Subgroups — #27273969

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
2	#20277122	C₂	1	1	yes	maximal, characteristic
3	#20272582	C₂	1	4	yes	maximal
4	C₅₃×#2285	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133176	—	2	1	no	—
6	#16810676	—	2	1	no	—
7	#16810676	—	2	1	no	—
8	#16810676	—	2	1	no	—
9	#16810676	—	2	1	no	—
10	C₂×C₄⋊#88322	C₄	1	2	yes	complemented
11	C₂×C₂²⋊#88322	C₂²	1	2	yes	—
12	C₂×C₄×#88322	C₄	1	2	yes	complemented
13	#16810493	—	2	1	no	—
14	#16810489	C₄	1	2	yes	complemented
15	#16810483	—	2	1	no	—
16	#16810483	C₂²	1	2	yes	—
17	C₄×C₄⋊Dic₅₃	—	2	1	no	—
18	C₄×C₄⋊Dic₅₃	C₄	1	2	yes	complemented
19	C₂×#117×C₅₃	—	2	1	no	—
20	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#384	—	2	1	no	—
22	C₅₃×#382	C₄	1	2	yes	complemented
23	C₅₃×#380	—	2	1	no	—
24	C₅₃×#380	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
26	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	complemented
27	C₂²×#88322	C₂×C₄	1	4	yes	—
28	#13133176	—	4	2	no	—
29	#13133176	—	4	2	no	—
30	C₂×C₄⋊Dic₅₃	—	2	2	no	—
31	C₂×C₄⋊Dic₅₃	—	2	2	no	—
32	C₂×C₄⋊Dic₅₃	—	2	2	no	—
33	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
34	C₂×C₄×Dic₅₃	—	2	2	no	—
35	C₄⋊#88322	—	2	2	no	—
36	C₄⋊#88322	—	2	2	no	—
37	C₂²⋊#88322	—	2	2	no	—
38	C₂²⋊#88322	—	2	2	no	—
39	C₄×#88322	—	2	2	no	—
40	C₄×#88322	—	2	2	no	—
41	#13133124	—	2	2	no	—
42	#13133124	C₂×C₄	1	4	yes	—
43	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
46	#117×C₅₃	—	4	2	no	—
47	#117×C₅₃	—	4	2	no	—
48	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
50	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
52	C₂×C₄²×C₅₃	—	2	2	no	—
53	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
65	#95×C₅₃	—	2	2	no	—
66	#95×C₅₃	C₂×C₄	1	4	yes	—
67	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	—	2	4	no	—
69	C₂²×C₄×C₅₃	—	4	2	no	—
70	C₂²×C₄×C₅₃	—	4	2	no	—
71	C₂²×C₄×C₅₃	C₂²⋊C₄	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	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₄⋊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	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	D₈	1	2	yes	—
84	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
85	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
86	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
87	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
93	C₂²×Dic₅₃	—	2	4	no	—
94	C₄⋊C₄×C₅₃	—	4	4	no	—
95	C₄⋊C₄×C₅₃	—	4	4	no	—
96	C₄⋊C₄×C₅₃	—	4	4	no	—
97	C₂²⋊C₄×C₅₃	—	4	2	no	—
98	C₂²⋊C₄×C₅₃	—	4	2	no	—
99	C₄²×C₅₃	—	4	4	no	—
100	C₄²×C₅₃	C₄²	1	1	yes	characteristic
101	C₄²×C₅₃	C₄²	1	1	yes	characteristic
102	C₄²×C₅₃	C₄²	1	1	yes	characteristic
103	C₄²×C₅₃	C₄²	1	1	yes	characteristic
104	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
105	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
115	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
116	C₂²⋊Dic₅₃	—	4	2	no	—
117	C₂²⋊Dic₅₃	—	4	2	no	—
118	C₄⋊Dic₅₃	—	4	4	no	—
119	C₄⋊Dic₅₃	—	4	4	no	—
120	C₄⋊Dic₅₃	—	4	4	no	—
121	C₄×Dic₅₃	—	4	4	no	—
122	C₂×#88322	—	2	4	no	—
123	C₂×#88322	—	4	4	no	—
124	C₂³×C₅₃	—	4	1	no	—
125	C₂³×C₅₃	—	4	1	no	—
126	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
130	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
131	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
132	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	4	1	no	—
140	C₂×C₄×C₅₃	—	4	1	no	—
141	C₂×C₄×C₅₃	—	4	1	no	—
142	C₂×C₄×C₅₃	—	4	1	no	—
143	C₂×C₄×C₅₃	—	4	1	no	—
144	C₂×C₄×C₅₃	—	4	1	no	—
