Subgroups — C₂×#20275614 #27388067

Order 13568.

1474 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810661	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810640	C₂	1	2	yes	maximal, complemented
3	C₂×#16810597	C₂	1	2	yes	maximal, complemented
4	C₂×#16810578	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275614	C₂	1	8	yes	maximal, direct product factor
6	C₂²×#135×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₂²	1	1	yes	characteristic
8	C₂×#13133176	C₂²	1	2	yes	—
9	C₂×#13133176	C₂²	1	1	yes	characteristic
10	#16810661	—	2	2	no	—
11	#16810661	C₂²	1	4	yes	complemented
12	C₂×#13133151	—	2	1	no	—
13	C₂×#13133149	—	2	1	no	—
14	#16810640	—	2	4	no	—
15	#16810640	C₂²	1	8	yes	complemented
16	C₂×#13133144	—	2	4	no	—
17	#16810597	—	2	4	no	—
18	#16810597	C₂²	1	8	yes	complemented
19	C₂×#13133132	—	2	4	no	—
20	#16810578	—	2	2	no	—
21	#16810578	C₂²	1	4	yes	complemented
22	C₂×#13133127	—	2	1	no	—
23	C₂×C₈×Dic₅₃	—	2	1	no	—
24	C₂²×#41×C₅₃	C₂²	1	2	yes	—
25	C₂×#135×C₅₃	—	2	1	no	—
26	C₂×#135×C₅₃	—	2	1	no	—
27	C₂×#135×C₅₃	—	2	2	no	—
28	C₂×#135×C₅₃	—	2	1	no	—
29	C₂×#135×C₅₃	—	2	1	no	—
30	C₂×#135×C₅₃	—	2	2	no	—
31	C₂×#135×C₅₃	C₄	1	4	yes	complemented
32	C₂×#135×C₅₃	C₂²	1	4	yes	—
33	C₂²×Q₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
34	C₂²×SD₁₆×C₅₃	C₄	1	2	yes	complemented
35	C₂²×D₈×C₅₃	C₄	1	1	yes	characteristic, complemented
36	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×#11719917	—	2	1	no	—
38	C₂×#11719917	—	2	1	no	—
39	C₂×#11719917	—	2	1	no	—
40	C₂×#11719917	D₄	1	2	yes	—
41	C₂×#11719917	D₄	1	2	yes	—
42	C₂×#11719917	C₂³	1	2	yes	—
43	C₂²×#88322	—	2	1	no	—
44	#13133176	—	2	1	no	—
45	#13133176	—	2	2	no	—
46	#13133176	—	2	2	no	—
47	#13133176	—	2	2	no	—
48	#13133176	—	2	1	no	—
49	#13133176	D₄	1	2	yes	—
50	#13133176	C₂³	1	4	yes	—
51	#13133176	C₂³	1	2	yes	—
52	C₂×C₄⋊Dic₅₃	—	2	1	no	—
53	C₂×C₄⋊Dic₅₃	—	2	2	no	—
54	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
55	C₂×C₂²⋊Dic₅₃	D₄	1	2	yes	—
56	C₂×C₄×Dic₅₃	—	2	1	no	—
57	C₂×C₄×Dic₅₃	—	2	2	no	—
58	#13133151	—	2	4	no	—
59	#13133149	—	2	4	no	—
60	#13133144	—	2	16	no	—
61	#13133132	—	2	16	no	—
62	#13133127	—	2	4	no	—
63	C₈×Dic₅₃	—	2	4	no	—
64	C₂×#41×C₅₃	—	2	2	no	—
65	C₂×#41×C₅₃	—	2	2	no	—
66	C₂×#41×C₅₃	—	2	2	no	—
67	C₂×#41×C₅₃	—	2	2	no	—
68	C₂×#41×C₅₃	—	2	2	no	—
69	C₂×#41×C₅₃	—	2	2	no	—
70	C₂×#41×C₅₃	—	2	2	no	—
71	C₂×#41×C₅₃	—	2	2	no	—
72	C₂×#41×C₅₃	—	2	2	no	—
73	C₂×#41×C₅₃	—	2	2	no	—
74	C₂×#41×C₅₃	C₂×C₄	1	4	yes	—
75	C₂×#41×C₅₃	C₂³	1	4	yes	—
76	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
77	C₂²×D₄×C₅₃	—	2	2	no	—
78	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
79	C₂³×C₄×C₅₃	—	2	2	no	—
80	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
81	#135×C₅₃	—	2	4	no	—
82	#135×C₅₃	—	2	4	no	—
83	#135×C₅₃	—	2	4	no	—
84	#135×C₅₃	—	2	4	no	—
85	#135×C₅₃	—	2	4	no	—
86	#135×C₅₃	—	2	4	no	—
87	#135×C₅₃	—	2	4	no	—
88	#135×C₅₃	—	2	4	no	—
89	C₂×Q₁₆×C₅₃	—	2	1	no	—
90	C₂×Q₁₆×C₅₃	—	2	1	no	—
91	C₂×Q₁₆×C₅₃	—	2	2	no	—
92	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
93	C₂×SD₁₆×C₅₃	—	2	2	no	—
94	C₂×SD₁₆×C₅₃	—	2	2	no	—
95	C₂×SD₁₆×C₅₃	—	2	4	no	—
96	C₂×SD₁₆×C₅₃	C₂×C₄	1	8	yes	complemented
97	C₂×D₈×C₅₃	—	2	1	no	—
98	C₂×D₈×C₅₃	—	2	1	no	—
99	C₂×D₈×C₅₃	—	2	2	no	—
