Subgroups — C₂×#20272829 #27299026

Order 13568.

570 conjugacy classes of subgroups.

The subgroup lattice has 416 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×#16810661	C₂	1	1	yes	maximal, characteristic
3	#20272829	C₂	1	4	yes	maximal, direct product factor
4	C₂²×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133176	—	2	1	no	—
6	C₂×#13133157	—	2	1	no	—
7	C₂×#13133157	—	2	1	no	—
8	C₂×#13133157	—	2	1	no	—
9	C₂×#13133157	C₄	1	2	yes	complemented
10	C₂×#13133157	C₄	1	2	yes	complemented
11	C₂×#13133157	C₂²	1	2	yes	—
12	C₂²×#11719908	—	2	1	no	—
13	#16810661	—	2	1	no	—
14	#16810661	—	2	1	no	—
15	#16810661	C₄	1	2	yes	—
16	#16810661	C₂²	1	2	yes	—
17	C₂×#13133151	—	2	1	no	—
18	C₂×#13133149	C₄	1	2	yes	—
19	C₂²×M₁₆×C₅₃	—	2	1	no	—
20	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#108×C₅₃	—	2	1	no	—
22	C₂×#108×C₅₃	—	2	1	no	—
23	C₂×#108×C₅₃	—	2	1	no	—
24	C₂×#108×C₅₃	C₄	1	2	yes	complemented
25	C₂×#108×C₅₃	C₄	1	2	yes	complemented
26	C₂×#108×C₅₃	C₂²	1	2	yes	—
27	#13133176	—	2	2	no	—
28	#13133176	—	2	2	no	—
29	#13133176	—	4	1	no	—
30	#13133176	—	4	1	no	—
31	C₂×C₄⋊Dic₅₃	—	2	2	no	—
32	C₂×C₂²⋊Dic₅₃	—	4	1	no	—
33	C₂×C₄×Dic₅₃	—	4	1	no	—
34	#13133157	—	2	2	no	—
35	#13133157	—	2	2	no	—
36	#13133157	—	2	2	no	—
37	#13133157	C₂×C₄	1	4	yes	complemented
38	C₂×#11719908	—	2	2	no	—
39	C₂×#11719908	—	2	2	no	—
40	C₂×#11719908	—	2	2	no	—
41	#13133151	—	2	2	no	—
42	#13133151	—	2	2	no	—
43	#13133149	—	2	2	no	—
44	#13133149	C₂×C₄	1	4	yes	—
45	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
46	C₂×M₁₆×C₅₃	—	2	2	no	—
47	C₂×M₁₆×C₅₃	—	2	2	no	—
48	C₂×M₁₆×C₅₃	—	2	2	no	—
49	C₂×M₁₆×C₅₃	—	4	1	no	—
50	C₂×M₁₆×C₅₃	—	4	1	no	—
51	C₂×M₁₆×C₅₃	—	4	1	no	—
52	C₂²×C₈×C₅₃	—	4	1	no	—
53	C₂²×C₈×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	Q₈	1	1	yes	characteristic
65	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
66	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	#108×C₅₃	—	2	2	no	—
68	#108×C₅₃	—	2	2	no	—
69	#108×C₅₃	—	2	2	no	—
70	#108×C₅₃	C₂×C₄	1	4	yes	complemented
71	C₂⁴×C₅₃	SD₁₆	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×Dic₅₃	—	4	2	no	—
87	M₁₆×C₅₃	—	2	4	no	—
88	M₁₆×C₅₃	—	4	2	no	—
89	M₁₆×C₅₃	—	4	2	no	—
90	M₁₆×C₅₃	—	4	2	no	—
91	C₂×C₈×C₅₃	—	4	2	no	—
92	C₂×C₈×C₅₃	—	4	2	no	—
93	C₂×C₈×C₅₃	—	4	2	no	—
94	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₄²	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	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₈×C₅₃	C₂²⋊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	C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
122	C₂²⋊Dic₅₃	—	4	2	no	—
123	C₂²⋊Dic₅₃	—	4	2	no	—
124	C₄⋊Dic₅₃	—	2	4	no	—
125	C₄⋊Dic₅₃	—	4	2	no	—
126	C₄×Dic₅₃	—	4	2	no	—
127	C₄×Dic₅₃	—	4	2	no	—
128	#11719908	—	2	4	no	—
129	C₂³×C₅₃	—	4	2	no	—
130	C₂³×C₅₃	#102	1	1	yes	characteristic
131	C₂³×C₅₃	#102	1	1	yes	characteristic
132	C₂³×C₅₃	#103	1	1	yes	characteristic
133	C₂³×C₅₃	#103	1	1	yes	characteristic
134	C₂³×C₅₃	#106	1	1	yes	characteristic
135	C₂³×C₅₃	#106	1	1	yes	characteristic
136	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
138	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
139	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
140	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
141	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
142	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
143	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
144	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
145	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
146	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
147	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
