Subgroups — #27273877

Order 13568.

674 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	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
2	#20277122	C₂	1	1	yes	maximal, characteristic
3	#20272575	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#423	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133176	—	2	1	no	—
6	#16810676	—	2	1	no	—
7	#16810676	—	2	1	no	—
8	#16810676	C₄	1	2	yes	complemented
9	#16810676	C₂²	1	2	yes	—
10	C₂×C₄⋊#88322	—	2	1	no	—
11	C₂×C₂²⋊#88322	—	2	1	no	—
12	C₂×C₄×#88322	C₄	1	2	yes	complemented
13	#16810493	—	2	1	no	—
14	#16810489	C₂²	1	2	yes	—
15	#16810483	—	2	1	no	—
16	#16810483	C₄	1	2	yes	—
17	C₄×C₄⋊Dic₅₃	—	2	1	no	—
18	C₄×C₄⋊Dic₅₃	C₄	1	2	yes	—
19	C₂²×M₁₆×C₅₃	—	2	1	no	—
20	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#423	—	2	1	no	—
22	C₅₃×#423	—	2	1	no	—
23	C₅₃×#423	C₄	1	2	yes	complemented
24	C₅₃×#423	C₂²	1	2	yes	—
25	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
26	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	complemented
27	C₂²×#88322	—	2	2	no	—
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	C₂×C₄	1	4	yes	—
41	#13133124	—	2	2	no	—
42	#13133124	C₂×C₄	1	4	yes	—
43	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
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	—	4	2	no	—
49	C₂×M₁₆×C₅₃	—	4	2	no	—
50	C₂²×C₈×C₅₃	—	2	2	no	—
51	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₅₃	—	2	2	no	—
64	C₄⋊C₈×C₅₃	—	2	2	no	—
65	C₄⋊C₈×C₅₃	—	2	2	no	—
66	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
67	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	—	4	2	no	—
69	C₂²×C₄×C₅₃	—	4	2	no	—
70	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₄²	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	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	D₈	1	2	yes	—
86	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
87	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
88	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
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×Dic₅₃	—	2	4	no	—
93	M₁₆×C₅₃	—	4	4	no	—
94	M₁₆×C₅₃	—	4	2	no	—
95	M₁₆×C₅₃	—	4	2	no	—
96	C₂×C₈×C₅₃	—	2	4	no	—
97	C₂×C₈×C₅₃	—	4	4	no	—
98	C₄²×C₅₃	C₄²	1	1	yes	characteristic
99	C₄²×C₅₃	C₄²	1	1	yes	characteristic
100	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
104	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
105	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₄⋊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	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
114	C₂²⋊Dic₅₃	—	4	4	no	—
115	C₄⋊Dic₅₃	—	4	4	no	—
116	C₄⋊Dic₅₃	—	4	4	no	—
117	C₄⋊Dic₅₃	—	4	2	no	—
118	C₄⋊Dic₅₃	—	4	2	no	—
119	C₄×Dic₅₃	—	4	4	no	—
120	C₂×#88322	—	2	4	no	—
121	C₂×#88322	—	4	4	no	—
122	C₂³×C₅₃	—	4	1	no	—
123	C₂³×C₅₃	—	4	1	no	—
124	C₂³×C₅₃	#95	1	1	yes	characteristic
125	C₂³×C₅₃	#95	1	1	yes	characteristic
126	C₂³×C₅₃	#102	1	2	yes	—
127	C₂³×C₅₃	#103	1	2	yes	—
128	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	4	1	no	—
136	C₂×C₄×C₅₃	—	4	1	no	—
137	C₂×C₄×C₅₃	—	4	1	no	—
138	C₂×C₄×C₅₃	—	4	1	no	—
139	C₂×C₄×C₅₃	—	4	1	no	—
140	C₂×C₄×C₅₃	—	4	1	no	—