100	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	complemented
101	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
102	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
103	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
104	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
105	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
106	C₂²×C₈×C₅₃	D₄	1	2	yes	—
107	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
108	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
109	C₂²×C₈×C₅₃	D₄	1	2	yes	—
110	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
111	C₂⁴×C₅₃	—	2	2	no	—
112	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
113	#41×C₅₃	—	2	4	no	—
114	#41×C₅₃	—	2	4	no	—
115	#41×C₅₃	—	2	4	no	—
116	#41×C₅₃	—	2	4	no	—
117	#41×C₅₃	—	2	4	no	—
118	#41×C₅₃	—	2	4	no	—
119	#41×C₅₃	—	2	4	no	—
120	#41×C₅₃	—	2	4	no	—
121	#41×C₅₃	—	4	2	no	—
122	#41×C₅₃	—	4	2	no	—
123	#41×C₅₃	—	4	2	no	—
124	#41×C₅₃	—	4	2	no	—
125	#41×C₅₃	—	4	2	no	—
126	#41×C₅₃	—	4	2	no	—
127	#41×C₅₃	—	4	2	no	—
128	#41×C₅₃	—	4	2	no	—
129	C₂×Q₈×C₅₃	—	2	2	no	—
130	C₂×Q₈×C₅₃	—	2	2	no	—
131	C₂×Q₈×C₅₃	—	2	2	no	—
132	C₂×Q₈×C₅₃	—	2	2	no	—
133	C₂×Q₈×C₅₃	—	2	2	no	—
134	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
135	C₂×D₄×C₅₃	—	2	4	no	—
136	C₂×D₄×C₅₃	—	2	4	no	—
137	C₂×D₄×C₅₃	—	2	2	no	—
138	C₂×D₄×C₅₃	—	2	2	no	—
139	C₂×D₄×C₅₃	—	2	2	no	—
140	C₂×D₄×C₅₃	—	2	2	no	—
141	C₂×D₄×C₅₃	—	2	2	no	—
142	C₂×D₄×C₅₃	—	4	2	no	—
143	C₂×D₄×C₅₃	—	4	2	no	—
144	C₂×D₄×C₅₃	—	4	2	no	—
145	C₂×D₄×C₅₃	—	4	2	no	—
146	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
147	C₂²×C₄×C₅₃	—	2	2	no	—
148	C₂²×C₄×C₅₃	—	2	4	no	—
149	C₂²×C₄×C₅₃	—	2	4	no	—
150	C₂²×C₄×C₅₃	—	4	2	no	—
151	C₂²×C₄×C₅₃	—	4	2	no	—
152	C₂²×C₄×C₅₃	—	4	2	no	—
153	C₂²×C₄×C₅₃	—	4	2	no	—
154	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
164	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
165	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
166	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
167	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
168	C₂²×Dic₅₃	—	2	2	no	—
169	C₂²×Dic₅₃	—	4	2	no	—
170	Q₁₆×C₅₃	—	2	4	no	—
171	Q₁₆×C₅₃	—	2	4	no	—
172	SD₁₆×C₅₃	—	2	8	no	—
173	SD₁₆×C₅₃	—	2	8	no	—
174	D₈×C₅₃	—	2	4	no	—
175	D₈×C₅₃	—	2	4	no	—
176	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
177	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
178	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
179	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
180	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
181	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
182	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
183	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
184	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
185	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
186	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
187	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
188	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
189	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
190	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
191	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
192	C₂²⋊Dic₅₃	—	2	2	no	—
193	C₂²⋊Dic₅₃	—	2	4	no	—