148	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
149	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
150	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
151	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
152	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
153	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
154	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
155	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
156	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
157	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
158	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
165	C₂×Dic₅₃	—	4	4	no	—
166	C₂×Dic₅₃	—	8	4	no	—
167	C₈×C₅₃	—	4	4	no	—
168	C₈×C₅₃	#95	1	1	yes	characteristic
169	C₈×C₅₃	#95	1	1	yes	characteristic
170	C₈×C₅₃	#95	1	1	yes	characteristic
171	C₈×C₅₃	#95	1	1	yes	characteristic
172	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
173	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
174	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂×#1389	—	53	1	no	maximal, Sylow
177	C₂²×C₅₃	—	2	2	no	—
178	C₂²×C₅₃	—	2	2	no	—
179	C₂²×C₅₃	—	2	2	no	—
180	C₂²×C₅₃	—	4	1	no	—
181	C₂²×C₅₃	—	4	1	no	—
182	C₂²×C₅₃	—	4	1	no	—
183	C₂²×C₅₃	—	4	1	no	—
184	C₂²×C₅₃	#340	1	1	yes	characteristic
185	C₂²×C₅₃	#340	1	1	yes	characteristic
186	C₂²×C₅₃	#340	1	1	yes	characteristic
187	C₂²×C₅₃	#340	1	1	yes	characteristic
188	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
189	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
190	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
191	C₄×C₅₃	#340	1	1	yes	characteristic
192	C₄×C₅₃	#340	1	1	yes	characteristic
193	C₄×C₅₃	#340	1	1	yes	characteristic
194	C₄×C₅₃	#340	1	1	yes	characteristic
195	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
196	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
197	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
198	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
199	Dic₅₃	—	8	4	no	—
200	C₂²×M₃₂	—	53	1	no	—
201	C₂²×#108	D₅₃	1	1	yes	characteristic
202	C₂×#427	—	53	1	no	—
203	#1389	—	53	4	no	—
204	C₂×C₅₃	—	2	1	no	—
205	C₂×C₅₃	—	2	1	no	—
206	C₂×C₅₃	—	2	1	no	—
207	C₂×C₅₃	—	2	1	no	—
208	C₂×C₅₃	—	2	1	no	—
209	C₂×C₅₃	—	2	1	no	—
210	C₂×C₅₃	#1389	1	2	yes	complemented
211	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
212	C₂²×M₁₆	—	2	1	no	—
213	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
214	C₂×#117	—	106	1	no	—
215	C₂×M₃₂	—	53	2	no	—
216	C₂×M₃₂	—	53	2	no	—
217	C₂×M₃₂	—	53	2	no	—
218	C₂×M₃₂	—	106	1	no	—
219	C₂×M₃₂	—	106	1	no	—
220	C₂×M₃₂	—	106	1	no	—
221	C₂²×C₁₆	—	106	1	no	—
222	C₂×#108	—	2	1	no	—
223	C₂×#108	—	2	1	no	—
224	C₂×#108	—	2	1	no	—
225	C₂×#108	Dic₅₃	1	2	yes	complemented
226	C₂×#108	Dic₅₃	1	2	yes	complemented
227	C₂×#108	C₂×D₅₃	1	2	yes	—
228	#427	—	53	2	no	—
229	#427	—	53	2	no	—
230	#427	—	106	1	no	—
231	#427	—	106	1	no	—
232	C₂×#107	—	106	1	no	—
233	C₂×#106	—	53	2	no	—
234	C₅₃	C₂×#1389	1	1	yes	characteristic, complemented, Sylow
235	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
236	C₂×M₁₆	—	2	2	no	—
237	C₂×M₁₆	—	2	2	no	—
238	C₂×M₁₆	—	2	2	no	—
239	C₂×M₁₆	—	4	1	no	—
240	C₂×M₁₆	—	4	1	no	—
241	C₂×M₁₆	—	4	1	no	—
242	C₂²×C₈	—	4	1	no	—
243	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
244	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
245	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
246	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
247	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
248	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
249	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
250	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
