Subgroups — C₂×#20276176 #27413030

Order 13568.

1006 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810671	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810628	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13133134	C₂	1	1	yes	maximal, characteristic
4	#20276176	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16810395	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810393	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16810379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#418	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11719917	C₂²	1	1	yes	characteristic
10	C₂²×#11719917	C₂²	1	1	yes	characteristic
11	C₂²×#11719917	C₂²	1	1	yes	characteristic
12	C₂×#13133176	C₂²	1	1	yes	characteristic
13	#16810671	—	2	2	no	—
14	#16810671	C₂²	1	4	yes	complemented
15	C₂×#13133152	—	2	2	no	—
16	#16810628	—	2	2	no	—
17	#16810628	C₂²	1	4	yes	—
18	C₂×#13133138	—	2	2	no	—
19	C₂×#13133134	—	2	2	no	—
20	C₂×#13133134	—	2	2	no	—
21	C₂×#13133134	C₂²	1	4	yes	—
22	#16810395	C₂²	1	4	yes	complemented
23	#16810393	C₂²	1	4	yes	complemented
24	#16810379	C₂²	1	4	yes	complemented
25	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#418	—	2	2	no	—
29	C₅₃×#418	C₂²	1	4	yes	—
30	C₂×#102×C₅₃	—	2	2	no	—
31	C₂×#11719917	—	2	1	no	—
32	C₂×#11719917	—	2	1	no	—
33	C₂×#11719917	—	2	1	no	—
34	C₂×#11719917	—	2	1	no	—
35	C₂×#11719917	—	2	1	no	—
36	C₂×#11719917	—	2	1	no	—
37	C₂×#11719917	—	2	1	no	—
38	C₂×#11719917	—	2	1	no	—
39	C₂×#11719917	—	2	1	no	—
40	C₂×#11719917	—	2	1	no	—
41	C₂×#11719917	—	2	1	no	—
42	C₂×#11719917	—	2	1	no	—
43	C₂×#11719917	—	2	1	no	—
44	C₂×#11719917	—	2	1	no	—
45	C₂×#11719917	D₄	1	2	yes	—
46	C₂×#11719917	C₂³	1	2	yes	—
47	C₂×#11719917	C₂³	1	2	yes	—
48	C₂×#11719917	C₂³	1	2	yes	—
49	C₂²×#88322	—	2	1	no	—
50	C₂²×#88322	—	2	1	no	—
51	C₂²×#88322	D₄	1	2	yes	—
52	#13133176	—	2	1	no	—
53	#13133176	—	2	1	no	—
54	#13133176	—	2	1	no	—
55	#13133176	C₂³	1	2	yes	—
56	C₂×C₄⋊Dic₅₃	—	2	1	no	—
57	C₂×C₂²⋊Dic₅₃	—	2	1	no	—
58	C₂×C₄×Dic₅₃	—	2	1	no	—
59	#13133152	—	2	8	no	—
60	#13133138	—	2	8	no	—
61	#13133134	—	2	8	no	—
62	C₂⁵×C₅₃	—	2	1	no	—
63	C₂²×D₄×C₅₃	—	2	1	no	—
64	C₂²×D₄×C₅₃	—	2	1	no	—
65	C₂²×D₄×C₅₃	—	2	1	no	—
66	C₂²×D₄×C₅₃	—	2	1	no	—
67	C₂²×D₄×C₅₃	—	2	1	no	—
68	C₂²×D₄×C₅₃	—	2	1	no	—
69	C₂²×D₄×C₅₃	—	2	1	no	—
70	C₂²×D₄×C₅₃	—	2	1	no	—
71	C₂²×D₄×C₅₃	—	2	1	no	—
72	C₂²×D₄×C₅₃	—	2	1	no	—
73	C₂²×D₄×C₅₃	—	2	1	no	—
74	C₂²×D₄×C₅₃	D₄	1	2	yes	—
75	C₂²×D₄×C₅₃	D₄	1	2	yes	—
76	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
77	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
78	C₂×M₁₆×C₅₃	—	2	1	no	—
79	C₂×M₁₆×C₅₃	—	2	1	no	—
80	C₂×M₁₆×C₅₃	—	2	1	no	—