194	C₂²⋊Dic₅₃	—	4	2	no	—
195	C₂²⋊Dic₅₃	C₂×D₄	1	4	yes	—
196	C₄⋊Dic₅₃	—	2	2	no	—
197	C₄⋊Dic₅₃	—	2	2	no	—
198	C₄⋊Dic₅₃	—	2	4	no	—
199	C₄⋊Dic₅₃	—	4	2	no	—
200	C₄×Dic₅₃	—	2	2	no	—
201	C₄×Dic₅₃	—	2	2	no	—
202	C₄×Dic₅₃	—	2	4	no	—
203	C₄×Dic₅₃	—	4	2	no	—
204	#11719917	—	2	2	no	—
205	#11719917	—	2	2	no	—
206	#11719917	—	2	2	no	—
207	#11719917	C₂×D₄	1	4	yes	—
208	C₂×#88322	—	2	2	no	—
209	C₂×#88322	—	2	2	no	—
210	C₂×#88322	—	2	2	no	—
211	C₂³×C₅₃	—	2	4	no	—
212	C₂³×C₅₃	—	4	2	no	—
213	C₂³×C₅₃	—	4	2	no	—
214	C₂³×C₅₃	—	4	2	no	—
215	C₂³×C₅₃	—	8	4	no	—
216	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
217	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
218	C₂³×C₅₃	#121	1	1	yes	characteristic
219	C₂³×C₅₃	#121	1	1	yes	characteristic
220	C₂³×C₅₃	#127	1	1	yes	characteristic
221	C₂³×C₅₃	#127	1	1	yes	characteristic
222	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
223	Q₈×C₅₃	—	2	4	no	—
224	Q₈×C₅₃	—	2	4	no	—
225	Q₈×C₅₃	—	4	2	no	—
226	Q₈×C₅₃	—	4	2	no	—
227	D₄×C₅₃	—	2	4	no	—
228	D₄×C₅₃	—	2	4	no	—
229	D₄×C₅₃	—	4	4	no	—
230	D₄×C₅₃	—	4	4	no	—
231	D₄×C₅₃	—	4	4	no	—
232	D₄×C₅₃	—	4	4	no	—
233	D₄×C₅₃	—	4	2	no	—
234	D₄×C₅₃	—	4	2	no	—
235	C₂×C₄×C₅₃	—	2	4	no	—
236	C₂×C₄×C₅₃	—	4	2	no	—
237	C₂×C₄×C₅₃	—	4	2	no	—
238	C₂×C₄×C₅₃	—	4	4	no	—
239	C₂×C₄×C₅₃	—	4	4	no	—
240	C₂×C₄×C₅₃	—	4	4	no	—
241	C₂×C₄×C₅₃	—	4	4	no	—
242	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
243	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
244	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
245	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
246	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
247	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
248	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
249	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
250	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
251	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
252	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
253	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
254	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
255	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
256	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
257	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
258	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
259	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
260	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
261	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
262	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
263	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
264	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
265	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
266	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
267	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
268	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
269	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
270	C₂×Dic₅₃	—	2	4	no	—
271	C₂×Dic₅₃	—	4	4	no	—
272	C₂×Dic₅₃	—	4	4	no	—
273	C₂×Dic₅₃	—	8	4	no	—
274	C₈×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
275	C₈×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