141	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
142	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
143	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
144	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
145	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
146	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
147	C₂×C₄×C₅₃	#102	1	2	yes	—
148	C₂×C₄×C₅₃	#102	1	2	yes	—
149	C₂×C₄×C₅₃	#102	1	2	yes	—
150	C₂×C₄×C₅₃	#103	1	2	yes	—
151	C₂×C₄×C₅₃	#103	1	2	yes	—
152	C₂×C₄×C₅₃	#103	1	2	yes	—
153	C₂×C₄×C₅₃	#106	1	2	yes	—
154	C₂×C₄×C₅₃	#106	1	2	yes	—
155	C₂×C₄×C₅₃	#107	1	2	yes	—
156	C₂×C₄×C₅₃	#107	1	2	yes	—
157	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
167	C₂×Dic₅₃	—	4	4	no	—
168	C₂×Dic₅₃	—	4	4	no	—
169	C₂×Dic₅₃	—	4	4	no	—
170	C₈×C₅₃	—	4	4	no	—
171	#88322	—	4	4	no	—
172	#7957	—	53	1	no	maximal, Sylow
173	C₂²×C₅₃	—	2	1	no	—
174	C₂²×C₅₃	—	2	1	no	—
175	C₂²×C₅₃	—	2	1	no	—
176	C₂²×C₅₃	—	2	1	no	—
177	C₂²×C₅₃	—	2	1	no	—
178	C₂²×C₅₃	—	2	1	no	—
179	C₂²×C₅₃	—	4	2	no	—
180	C₂²×C₅₃	—	4	2	no	—
181	C₂²×C₅₃	#340	1	2	yes	—
182	C₂²×C₅₃	#340	1	2	yes	—
183	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
184	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
185	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
186	C₄×C₅₃	—	2	1	no	—
187	C₄×C₅₃	—	2	1	no	—
188	C₄×C₅₃	—	2	1	no	—
189	C₄×C₅₃	—	2	1	no	—
190	C₄×C₅₃	—	2	1	no	—
191	C₄×C₅₃	—	2	1	no	—
192	C₄×C₅₃	H(3,ℤ/4)	1	2	yes	—
193	C₄×C₅₃	#340	1	2	yes	—
194	C₄×C₅₃	#340	1	2	yes	—
195	C₄×C₅₃	#344	1	2	yes	—
196	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
197	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
198	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
199	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
200	Dic₅₃	—	8	4	no	—
201	C₂×#431	—	53	1	no	—
202	C₂×#423	D₅₃	1	1	yes	characteristic
203	#2285	—	53	1	no	—
204	#1286	—	53	4	no	—
205	C₂×C₅₃	—	2	1	no	—
206	C₂×C₅₃	—	2	2	no	—
207	C₂×C₅₃	—	2	2	no	—
208	C₂×C₅₃	—	2	1	no	—
209	C₂×C₅₃	C₂×H(3,ℤ/4)	1	1	yes	characteristic
210	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
211	C₂×C₅₃	C₂×#344	1	1	yes	characteristic
212	C₂²×M₁₆	—	2	1	no	—
213	C₂×#117	—	106	1	no	—
214	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
215	#431	—	53	2	no	—
216	#431	—	53	2	no	—
217	#431	—	106	1	no	—
218	#431	—	106	1	no	—
219	#423	—	2	1	no	—
220	#423	—	2	1	no	—
221	#423	Dic₅₃	1	2	yes	complemented
222	#423	C₂×D₅₃	1	2	yes	—
223	C₂×C₄⋊C₈	—	2	1	no	—
224	C₂×C₄⋊C₈	—	106	1	no	—
225	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	complemented
226	C₂×C₂²⋊C₈	—	106	1	no	—
227	C₂×C₄×C₈	—	53	2	no	—
228	#384	—	106	1	no	—
229	#382	—	53	2	no	—
230	#380	—	53	2	no	—
231	#380	—	106	1	no	—
232	C₄×C₄⋊C₄	—	53	2	no	—
233	C₄×C₄⋊C₄	—	106	1	no	—
234	C₅₃	#7957	1	1	yes	characteristic, complemented, Sylow
235	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
236	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
237	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
238	C₂×M₁₆	—	2	2	no	—
239	C₂×M₁₆	—	2	2	no	—
240	C₂×M₁₆	—	4	2	no	—
241	C₂×M₁₆	—	4	2	no	—
242	C₂²×C₈	—	2	2	no	—
243	C₂²×C₈	—	106	2	no	—
244	#117	—	212	2	no	—
245	#117	—	212	2	no	—
