Subgroups — C₂×#20276112 #27412808

Order 13568.

1078 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810672	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810671	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#13133133	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13133133	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13133133	C₂	1	1	yes	maximal, characteristic
6	#20276112	C₂	1	8	yes	maximal, direct product factor
7	C₂×#16810381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#430	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic
10	C₂²×#11719917	C₂²	1	1	yes	characteristic
11	C₂²×#11719917	C₂²	1	1	yes	characteristic
12	C₂²×#11719917	C₂²	1	1	yes	characteristic
13	#16810672	—	2	2	no	—
14	#16810672	C₂²	1	4	yes	complemented
15	#16810671	—	2	2	no	—
16	#16810671	C₂²	1	4	yes	complemented
17	C₂×#13133153	—	2	2	no	—
18	C₂×#13133152	—	2	2	no	—
19	C₂×#13133133	—	2	2	no	—
20	C₂×#13133133	—	2	2	no	—
21	C₂×#13133133	—	2	2	no	—
22	C₂×#13133133	—	2	2	no	—
23	C₂×#13133133	—	2	2	no	—
24	C₂×#13133133	—	2	2	no	—
25	C₂×#13133133	C₂²	1	4	yes	—
26	C₂×#13133133	C₂²	1	4	yes	—
27	C₂×#13133133	C₂²	1	4	yes	—
28	#16810381	C₂²	1	4	yes	complemented
29	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
30	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
31	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#430	—	2	2	no	—
33	C₅₃×#430	C₂²	1	4	yes	complemented
34	C₂×#108×C₅₃	—	2	2	no	—
35	C₂⁴×D₅₃	Q₈	1	2	yes	—
36	C₂²×D₂₁₂	—	2	1	no	—
37	C₂²×D₂₁₂	—	2	1	no	—
38	C₂²×D₂₁₂	—	2	1	no	—
39	C₂²×D₂₁₂	—	2	1	no	—
40	C₂²×D₂₁₂	—	2	1	no	—
41	C₂²×D₂₁₂	—	2	1	no	—
42	C₂²×D₂₁₂	—	2	1	no	—
43	C₂²×D₂₁₂	—	2	1	no	—
44	C₂²×D₂₁₂	—	2	1	no	—
45	C₂²×D₂₁₂	—	2	1	no	—
46	C₂²×D₂₁₂	—	2	1	no	—
47	C₂²×D₂₁₂	—	2	1	no	—
48	C₂²×D₂₁₂	Q₈	1	2	yes	—
49	C₂²×D₂₁₂	C₂³	1	2	yes	—
50	C₂×#11719917	—	2	1	no	—
51	C₂×#11719917	—	2	1	no	—
52	C₂×#11719917	—	2	1	no	—
53	C₂×#11719917	—	2	1	no	—
54	C₂×#11719917	—	2	1	no	—
55	C₂×#11719917	—	2	1	no	—
56	C₂×#11719917	—	2	1	no	—
57	C₂×#11719917	—	2	1	no	—
58	C₂×#11719917	—	2	1	no	—
59	C₂×#11719917	—	2	1	no	—
60	C₂×#11719917	—	2	1	no	—
61	C₂×#11719917	—	2	1	no	—
62	C₂×#11719917	—	2	1	no	—
63	C₂×#11719917	D₄	1	2	yes	—
64	C₂×#11719917	D₄	1	2	yes	—
65	C₂×#11719917	C₂³	1	2	yes	—
66	C₂×#11719917	C₂³	1	2	yes	—
67	C₂×#11719917	C₂³	1	2	yes	—
68	C₂²×#88322	—	2	1	no	—
69	C₂²×#88322	—	2	1	no	—
70	C₂²×#88322	—	2	1	no	—
71	#13133153	—	2	8	no	—
72	#13133152	—	2	8	no	—
73	#13133133	—	2	8	no	—
74	#13133133	—	2	8	no	—
75	#13133133	—	2	8	no	—
76	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
77	C₂×M₁₆×C₅₃	—	2	1	no	—
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	—	2	1	no	—
84	C₂×M₁₆×C₅₃	—	2	1	no	—