145	C₂×C₄×C₅₃	—	4	4	no	—
146	C₂×C₄×C₅₃	—	4	4	no	—
147	C₂×C₄×C₅₃	—	4	4	no	—
148	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
149	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
150	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
151	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
152	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
153	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
154	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
155	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
156	C₂×C₄×C₅₃	#102	1	2	yes	—
157	C₂×C₄×C₅₃	#102	1	2	yes	—
158	C₂×C₄×C₅₃	#102	1	2	yes	—
159	C₂×C₄×C₅₃	#102	1	2	yes	—
160	C₂×C₄×C₅₃	#103	1	2	yes	—
161	C₂×C₄×C₅₃	#103	1	2	yes	—
162	C₂×C₄×C₅₃	#103	1	2	yes	—
163	C₂×C₄×C₅₃	#103	1	2	yes	—
164	C₂×C₄×C₅₃	#106	1	2	yes	—
165	C₂×C₄×C₅₃	#106	1	2	yes	—
166	C₂×C₄×C₅₃	#107	1	2	yes	—
167	C₂×C₄×C₅₃	#107	1	2	yes	—
168	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
169	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
170	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂×Dic₅₃	—	4	4	no	—
173	C₂×Dic₅₃	—	4	4	no	—
174	C₂×Dic₅₃	—	4	4	no	—
175	#88322	—	4	4	no	—
176	#7981	—	53	1	no	maximal, Sylow
177	C₂²×C₅₃	—	2	1	no	—
178	C₂²×C₅₃	—	2	1	no	—
179	C₂²×C₅₃	—	2	1	no	—
180	C₂²×C₅₃	—	2	1	no	—
181	C₂²×C₅₃	—	2	1	no	—
182	C₂²×C₅₃	—	2	1	no	—
183	C₂²×C₅₃	—	4	1	no	—
184	C₂²×C₅₃	—	4	1	no	—
185	C₂²×C₅₃	—	4	1	no	—
186	C₂²×C₅₃	—	4	1	no	—
187	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
188	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
189	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
190	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
191	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
192	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
193	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
194	C₄×C₅₃	—	2	1	no	—
195	C₄×C₅₃	—	2	1	no	—
196	C₄×C₅₃	—	2	1	no	—
197	C₄×C₅₃	—	2	1	no	—
198	C₄×C₅₃	—	2	1	no	—
199	C₄×C₅₃	—	2	1	no	—
200	C₄×C₅₃	—	8	4	no	—
201	C₄×C₅₃	H(3,ℤ/4)	1	2	yes	—
202	C₄×C₅₃	#340	1	2	yes	—
203	C₄×C₅₃	#340	1	2	yes	—
204	C₄×C₅₃	#340	1	2	yes	—
205	C₄×C₅₃	#340	1	2	yes	—
206	C₄×C₅₃	#410	1	2	yes	—
207	Dic₅₃	—	8	4	no	—
208	C₂×#431	—	53	1	no	—
209	#2285	—	53	1	no	—
210	#2285	D₅₃	1	1	yes	characteristic
211	#1289	—	53	4	no	—
212	C₂×C₅₃	—	2	1	no	—
213	C₂×C₅₃	—	2	1	no	—
214	C₂×C₅₃	—	2	1	no	—
215	C₂×C₅₃	—	2	1	no	—
216	C₂×C₅₃	—	2	1	no	—
217	C₂×C₅₃	—	2	1	no	—
218	C₂×C₅₃	C₂×H(3,ℤ/4)	1	1	yes	characteristic
219	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
220	C₂×C₅₃	#1740	1	1	yes	characteristic
221	C₂×#117	—	2	1	no	—
222	C₂×#117	—	106	1	no	—
223	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
224	#431	—	106	1	no	—
225	#431	—	106	1	no	—
226	#431	—	106	1	no	—
227	#431	—	106	1	no	—
228	C₂×C₄⋊C₈	—	53	2	no	—
229	C₂×C₂²⋊C₈	—	53	2	no	—
230	C₂×C₄×C₈	—	53	2	no	—
231	#384	—	2	1	no	—
232	#384	—	106	1	no	—
233	#382	—	53	2	no	—
234	#382	Dic₅₃	1	2	yes	complemented
235	#380	—	2	1	no	—
236	#380	—	53	2	no	—
237	#380	—	106	1	no	—
238	#380	C₂×D₅₃	1	2	yes	—
239	C₄×C₄⋊C₄	—	2	1	no	—
240	C₄×C₄⋊C₄	—	53	2	no	—
241	C₄×C₄⋊C₄	—	106	1	no	—
242	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	complemented
243	C₅₃	#7981	1	1	yes	characteristic, complemented, Sylow
244	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
245	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
246	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
247	C₂²×C₈	—	53	4	no	—
248	#117	—	4	2	no	—