81	C₂×M₁₆×C₅₃	—	2	1	no	—
82	C₂×M₁₆×C₅₃	—	2	1	no	—
83	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
84	C₂²×C₈×C₅₃	—	2	1	no	—
85	#117×C₅₃	—	2	1	no	—
86	#117×C₅₃	—	2	1	no	—
87	#117×C₅₃	—	2	1	no	—
88	#117×C₅₃	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
91	C₂×C₄²×C₅₃	—	2	1	no	—
92	#102×C₅₃	—	2	8	no	—
93	C₂⁴×C₅₃	—	2	2	no	—
94	C₂⁴×C₅₃	—	2	2	no	—
95	C₂⁴×C₅₃	—	2	2	no	—
96	C₂⁴×C₅₃	—	4	1	no	—
97	C₂⁴×C₅₃	—	4	1	no	—
98	C₂⁴×C₅₃	—	4	1	no	—
99	C₂⁴×C₅₃	—	4	1	no	—
100	C₂⁴×C₅₃	—	8	4	no	—
101	C₂⁴×C₅₃	#41	1	1	yes	characteristic
102	C₂×D₄×C₅₃	—	2	2	no	—
103	C₂×D₄×C₅₃	—	2	2	no	—
104	C₂×D₄×C₅₃	—	2	2	no	—
105	C₂×D₄×C₅₃	—	2	2	no	—
106	C₂×D₄×C₅₃	—	2	2	no	—
107	C₂×D₄×C₅₃	—	2	2	no	—
108	C₂×D₄×C₅₃	—	2	2	no	—
109	C₂×D₄×C₅₃	—	2	2	no	—
110	C₂×D₄×C₅₃	—	2	2	no	—
111	C₂×D₄×C₅₃	—	2	2	no	—
112	C₂×D₄×C₅₃	—	2	2	no	—
113	C₂×D₄×C₅₃	—	2	2	no	—
114	C₂×D₄×C₅₃	—	2	2	no	—
115	C₂×D₄×C₅₃	—	2	2	no	—
116	C₂×D₄×C₅₃	—	2	2	no	—
117	C₂×D₄×C₅₃	—	4	1	no	—
118	C₂×D₄×C₅₃	—	4	1	no	—
119	C₂×D₄×C₅₃	—	4	1	no	—
120	C₂×D₄×C₅₃	—	4	1	no	—
121	C₂×D₄×C₅₃	—	4	1	no	—
122	C₂×D₄×C₅₃	—	4	1	no	—
123	C₂×D₄×C₅₃	—	4	1	no	—
124	C₂×D₄×C₅₃	—	4	1	no	—
125	C₂×D₄×C₅₃	—	4	1	no	—
126	C₂×D₄×C₅₃	—	4	1	no	—
127	C₂×D₄×C₅₃	—	4	1	no	—
128	C₂×D₄×C₅₃	—	4	1	no	—
129	C₂×D₄×C₅₃	C₂×D₄	1	4	yes	—
130	C₂²×C₄×C₅₃	—	4	1	no	—
131	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
136	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
137	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
138	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
139	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
140	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
141	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
142	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
143	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
144	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
145	C₂²×Dic₅₃	—	4	1	no	—
146	M₁₆×C₅₃	—	2	2	no	—
147	M₁₆×C₅₃	—	2	2	no	—
148	M₁₆×C₅₃	—	4	1	no	—
149	M₁₆×C₅₃	—	4	1	no	—
150	C₂×C₈×C₅₃	—	2	2	no	—
151	C₂×C₈×C₅₃	—	4	1	no	—
152	C₂×C₈×C₅₃	—	4	1	no	—
153	C₄⋊C₄×C₅₃	—	2	2	no	—
154	C₄⋊C₄×C₅₃	—	4	1	no	—
155	C₂²⋊C₄×C₅₃	—	2	2	no	—
156	C₂²⋊C₄×C₅₃	—	4	1	no	—
157	C₄²×C₅₃	—	2	2	no	—
158	C₄²×C₅₃	—	4	1	no	—
159	C₂²⋊Dic₅₃	—	2	2	no	—
160	C₂²⋊Dic₅₃	—	4	1	no	—
161	C₄⋊Dic₅₃	—	2	2	no	—
162	C₄⋊Dic₅₃	—	4	1	no	—
163	C₄×Dic₅₃	—	2	2	no	—
164	C₄×Dic₅₃	—	4	1	no	—
165	#11719917	—	2	2	no	—
166	#11719917	—	2	2	no	—
167	#11719917	—	2	2	no	—
168	#11719917	—	2	2	no	—
169	#11719917	—	2	2	no	—
170	#11719917	—	2	2	no	—
171	#11719917	—	2	2	no	—
172	#11719917	—	2	2	no	—
173	#11719917	—	4	1	no	—
174	#11719917	—	4	1	no	—
175	#11719917	—	4	1	no	—