85	C₂×M₁₆×C₅₃	—	2	1	no	—
86	C₂×M₁₆×C₅₃	—	2	1	no	—
87	C₂×M₁₆×C₅₃	—	2	1	no	—
88	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
89	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
90	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
91	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
92	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
93	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
94	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
95	C₂²×C₈×C₅₃	—	2	1	no	—
96	C₂²×C₈×C₅₃	—	2	1	no	—
97	C₂²×C₈×C₅₃	—	2	1	no	—
98	#108×C₅₃	—	2	8	no	—
99	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
100	C₂³×D₅₃	—	2	2	no	—
101	C₂³×D₅₃	—	2	2	no	—
102	C₂³×D₅₃	—	2	2	no	—
103	C₂³×D₅₃	—	2	2	no	—
104	C₂³×D₅₃	—	2	2	no	—
105	C₂³×D₅₃	—	2	2	no	—
106	C₂³×D₅₃	—	8	4	no	—
107	C₂³×D₅₃	C₂×Q₈	1	4	yes	—
108	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
115	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
116	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
117	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
118	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
119	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
120	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
121	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
122	C₂×D₂₁₂	—	2	2	no	—
123	C₂×D₂₁₂	—	2	2	no	—
124	C₂×D₂₁₂	—	2	2	no	—
125	C₂×D₂₁₂	—	2	2	no	—
126	C₂×D₂₁₂	—	2	2	no	—
127	C₂×D₂₁₂	—	2	2	no	—
128	C₂×D₂₁₂	—	2	2	no	—
129	C₂×D₂₁₂	—	2	2	no	—
130	C₂×D₂₁₂	—	2	2	no	—
131	C₂×D₂₁₂	—	2	2	no	—
132	C₂×D₂₁₂	—	2	2	no	—
133	C₂×D₂₁₂	—	2	2	no	—
134	C₂×D₂₁₂	—	4	1	no	—
135	C₂×D₂₁₂	—	4	1	no	—
136	C₂×D₂₁₂	—	4	1	no	—
137	C₂×D₂₁₂	—	4	1	no	—
138	C₂×D₂₁₂	—	4	1	no	—
139	C₂×D₂₁₂	—	4	1	no	—
140	C₂×D₂₁₂	—	4	1	no	—
141	C₂×D₂₁₂	—	4	1	no	—
142	C₂×D₂₁₂	—	4	1	no	—
143	C₂×D₂₁₂	—	4	1	no	—
144	C₂×D₂₁₂	—	4	1	no	—
145	C₂×D₂₁₂	—	4	1	no	—
146	C₂×D₂₁₂	—	4	1	no	—
147	C₂×D₂₁₂	—	4	1	no	—
148	C₂×D₂₁₂	—	4	1	no	—
149	C₂×D₂₁₂	—	4	1	no	—
150	M₁₆×C₅₃	—	2	2	no	—
151	M₁₆×C₅₃	—	2	2	no	—
152	M₁₆×C₅₃	—	2	2	no	—
153	M₁₆×C₅₃	—	2	2	no	—
154	M₁₆×C₅₃	—	2	2	no	—
155	M₁₆×C₅₃	—	2	2	no	—
156	M₁₆×C₅₃	—	2	2	no	—
157	M₁₆×C₅₃	—	2	2	no	—
158	M₁₆×C₅₃	—	4	1	no	—
159	M₁₆×C₅₃	—	4	1	no	—
160	M₁₆×C₅₃	C₂×D₄	1	4	yes	—
161	M₁₆×C₅₃	C₂×D₄	1	4	yes	—
162	C₂×C₈×C₅₃	—	2	2	no	—
163	C₂×C₈×C₅₃	—	2	2	no	—
164	C₂×C₈×C₅₃	—	2	2	no	—
165	C₂×C₈×C₅₃	—	2	2	no	—
166	C₂×C₈×C₅₃	—	2	2	no	—
167	C₂×C₈×C₅₃	—	2	2	no	—
168	C₂×C₈×C₅₃	—	2	2	no	—
169	C₂×C₈×C₅₃	—	4	1	no	—
170	C₂×C₈×C₅₃	—	4	1	no	—
171	#11719917	—	2	2	no	—
172	#11719917	—	2	2	no	—
173	#11719917	—	2	2	no	—
174	#11719917	—	2	2	no	—
175	#11719917	—	2	2	no	—
176	#11719917	—	2	2	no	—
177	#11719917	—	2	2	no	—
178	#11719917	—	4	1	no	—