276	C₈×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
277	C₈×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
278	#88322	—	2	4	no	—
279	C₂×#1976	—	53	1	no	maximal, Sylow
280	C₂²×C₅₃	—	2	2	no	—
281	C₂²×C₅₃	—	2	2	no	—
282	C₂²×C₅₃	—	2	2	no	—
283	C₂²×C₅₃	—	4	1	no	—
284	C₂²×C₅₃	—	4	1	no	—
285	C₂²×C₅₃	—	4	1	no	—
286	C₂²×C₅₃	—	4	1	no	—
287	C₂²×C₅₃	—	4	4	no	—
288	C₂²×C₅₃	—	8	4	no	—
289	C₂²×C₅₃	—	8	4	no	—
290	C₂²×C₅₃	—	8	4	no	—
291	C₂²×C₅₃	#390	1	1	yes	characteristic
292	C₂²×C₅₃	#390	1	1	yes	characteristic
293	C₂²×C₅₃	#390	1	1	yes	characteristic
294	C₂²×C₅₃	#390	1	1	yes	characteristic
295	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
296	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
297	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
298	C₄×C₅₃	—	4	4	no	—
299	C₄×C₅₃	#390	1	1	yes	characteristic
300	C₄×C₅₃	#390	1	1	yes	characteristic
301	C₄×C₅₃	#390	1	1	yes	characteristic
302	C₄×C₅₃	#390	1	1	yes	characteristic
303	C₄×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
304	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
305	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
306	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
307	Dic₅₃	—	4	4	no	—
308	Dic₅₃	—	8	4	no	—
309	C₂²×#135	D₅₃	1	1	yes	characteristic
310	C₂×#427	—	53	1	no	—
311	C₂×#421	—	53	2	no	—
312	C₂×#410	—	53	2	no	—
313	C₂×#405	—	53	1	no	—
314	#1976	—	53	8	no	—
315	C₂×C₅₃	—	2	1	no	—
316	C₂×C₅₃	—	2	1	no	—
317	C₂×C₅₃	—	2	1	no	—
318	C₂×C₅₃	—	2	1	no	—
319	C₂×C₅₃	—	2	1	no	—
320	C₂×C₅₃	—	2	1	no	—
321	C₂×C₅₃	—	8	4	no	—
322	C₂×C₅₃	#1976	1	2	yes	complemented
323	C₂×C₅₃	C₂×#390	1	1	yes	characteristic
324	C₂²×#41	C₂×D₅₃	1	2	yes	—
325	C₂×#135	—	2	1	no	—
326	C₂×#135	—	2	1	no	—
327	C₂×#135	—	2	2	no	—
328	C₂×#135	—	2	1	no	—
329	C₂×#135	—	2	1	no	—
330	C₂×#135	—	2	2	no	—
331	C₂×#135	Dic₅₃	1	4	yes	complemented
332	C₂×#135	C₂×D₅₃	1	4	yes	—
333	C₂²×Q₁₆	Dic₅₃	1	1	yes	characteristic, complemented
334	C₂²×SD₁₆	Dic₅₃	1	2	yes	complemented
335	C₂²×D₈	Dic₅₃	1	1	yes	characteristic, complemented
336	C₂²×M₁₆	—	53	1	no	—
337	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
338	C₂×#117	—	53	1	no	—
339	C₂×#117	—	53	2	no	—
340	#427	—	53	4	no	—
341	#427	—	106	2	no	—
342	C₂×#107	—	106	1	no	—
343	C₂×#106	—	106	1	no	—
344	#421	—	53	8	no	—
345	#421	—	106	4	no	—
346	C₂×#104	—	106	4	no	—
347	#410	—	53	8	no	—
348	#410	—	106	4	no	—
349	C₂×#99	—	106	4	no	—
350	#405	—	53	4	no	—
351	#405	—	106	2	no	—
352	C₂×#97	—	106	1	no	—
353	C₂×C₄×C₈	—	106	1	no	—
354	C₅₃	C₂×#1976	1	1	yes	characteristic, complemented, Sylow
355	C₂×#41	—	2	2	no	—
356	C₂×#41	—	2	2	no	—
357	C₂×#41	—	2	2	no	—
358	C₂×#41	—	2	2	no	—
359	C₂×#41	—	2	2	no	—
360	C₂×#41	—	2	2	no	—
361	C₂×#41	—	2	2	no	—
362	C₂×#41	—	2	2	no	—
363	C₂×#41	—	2	2	no	—
364	C₂×#41	—	2	2	no	—
365	C₂×#41	C₂×Dic₅₃	1	4	yes	—
366	C₂×#41	C₂²×D₅₃	1	4	yes	—
367	C₂²×Q₈	C₂×Dic₅₃	1	2	yes	—
368	C₂²×D₄	—	2	2	no	—
369	C₂²×D₄	C₂×Dic₅₃	1	2	yes	—
370	C₂³×C₄	—	2	2	no	—
371	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
372	#135	—	2	4	no	—
373	#135	—	2	4	no	—
374	#135	—	2	4	no	—
375	#135	—	2	4	no	—
376	#135	—	2	4	no	—
377	#135	—	2	4	no	—
378	#135	—	2	4	no	—
379	#135	—	2	4	no	—
380	C₂×Q₁₆	—	2	1	no	—
381	C₂×Q₁₆	—	2	1	no	—
382	C₂×Q₁₆	—	2	2	no	—
383	C₂×Q₁₆	C₂×Dic₅₃	1	4	yes	complemented
