Subgroups — #27402901

Order 13568.

832 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810751	C₂	1	2	yes	maximal
2	C₄²×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
3	#20275438	C₂	1	2	yes	maximal
4	#20275437	C₂	1	2	yes	maximal
5	#20275432	C₂	1	2	yes	maximal
6	#20274867	C₂	1	1	yes	maximal, characteristic
7	#20274866	C₂	1	1	yes	maximal, characteristic
8	#20274866	C₂	1	1	yes	maximal, characteristic
9	#20274570	C₂	1	1	yes	maximal, characteristic
10	C₄⋊C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1767	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
15	#16810751	—	2	8	no	—
16	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊#88322	C₂²	1	2	yes	—
19	C₂×C₄⋊#88322	C₂²	1	2	yes	—
20	C₂×#13133146	C₂²	1	2	yes	—
21	C₂×#13133146	C₂²	1	2	yes	—
22	C₂×#13133146	C₂²	1	2	yes	—
23	C₂×#13133146	C₂²	1	2	yes	—
24	C₂×#13133127	C₂²	1	2	yes	—
25	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
26	C₄×#11719919	C₄	1	1	yes	characteristic
27	C₄×#11719919	C₄	1	1	yes	characteristic
28	C₄×#11719919	C₄	1	1	yes	characteristic
29	C₄×#11719919	C₄	1	1	yes	characteristic
30	C₄×#11719919	C₂²	1	1	yes	characteristic
31	C₄×#11719919	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
33	C₄×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
34	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
35	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
36	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
37	#16810377	C₂²	1	2	yes	—
38	#16810377	C₂²	1	2	yes	—
39	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×#97×C₅₃	C₂²	1	2	yes	—
42	C₄³×C₅₃	C₂²	1	1	yes	characteristic
43	C₅₃×#336	C₂²	1	2	yes	—
44	C₂²×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
45	C₂²×#88322	C₂³	1	2	yes	—
46	C₂²×#88322	C₂³	1	2	yes	—
47	C₄×Dic₁₀₆	—	2	4	no	—
48	C₄×Dic₁₀₆	—	2	4	no	—
49	C₄×Dic₁₀₆	D₄	1	4	yes	—
50	C₄×Dic₁₀₆	Q₈	1	4	yes	—
51	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
55	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
56	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
57	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
58	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
59	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
70	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
71	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
72	C₄⋊#88322	—	2	4	no	—
73	C₄⋊#88322	—	2	2	no	—
74	C₄⋊#88322	—	2	2	no	—
75	C₄⋊#88322	—	4	2	no	—
76	#13133146	—	2	4	no	—
77	#13133146	—	2	4	no	—
78	#13133146	—	4	2	no	—
79	#13133146	—	4	2	no	—
80	#13133127	—	2	4	no	—
81	C₈×Dic₅₃	—	2	4	no	—
82	#13133124	C₂×C₄	1	1	yes	characteristic
83	#13133124	C₂×C₄	1	1	yes	characteristic
84	#13133124	C₂×C₄	1	1	yes	characteristic
85	#13133124	C₂×C₄	1	1	yes	characteristic
86	#13133124	C₂×C₄	1	1	yes	characteristic
87	#13133124	C₂×C₄	1	1	yes	characteristic
88	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
89	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
90	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
95	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
96	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
97	C₄⋊C₈×C₅₃	—	2	2	no	—