179	#11719917	—	4	1	no	—
180	#11719917	—	4	1	no	—
181	#11719917	—	4	1	no	—
182	#11719917	C₂×D₄	1	4	yes	—
183	C₂×#88322	—	2	2	no	—
184	C₂×#88322	—	2	2	no	—
185	C₂×#88322	—	2	2	no	—
186	C₂×#88322	—	2	2	no	—
187	C₂×#88322	—	2	2	no	—
188	C₂×#88322	—	4	1	no	—
189	C₂×#88322	—	4	1	no	—
190	C₂×#88322	—	4	1	no	—
191	C₂×#88322	—	4	1	no	—
192	C₂³×C₅₃	—	4	2	no	—
193	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
194	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
195	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
196	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
197	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
198	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
199	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
200	C₂²×D₅₃	—	2	4	no	—
201	C₂²×D₅₃	—	2	4	no	—
202	C₂²×D₅₃	—	2	4	no	—
203	C₂²×D₅₃	—	4	2	no	—
204	C₂²×D₅₃	—	4	2	no	—
205	C₂²×D₅₃	—	4	2	no	—
206	C₂²×D₅₃	—	4	2	no	—
207	C₂²×D₅₃	—	8	4	no	—
208	C₂²×D₅₃	—	8	4	no	—
209	C₂²×D₅₃	—	8	4	no	—
210	C₂²×D₅₃	—	8	4	no	—
211	C₂²×D₅₃	—	8	4	no	—
212	C₂²×D₅₃	—	8	4	no	—
213	C₂²×D₅₃	—	8	4	no	—
214	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
215	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
216	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
217	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
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	#121	1	1	yes	characteristic
223	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
224	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
225	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
226	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
227	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
228	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
229	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
230	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
231	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
232	C₂×C₄×C₅₃	#126	1	1	yes	characteristic
233	C₂×C₄×C₅₃	#126	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	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
239	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
240	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
241	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
242	D₂₁₂	—	4	2	no	—
243	D₂₁₂	—	4	2	no	—
244	D₂₁₂	—	4	2	no	—
245	D₂₁₂	—	4	2	no	—
246	D₂₁₂	—	4	2	no	—
247	D₂₁₂	—	4	2	no	—
248	D₂₁₂	—	4	2	no	—
249	D₂₁₂	—	4	2	no	—
250	C₈×C₅₃	—	2	4	no	—
251	C₈×C₅₃	—	2	4	no	—
252	C₈×C₅₃	—	4	2	no	—
253	#88322	—	2	4	no	—
254	#88322	—	4	2	no	—
255	#88322	—	4	2	no	—
256	C₂×#2069	—	53	1	no	maximal, Sylow
257	C₂²×C₅₃	—	2	2	no	—
258	C₂²×C₅₃	—	2	2	no	—
259	C₂²×C₅₃	—	2	2	no	—