384	C₂×SD₁₆	—	2	2	no	—
385	C₂×SD₁₆	—	2	2	no	—
386	C₂×SD₁₆	—	2	4	no	—
387	C₂×SD₁₆	C₂×Dic₅₃	1	8	yes	complemented
388	C₂×D₈	—	2	1	no	—
389	C₂×D₈	—	2	1	no	—
390	C₂×D₈	—	2	2	no	—
391	C₂×D₈	C₂×Dic₅₃	1	4	yes	complemented
392	C₂×M₁₆	—	53	2	no	—
393	C₂×M₁₆	—	53	2	no	—
394	C₂×M₁₆	—	53	2	no	—
395	C₂×M₁₆	—	106	1	no	—
396	C₂×M₁₆	—	106	1	no	—
397	C₂×M₁₆	—	106	1	no	—
398	C₂²×C₈	—	106	1	no	—
399	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
400	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
401	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
402	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
403	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
404	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
405	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
406	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
407	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
408	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
409	#117	—	53	2	no	—
410	#117	—	53	2	no	—
411	#117	—	53	4	no	—
412	#117	—	106	1	no	—
413	#117	—	106	1	no	—
414	#117	—	106	2	no	—
415	#117	—	106	2	no	—
416	#117	—	106	2	no	—
417	C₂×C₄⋊C₄	—	106	1	no	—
418	C₂×C₄⋊C₄	—	106	2	no	—
419	C₂×C₂²⋊C₄	—	53	2	no	—
420	C₂×C₂²⋊C₄	—	106	2	no	—
421	C₂×C₄²	—	106	1	no	—
422	C₂×C₄²	—	106	2	no	—
423	#107	—	106	4	no	—
424	#106	—	106	4	no	—
425	#104	—	106	16	no	—
426	#99	—	106	16	no	—
427	#97	—	106	4	no	—
428	C₄×C₈	—	106	4	no	—
429	C₂⁴	—	2	2	no	—
430	C₂⁴	D₄×D₅₃	1	1	yes	characteristic
431	#41	—	2	4	no	—
432	#41	—	2	4	no	—
433	#41	—	2	4	no	—
434	#41	—	2	4	no	—
435	#41	—	2	4	no	—
436	#41	—	2	4	no	—
437	#41	—	2	4	no	—
438	#41	—	2	4	no	—
439	#41	—	4	2	no	—
440	#41	—	4	2	no	—
441	#41	—	4	2	no	—
442	#41	—	4	2	no	—
443	#41	—	4	2	no	—
444	#41	—	4	2	no	—
445	#41	—	4	2	no	—
446	#41	—	4	2	no	—
447	C₂×Q₈	—	2	2	no	—
448	C₂×Q₈	—	2	2	no	—
449	C₂×Q₈	—	2	2	no	—
450	C₂×Q₈	—	2	2	no	—
451	C₂×Q₈	—	2	2	no	—
452	C₂×Q₈	C₂²×Dic₅₃	1	4	yes	—
453	C₂×D₄	—	2	4	no	—
454	C₂×D₄	—	2	4	no	—
455	C₂×D₄	—	2	2	no	—
456	C₂×D₄	—	2	2	no	—
457	C₂×D₄	—	2	2	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	—	4	2	no	—
461	C₂×D₄	—	4	2	no	—
462	C₂×D₄	—	4	2	no	—
463	C₂×D₄	—	4	2	no	—
464	C₂×D₄	C₂²×Dic₅₃	1	4	yes	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	4	no	—
467	C₂²×C₄	—	2	4	no	—
468	C₂²×C₄	—	4	2	no	—
469	C₂²×C₄	—	4	2	no	—
470	C₂²×C₄	—	4	2	no	—
471	C₂²×C₄	—	4	2	no	—
472	C₂²×C₄	—	106	2	no	—
473	C₂²×C₄	—	212	2	no	—
474	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
475	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
476	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
477	C₂²×C₄	#11719947	1	1	yes	characteristic
478	C₂²×C₄	#11719947	1	1	yes	characteristic
479	C₂²×C₄	#11719947	1	1	yes	characteristic
480	C₂²×C₄	#11719947	1	1	yes	characteristic
481	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
482	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
483	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
484	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
485	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
486	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
487	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
488	Q₁₆	—	2	4	no	—