249	#117	—	4	2	no	—
250	#117	—	212	2	no	—
251	#117	—	212	2	no	—
252	C₂×C₄⋊C₄	—	2	2	no	—
253	C₂×C₄⋊C₄	—	2	2	no	—
254	C₂×C₄⋊C₄	—	2	2	no	—
255	C₂×C₄⋊C₄	—	106	2	no	—
256	C₂×C₄⋊C₄	—	106	2	no	—
257	C₂×C₄⋊C₄	—	106	2	no	—
258	C₂×C₂²⋊C₄	—	2	2	no	—
259	C₂×C₂²⋊C₄	—	106	2	no	—
260	C₂×C₄²	—	2	2	no	—
261	C₂×C₄²	—	106	2	no	—
262	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
263	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
264	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
265	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
266	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
267	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
268	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
269	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
270	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
271	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
272	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
273	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
274	C₄⋊C₈	—	106	2	no	—
275	C₄⋊C₈	—	106	2	no	—
276	C₂²⋊C₈	—	106	2	no	—
277	C₂²⋊C₈	—	106	2	no	—
278	C₄×C₈	—	106	2	no	—
279	C₄×C₈	—	106	2	no	—
280	#95	—	2	2	no	—
281	#95	—	53	4	no	—
282	#95	—	106	2	no	—
283	#95	C₂×Dic₅₃	1	4	yes	—
284	C₂⁴	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
285	C₂²×C₄	—	2	4	no	—
286	C₂²×C₄	—	4	2	no	—
287	C₂²×C₄	—	4	2	no	—
288	C₂²×C₄	—	106	4	no	—
289	C₂²×C₄	#11719919	1	1	yes	characteristic
290	C₂²×C₄	#11719919	1	1	yes	characteristic
291	C₂²×C₄	#11719919	1	1	yes	characteristic
292	C₂²×C₄	#11719919	1	1	yes	characteristic
293	C₂²×C₄	#11719921	1	1	yes	characteristic
294	C₂²×C₄	#11719921	1	1	yes	characteristic
295	C₂²×C₄	#11719921	1	1	yes	characteristic
296	C₂²×C₄	#11719921	1	1	yes	characteristic
297	C₂²×C₄	#11719922	1	2	yes	—
298	C₂²×C₄	#11719923	1	2	yes	—
299	C₂²×C₄	#11719924	1	2	yes	—
300	C₂²×C₄	#11719925	1	2	yes	—
301	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
302	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
303	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
304	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
305	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
306	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
307	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
308	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
309	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
310	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
311	C₂×C₈	—	106	4	no	—
312	C₂×C₈	—	212	4	no	—
313	C₄⋊C₄	—	4	4	no	—
314	C₄⋊C₄	—	4	4	no	—
315	C₄⋊C₄	—	4	4	no	—
316	C₄⋊C₄	—	212	4	no	—
317	C₄⋊C₄	—	212	4	no	—
318	C₄⋊C₄	—	212	4	no	—
319	C₂²⋊C₄	—	4	2	no	—
320	C₂²⋊C₄	—	4	2	no	—
321	C₂²⋊C₄	—	212	2	no	—
322	C₂²⋊C₄	—	212	2	no	—
323	C₄²	—	4	4	no	—
324	C₄²	—	212	4	no	—
325	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
326	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
327	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
328	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
329	C₄²	#11719919	1	1	yes	characteristic
330	C₄²	#11719919	1	1	yes	characteristic
331	C₄²	#11719919	1	1	yes	characteristic
332	C₄²	#11719919	1	1	yes	characteristic
333	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
334	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
335	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
336	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