260	C₂²×C₅₃	—	4	1	no	—
261	C₂²×C₅₃	—	4	1	no	—
262	C₂²×C₅₃	—	4	1	no	—
263	C₂²×C₅₃	—	4	1	no	—
264	C₂²×C₅₃	#396	1	1	yes	characteristic
265	C₂²×C₅₃	#396	1	1	yes	characteristic
266	C₂²×C₅₃	#396	1	1	yes	characteristic
267	C₂²×C₅₃	#396	1	1	yes	characteristic
268	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
269	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
270	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
271	C₂×D₅₃	—	4	4	no	—
272	C₂×D₅₃	—	8	4	no	—
273	C₂×D₅₃	—	8	4	no	—
274	C₂×D₅₃	—	8	4	no	—
275	C₂×D₅₃	—	8	4	no	—
276	C₂×D₅₃	—	8	4	no	—
277	C₂×D₅₃	—	8	4	no	—
278	C₂×D₅₃	—	8	4	no	—
279	C₄×C₅₃	#396	1	1	yes	characteristic
280	C₄×C₅₃	#396	1	1	yes	characteristic
281	C₄×C₅₃	#396	1	1	yes	characteristic
282	C₄×C₅₃	#396	1	1	yes	characteristic
283	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
284	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
285	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
286	C₄×C₅₃	C₂×#126	1	1	yes	characteristic
287	C₂×#430	—	53	1	no	—
288	C₂×#430	—	53	1	no	—
289	C₂×#430	D₅₃	1	1	yes	characteristic, complemented
290	C₂²×#100	—	53	1	no	—
291	C₂²×#100	—	53	1	no	—
292	C₂²×#100	—	53	1	no	—
293	#2069	—	53	8	no	—
294	C₂×#338	—	53	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	—	2	1	no	—
300	C₂×C₅₃	—	2	1	no	—
301	C₂×C₅₃	#2069	1	2	yes	complemented
302	C₂×C₅₃	C₂×#396	1	1	yes	characteristic
303	D₅₃	—	8	4	no	—
304	C₂³×D₄	—	53	1	no	—
305	C₂²×M₁₆	—	53	1	no	—
306	C₂²×M₁₆	—	53	1	no	—
307	C₂²×M₁₆	—	53	1	no	—
308	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
309	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
310	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
311	#430	—	2	2	no	—
312	#430	—	53	4	no	—
313	#430	—	53	4	no	—
314	#430	—	106	2	no	—
315	#430	—	106	2	no	—
316	#430	C₂×D₅₃	1	4	yes	complemented
317	C₂×#108	—	2	2	no	—
318	C₂×#108	—	106	2	no	—
319	C₂×#108	—	106	2	no	—
320	C₂×#100	—	53	4	no	—
321	C₂×#100	—	53	4	no	—
322	C₂×#100	—	53	4	no	—
323	C₂×#100	—	106	2	no	—
324	C₂×#100	—	106	2	no	—
325	C₂×#100	—	106	2	no	—
326	C₂×#100	—	106	2	no	—
327	C₂×#100	—	106	2	no	—
328	C₂×#100	—	106	2	no	—
329	#338	—	53	4	no	—
330	C₅₃	C₂×#2069	1	1	yes	characteristic, complemented, Sylow
331	C₂⁵	—	53	2	no	—
332	C₂²×D₄	—	53	2	no	—
333	C₂²×D₄	—	53	2	no	—
334	C₂²×D₄	—	106	1	no	—
335	C₂²×D₄	—	106	1	no	—
336	C₂²×D₄	—	106	1	no	—
337	C₂²×D₄	—	106	1	no	—
338	C₂²×D₄	—	106	1	no	—
339	C₂²×D₄	—	106	1	no	—
340	C₂²×D₄	—	106	1	no	—
341	C₂²×D₄	—	106	1	no	—
342	C₂²×D₄	—	106	1	no	—
343	C₂²×D₄	—	106	1	no	—
344	C₂²×D₄	—	106	1	no	—
345	C₂²×D₄	—	106	1	no	—
346	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
347	C₂×M₁₆	—	2	1	no	—
348	C₂×M₁₆	—	2	1	no	—
349	C₂×M₁₆	—	2	1	no	—