251	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
252	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
253	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
254	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
255	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
256	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
257	#117	—	106	2	no	—
258	#117	—	106	2	no	—
259	#117	—	212	1	no	—
260	#117	—	212	1	no	—
261	C₂×C₄⋊C₄	—	106	2	no	—
262	C₂×C₂²⋊C₄	—	212	1	no	—
263	C₂×C₄²	—	212	1	no	—
264	M₃₂	—	53	4	no	—
265	M₃₂	—	106	2	no	—
266	M₃₂	—	106	2	no	—
267	M₃₂	—	106	2	no	—
268	C₂×C₁₆	—	106	2	no	—
269	C₂×C₁₆	—	106	2	no	—
270	C₂×C₁₆	—	106	2	no	—
271	#108	—	2	2	no	—
272	#108	—	2	2	no	—
273	#108	—	2	2	no	—
274	#108	C₂×Dic₅₃	1	4	yes	complemented
275	#107	—	106	2	no	—
276	#107	—	106	2	no	—
277	#106	—	53	4	no	—
278	#106	—	106	2	no	—
279	C₂⁴	#11719924	1	1	yes	characteristic
280	C₂²×C₄	—	212	2	no	—
281	C₂²×C₄	#11719919	1	1	yes	characteristic
282	C₂²×C₄	#11719919	1	1	yes	characteristic
283	C₂²×C₄	#11719921	1	1	yes	characteristic
284	C₂²×C₄	#11719921	1	1	yes	characteristic
285	C₂²×C₄	#11719922	1	1	yes	characteristic
286	C₂²×C₄	#11719922	1	1	yes	characteristic
287	C₂²×C₄	#11719923	1	1	yes	characteristic
288	C₂²×C₄	#11719923	1	1	yes	characteristic
289	C₂²×C₄	#11719924	1	1	yes	characteristic
290	C₂²×C₄	#11719925	1	1	yes	characteristic
291	C₂²×C₄	#11719925	1	1	yes	characteristic
292	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
293	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
294	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
295	M₁₆	—	2	4	no	—
296	M₁₆	—	4	2	no	—
297	M₁₆	—	4	2	no	—
298	M₁₆	—	4	2	no	—
299	C₂×C₈	—	4	2	no	—
300	C₂×C₈	—	4	2	no	—
301	C₂×C₈	—	4	2	no	—
302	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
303	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
304	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
305	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
306	C₂×C₈	#11719919	1	1	yes	characteristic
307	C₂×C₈	#11719919	1	1	yes	characteristic
308	C₂×C₈	#11719919	1	1	yes	characteristic
309	C₂×C₈	#11719919	1	1	yes	characteristic
310	C₂×C₈	#11719919	1	1	yes	characteristic
311	C₂×C₈	#11719919	1	1	yes	characteristic
312	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
313	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
314	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
315	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
316	C₂×C₈	#11719921	1	1	yes	characteristic
317	C₂×C₈	#11719921	1	1	yes	characteristic
318	C₂×C₈	#11719921	1	1	yes	characteristic
319	C₂×C₈	#11719921	1	1	yes	characteristic
320	C₂×C₈	#11719921	1	1	yes	characteristic
321	C₂×C₈	#11719921	1	1	yes	characteristic
322	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
323	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
324	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
325	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
326	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
327	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
328	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
329	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
330	C₄⋊C₄	—	106	4	no	—
331	C₄⋊C₄	—	212	2	no	—
332	C₂²⋊C₄	—	212	2	no	—
333	C₂²⋊C₄	—	212	2	no	—
334	C₄²	—	212	2	no	—
335	C₄²	—	212	2	no	—
336	C₁₆	—	106	4	no	—
337	C₂³	—	4	2	no	—
338	C₂³	#13133137	1	1	yes	characteristic
339	C₂³	#13133137	1	1	yes	characteristic
340	C₂³	#13133141	1	1	yes	characteristic
341	C₂³	#13133141	1	1	yes	characteristic
342	C₂³	#13133148	1	1	yes	characteristic
343	C₂³	#13133148	1	1	yes	characteristic