98	C₄⋊C₈×C₅₃	Q₈	1	4	yes	complemented
99	#97×C₅₃	—	2	4	no	—
100	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
107	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×Dic₁₀₆	—	2	2	no	—
116	C₂×Dic₁₀₆	—	2	2	no	—
117	C₂×Dic₁₀₆	C₄⋊C₄	1	4	yes	—
118	C₂×C₈×C₅₃	—	2	2	no	—
119	C₂×C₈×C₅₃	—	2	2	no	—
120	C₂×C₈×C₅₃	—	2	2	no	—
121	C₂×C₈×C₅₃	—	2	2	no	—
122	C₂×C₈×C₅₃	—	2	2	no	—
123	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
124	C₄²×C₅₃	—	2	2	no	—
125	C₄²×C₅₃	—	2	2	no	—
126	C₄²×C₅₃	—	2	2	no	—
127	C₄²×C₅₃	—	2	1	no	—
128	C₄²×C₅₃	—	2	1	no	—
129	C₄²×C₅₃	—	4	1	no	—
130	C₄²×C₅₃	C₂×D₄	1	2	yes	—
131	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
132	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
133	C₄²×C₅₃	#41	1	2	yes	—
134	C₄⋊Dic₅₃	—	2	2	no	—
135	C₄⋊Dic₅₃	—	2	2	no	—
136	C₄⋊Dic₅₃	C₄⋊C₄	1	4	yes	—
137	#11719919	—	2	2	no	—
138	#11719919	—	2	2	no	—
139	#11719919	—	2	2	no	—
140	#11719919	C₄⋊C₄	1	4	yes	—
141	#11719919	C₄⋊C₄	1	4	yes	—
142	#11719919	M₁₆	1	4	yes	—
143	C₄×Dic₅₃	—	2	2	no	—
144	C₄×Dic₅₃	—	2	2	no	—
145	C₄×Dic₅₃	—	2	2	no	—
146	C₄×Dic₅₃	—	2	1	no	—
147	C₄×Dic₅₃	—	2	1	no	—
148	C₄×Dic₅₃	—	2	1	no	—
149	C₄×Dic₅₃	—	2	1	no	—
150	C₄×Dic₅₃	—	2	2	no	—
151	C₄×Dic₅₃	—	2	2	no	—
152	C₄×Dic₅₃	—	4	1	no	—
153	C₄×Dic₅₃	—	4	1	no	—
154	C₄×Dic₅₃	M₁₆	1	4	yes	—
155	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
156	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
157	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
158	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
159	C₂×#88322	—	2	2	no	—
160	C₂×#88322	—	2	2	no	—
161	C₂×#88322	—	2	2	no	—
162	C₂×#88322	—	2	2	no	—
163	C₂×#88322	—	2	2	no	—
164	C₂×#88322	—	2	2	no	—
165	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
166	C₂×C₄×C₅₃	—	2	1	no	—
167	C₂×C₄×C₅₃	—	2	1	no	—
168	C₂×C₄×C₅₃	—	2	1	no	—
169	C₂×C₄×C₅₃	—	2	1	no	—
170	C₂×C₄×C₅₃	—	2	1	no	—
171	C₂×C₄×C₅₃	—	2	1	no	—
172	C₂×C₄×C₅₃	—	2	1	no	—
173	C₂×C₄×C₅₃	—	2	1	no	—
174	C₂×C₄×C₅₃	—	2	1	no	—
175	C₂×C₄×C₅₃	—	2	1	no	—
176	C₂×C₄×C₅₃	—	2	1	no	—
177	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
178	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
179	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
180	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
181	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
182	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
183	C₂×C₄×C₅₃	#131	1	2	yes	—
184	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
186	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
187	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
188	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
189	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
190	C₂×Dic₅₃	—	2	1	no	—
191	C₂×Dic₅₃	—	2	1	no	—
192	C₂×Dic₅₃	—	2	1	no	—
193	C₂×Dic₅₃	—	2	1	no	—
194	C₂×Dic₅₃	—	2	1	no	—
195	C₂×Dic₅₃	—	2	1	no	—
196	C₂×Dic₅₃	—	2	1	no	—
197	C₂×Dic₅₃	—	2	1	no	—
198	C₂×Dic₅₃	—	2	1	no	—
199	C₂×Dic₅₃	—	2	1	no	—
200	C₂×Dic₅₃	—	2	1	no	—
201	C₂×Dic₅₃	—	2	1	no	—
202	C₂×Dic₅₃	—	2	1	no	—
203	C₂×Dic₅₃	—	2	1	no	—
204	C₂×Dic₅₃	—	2	1	no	—
205	C₂×Dic₅₃	—	2	1	no	—
206	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