489	Q₁₆	—	2	4	no	—
490	SD₁₆	—	2	8	no	—
491	SD₁₆	—	2	8	no	—
492	D₈	—	2	4	no	—
493	D₈	—	2	4	no	—
494	M₁₆	—	53	4	no	—
495	M₁₆	—	106	2	no	—
496	M₁₆	—	106	2	no	—
497	M₁₆	—	106	2	no	—
498	C₂×C₈	—	106	2	no	—
499	C₂×C₈	—	106	2	no	—
500	C₂×C₈	—	106	2	no	—
501	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
502	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
503	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
504	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
505	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
506	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
507	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
508	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
509	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
510	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
511	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
512	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
513	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
514	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
515	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
516	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
517	C₄⋊C₄	—	106	2	no	—
518	C₄⋊C₄	—	106	2	no	—
519	C₄⋊C₄	—	106	4	no	—
520	C₄⋊C₄	—	212	2	no	—
521	C₂²⋊C₄	—	53	4	no	—
522	C₂²⋊C₄	—	106	2	no	—
523	C₂²⋊C₄	—	106	4	no	—
524	C₂²⋊C₄	—	212	2	no	—
525	C₄²	—	106	2	no	—
526	C₄²	—	106	2	no	—
527	C₄²	—	106	4	no	—
528	C₄²	—	212	2	no	—
529	C₂³	—	2	4	no	—
530	C₂³	—	4	2	no	—
531	C₂³	—	4	2	no	—
532	C₂³	—	4	2	no	—
533	C₂³	—	8	4	no	—
534	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
535	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
536	C₂³	#13133188	1	1	yes	characteristic
537	C₂³	#13133188	1	1	yes	characteristic
538	C₂³	#13133213	1	1	yes	characteristic
539	C₂³	#13133213	1	1	yes	characteristic
540	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
541	Q₈	—	2	4	no	—
542	Q₈	—	2	4	no	—
543	Q₈	—	4	2	no	—
544	Q₈	—	4	2	no	—
545	D₄	—	2	4	no	—
546	D₄	—	2	4	no	—
547	D₄	—	4	4	no	—
548	D₄	—	4	4	no	—
549	D₄	—	4	4	no	—
550	D₄	—	4	4	no	—
551	D₄	—	4	2	no	—
552	D₄	—	4	2	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	4	2	no	—
555	C₂×C₄	—	4	2	no	—
556	C₂×C₄	—	4	4	no	—
557	C₂×C₄	—	4	4	no	—
558	C₂×C₄	—	4	4	no	—
559	C₂×C₄	—	4	4	no	—
560	C₂×C₄	—	106	4	no	—
561	C₂×C₄	—	424	4	no	—
562	C₂×C₄	—	212	4	no	—
563	C₂×C₄	—	212	4	no	—
564	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
565	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
566	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
567	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
568	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
569	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
570	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
571	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
572	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
573	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
574	C₂×C₄	#13133188	1	1	yes	characteristic
575	C₂×C₄	#13133188	1	1	yes	characteristic
576	C₂×C₄	#13133188	1	1	yes	characteristic
577	C₂×C₄	#13133188	1	1	yes	characteristic
578	C₂×C₄	#13133188	1	1	yes	characteristic
579	C₂×C₄	#13133188	1	1	yes	characteristic
