Subgroups — C₂²×#16811420 #28297895

Order 13568.

11294 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×D₅₃	C₂	1	2	yes	maximal, complemented
2	C₂×#16811420	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#13133249	C₂	1	2	yes	maximal, complemented
4	C₂²×#13133246	C₂	1	2	yes	maximal, complemented
5	C₂²×#13133244	C₂	1	2	yes	maximal, complemented
6	C₂²×#13133241	C₂	1	2	yes	maximal, complemented
7	C₂²×#13133239	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#13133237	C₂	1	2	yes	maximal, complemented
9	C₂³×#11719912	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#135×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#11719949	C₂²	1	2	yes	—
12	C₂²×#11719945	—	2	2	no	—
13	C₂²×#11719945	C₂²	1	2	yes	complemented
14	C₂²×#11719947	—	2	2	no	—
15	C₂²×#11719947	C₂²	1	2	yes	—
16	C₂×#41×D₅₃	—	2	24	no	—
17	C₂×#41×D₅₃	C₂²	1	48	yes	complemented
18	C₂²×Q₈×D₅₃	C₂²	1	2	yes	complemented
19	C₂²×D₄×D₅₃	—	2	2	no	—
20	C₂²×D₄×D₅₃	C₂²	1	2	yes	complemented
21	C₂³×C₄×D₅₃	—	2	2	no	—
22	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
23	#16811420	C₂²	1	256	yes	direct product factor
24	C₂×#13133249	C₂²	1	48	yes	complemented
25	C₂×#13133246	C₂²	1	48	yes	complemented
26	C₂×#13133244	C₂²	1	48	yes	complemented
27	C₂×#13133241	C₂²	1	48	yes	complemented
28	C₂×#13133239	C₂²	1	24	yes	complemented
29	C₂×#13133237	C₂²	1	48	yes	complemented
30	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic, complemented
31	C₂²×#11719925	C₂²	1	2	yes	complemented
32	C₂²×#11719913	C₂²	1	2	yes	complemented
33	C₂²×#11719923	C₂²	1	2	yes	complemented
34	C₂²×#11719924	C₂²	1	2	yes	complemented
35	C₂²×D₄₂₄	C₂²	1	1	yes	characteristic, complemented
36	C₂²×#11719922	C₂²	1	2	yes	complemented
37	C₂²×#11719912	C₂²	1	2	yes	complemented
38	C₂²×#11719912	C₂²	1	2	yes	complemented
39	C₂²×#11719912	C₂²	1	24	yes	complemented
40	C₂³×#88322	C₂²	1	1	yes	characteristic, complemented
41	C₂²×#41×C₅₃	C₂²	1	2	yes	complemented
42	C₂×#135×C₅₃	C₂²	1	24	yes	complemented
43	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
44	C₂²×SD₁₆×C₅₃	C₂²	1	2	yes	complemented
45	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
46	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
47	C₂⁴×D₅₃	—	2	2	no	—
48	C₂⁴×D₅₃	D₄	1	1	yes	characteristic
49	C₂×#11719949	—	2	12	no	—