344	C₂³	C₂×#11719924	1	1	yes	characteristic
345	C₂×C₄	—	424	4	no	—
346	C₂×C₄	—	212	4	no	—
347	C₂×C₄	#13133124	1	1	yes	characteristic
348	C₂×C₄	#13133124	1	1	yes	characteristic
349	C₂×C₄	#13133124	1	1	yes	characteristic
350	C₂×C₄	#13133124	1	1	yes	characteristic
351	C₂×C₄	#13133137	1	1	yes	characteristic
352	C₂×C₄	#13133137	1	1	yes	characteristic
353	C₂×C₄	#13133138	1	1	yes	characteristic
354	C₂×C₄	#13133138	1	1	yes	characteristic
355	C₂×C₄	#13133138	1	1	yes	characteristic
356	C₂×C₄	#13133138	1	1	yes	characteristic
357	C₂×C₄	#13133140	1	1	yes	characteristic
358	C₂×C₄	#13133140	1	1	yes	characteristic
359	C₂×C₄	#13133140	1	1	yes	characteristic
360	C₂×C₄	#13133140	1	1	yes	characteristic
361	C₂×C₄	#13133141	1	1	yes	characteristic
362	C₂×C₄	#13133141	1	1	yes	characteristic
363	C₂×C₄	#13133148	1	1	yes	characteristic
364	C₂×C₄	#13133148	1	1	yes	characteristic
365	C₂×C₄	#13133150	1	1	yes	characteristic
366	C₂×C₄	#13133150	1	1	yes	characteristic
367	C₂×C₄	#13133150	1	1	yes	characteristic
368	C₂×C₄	#13133150	1	1	yes	characteristic
369	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
370	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
371	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
372	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
373	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
374	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
375	C₈	—	4	4	no	—
376	C₈	#13133124	1	1	yes	characteristic
377	C₈	#13133124	1	1	yes	characteristic
378	C₈	#13133124	1	1	yes	characteristic
379	C₈	#13133124	1	1	yes	characteristic
380	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
381	C₈	C₂×#11719921	1	1	yes	characteristic
382	C₈	C₂×#11719919	1	1	yes	characteristic
383	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
384	C₂²	—	2	2	no	—
385	C₂²	—	2	2	no	—
386	C₂²	—	2	2	no	—
387	C₂²	—	4	1	no	—
388	C₂²	—	4	1	no	—
389	C₂²	—	4	1	no	—
390	C₂²	—	4	1	no	—
391	C₂²	#16810385	1	1	yes	characteristic
392	C₂²	#16810385	1	1	yes	characteristic
393	C₂²	#16810385	1	1	yes	characteristic
394	C₂²	#16810385	1	1	yes	characteristic
395	C₂²	C₂×#13133137	1	1	yes	characteristic
396	C₂²	C₂×#13133141	1	1	yes	characteristic
397	C₂²	C₂×#13133148	1	1	yes	characteristic
398	C₄	—	424	4	no	—
399	C₄	#16810385	1	1	yes	characteristic
400	C₄	#16810385	1	1	yes	characteristic
401	C₄	#16810385	1	1	yes	characteristic
402	C₄	#16810385	1	1	yes	characteristic
403	C₄	C₂×#13133124	1	1	yes	characteristic
404	C₄	C₂×#13133138	1	1	yes	characteristic
405	C₄	C₂×#13133140	1	1	yes	characteristic
406	C₄	C₂×#13133150	1	1	yes	characteristic
407	C₂	—	2	1	no	—
408	C₂	—	2	1	no	—
409	C₂	—	2	1	no	—
410	C₂	—	2	1	no	—
411	C₂	—	2	1	no	—
412	C₂	—	2	1	no	—
413	C₂	#20272829	1	2	yes	direct product factor
414	C₂	C₂×#16810385	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133138
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂²×#108×C₅₃	C₂
socle	C₂²×C₅₃	#340
prime radical	C₂⁴×C₅₃	SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×#108	D₅₃
hypocenter	C₅₃	C₂×#1389
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
3. #20272829	C₂	direct (N × H)
9. C₂×#13133157	C₄	semidirect
10. C₂×#13133157	C₄	semidirect
24. C₂×#108×C₅₃	C₄	semidirect
25. C₂×#108×C₅₃	C₄	semidirect
37. #13133157	C₂×C₄	semidirect
70. #108×C₅₃	C₂×C₄	semidirect
210. C₂×C₅₃	#1389	semidirect
225. C₂×#108	Dic₅₃	semidirect
226. C₂×#108	Dic₅₃	semidirect
234. C₅₃	C₂×#1389	semidirect
274. #108	C₂×Dic₅₃	semidirect
413. C₂	#20272829	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×#16810661	2
#20272829	2
C₂²×#108×C₅₃	2
C₂×#1389	53

Subgroups — C2×#20272829 #27299026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272829 #27299026