176	#11719917	—	4	1	no	—
177	C₂×#88322	—	2	2	no	—
178	C₂×#88322	—	2	2	no	—
179	C₂×#88322	—	2	2	no	—
180	C₂×#88322	—	2	2	no	—
181	C₂×#88322	—	4	1	no	—
182	C₂×#88322	—	4	1	no	—
183	C₂×#88322	—	4	1	no	—
184	C₂×#88322	—	4	1	no	—
185	C₂×#88322	C₂×D₄	1	4	yes	—
186	C₂³×C₅₃	—	2	4	no	—
187	C₂³×C₅₃	—	4	2	no	—
188	C₂³×C₅₃	—	4	2	no	—
189	C₂³×C₅₃	—	4	2	no	—
190	C₂³×C₅₃	—	4	2	no	—
191	C₂³×C₅₃	—	4	2	no	—
192	C₂³×C₅₃	—	4	2	no	—
193	C₂³×C₅₃	—	4	2	no	—
194	C₂³×C₅₃	—	8	4	no	—
195	C₂³×C₅₃	—	8	4	no	—
196	C₂³×C₅₃	—	8	4	no	—
197	C₂³×C₅₃	—	8	4	no	—
198	C₂³×C₅₃	—	8	4	no	—
199	C₂³×C₅₃	—	8	4	no	—
200	C₂³×C₅₃	—	8	4	no	—
201	C₂³×C₅₃	#123	1	1	yes	characteristic
202	C₂³×C₅₃	#123	1	1	yes	characteristic
203	C₂³×C₅₃	#124	1	1	yes	characteristic
204	C₂³×C₅₃	#124	1	1	yes	characteristic
205	C₂³×C₅₃	#126	1	1	yes	characteristic
206	C₂³×C₅₃	#126	1	1	yes	characteristic
207	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
208	D₄×C₅₃	—	2	4	no	—
209	D₄×C₅₃	—	2	4	no	—
210	D₄×C₅₃	—	4	2	no	—
211	D₄×C₅₃	—	4	2	no	—
212	D₄×C₅₃	—	4	2	no	—
213	D₄×C₅₃	—	4	2	no	—
214	D₄×C₅₃	—	4	2	no	—
215	D₄×C₅₃	—	4	2	no	—
216	C₂×C₄×C₅₃	—	4	2	no	—
217	C₂×C₄×C₅₃	—	8	2	no	—
218	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
219	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
220	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
221	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
222	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
223	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
224	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
225	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
226	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
227	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
228	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
229	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
230	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
231	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
232	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
233	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
234	C₂×C₄×C₅₃	#126	1	1	yes	characteristic
235	C₂×C₄×C₅₃	#126	1	1	yes	characteristic
236	C₂×C₄×C₅₃	#126	1	1	yes	characteristic
237	C₂×C₄×C₅₃	#126	1	1	yes	characteristic
238	C₂×C₄×C₅₃	#126	1	1	yes	characteristic
239	C₂×C₄×C₅₃	#126	1	1	yes	characteristic
240	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
243	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
244	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
245	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
246	C₂×Dic₅₃	—	4	2	no	—
247	C₂×Dic₅₃	—	8	2	no	—
248	C₈×C₅₃	—	4	2	no	—