246	C₂×C₄⋊C₄	—	106	2	no	—
247	C₂×C₄⋊C₄	—	106	2	no	—
248	C₂×C₄⋊C₄	—	106	2	no	—
249	C₂×C₂²⋊C₄	—	106	2	no	—
250	C₂×C₄²	—	106	2	no	—
251	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
252	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
253	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
254	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
255	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
256	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
257	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
258	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
259	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
260	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
261	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
262	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
263	C₄⋊C₈	—	2	2	no	—
264	C₄⋊C₈	—	2	2	no	—
265	C₄⋊C₈	—	2	2	no	—
266	C₄⋊C₈	—	106	2	no	—
267	C₄⋊C₈	—	106	2	no	—
268	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
269	C₂²⋊C₈	—	106	2	no	—
270	C₂²⋊C₈	—	106	2	no	—
271	C₄×C₈	—	53	4	no	—
272	C₄×C₈	—	106	2	no	—
273	#95	—	53	4	no	—
274	#95	—	106	2	no	—
275	C₂⁴	#11719921	1	1	yes	characteristic
276	C₂²×C₄	—	4	2	no	—
277	C₂²×C₄	—	4	2	no	—
278	C₂²×C₄	—	106	4	no	—
279	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
280	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
281	C₂²×C₄	#11719919	1	1	yes	characteristic
282	C₂²×C₄	#11719919	1	1	yes	characteristic
283	C₂²×C₄	#11719919	1	1	yes	characteristic
284	C₂²×C₄	#11719919	1	1	yes	characteristic
285	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
286	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
287	C₂²×C₄	#11719921	1	1	yes	characteristic
288	C₂²×C₄	#11719921	1	1	yes	characteristic
289	C₂²×C₄	#11719921	1	1	yes	characteristic
290	C₂²×C₄	#11719921	1	1	yes	characteristic
291	C₂²×C₄	#11719921	1	1	yes	characteristic
292	C₂²×C₄	#11719922	1	2	yes	—
293	C₂²×C₄	#11719923	1	2	yes	—
294	C₂²×C₄	#11719924	1	2	yes	—
295	C₂²×C₄	#11719925	1	2	yes	—
296	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
297	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
298	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
299	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
300	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
301	M₁₆	—	4	4	no	—
302	M₁₆	—	4	2	no	—
303	M₁₆	—	4	2	no	—
304	C₂×C₈	—	2	4	no	—
305	C₂×C₈	—	4	4	no	—
306	C₂×C₈	—	106	4	no	—
307	C₂×C₈	—	212	4	no	—
308	C₄⋊C₄	—	212	4	no	—
309	C₄⋊C₄	—	212	4	no	—
310	C₄⋊C₄	—	212	2	no	—
311	C₄⋊C₄	—	212	2	no	—
312	C₂²⋊C₄	—	212	4	no	—
313	C₄²	—	212	4	no	—
314	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
315	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
316	C₄²	#11719919	1	1	yes	characteristic
317	C₄²	#11719919	1	1	yes	characteristic
318	C₄²	#11719919	1	1	yes	characteristic
319	C₄²	#11719919	1	1	yes	characteristic
320	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
321	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
322	C₄²	#11719921	1	1	yes	characteristic
323	C₄²	#11719921	1	1	yes	characteristic
324	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