350	C₂×M₁₆	—	2	1	no	—
351	C₂×M₁₆	—	2	1	no	—
352	C₂×M₁₆	—	2	1	no	—
353	C₂×M₁₆	—	2	1	no	—
354	C₂×M₁₆	—	2	1	no	—
355	C₂×M₁₆	—	2	1	no	—
356	C₂×M₁₆	—	2	1	no	—
357	C₂×M₁₆	—	2	1	no	—
358	C₂×M₁₆	—	53	2	no	—
359	C₂×M₁₆	—	53	2	no	—
360	C₂×M₁₆	—	53	2	no	—
361	C₂×M₁₆	—	53	2	no	—
362	C₂×M₁₆	—	53	2	no	—
363	C₂×M₁₆	—	106	1	no	—
364	C₂×M₁₆	—	106	1	no	—
365	C₂×M₁₆	—	106	1	no	—
366	C₂×M₁₆	—	106	1	no	—
367	C₂×M₁₆	—	106	1	no	—
368	C₂×M₁₆	—	106	1	no	—
369	C₂×M₁₆	—	106	1	no	—
370	C₂×M₁₆	—	106	1	no	—
371	C₂×M₁₆	—	106	1	no	—
372	C₂×M₁₆	—	106	1	no	—
373	C₂×M₁₆	—	106	1	no	—
374	C₂×M₁₆	—	106	1	no	—
375	C₂×M₁₆	—	106	1	no	—
376	C₂×M₁₆	D₂₁₂	1	2	yes	—
377	C₂×M₁₆	D₂₁₂	1	2	yes	—
378	C₂×M₁₆	C₂²⋊D₅₃	1	2	yes	—
379	C₂×M₁₆	C₂²⋊D₅₃	1	2	yes	—
380	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
381	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
382	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
383	C₂²×C₈	—	2	1	no	—
384	C₂²×C₈	—	2	1	no	—
385	C₂²×C₈	—	2	1	no	—
386	C₂²×C₈	—	106	1	no	—
387	C₂²×C₈	—	106	1	no	—
388	C₂²×C₈	—	106	1	no	—
389	#108	—	2	8	no	—
390	#108	—	106	8	no	—
391	#108	—	106	8	no	—
392	#100	—	106	8	no	—
393	#100	—	106	8	no	—
394	#100	—	106	8	no	—
395	C₂⁴	—	53	4	no	—
396	C₂⁴	—	106	2	no	—
397	C₂⁴	—	106	2	no	—
398	C₂⁴	—	106	2	no	—
399	C₂⁴	—	106	2	no	—
400	C₂⁴	—	106	2	no	—
401	C₂⁴	—	106	2	no	—
402	C₂⁴	—	424	4	no	—
403	C₂⁴	Q₈×D₅₃	1	1	yes	characteristic
404	C₂×D₄	—	106	2	no	—
405	C₂×D₄	—	106	2	no	—
406	C₂×D₄	—	106	2	no	—
407	C₂×D₄	—	106	2	no	—
408	C₂×D₄	—	106	2	no	—
409	C₂×D₄	—	106	2	no	—
410	C₂×D₄	—	106	2	no	—
411	C₂×D₄	—	106	2	no	—
412	C₂×D₄	—	106	2	no	—
413	C₂×D₄	—	106	2	no	—
414	C₂×D₄	—	106	2	no	—
415	C₂×D₄	—	106	2	no	—
416	C₂×D₄	—	212	1	no	—
417	C₂×D₄	—	212	1	no	—
418	C₂×D₄	—	212	1	no	—
419	C₂×D₄	—	212	1	no	—
420	C₂×D₄	—	212	1	no	—
421	C₂×D₄	—	212	1	no	—
422	C₂×D₄	—	212	1	no	—
423	C₂×D₄	—	212	1	no	—
424	C₂×D₄	—	212	1	no	—
425	C₂×D₄	—	212	1	no	—
426	C₂×D₄	—	212	1	no	—
427	C₂×D₄	—	212	1	no	—
428	C₂×D₄	—	212	1	no	—
429	C₂×D₄	—	212	1	no	—
430	C₂×D₄	—	212	1	no	—
431	C₂×D₄	—	212	1	no	—
432	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
433	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
434	C₂²×C₄	#11719945	1	1	yes	characteristic
435	C₂²×C₄	#11719945	1	1	yes	characteristic
436	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
437	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
438	C₂²×C₄	#11719947	1	1	yes	characteristic
439	C₂²×C₄	#11719947	1	1	yes	characteristic
440	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
441	C₂²×C₄	#11719949	1	1	yes	characteristic
442	C₂²×C₄	#11719949	1	1	yes	characteristic
443	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
444	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