50	C₂×#11719949	C₂³	1	24	yes	—
51	C₂×#11719945	—	2	12	no	—
52	C₂×#11719945	—	2	48	no	—
53	C₂×#11719945	C₂³	1	24	yes	complemented
54	C₂×#11719947	—	2	48	no	—
55	C₂×#11719947	—	2	12	no	—
56	C₂×#11719947	C₂³	1	24	yes	—
57	#41×D₅₃	—	2	192	no	—
58	#41×D₅₃	C₂³	1	128	yes	complemented
59	C₂²×Dic₁₀₆	—	2	2	no	—
60	C₂²×Dic₁₀₆	C₂³	1	2	yes	—
61	C₂×Q₈×D₅₃	—	2	12	no	—
62	C₂×Q₈×D₅₃	C₂³	1	24	yes	complemented
63	C₂²×D₂₁₂	—	2	2	no	—
64	C₂²×D₂₁₂	C₂³	1	2	yes	—
65	C₂²×C₂²⋊D₅₃	—	2	2	no	—
66	C₂²×C₂²⋊D₅₃	—	2	2	no	—
67	C₂²×C₂²⋊D₅₃	—	2	2	no	—
68	C₂×D₄×D₅₃	—	2	48	no	—
69	C₂×D₄×D₅₃	—	2	12	no	—
70	C₂×D₄×D₅₃	C₂³	1	24	yes	complemented
71	C₂³×Dic₅₃	—	2	2	no	—
72	C₂³×Dic₅₃	D₄	1	1	yes	characteristic
73	C₂²×C₄×D₅₃	—	2	2	no	—
74	C₂²×C₄×D₅₃	—	2	2	no	—
75	C₂²×C₄×D₅₃	—	2	2	no	—
76	C₂²×C₄×D₅₃	—	2	48	no	—
77	C₂²×C₄×D₅₃	D₄	1	2	yes	—
78	C₂²×C₄×D₅₃	D₄	1	12	yes	—
79	C₂²×C₄×D₅₃	C₂³	1	2	yes	—
80	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
81	#13133249	C₂³	1	128	yes	complemented
82	#13133246	C₂³	1	128	yes	complemented
83	#13133244	C₂³	1	128	yes	complemented
84	#13133241	C₂³	1	128	yes	complemented
85	#13133239	C₂³	1	64	yes	complemented
86	#13133237	C₂³	1	128	yes	complemented
87	C₂×Dic₂₁₂	C₂³	1	12	yes	complemented
88	C₂×#11719925	C₂³	1	24	yes	complemented
89	C₂×#11719913	C₂³	1	24	yes	complemented
90	C₂×#11719923	C₂³	1	24	yes	complemented
91	C₂×#11719924	C₂³	1	24	yes	complemented
92	C₂×D₄₂₄	C₂³	1	12	yes	complemented
93	C₂×#11719922	C₂³	1	24	yes	complemented
94	C₂×#11719912	C₂³	1	24	yes	complemented
95	C₂×#11719912	C₂³	1	24	yes	complemented
96	C₂×#11719912	C₂³	1	64	yes	complemented
97	C₂²×#88322	C₂³	1	2	yes	—
98	C₂²×#88322	C₂³	1	12	yes	complemented
99	C₂×#41×C₅₃	—	2	12	no	—
100	C₂×#41×C₅₃	C₂³	1	24	yes	complemented
101	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
102	C₂²×D₄×C₅₃	—	2	2	no	—
103	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
104	C₂³×C₄×C₅₃	—	2	2	no	—
105	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
106	#135×C₅₃	C₂³	1	64	yes	complemented
107	C₂×Q₁₆×C₅₃	C₂³	1	12	yes	complemented
108	C₂×SD₁₆×C₅₃	C₂³	1	24	yes	complemented
109	C₂×D₈×C₅₃	C₂³	1	12	yes	complemented
110	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
111	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
112	C₂²×C₈×C₅₃	C₂³	1	12	yes	complemented
113	C₂⁴×C₅₃	—	2	2	no	—
