₃₂	—	2	2	no	—
272	M₃₂	—	2	2	no	—
273	M₃₂	—	106	2	no	—
274	M₃₂	—	106	2	no	—
275	M₃₂	—	106	2	no	—
276	M₃₂	—	106	2	no	—
277	M₃₂	C₂×Dic₅₃	1	4	yes	complemented
278	C₂×C₁₆	—	2	2	no	—
279	C₂×C₁₆	—	2	2	no	—
280	C₂×C₁₆	—	2	2	no	—
281	C₂×C₁₆	—	53	4	no	—
282	C₂×C₁₆	—	106	2	no	—
283	C₂×C₁₆	—	106	2	no	—
284	#97	—	106	2	no	—
285	#97	—	106	2	no	—
286	C₄×C₈	—	106	2	no	—
287	C₄×C₈	—	106	2	no	—
288	C₂⁴	C₈×D₅₃	1	1	yes	characteristic
289	C₂²×C₄	—	106	2	no	—
290	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
291	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
292	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
293	C₂²×C₄	#11719912	1	1	yes	characteristic
294	C₂²×C₄	#11719912	1	1	yes	characteristic
295	C₂²×C₄	#11719912	1	1	yes	characteristic
296	C₂²×C₄	#11719912	1	1	yes	characteristic
297	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
298	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
299	C₂²×C₄	#11719919	1	1	yes	characteristic
300	C₂²×C₄	#11719919	1	1	yes	characteristic
301	C₂²×C₄	#11719921	1	1	yes	characteristic
302	C₂²×C₄	#11719921	1	1	yes	characteristic
303	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
304	M₁₆	—	53	4	no	—
305	M₁₆	—	106	2	no	—
306	M₁₆	—	106	2	no	—
307	M₁₆	—	106	2	no	—
308	C₂×C₈	—	106	2	no	—
309	C₂×C₈	—	106	2	no	—
310	C₂×C₈	—	106	2	no	—
311	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
312	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
313	C₂×C₈	#11719919	1	1	yes	characteristic
314	C₂×C₈	#11719919	1	1	yes	characteristic
315	C₂×C₈	#11719919	1	1	yes	characteristic
316	C₂×C₈	#11719919	1	1	yes	characteristic
317	C₂×C₈	#11719919	1	1	yes	characteristic
318	C₂×C₈	#11719919	1	1	yes	characteristic
319	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
320	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
321	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
322	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
323	C₂×C₈	#11719921	1	1	yes	characteristic
324	C₂×C₈	#11719921	1	1	yes	characteristic
325	C₂×C₈	#11719921	1	1	yes	characteristic
326	C₂×C₈	#11719921	1	1	yes	characteristic
327	C₂×C₈	#11719921	1	1	yes	characteristic
328	C₂×C₈	#11719921	1	1	yes	characteristic
329	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
330	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
331	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
332	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
333	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
334	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
335	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
336	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
337	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
338	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
339	C₄⋊C₄	—	106	2	no	—
340	C₄⋊C₄	—	106	2	no	—
341	C₂²⋊C₄	—	53	4	no	—
342	C₂²⋊C₄	—	106	2	no	—
343	C₄²	—	106	2	no	—
344	C₄²	—	106	2	no	—
345	C₁₆	—	2	4	no	—
346	C₁₆	—	106	4	no	—
347	C₂³	—	4	2	no	—
348	C₂³	C₈×Dic₅₃	1	1	yes	characteristic
349	C₂³	C₈×Dic₅₃	1	1	yes	characteristic
350	C₂³	#13133129	1	1	yes	characteristic
351	C₂³	#13133129	1	1	yes	characteristic
352	C₂³	#13133146	1	1	yes	characteristic
353	C₂³	#13133146	1	1	yes	characteristic
354	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
355	C₂×C₄	—	106	4	no	—
356	C₂×C₄	—	212	4	no	—
357	C₂×C₄	#13133124	1	1	yes	characteristic
358	C₂×C₄	#13133124	1	1	yes	characteristic
359	C₂×C₄	#13133124	1	1	yes	characteristic