207	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
208	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
209	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
210	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
211	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
212	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
213	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
214	Dic₁₀₆	—	2	8	no	—
215	C₈×C₅₃	—	2	4	no	—
216	C₈×C₅₃	—	4	2	no	—
217	#88322	—	4	2	no	—
218	#88322	—	4	2	no	—
219	#22722	—	53	1	no	maximal, Sylow
220	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
221	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
222	C₂²×C₅₃	#520	1	1	yes	characteristic
223	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
224	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
225	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
226	C₂²×C₅₃	#568	1	1	yes	characteristic
227	C₄×C₅₃	—	2	2	no	—
228	C₄×C₅₃	—	2	2	no	—
229	C₄×C₅₃	—	2	2	no	—
230	C₄×C₅₃	—	2	2	no	—
231	C₄×C₅₃	—	4	1	no	—
232	C₄×C₅₃	#446	1	4	yes	complemented
233	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
234	C₄×C₅₃	#550	1	1	yes	characteristic
235	C₄×C₅₃	#557	1	1	yes	characteristic
236	C₄×C₅₃	Q₈²	1	1	yes	characteristic
237	Dic₅₃	—	2	2	no	—
238	Dic₅₃	—	2	2	no	—
239	Dic₅₃	—	2	2	no	—
240	Dic₅₃	—	2	2	no	—
241	Dic₅₃	—	2	2	no	—
242	Dic₅₃	—	4	1	no	—
243	Dic₅₃	—	4	1	no	—
244	Dic₅₃	#423	1	4	yes	—
245	C₂×#446	—	53	2	no	—
246	C₄²×Q₈	—	53	1	no	—
247	#1940	—	53	2	no	—
248	#1940	—	53	2	no	—
249	#1939	—	53	2	no	—
250	#1842	—	53	1	no	—
251	#1842	—	53	1	no	—
252	#1842	—	53	1	no	—
253	#1767	—	53	1	no	—
254	#1767	D₅₃	1	1	yes	characteristic
255	C₄×C₄⋊C₈	—	53	1	no	—
256	C₂×C₅₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
257	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
258	C₂×C₅₃	#2906	1	1	yes	characteristic
259	C₂×C₅₃	C₂×#446	1	1	yes	characteristic
260	C₂×C₅₃	Q₈×M₁₆	1	1	yes	characteristic
261	C₂×C₅₃	#2989	1	1	yes	characteristic
262	C₂×C₅₃	#2990	1	1	yes	characteristic
263	C₂×C₄×Q₈	—	53	1	no	—
264	C₂×C₄×Q₈	—	53	1	no	—
265	C₂×C₄×Q₈	—	53	1	no	—
266	#446	—	106	8	no	—
267	C₂×C₄⋊C₈	—	53	2	no	—
268	C₂×C₄⋊C₈	—	53	2	no	—
269	C₂×C₄⋊C₈	—	53	1	no	—
270	C₂×C₄⋊C₈	—	53	1	no	—
271	C₂×C₄⋊C₈	—	53	2	no	—
272	C₂×C₄⋊C₈	—	53	2	no	—
273	C₂×C₄⋊C₈	—	53	2	no	—
274	C₂×C₄⋊C₈	—	53	2	no	—
275	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
276	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
277	C₂×#97	—	53	2	no	—
278	C₂×#97	C₂×D₅₃	1	2	yes	—
279	C₂×C₄×C₈	—	53	2	no	—
280	C₄×C₄⋊C₄	—	53	1	no	—
281	C₄×C₄⋊C₄	—	53	1	no	—
282	C₄×C₄⋊C₄	—	53	1	no	—
283	C₄×C₄⋊C₄	—	53	1	no	—
284	C₄×C₄⋊C₄	—	53	1	no	—
285	C₄×C₄⋊C₄	—	53	1	no	—
286	C₄×C₄⋊C₄	—	53	1	no	—
287	C₄×C₄⋊C₄	—	53	1	no	—
288	C₄×C₄⋊C₄	—	53	1	no	—
289	C₄³	—	53	1	no	—
290	C₄³	—	53	1	no	—
291	C₄³	C₂×D₅₃	1	1	yes	characteristic
292	#336	—	53	2	no	—
293	#336	—	53	2	no	—
294	#336	C₂×D₅₃	1	2	yes	—
295	C₅₃	#22722	1	1	yes	characteristic, complemented, Sylow
296	C₂²×Q₈	—	53	1	no	—
297	C₂²×C₈	—	53	2	no	—
298	C₂²×C₈	—	53	2	no	—
299	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
300	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