445	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
446	M₁₆	—	2	2	no	—
447	M₁₆	—	2	2	no	—
448	M₁₆	—	2	2	no	—
449	M₁₆	—	2	2	no	—
450	M₁₆	—	2	2	no	—
451	M₁₆	—	2	2	no	—
452	M₁₆	—	2	2	no	—
453	M₁₆	—	2	2	no	—
454	M₁₆	—	4	1	no	—
455	M₁₆	—	4	1	no	—
456	M₁₆	—	53	4	no	—
457	M₁₆	—	106	2	no	—
458	M₁₆	—	106	2	no	—
459	M₁₆	—	106	2	no	—
460	M₁₆	—	106	2	no	—
461	M₁₆	—	106	2	no	—
462	M₁₆	—	106	2	no	—
463	M₁₆	—	106	2	no	—
464	M₁₆	—	212	1	no	—
465	M₁₆	—	212	1	no	—
466	M₁₆	—	212	1	no	—
467	M₁₆	—	212	1	no	—
468	M₁₆	C₂×D₂₁₂	1	4	yes	—
469	M₁₆	C₂×C₂²⋊D₅₃	1	4	yes	—
470	C₂×C₈	—	2	2	no	—
471	C₂×C₈	—	2	2	no	—
472	C₂×C₈	—	2	2	no	—
473	C₂×C₈	—	2	2	no	—
474	C₂×C₈	—	2	2	no	—
475	C₂×C₈	—	2	2	no	—
476	C₂×C₈	—	2	2	no	—
477	C₂×C₈	—	4	1	no	—
478	C₂×C₈	—	4	1	no	—
479	C₂×C₈	—	106	2	no	—
480	C₂×C₈	—	106	2	no	—
481	C₂×C₈	—	106	2	no	—
482	C₂×C₈	—	106	2	no	—
483	C₂×C₈	—	106	2	no	—
484	C₂×C₈	—	212	1	no	—
485	C₂×C₈	—	212	1	no	—
486	C₂×C₈	—	212	1	no	—
487	C₂×C₈	—	212	1	no	—
488	C₂³	—	4	2	no	—
489	C₂³	—	106	4	no	—
490	C₂³	—	106	4	no	—
491	C₂³	—	106	4	no	—
492	C₂³	—	424	4	no	—
493	C₂³	—	424	4	no	—
494	C₂³	—	424	4	no	—
495	C₂³	—	424	4	no	—
496	C₂³	—	424	4	no	—
497	C₂³	—	424	4	no	—
498	C₂³	—	424	4	no	—
499	C₂³	—	212	2	no	—
500	C₂³	—	212	2	no	—
501	C₂³	—	212	2	no	—
502	C₂³	—	212	2	no	—
503	C₂³	#13133193	1	1	yes	characteristic
504	C₂³	#13133193	1	1	yes	characteristic
505	C₂³	#13133195	1	1	yes	characteristic
506	C₂³	#13133195	1	1	yes	characteristic
507	C₂³	#13133197	1	1	yes	characteristic
508	C₂³	#13133197	1	1	yes	characteristic
509	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
510	D₄	—	212	2	no	—
511	D₄	—	212	2	no	—
512	D₄	—	212	2	no	—
513	D₄	—	212	2	no	—
514	D₄	—	212	2	no	—
515	D₄	—	212	2	no	—
516	D₄	—	212	2	no	—
517	D₄	—	212	2	no	—
518	C₂×C₄	#13133189	1	1	yes	characteristic
519	C₂×C₄	#13133189	1	1	yes	characteristic
520	C₂×C₄	#13133189	1	1	yes	characteristic
521	C₂×C₄	#13133189	1	1	yes	characteristic
522	C₂×C₄	#13133191	1	1	yes	characteristic
523	C₂×C₄	#13133191	1	1	yes	characteristic
524	C₂×C₄	#13133191	1	1	yes	characteristic
525	C₂×C₄	#13133191	1	1	yes	characteristic
526	C₂×C₄	#13133192	1	1	yes	characteristic
527	C₂×C₄	#13133192	1	1	yes	characteristic
528	C₂×C₄	#13133192	1	1	yes	characteristic
529	C₂×C₄	#13133192	1	1	yes	characteristic
530	C₂×C₄	#13133193	1	1	yes	characteristic
531	C₂×C₄	#13133193	1	1	yes	characteristic
532	C₂×C₄	#13133195	1	1	yes	characteristic
533	C₂×C₄	#13133195	1	1	yes	characteristic
534	C₂×C₄	#13133197	1	1	yes	characteristic
535	C₂×C₄	#13133197	1	1	yes	characteristic
536	C₂×C₄	#13133212	1	1	yes	characteristic
537	C₂×C₄	#13133212	1	1	yes	characteristic
538	C₂×C₄	#13133212	1	1	yes	characteristic