114	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₂³×D₅₃	—	2	2	no	—
116	C₂³×D₅₃	—	2	24	no	—
117	C₂³×D₅₃	—	4	4	no	—
118	C₂³×D₅₃	—	4	16	no	—
119	C₂³×D₅₃	C₂×D₄	1	2	yes	—
120	C₂³×D₅₃	C₂×D₄	1	12	yes	—
121	#41×C₅₃	—	2	48	no	—
122	#41×C₅₃	C₂⁴	1	32	yes	complemented
123	C₂×Q₈×C₅₃	—	2	6	no	—
124	C₂×Q₈×C₅₃	C₂⁴	1	12	yes	—
125	C₂×D₄×C₅₃	—	2	24	no	—
126	C₂×D₄×C₅₃	—	2	6	no	—
127	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
128	C₂²×C₄×C₅₃	—	2	2	no	—
129	C₂²×C₄×C₅₃	—	2	24	no	—
130	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
131	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
133	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
134	C₂×C₂²⋊D₅₃	—	2	24	no	—
135	C₂×C₂²⋊D₅₃	—	2	24	no	—
136	C₂×C₂²⋊D₅₃	—	2	24	no	—
137	C₂²×Dic₅₃	—	2	2	no	—
138	C₂²×Dic₅₃	—	2	24	no	—
139	C₂²×Dic₅₃	C₂×D₄	1	2	yes	—
140	C₂²×Dic₅₃	C₂×D₄	1	12	yes	—
141	#11719949	—	2	48	no	—
142	#11719949	C₂⁴	1	32	yes	—
143	Q₈×D₅₃	—	2	48	no	—
144	Q₈×D₅₃	C₂⁴	1	32	yes	complemented
145	#11719947	—	2	128	no	—
146	#11719947	—	2	48	no	—
147	#11719947	C₂⁴	1	32	yes	—
148	D₄×D₅₃	—	2	128	no	—
149	D₄×D₅₃	—	2	48	no	—
150	D₄×D₅₃	C₂⁴	1	32	yes	complemented
151	#11719945	—	2	48	no	—
152	#11719945	—	2	128	no	—
153	#11719945	C₂⁴	1	32	yes	complemented
154	C₂×D₂₁₂	—	2	24	no	—
155	C₂×D₂₁₂	—	2	6	no	—
156	C₂×D₂₁₂	C₂⁴	1	12	yes	—
157	C₂×C₄×D₅₃	—	2	24	no	—
158	C₂×C₄×D₅₃	—	2	24	no	—
159	C₂×C₄×D₅₃	—	2	24	no	—
160	C₂×C₄×D₅₃	—	2	128	no	—
161	C₂×C₄×D₅₃	C₂×D₄	1	24	yes	—
162	C₂×C₄×D₅₃	C₂×D₄	1	12	yes	—
163	C₂×C₄×D₅₃	C₂×D₄	1	48	yes	—
164	C₂×C₄×D₅₃	C₂⁴	1	12	yes	—
165	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
166	C₂×Dic₁₀₆	—	2	6	no	—
167	C₂×Dic₁₀₆	—	2	24	no	—
168	C₂×Dic₁₀₆	C₂⁴	1	12	yes	—
169	Q₁₆×C₅₃	C₂⁴	1	16	yes	complemented
170	SD₁₆×C₅₃	C₂⁴	1	32	yes	complemented
171	D₈×C₅₃	C₂⁴	1	16	yes	complemented
172	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
173	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
174	C₂×C₈×C₅₃	C₂⁴	1	16	yes	complemented
175	#11719925	C₂⁴	1	32	yes	complemented
176	#11719924	C₂⁴	1	32	yes	complemented
177	#11719923	C₂⁴	1	32	yes	complemented
178	#11719922	C₂⁴	1	32	yes	complemented
179	C₂×#88322	C₂⁴	1	12	yes	—
180	C₂×#88322	C₂⁴	1	16	yes	complemented
181	Dic₂₁₂	C₂⁴	1	16	yes	complemented
182	D₄₂₄	C₂⁴	1	16	yes	complemented