337	C₄²	#11719921	1	1	yes	characteristic
338	C₄²	#11719921	1	1	yes	characteristic
339	C₄²	#11719921	1	1	yes	characteristic
340	C₄²	#11719921	1	1	yes	characteristic
341	C₂³	—	4	1	no	—
342	C₂³	—	4	1	no	—
343	C₂³	C₂×#11719921	1	1	yes	characteristic
344	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
345	C₂³	C₂×#11719919	1	1	yes	characteristic
346	C₂³	C₂×#11719922	1	1	yes	characteristic
347	C₂³	C₂×#11719924	1	1	yes	characteristic
348	C₂³	C₂×#11719923	1	1	yes	characteristic
349	C₂³	C₂×#11719925	1	1	yes	characteristic
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	4	1	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	4	1	no	—
359	C₂×C₄	—	4	1	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	4	no	—
363	C₂×C₄	—	4	4	no	—
364	C₂×C₄	—	4	4	no	—
365	C₂×C₄	—	212	4	no	—
366	C₂×C₄	—	212	4	no	—
367	C₂×C₄	—	212	4	no	—
368	C₂×C₄	#13133124	1	1	yes	characteristic
369	C₂×C₄	#13133124	1	1	yes	characteristic
370	C₂×C₄	#13133124	1	1	yes	characteristic
371	C₂×C₄	#13133124	1	1	yes	characteristic
372	C₂×C₄	#13133124	1	1	yes	characteristic
373	C₂×C₄	#13133124	1	1	yes	characteristic
374	C₂×C₄	#13133124	1	1	yes	characteristic
375	C₂×C₄	#13133124	1	1	yes	characteristic
376	C₂×C₄	#13133137	1	2	yes	—
377	C₂×C₄	#13133137	1	2	yes	—
378	C₂×C₄	#13133138	1	2	yes	—
379	C₂×C₄	#13133138	1	2	yes	—
380	C₂×C₄	#13133140	1	2	yes	—
381	C₂×C₄	#13133140	1	2	yes	—
382	C₂×C₄	#13133141	1	2	yes	—
383	C₂×C₄	#13133141	1	2	yes	—
384	C₂×C₄	#13133148	1	2	yes	—
385	C₂×C₄	#13133148	1	2	yes	—
386	C₂×C₄	#13133150	1	2	yes	—
387	C₂×C₄	#13133150	1	2	yes	—
388	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
389	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
390	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
391	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
392	C₈	—	212	4	no	—
393	C₂²	—	2	1	no	—
394	C₂²	—	2	1	no	—
395	C₂²	—	2	1	no	—
396	C₂²	—	2	1	no	—
397	C₂²	—	2	1	no	—
398	C₂²	—	2	1	no	—
399	C₂²	—	4	1	no	—
400	C₂²	—	4	1	no	—
401	C₂²	—	4	1	no	—
402	C₂²	—	4	1	no	—
403	C₂²	C₂×#13133124	1	1	yes	characteristic
404	C₂²	C₂×#13133137	1	1	yes	characteristic
405	C₂²	C₂×#13133138	1	1	yes	characteristic
406	C₂²	C₂×#13133140	1	1	yes	characteristic
407	C₂²	C₂×#13133141	1	1	yes	characteristic
408	C₂²	C₂×#13133148	1	1	yes	characteristic
409	C₂²	C₂×#13133150	1	1	yes	characteristic
410	C₄	—	2	1	no	—
411	C₄	—	2	1	no	—
412	C₄	—	2	1	no	—
413	C₄	—	2	1	no	—
414	C₄	—	2	1	no	—
415	C₄	—	2	1	no	—
416	C₄	—	8	4	no	—
417	C₄	—	424	4	no	—
418	C₄	#16810379	1	2	yes	—
419	C₄	#16810385	1	2	yes	—
420	C₄	#16810385	1	2	yes	—
421	C₄	#16810385	1	2	yes	—
422	C₄	#16810385	1	2	yes	—
423	C₄	#16810596	1	2	yes	—
424	C₂	—	2	1	no	—
425	C₂	—	2	1	no	—
426	C₂	—	2	1	no	—
427	C₂	—	2	1	no	—
428	C₂	—	2	1	no	—
429	C₂	—	2	1	no	—
430	C₂	C₂×#16810379	1	1	yes	characteristic
431	C₂	C₂×#16810385	1	1	yes	characteristic
432	C₂	#20274447	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133124
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#2285	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#2285	D₅₃
hypocenter	C₅₃	#7981
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
10. C₂×C₄⋊#88322	C₄	semidirect
12. C₂×C₄×#88322	C₄	semidirect
14. #16810489	C₄	semidirect
18. C₄×C₄⋊Dic₅₃	C₄	semidirect
22. C₅₃×#382	C₄	semidirect
26. C₄×C₄⋊C₄×C₅₃	C₄	semidirect
234. #382	Dic₅₃	semidirect
242. C₄×C₄⋊C₄	Dic₅₃	semidirect
243. C₅₃	#7981	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810676	2
#20277122	2
#20272582	2
C₅₃×#2285	2
#7981	53