580	C₂×C₄	#13133204	1	1	yes	characteristic
581	C₂×C₄	#13133204	1	1	yes	characteristic
582	C₂×C₄	#13133204	1	1	yes	characteristic
583	C₂×C₄	#13133204	1	1	yes	characteristic
584	C₂×C₄	#13133213	1	1	yes	characteristic
585	C₂×C₄	#13133213	1	1	yes	characteristic
586	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
587	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
588	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
589	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
590	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
591	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
592	C₈	—	106	4	no	—
593	C₈	C₂×C₂²⋊Dic₅₃	1	2	yes	—
594	C₈	C₂×C₂²⋊Dic₅₃	1	2	yes	—
595	C₈	C₂×C₂²⋊Dic₅₃	1	2	yes	—
596	C₈	C₂×C₂²⋊Dic₅₃	1	2	yes	—
597	C₂²	—	2	2	no	—
598	C₂²	—	2	2	no	—
599	C₂²	—	2	2	no	—
600	C₂²	—	4	1	no	—
601	C₂²	—	4	1	no	—
602	C₂²	—	4	1	no	—
603	C₂²	—	4	1	no	—
604	C₂²	—	4	4	no	—
605	C₂²	—	8	4	no	—
606	C₂²	—	8	4	no	—
607	C₂²	—	8	4	no	—
608	C₂²	#16810526	1	1	yes	characteristic
609	C₂²	#16810526	1	1	yes	characteristic
610	C₂²	#16810526	1	1	yes	characteristic
611	C₂²	#16810526	1	1	yes	characteristic
612	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
613	C₂²	C₂×#13133188	1	1	yes	characteristic
614	C₂²	C₂×#13133213	1	1	yes	characteristic
615	C₄	—	4	4	no	—
616	C₄	—	424	4	no	—
617	C₄	—	212	4	no	—
618	C₄	#16810526	1	1	yes	characteristic
619	C₄	#16810526	1	1	yes	characteristic
620	C₄	#16810526	1	1	yes	characteristic
621	C₄	#16810526	1	1	yes	characteristic
622	C₄	C₂²×C₂²⋊Dic₅₃	1	1	yes	characteristic
623	C₄	C₂×D₄×Dic₅₃	1	1	yes	characteristic
624	C₄	C₂×#13133188	1	1	yes	characteristic
625	C₄	C₂×#13133204	1	1	yes	characteristic
626	C₂	—	2	1	no	—
627	C₂	—	2	1	no	—
628	C₂	—	2	1	no	—
629	C₂	—	2	1	no	—
630	C₂	—	2	1	no	—
631	C₂	—	2	1	no	—
632	C₂	—	8	4	no	—
633	C₂	#20275614	1	2	yes	direct product factor
634	C₂	C₂×#16810526	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133188
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting, prime radical	C₂²×#135×C₅₃	C₂
socle	C₂²×C₅₃	#390
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂²×#135	D₅₃
hypocenter	C₅₃	C₂×#1976
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#16810661	C₂	semidirect
2. C₂×#16810640	C₂	semidirect
3. C₂×#16810597	C₂	semidirect
4. C₂×#16810578	C₂	semidirect
5. #20275614	C₂	direct (N × H)
11. #16810661	C₂²	semidirect
15. #16810640	C₂²	semidirect
18. #16810597	C₂²	semidirect
21. #16810578	C₂²	semidirect
31. C₂×#135×C₅₃	C₄	semidirect
33. C₂²×Q₁₆×C₅₃	C₄	semidirect
34. C₂²×SD₁₆×C₅₃	C₄	semidirect
35. C₂²×D₈×C₅₃	C₄	semidirect
92. C₂×Q₁₆×C₅₃	C₂×C₄	semidirect
96. C₂×SD₁₆×C₅₃	C₂×C₄	semidirect
100. C₂×D₈×C₅₃	C₂×C₄	semidirect
322. C₂×C₅₃	#1976	semidirect
331. C₂×#135	Dic₅₃	semidirect
333. C₂²×Q₁₆	Dic₅₃	semidirect
334. C₂²×SD₁₆	Dic₅₃	semidirect
335. C₂²×D₈	Dic₅₃	semidirect
354. C₅₃	C₂×#1976	semidirect
383. C₂×Q₁₆	C₂×Dic₅₃	semidirect
387. C₂×SD₁₆	C₂×Dic₅₃	semidirect
391. C₂×D₈	C₂×Dic₅₃	semidirect
633. C₂	#20275614	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×#16810661	2
C₂×#16810640	2
C₂×#16810597	2
C₂×#16810578	2
#20275614	2
C₂²×#135×C₅₃	2
C₂×#1976	53

Subgroups — C2×#20275614 #27388067

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20275614 #27388067