183	#11719913	C₂⁴	1	32	yes	complemented
184	#11719912	C₂⁴	1	32	yes	complemented
185	#11719912	C₂⁴	1	32	yes	complemented
186	C₂³×C₅₃	—	2	4	no	—
187	C₂³×C₅₃	—	2	12	no	—
188	C₂³×C₅₃	—	4	8	no	—
189	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
190	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
191	C₂²×D₅₃	—	2	8	no	—
192	C₂²×D₅₃	—	2	12	no	—
193	C₂²×D₅₃	—	2	32	no	—
194	C₂²×D₅₃	—	4	24	no	—
195	C₂²×D₅₃	—	4	96	no	—
196	C₂²×D₅₃	—	4	8	no	—
197	C₂²×D₅₃	C₂²×D₄	1	12	yes	—
198	C₂²×D₅₃	C₂²×D₄	1	16	yes	—
199	Q₈×C₅₃	—	2	12	no	—
200	Q₈×C₅₃	C₂⁵	1	8	yes	—
201	D₄×C₅₃	—	2	32	no	—
202	D₄×C₅₃	—	2	12	no	—
203	D₄×C₅₃	C₂⁵	1	8	yes	—
204	C₂×C₄×C₅₃	—	2	12	no	—
205	C₂×C₄×C₅₃	—	2	32	no	—
206	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
207	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
208	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
209	C₂×C₄×C₅₃	C₂⁵	1	3	yes	—
210	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
211	C₂²⋊D₅₃	—	2	32	no	—
212	C₂²⋊D₅₃	—	2	32	no	—
213	C₂²⋊D₅₃	—	2	32	no	—
214	C₂×Dic₅₃	—	2	12	no	—
215	C₂×Dic₅₃	—	2	32	no	—
216	C₂×Dic₅₃	C₂²×D₄	1	12	yes	—
217	C₂×Dic₅₃	C₂²×D₄	1	16	yes	—
218	D₂₁₂	—	2	32	no	—
219	D₂₁₂	—	2	12	no	—
220	D₂₁₂	C₂⁵	1	8	yes	—
221	C₄×D₅₃	—	2	32	no	—
222	C₄×D₅₃	—	2	32	no	—
223	C₄×D₅₃	—	2	32	no	—
224	C₄×D₅₃	C₂²×D₄	1	32	yes	—
225	C₄×D₅₃	C₂²×D₄	1	24	yes	—
226	C₄×D₅₃	C₂⁵	1	8	yes	—
227	Dic₁₀₆	—	2	12	no	—
228	Dic₁₀₆	—	2	32	no	—
229	Dic₁₀₆	C₂⁵	1	8	yes	—
230	C₈×C₅₃	C₂⁵	1	4	yes	—
231	C₈×C₅₃	C₂⁵	1	4	yes	—
232	#88322	C₂⁵	1	8	yes	—
233	C₂²×#575	—	53	1	no	maximal, Sylow
234	C₂²×C₅₃	—	2	12	no	—
235	C₂²×C₅₃	—	2	8	no	—
236	C₂²×C₅₃	—	4	24	no	—
237	C₂²×C₅₃	#575	1	4	yes	complemented
238	C₂²×C₅₃	C₂³×D₄	1	4	yes	—
239	C₂²×C₅₃	C₂³×D₄	1	3	yes	—
240	C₂×D₅₃	—	2	24	no	—
241	C₂×D₅₃	—	2	8	no	—
242	C₂×D₅₃	—	4	16	no	—
243	C₂×D₅₃	—	4	64	no	—
244	C₂×D₅₃	—	4	24	no	—
245	C₂×D₅₃	C₂³×D₄	1	8	yes	—
246	C₄×C₅₃	—	2	8	no	—
247	C₄×C₅₃	C₂³×D₄	1	3	yes	—
248	C₄×C₅₃	C₂³×D₄	1	4	yes	—
249	C₄×C₅₃	C₂⁶	1	1	yes	characteristic
250	Dic₅₃	—	2	8	no	—
251	Dic₅₃	C₂³×D₄	1	8	yes	—
252	C₂³×#41	—	53	2	no	—
253	C₂×#575	—	53	48	no	—
254	C₂²×#137	—	53	2	no	—
255	C₂²×#137	—	53	2	no	—
256	C₂²×AGL(1,ℤ/8)	—	53	2	no	—
257	C₂²×AGL(1,ℤ/8)	—	53	2	no	—
258	C₂²×#135	—	53	2	no	—
259	C₂²×#135	—	53	1	no	—
260	C₂²×#135	D₅₃	1	1	yes	characteristic, complemented