301	C₄×Q₈	—	53	4	no	—
302	C₄×Q₈	—	53	4	no	—
303	C₄×Q₈	—	106	4	no	—
304	C₄×Q₈	—	106	4	no	—
305	C₂×C₄⋊C₄	—	53	1	no	—
306	C₂×C₄⋊C₄	—	53	1	no	—
307	C₂×C₄⋊C₄	—	53	1	no	—
308	C₂×C₄⋊C₄	—	53	1	no	—
309	C₂×C₄⋊C₄	—	53	1	no	—
310	C₂×C₄⋊C₄	—	53	1	no	—
311	C₂×C₄⋊C₄	—	53	1	no	—
312	C₂×C₄⋊C₄	—	53	1	no	—
313	C₂×C₄⋊C₄	—	53	1	no	—
314	C₂×C₄²	—	53	1	no	—
315	C₂×C₄²	—	53	1	no	—
316	C₂×C₄²	—	53	1	no	—
317	C₂×C₄²	—	53	1	no	—
318	C₂×C₄²	—	53	1	no	—
319	C₂×C₄²	—	53	1	no	—
320	C₂×C₄²	—	53	1	no	—
321	C₂×C₄²	—	53	1	no	—
322	C₂×C₄²	—	53	1	no	—
323	C₂×C₄²	—	53	1	no	—
324	C₂×C₄²	—	53	1	no	—
325	C₂×C₄²	—	53	1	no	—
326	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
327	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
328	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
329	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
330	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
331	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
332	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
333	C₄⋊C₈	—	2	2	no	—
334	C₄⋊C₈	—	106	4	no	—
335	C₄⋊C₈	—	106	4	no	—
336	C₄⋊C₈	—	106	2	no	—
337	C₄⋊C₈	—	106	2	no	—
338	C₄⋊C₈	—	106	4	no	—
339	C₄⋊C₈	—	212	2	no	—
340	C₄⋊C₈	—	212	2	no	—
341	C₄⋊C₈	—	212	2	no	—
342	C₄⋊C₈	Dic₁₀₆	1	4	yes	complemented
343	#97	—	2	4	no	—
344	#97	—	106	4	no	—
345	C₄×C₈	—	106	4	no	—
346	#95	—	53	1	no	—
347	#95	—	53	1	no	—
348	#95	—	53	1	no	—
349	#95	—	53	1	no	—
350	#95	—	53	1	no	—
351	#95	—	53	1	no	—
352	C₂×Q₈	—	53	4	no	—
353	C₂×Q₈	—	106	2	no	—
354	C₂×Q₈	—	106	2	no	—
355	C₂²×C₄	—	53	1	no	—
356	C₂²×C₄	—	53	1	no	—
357	C₂²×C₄	—	53	1	no	—
358	C₂²×C₄	—	53	1	no	—
359	C₂²×C₄	—	53	1	no	—
360	C₂²×C₄	—	53	1	no	—
361	C₂²×C₄	—	53	1	no	—
362	C₂²×C₄	—	53	1	no	—
363	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
364	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
365	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
366	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
367	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
368	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
369	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	2	2	no	—
372	C₂×C₈	—	2	2	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	106	2	no	—
376	C₂×C₈	—	106	2	no	—
377	C₂×C₈	—	106	2	no	—
378	C₂×C₈	—	106	2	no	—
379	C₂×C₈	—	106	2	no	—
380	C₂×C₈	—	106	2	no	—
381	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
382	C₄⋊C₄	—	53	4	no	—
383	C₄⋊C₄	—	53	4	no	—
384	C₄⋊C₄	—	53	4	no	—
385	C₄⋊C₄	—	53	4	no	—
386	C₄⋊C₄	—	106	2	no	—
387	C₄⋊C₄	—	106	2	no	—
388	C₄⋊C₄	—	106	2	no	—
389	C₄⋊C₄	—	106	2	no	—
390	C₄⋊C₄	—	106	2	no	—
391	C₄²	—	2	2	no	—
392	C₄²	—	2	2	no	—
393	C₄²	—	2	2	no	—
394	C₄²	—	2	1	no	—
395	C₄²	—	2	1	no	—
396	C₄²	—	4	1	no	—
397	C₄²	—	53	4	no	—
398	C₄²	—	53	2	no	—
399	C₄²	—	53	2	no	—
400	C₄²	—	53	2	no	—
401	C₄²	—	53	2	no	—
402	C₄²	—	106	2	no	—
403	C₄²	—	106	2	no	—
404	C₄²	—	106	2	no	—
405	C₄²	—	106	2	no	—
406	C₄²	—	106	2	no	—
407	C₄²	—	106	1	no	—
408	C₄²	—	106	1	no	—
409	C₄²	—	106	1	no	—
410	C₄²	—	106	1	no	—
411	C₄²	—	212	1	no	—
412	C₄²	—	212	1	no	—
413	C₄²	C₂×Dic₁₀₆	1	2	yes	—
414	C₄²	#11719945	1	2	yes	—
415	C₄²	D₄×D₅₃	1	2	yes	—
416	C₄²	Q₈×D₅₃	1	2	yes	—