539	C₂×C₄	#13133212	1	1	yes	characteristic
540	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
541	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
542	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
543	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
544	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
545	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
546	C₈	—	2	4	no	—
547	C₈	—	2	4	no	—
548	C₈	—	4	2	no	—
549	C₈	—	106	4	no	—
550	C₈	—	212	2	no	—
551	C₈	—	212	2	no	—
552	C₂²	—	2	2	no	—
553	C₂²	—	2	2	no	—
554	C₂²	—	2	2	no	—
555	C₂²	—	4	1	no	—
556	C₂²	—	4	1	no	—
557	C₂²	—	4	1	no	—
558	C₂²	—	4	1	no	—
559	C₂²	—	424	4	no	—
560	C₂²	—	424	4	no	—
561	C₂²	—	424	4	no	—
562	C₂²	—	424	4	no	—
563	C₂²	—	424	4	no	—
564	C₂²	—	424	4	no	—
565	C₂²	—	424	4	no	—
566	C₂²	—	212	4	no	—
567	C₂²	#16810545	1	1	yes	characteristic
568	C₂²	#16810545	1	1	yes	characteristic
569	C₂²	#16810545	1	1	yes	characteristic
570	C₂²	#16810545	1	1	yes	characteristic
571	C₂²	C₂×#13133193	1	1	yes	characteristic
572	C₂²	C₂×#13133195	1	1	yes	characteristic
573	C₂²	C₂×#13133197	1	1	yes	characteristic
574	C₄	#16810545	1	1	yes	characteristic
575	C₄	#16810545	1	1	yes	characteristic
576	C₄	#16810545	1	1	yes	characteristic
577	C₄	#16810545	1	1	yes	characteristic
578	C₄	C₂×#13133189	1	1	yes	characteristic
579	C₄	C₂×#13133191	1	1	yes	characteristic
580	C₄	C₂×#13133192	1	1	yes	characteristic
581	C₄	C₂×#13133212	1	1	yes	characteristic
582	C₂	—	2	1	no	—
583	C₂	—	2	1	no	—
584	C₂	—	2	1	no	—
585	C₂	—	2	1	no	—
586	C₂	—	2	1	no	—
587	C₂	—	2	1	no	—
588	C₂	—	424	4	no	—
589	C₂	#20276112	1	2	yes	direct product factor
590	C₂	C₂×#16810545	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810545
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#430	C₂
socle	C₂²×C₅₃	#396
prime radical	C₂³×D₂₁₂	C₂²
hypercenter	C₂×#430	D₅₃
hypocenter	C₅₃	C₂×#2069
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#430

Complements

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

N	H ≅ G/N	type
1. C₂×#16810672	C₂	semidirect
2. C₂×#16810671	C₂	semidirect
6. #20276112	C₂	direct (N × H)
7. C₂×#16810381	C₂	semidirect
8. C₂×C₅₃×#430	C₂	semidirect
14. #16810672	C₂²	semidirect
16. #16810671	C₂²	semidirect
28. #16810381	C₂²	semidirect
33. C₅₃×#430	C₂²	semidirect
289. C₂×#430	D₅₃	semidirect
301. C₂×C₅₃	#2069	semidirect
316. #430	C₂×D₅₃	semidirect
330. C₅₃	C₂×#2069	semidirect
589. C₂	#20276112	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×#16810672	2
C₂×#16810671	2
C₂²×#13133133	2
#20276112	2
C₂×#16810381	2
C₂×C₅₃×#430	2
C₂×#2069	53

Subgroups — C2×#20276112 #27412808

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276112 #27412808