249	#88322	—	2	4	no	—
250	#88322	—	4	2	no	—
251	#88322	—	4	2	no	—
252	C₂×#2081	—	53	1	no	maximal, Sylow
253	C₂²×C₅₃	—	2	2	no	—
254	C₂²×C₅₃	—	2	2	no	—
255	C₂²×C₅₃	—	2	2	no	—
256	C₂²×C₅₃	—	4	1	no	—
257	C₂²×C₅₃	—	4	1	no	—
258	C₂²×C₅₃	—	4	1	no	—
259	C₂²×C₅₃	—	4	1	no	—
260	C₂²×C₅₃	—	4	4	no	—
261	C₂²×C₅₃	—	8	4	no	—
262	C₂²×C₅₃	—	8	4	no	—
263	C₂²×C₅₃	—	8	4	no	—
264	C₂²×C₅₃	—	8	4	no	—
265	C₂²×C₅₃	—	8	4	no	—
266	C₂²×C₅₃	—	8	4	no	—
267	C₂²×C₅₃	—	8	4	no	—
268	C₂²×C₅₃	#397	1	1	yes	characteristic
269	C₂²×C₅₃	#397	1	1	yes	characteristic
270	C₂²×C₅₃	#397	1	1	yes	characteristic
271	C₂²×C₅₃	#397	1	1	yes	characteristic
272	C₂²×C₅₃	C₂×#123	1	1	yes	characteristic
273	C₂²×C₅₃	C₂×#124	1	1	yes	characteristic
274	C₂²×C₅₃	C₂×#126	1	1	yes	characteristic
275	C₄×C₅₃	—	8	2	no	—
276	C₄×C₅₃	#397	1	1	yes	characteristic
277	C₄×C₅₃	#397	1	1	yes	characteristic
278	C₄×C₅₃	#397	1	1	yes	characteristic
279	C₄×C₅₃	#397	1	1	yes	characteristic
280	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
281	C₄×C₅₃	C₂×#123	1	1	yes	characteristic
282	C₄×C₅₃	C₂×#123	1	1	yes	characteristic
283	C₄×C₅₃	C₂×#126	1	1	yes	characteristic
284	Dic₅₃	—	8	2	no	—
285	C₂×#430	—	53	1	no	—
286	C₂×#418	—	53	1	no	—
287	C₂×#418	D₅₃	1	1	yes	characteristic
288	C₂²×#100	—	53	1	no	—
289	#2081	—	53	8	no	—
290	C₂×#344	—	53	1	no	—
291	C₂×#344	—	53	1	no	—
292	C₂×H(3,ℤ/4)	—	53	1	no	—
293	C₂×C₅₃	—	2	1	no	—
294	C₂×C₅₃	—	2	1	no	—
295	C₂×C₅₃	—	2	1	no	—
296	C₂×C₅₃	—	2	1	no	—
297	C₂×C₅₃	—	2	1	no	—
298	C₂×C₅₃	—	2	1	no	—
299	C₂×C₅₃	—	8	4	no	—
300	C₂×C₅₃	#2081	1	2	yes	complemented
301	C₂×C₅₃	C₂×#397	1	1	yes	characteristic
302	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic
303	C₂²×M₁₆	—	53	1	no	—
304	C₂²×M₁₆	—	53	1	no	—
305	C₂²×M₁₆	—	53	1	no	—
306	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
307	C₂×#117	—	53	1	no	—
308	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
309	#430	—	53	4	no	—
310	#430	—	106	2	no	—
311	C₂×#108	—	106	2	no	—
312	#418	—	2	2	no	—
313	#418	—	53	4	no	—
314	#418	—	106	2	no	—
315	#418	C₂×D₅₃	1	4	yes	—
316	C₂×#102	—	2	2	no	—
317	C₂×#102	—	106	2	no	—
318	C₂×#100	—	53	4	no	—
319	C₂×#100	—	106	2	no	—
320	C₂×#100	—	106	2	no	—
321	#344	—	53	4	no	—
322	#344	—	53	4	no	—
323	H(3,ℤ/4)	—	53	4	no	—
324	C₅₃	C₂×#2081	1	1	yes	characteristic, complemented, Sylow
325	C₂⁵	—	2	1	no	—
326	C₂²×D₄	—	2	1	no	—
327	C₂²×D₄	—	2	1	no	—
328	C₂²×D₄	—	2	1	no	—
329	C₂²×D₄	—	2	1	no	—
330	C₂²×D₄	—	2	1	no	—
331	C₂²×D₄	—	2	1	no	—
332	C₂²×D₄	—	2	1	no	—
333	C₂²×D₄	—	2	1	no	—
334	C₂²×D₄	—	2	1	no	—
335	C₂²×D₄	—	2	1	no	—
336	C₂²×D₄	—	2	1	no	—
337	C₂²×D₄	C₂²⋊D₅₃	1	2	yes	—
338	C₂²×D₄	C₂²⋊D₅₃	1	2	yes	—
339	C₂²×D₄	C₂²×D₅₃	1	2	yes	—