417	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
418	Q₈	—	106	8	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	53	2	no	—
431	C₂×C₄	—	53	2	no	—
432	C₂×C₄	—	53	2	no	—
433	C₂×C₄	—	53	2	no	—
434	C₂×C₄	—	53	2	no	—
435	C₂×C₄	—	53	2	no	—
436	C₂×C₄	—	53	2	no	—
437	C₂×C₄	—	53	2	no	—
438	C₂×C₄	—	106	1	no	—
439	C₂×C₄	—	106	1	no	—
440	C₂×C₄	—	106	1	no	—
441	C₂×C₄	—	106	1	no	—
442	C₂×C₄	—	106	1	no	—
443	C₂×C₄	—	106	1	no	—
444	C₂×C₄	—	106	1	no	—
445	C₂×C₄	—	106	1	no	—
446	C₂×C₄	—	106	1	no	—
447	C₂×C₄	—	106	1	no	—
448	C₂×C₄	—	106	1	no	—
449	C₂×C₄	—	106	1	no	—
450	C₂×C₄	—	106	1	no	—
451	C₂×C₄	—	106	1	no	—
452	C₂×C₄	—	106	1	no	—
453	C₂×C₄	—	106	1	no	—
454	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
455	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
456	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
457	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
458	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
459	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
460	C₂×C₄	#13133225	1	2	yes	—
461	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
462	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
463	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
464	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
465	C₂×C₄	#13133263	1	1	yes	characteristic
466	C₂×C₄	#13133264	1	1	yes	characteristic
467	C₈	—	2	4	no	—
468	C₈	—	4	2	no	—
469	C₈	—	212	2	no	—
470	C₈	—	212	2	no	—
471	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
472	C₂²	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
473	C₂²	#16811103	1	1	yes	characteristic
474	C₂²	C₂×M₁₆×D₅₃	1	1	yes	characteristic
475	C₂²	C₂×M₁₆×D₅₃	1	1	yes	characteristic
476	C₂²	C₂×#13133225	1	1	yes	characteristic
477	C₂²	#16811389	1	1	yes	characteristic
478	C₄	—	2	2	no	—
479	C₄	—	2	2	no	—
480	C₄	—	2	2	no	—
481	C₄	—	2	2	no	—
482	C₄	—	4	1	no	—
483	C₄	—	53	4	no	—
484	C₄	—	106	2	no	—
485	C₄	—	106	2	no	—
486	C₄	—	106	2	no	—
487	C₄	—	106	2	no	—
488	C₄	—	106	2	no	—
489	C₄	—	212	1	no	—
490	C₄	—	212	1	no	—
491	C₄	#16810751	1	4	yes	complemented
492	C₄	D₄×Dic₁₀₆	1	1	yes	characteristic
493	C₄	#16811294	1	1	yes	characteristic
494	C₄	#16811351	1	1	yes	characteristic
495	C₄	Q₈×Dic₁₀₆	1	1	yes	characteristic
496	C₂	C₄⋊C₄×Dic₁₀₆	1	1	yes	characteristic
497	C₂	#423×D₅₃	1	1	yes	characteristic
498	C₂	#20283284	1	1	yes	characteristic
499	C₂	C₂×#16810751	1	1	yes	characteristic
500	C₂	M₁₆×Dic₁₀₆	1	1	yes	characteristic
501	C₂	#20283834	1	1	yes	characteristic
502	C₂	#20283842	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂³×C₄
Fitting	C₅₃×#1767	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1767	D₅₃
hypocenter	C₅₃	#22722
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1767

Complements

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

N	H ≅ G/N	type
98. C₄⋊C₈×C₅₃	Q₈	semidirect
232. C₄×C₅₃	#446	semidirect
295. C₅₃	#22722	semidirect
342. C₄⋊C₈	Dic₁₀₆	semidirect
491. C₄	#16810751	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810751	2
C₄²×Dic₁₀₆	2
#20275438	2
#20275437	2
#20275432	2
#20274867	2
#20274866	2
#20274570	2
C₄⋊C₈×Dic₅₃	2
C₅₃×#1767	2
#22722	53