360	C₂×C₄	#13133124	1	1	yes	characteristic
361	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
362	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
363	C₂×C₄	#13133127	1	1	yes	characteristic
364	C₂×C₄	#13133127	1	1	yes	characteristic
365	C₂×C₄	#13133127	1	1	yes	characteristic
366	C₂×C₄	#13133127	1	1	yes	characteristic
367	C₂×C₄	#13133129	1	1	yes	characteristic
368	C₂×C₄	#13133129	1	1	yes	characteristic
369	C₂×C₄	#13133129	1	1	yes	characteristic
370	C₂×C₄	#13133129	1	1	yes	characteristic
371	C₂×C₄	#13133129	1	1	yes	characteristic
372	C₂×C₄	#13133129	1	1	yes	characteristic
373	C₂×C₄	#13133146	1	1	yes	characteristic
374	C₂×C₄	#13133146	1	1	yes	characteristic
375	C₂×C₄	#13133146	1	1	yes	characteristic
376	C₂×C₄	#13133146	1	1	yes	characteristic
377	C₂×C₄	#13133146	1	1	yes	characteristic
378	C₂×C₄	#13133146	1	1	yes	characteristic
379	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
380	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
381	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
382	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
383	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
384	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
385	C₈	—	106	4	no	—
386	C₈	#13133124	1	1	yes	characteristic
387	C₈	#13133124	1	1	yes	characteristic
388	C₈	#13133124	1	1	yes	characteristic
389	C₈	#13133124	1	1	yes	characteristic
390	C₈	C₂×#11719921	1	1	yes	characteristic
391	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
392	C₈	C₂×#11719919	1	1	yes	characteristic
393	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
394	C₂²	—	2	2	no	—
395	C₂²	—	2	2	no	—
396	C₂²	—	2	2	no	—
397	C₂²	—	4	1	no	—
398	C₂²	—	4	1	no	—
399	C₂²	—	4	1	no	—
400	C₂²	—	4	1	no	—
401	C₂²	#16810377	1	1	yes	characteristic
402	C₂²	#16810377	1	1	yes	characteristic
403	C₂²	#16810377	1	1	yes	characteristic
404	C₂²	#16810377	1	1	yes	characteristic
405	C₂²	C₂×C₈×Dic₅₃	1	1	yes	characteristic
406	C₂²	C₂×#13133129	1	1	yes	characteristic
407	C₂²	C₂×#13133146	1	1	yes	characteristic
408	C₄	—	212	4	no	—
409	C₄	#16810377	1	1	yes	characteristic
410	C₄	#16810377	1	1	yes	characteristic
411	C₄	#16810377	1	1	yes	characteristic
412	C₄	#16810377	1	1	yes	characteristic
413	C₄	C₂×#13133124	1	1	yes	characteristic
414	C₄	C₂×#13133127	1	1	yes	characteristic
415	C₄	C₂×#13133129	1	1	yes	characteristic
416	C₄	C₂×#13133146	1	1	yes	characteristic
417	C₂	—	2	1	no	—
418	C₂	—	2	1	no	—
419	C₂	—	2	1	no	—
420	C₂	—	2	1	no	—
421	C₂	—	2	1	no	—
422	C₂	—	2	1	no	—
423	C₂	#20272802	1	2	yes	direct product factor
424	C₂	C₂×#16810377	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#11719921
commutator	C₂²×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	#336
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	C₂×#1378
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20272802	C₂	direct (N × H)
11. C₂×#13133157	C₄	semidirect
13. C₂²×#11719908	C₄	semidirect
24. C₂×M₃₂×C₅₃	C₄	semidirect
25. C₂×M₃₂×C₅₃	C₄	semidirect
48. C₂×#11719908	C₂×C₄	semidirect
71. M₃₂×C₅₃	C₂×C₄	semidirect
215. C₂×C₅₃	#1378	semidirect
229. C₂×M₃₂	Dic₅₃	semidirect
230. C₂×M₃₂	Dic₅₃	semidirect
240. C₅₃	C₂×#1378	semidirect
277. M₃₂	C₂×Dic₅₃	semidirect
423. C₂	#20272802	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133157	2
C₂×#16810578	2
#20272802	2
C₂²×M₃₂×C₅₃	2
C₂×#1378	53

Subgroups — C2×#20272802 #27298958

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272802 #27298958