340	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
341	C₂×M₁₆	—	2	1	no	—
342	C₂×M₁₆	—	2	1	no	—
343	C₂×M₁₆	—	2	1	no	—
344	C₂×M₁₆	—	2	1	no	—
345	C₂×M₁₆	—	2	1	no	—
346	C₂×M₁₆	—	53	2	no	—
347	C₂×M₁₆	—	53	2	no	—
348	C₂×M₁₆	—	53	2	no	—
349	C₂×M₁₆	—	53	2	no	—
350	C₂×M₁₆	—	106	1	no	—
351	C₂×M₁₆	—	106	1	no	—
352	C₂×M₁₆	—	106	1	no	—
353	C₂×M₁₆	—	106	1	no	—
354	C₂×M₁₆	—	106	1	no	—
355	C₂×M₁₆	—	106	1	no	—
356	C₂×M₁₆	—	106	1	no	—
357	C₂×M₁₆	—	106	1	no	—
358	C₂×M₁₆	—	106	1	no	—
359	C₂×M₁₆	—	106	1	no	—
360	C₂×M₁₆	—	106	1	no	—
361	C₂×M₁₆	—	106	1	no	—
362	C₂×M₁₆	—	106	1	no	—
363	C₂×M₁₆	—	106	1	no	—
364	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
365	C₂²×C₈	—	2	1	no	—
366	C₂²×C₈	—	53	2	no	—
367	C₂²×C₈	—	106	1	no	—
368	C₂²×C₈	—	106	1	no	—
369	#117	—	2	1	no	—
370	#117	—	2	1	no	—
371	#117	—	2	1	no	—
372	#117	—	53	2	no	—
373	#117	—	106	1	no	—
374	#117	—	106	1	no	—
375	#117	—	106	1	no	—
376	#117	C₂²×D₅₃	1	2	yes	—
377	C₂×C₄⋊C₄	—	2	1	no	—
378	C₂×C₄⋊C₄	—	106	1	no	—
379	C₂×C₂²⋊C₄	—	2	1	no	—
380	C₂×C₂²⋊C₄	—	106	1	no	—
381	C₂×C₄²	—	2	1	no	—
382	C₂×C₄²	—	106	1	no	—
383	#108	—	106	8	no	—
384	#102	—	2	8	no	—
385	#102	—	106	8	no	—
386	#100	—	106	8	no	—
387	C₂⁴	—	2	2	no	—
388	C₂⁴	—	2	2	no	—
389	C₂⁴	—	2	2	no	—
390	C₂⁴	—	4	1	no	—
391	C₂⁴	—	4	1	no	—
392	C₂⁴	—	4	1	no	—
393	C₂⁴	—	4	1	no	—
394	C₂⁴	—	8	4	no	—
395	C₂⁴	#11719947	1	1	yes	characteristic
396	C₂×D₄	—	2	2	no	—
397	C₂×D₄	—	2	2	no	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	—	2	2	no	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	2	2	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	2	2	no	—
409	C₂×D₄	—	2	2	no	—
410	C₂×D₄	—	2	2	no	—
411	C₂×D₄	—	4	1	no	—
412	C₂×D₄	—	4	1	no	—
413	C₂×D₄	—	4	1	no	—
414	C₂×D₄	—	4	1	no	—
415	C₂×D₄	—	4	1	no	—
416	C₂×D₄	—	4	1	no	—
417	C₂×D₄	—	4	1	no	—
418	C₂×D₄	—	4	1	no	—
419	C₂×D₄	—	4	1	no	—
420	C₂×D₄	—	4	1	no	—
421	C₂×D₄	—	4	1	no	—
422	C₂×D₄	—	4	1	no	—
423	C₂×D₄	C₂×C₂²⋊D₅₃	1	4	yes	—
424	C₂²×C₄	—	4	1	no	—
425	C₂²×C₄	—	212	1	no	—
426	C₂²×C₄	#11719945	1	1	yes	characteristic
427	C₂²×C₄	#11719945	1	1	yes	characteristic
428	C₂²×C₄	#11719945	1	1	yes	characteristic
429	C₂²×C₄	#11719945	1	1	yes	characteristic
430	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
431	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
432	C₂²×C₄	#11719947	1	1	yes	characteristic
433	C₂²×C₄	#11719947	1	1	yes	characteristic
434	C₂²×C₄	#11719947	1	1	yes	characteristic
435	C₂²×C₄	#11719947	1	1	yes	characteristic
436	C₂²×C₄	#11719947	1	1	yes	characteristic
437	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
438	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
439	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