325	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
326	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
327	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
328	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
329	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
330	C₂³	—	4	1	no	—
331	C₂³	—	4	1	no	—
332	C₂³	#13133124	1	1	yes	characteristic
333	C₂³	#13133124	1	1	yes	characteristic
334	C₂³	#13133137	1	2	yes	—
335	C₂³	#13133140	1	2	yes	—
336	C₂³	C₂×#11719921	1	1	yes	characteristic
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	212	4	no	—
350	C₂×C₄	—	212	4	no	—
351	C₂×C₄	—	212	4	no	—
352	C₂×C₄	#13133124	1	1	yes	characteristic
353	C₂×C₄	#13133124	1	1	yes	characteristic
354	C₂×C₄	#13133124	1	1	yes	characteristic
355	C₂×C₄	#13133124	1	1	yes	characteristic
356	C₂×C₄	#13133124	1	1	yes	characteristic
357	C₂×C₄	#13133124	1	1	yes	characteristic
358	C₂×C₄	#13133137	1	2	yes	—
359	C₂×C₄	#13133138	1	2	yes	—
360	C₂×C₄	#13133138	1	2	yes	—
361	C₂×C₄	#13133140	1	2	yes	—
362	C₂×C₄	#13133141	1	2	yes	—
363	C₂×C₄	#13133141	1	2	yes	—
364	C₂×C₄	#13133148	1	2	yes	—
365	C₂×C₄	#13133148	1	2	yes	—
366	C₂×C₄	#13133150	1	2	yes	—
367	C₂×C₄	#13133150	1	2	yes	—
368	C₂×C₄	C₂×C₄×Dic₅₃	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×#11719919	1	1	yes	characteristic
373	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
374	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
375	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
376	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
377	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
378	C₈	—	4	4	no	—
379	C₈	—	212	4	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	1	no	—
385	C₂²	—	2	1	no	—
386	C₂²	—	4	2	no	—
387	C₂²	—	4	2	no	—
388	C₂²	#16810385	1	2	yes	—
389	C₂²	#16810385	1	2	yes	—
390	C₂²	C₂×#13133124	1	1	yes	characteristic
391	C₂²	C₂×#13133137	1	1	yes	characteristic
392	C₂²	C₂×#13133140	1	1	yes	characteristic
393	C₄	—	2	1	no	—
394	C₄	—	2	1	no	—
395	C₄	—	2	1	no	—
396	C₄	—	2	1	no	—
397	C₄	—	2	1	no	—
398	C₄	—	2	1	no	—
399	C₄	—	424	4	no	—
400	C₄	#16810380	1	2	yes	—
401	C₄	#16810385	1	2	yes	—
402	C₄	#16810385	1	2	yes	—
403	C₄	#16810395	1	2	yes	—
404	C₄	C₂×#13133138	1	1	yes	characteristic
405	C₄	C₂×#13133141	1	1	yes	characteristic
406	C₄	C₂×#13133148	1	1	yes	characteristic
407	C₄	C₂×#13133150	1	1	yes	characteristic
408	C₂	—	2	1	no	—
409	C₂	—	2	1	no	—
410	C₂	—	2	2	no	—
411	C₂	—	2	2	no	—
412	C₂	C₂×#16810380	1	1	yes	characteristic
413	C₂	C₂×#16810385	1	1	yes	characteristic
414	C₂	C₂×#16810395	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133124
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#423	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#423	D₅₃
hypocenter	C₅₃	#7957
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #20272575	C₂	semidirect
8. #16810676	C₄	semidirect
12. C₂×C₄×#88322	C₄	semidirect
23. C₅₃×#423	C₄	semidirect
26. C₂×C₄⋊C₈×C₅₃	C₄	semidirect
221. #423	Dic₅₃	semidirect
225. C₂×C₄⋊C₈	Dic₅₃	semidirect
234. C₅₃	#7957	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
#20272575	2
C₂×C₅₃×#423	2
#7957	53