440	M₁₆	—	2	2	no	—
441	M₁₆	—	2	2	no	—
442	M₁₆	—	4	1	no	—
443	M₁₆	—	4	1	no	—
444	M₁₆	—	106	2	no	—
445	M₁₆	—	106	2	no	—
446	M₁₆	—	106	2	no	—
447	M₁₆	—	106	2	no	—
448	M₁₆	—	106	2	no	—
449	M₁₆	—	106	2	no	—
450	M₁₆	—	106	2	no	—
451	M₁₆	—	106	2	no	—
452	M₁₆	—	212	1	no	—
453	M₁₆	—	212	1	no	—
454	M₁₆	—	212	1	no	—
455	M₁₆	—	212	1	no	—
456	C₂×C₈	—	2	2	no	—
457	C₂×C₈	—	4	1	no	—
458	C₂×C₈	—	4	1	no	—
459	C₂×C₈	—	53	4	no	—
460	C₂×C₈	—	106	2	no	—
461	C₂×C₈	—	106	2	no	—
462	C₂×C₈	—	106	2	no	—
463	C₂×C₈	—	106	2	no	—
464	C₂×C₈	—	212	1	no	—
465	C₂×C₈	—	212	1	no	—
466	C₂×C₈	—	212	1	no	—
467	C₂×C₈	—	212	1	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	4	1	no	—
470	C₄⋊C₄	—	106	2	no	—
471	C₄⋊C₄	—	212	1	no	—
472	C₂²⋊C₄	—	2	2	no	—
473	C₂²⋊C₄	—	4	1	no	—
474	C₂²⋊C₄	—	106	2	no	—
475	C₂²⋊C₄	—	212	1	no	—
476	C₄²	—	2	2	no	—
477	C₄²	—	4	1	no	—
478	C₄²	—	106	2	no	—
479	C₄²	—	212	1	no	—
480	C₂³	—	2	4	no	—
481	C₂³	—	4	2	no	—
482	C₂³	—	4	2	no	—
483	C₂³	—	4	2	no	—
484	C₂³	—	4	2	no	—
485	C₂³	—	4	2	no	—
486	C₂³	—	4	2	no	—
487	C₂³	—	4	2	no	—
488	C₂³	—	8	4	no	—
489	C₂³	—	8	4	no	—
490	C₂³	—	8	4	no	—
491	C₂³	—	8	4	no	—
492	C₂³	—	8	4	no	—
493	C₂³	—	8	4	no	—
494	C₂³	—	8	4	no	—
495	C₂³	#13133198	1	1	yes	characteristic
496	C₂³	#13133198	1	1	yes	characteristic
497	C₂³	#13133203	1	1	yes	characteristic
498	C₂³	#13133203	1	1	yes	characteristic
499	C₂³	#13133210	1	1	yes	characteristic
500	C₂³	#13133210	1	1	yes	characteristic
501	C₂³	C₂×#11719947	1	1	yes	characteristic
502	D₄	—	2	4	no	—
503	D₄	—	2	4	no	—
504	D₄	—	4	2	no	—
505	D₄	—	4	2	no	—
506	D₄	—	4	2	no	—
507	D₄	—	4	2	no	—
508	D₄	—	4	2	no	—
509	D₄	—	4	2	no	—
510	C₂×C₄	—	4	2	no	—
511	C₂×C₄	—	8	2	no	—
512	C₂×C₄	—	424	2	no	—
513	C₂×C₄	—	212	2	no	—
514	C₂×C₄	#13133189	1	1	yes	characteristic
515	C₂×C₄	#13133189	1	1	yes	characteristic
516	C₂×C₄	#13133189	1	1	yes	characteristic
517	C₂×C₄	#13133189	1	1	yes	characteristic
518	C₂×C₄	#13133198	1	1	yes	characteristic
519	C₂×C₄	#13133198	1	1	yes	characteristic
520	C₂×C₄	#13133199	1	1	yes	characteristic
521	C₂×C₄	#13133199	1	1	yes	characteristic
522	C₂×C₄	#13133199	1	1	yes	characteristic
523	C₂×C₄	#13133199	1	1	yes	characteristic
524	C₂×C₄	#13133200	1	1	yes	characteristic
525	C₂×C₄	#13133200	1	1	yes	characteristic
526	C₂×C₄	#13133200	1	1	yes	characteristic
527	C₂×C₄	#13133200	1	1	yes	characteristic
528	C₂×C₄	#13133203	1	1	yes	characteristic
529	C₂×C₄	#13133203	1	1	yes	characteristic
530	C₂×C₄	#13133210	1	1	yes	characteristic
531	C₂×C₄	#13133210	1	1	yes	characteristic
532	C₂×C₄	#13133210	1	1	yes	characteristic
533	C₂×C₄	#13133210	1	1	yes	characteristic
534	C₂×C₄	#13133210	1	1	yes	characteristic
535	C₂×C₄	#13133210	1	1	yes	characteristic
536	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
537	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
538	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
539	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
540	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
541	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
542	C₈	—	4	2	no	—
543	C₈	—	106	4	no	—
544	C₈	—	212	2	no	—
545	C₈	—	212	2	no	—
546	C₂²	—	2	2	no	—
547	C₂²	—	2	2	no	—
548	C₂²	—	2	2	no	—
549	C₂²	—	4	1	no	—
550	C₂²	—	4	1	no	—
551	C₂²	—	4	1	no	—
552	C₂²	—	4	1	no	—
553	C₂²	—	4	4	no	—
554	C₂²	—	8	4	no	—
555	C₂²	—	8	4	no	—
556	C₂²	—	8	4	no	—
557	C₂²	—	8	4	no	—
558	C₂²	—	8	4	no	—
559	C₂²	—	8	4	no	—
560	C₂²	—	8	4	no	—
561	C₂²	#16810551	1	1	yes	characteristic
562	C₂²	#16810551	1	1	yes	characteristic
563	C₂²	#16810551	1	1	yes	characteristic
564	C₂²	#16810551	1	1	yes	characteristic
565	C₂²	C₂×#13133198	1	1	yes	characteristic
566	C₂²	C₂×#13133203	1	1	yes	characteristic
567	C₂²	C₂×#13133210	1	1	yes	characteristic
568	C₄	—	8	2	no	—
569	C₄	—	424	2	no	—
570	C₄	#16810551	1	1	yes	characteristic
571	C₄	#16810551	1	1	yes	characteristic
572	C₄	#16810551	1	1	yes	characteristic
573	C₄	#16810551	1	1	yes	characteristic
574	C₄	C₂×#13133189	1	1	yes	characteristic
575	C₄	C₂×#13133199	1	1	yes	characteristic
576	C₄	C₂×#13133200	1	1	yes	characteristic
577	C₄	C₂×#13133210	1	1	yes	characteristic
578	C₂	—	2	1	no	—
579	C₂	—	2	1	no	—
580	C₂	—	2	1	no	—
581	C₂	—	2	1	no	—
582	C₂	—	2	1	no	—
583	C₂	—	2	1	no	—
584	C₂	—	8	4	no	—
585	C₂	#20276176	1	2	yes	direct product factor
586	C₂	C₂×#16810551	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810551
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#418	C₂
socle	C₂²×C₅₃	#397
prime radical	C₂³×D₄×C₅₃	C₂²
hypercenter	C₂×#418	D₅₃
hypocenter	C₅₃	C₂×#2081
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#418

Complements

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

N	H ≅ G/N	type
1. C₂×#16810671	C₂	semidirect
4. #20276176	C₂	direct (N × H)
5. C₂×#16810395	C₂	semidirect
6. C₂×#16810393	C₂	semidirect
7. C₂×#16810379	C₂	semidirect
14. #16810671	C₂²	semidirect
22. #16810395	C₂²	semidirect
23. #16810393	C₂²	semidirect
24. #16810379	C₂²	semidirect
300. C₂×C₅₃	#2081	semidirect
324. C₅₃	C₂×#2081	semidirect
585. C₂	#20276176	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×#16810671	2
C₂×#16810628	2
C₂²×#13133134	2
#20276176	2
C₂×#16810395	2
C₂×#16810393	2
C₂×#16810379	2
C₂×C₅₃×#418	2
C₂×#2081	53

Subgroups — C2×#20276176 #27413030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276176 #27413030