261	C₂³×M₁₆	—	53	1	no	—
262	C₂×C₅₃	—	2	4	no	—
263	C₂×C₅₃	—	4	8	no	—
264	C₂×C₅₃	C₂×#575	1	6	yes	complemented
265	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
266	D₅₃	—	2	8	no	—
267	D₅₃	—	4	8	no	—
268	C₂²×#41	—	53	48	no	—
269	C₂²×#41	—	53	2	no	—
270	C₂²×#41	—	53	2	no	—
271	C₂²×#41	—	53	2	no	—
272	C₂²×#41	—	106	2	no	—
273	C₂²×#41	—	106	2	no	—
274	C₂²×#41	—	106	24	no	—
275	C₂²×#41	C₂×D₅₃	1	2	yes	complemented
276	C₂³×Q₈	—	53	2	no	—
277	C₂³×D₄	—	53	2	no	—
278	C₂³×D₄	—	106	2	no	—
279	C₂⁴×C₄	—	53	1	no	—
280	C₂⁴×C₄	—	106	2	no	—
281	#575	—	53	256	no	—
282	C₂×#137	—	53	48	no	—
283	C₂×#137	—	53	48	no	—
284	C₂×AGL(1,ℤ/8)	—	53	48	no	—
285	C₂×AGL(1,ℤ/8)	—	53	48	no	—
286	C₂×#135	—	53	48	no	—
287	C₂×#135	—	53	24	no	—
288	C₂×#135	C₂×D₅₃	1	24	yes	complemented
289	C₂²×Q₁₆	—	53	2	no	—
290	C₂²×Q₁₆	—	53	1	no	—
291	C₂²×Q₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
292	C₂²×SD₁₆	—	53	2	no	—
293	C₂²×SD₁₆	—	53	2	no	—
294	C₂²×SD₁₆	—	53	2	no	—
295	C₂²×SD₁₆	C₂×D₅₃	1	2	yes	complemented
296	C₂²×D₈	—	53	2	no	—
297	C₂²×D₈	—	53	1	no	—
298	C₂²×D₈	C₂×D₅₃	1	1	yes	characteristic, complemented
299	C₂²×M₁₆	—	53	2	no	—
300	C₂²×M₁₆	—	53	2	no	—
301	C₂²×M₁₆	—	53	24	no	—
302	C₂³×C₈	—	53	1	no	—
303	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic, complemented
304	C₅₃	C₂²×#575	1	1	yes	characteristic, complemented, Sylow
305	C₂⁵	—	53	1	no	—
306	C₂⁵	—	106	2	no	—
307	C₂×#41	—	2	12	no	—
308	C₂×#41	—	53	128	no	—
309	C₂×#41	—	53	24	no	—
310	C₂×#41	—	53	24	no	—
311	C₂×#41	—	53	24	no	—
312	C₂×#41	—	106	48	no	—
313	C₂×#41	—	106	48	no	—
314	C₂×#41	—	106	12	no	—
315	C₂×#41	—	106	12	no	—
316	C₂×#41	—	106	192	no	—
317	C₂×#41	—	106	12	no	—
318	C₂×#41	C₂²×D₅₃	1	24	yes	complemented
319	C₂²×Q₈	—	53	24	no	—
320	C₂²×Q₈	—	53	2	no	—
321	C₂²×Q₈	—	106	2	no	—
322	C₂²×Q₈	—	106	12	no	—
323	C₂²×Q₈	C₂²×D₅₃	1	2	yes	—
324	C₂²×D₄	—	2	2	no	—
325	C₂²×D₄	—	53	24	no	—
326	C₂²×D₄	—	53	2	no	—
327	C₂²×D₄	—	106	2	no	—
328	C₂²×D₄	—	106	12	no	—
329	C₂²×D₄	—	106	2	no	—
330	C₂²×D₄	—	106	48	no	—
331	C₂²×D₄	—	106	2	no	—
332	C₂²×D₄	—	106	2	no	—
333	C₂²×D₄	C₂²×D₅₃	1	2	yes	—
334	C₂³×C₄	—	2	2	no	—
335	C₂³×C₄	—	53	1	no	—
336	C₂³×C₄	—	53	12	no	—
337	C₂³×C₄	—	53	2	no	—
338	C₂³×C₄	—	53	2	no	—
339	C₂³×C₄	—	53	12	no	—
340	C₂³×C₄	—	106	2	no	—
341	C₂³×C₄	—	106	48	no	—
342	C₂³×C₄	—	106	2	no	—
343	C₂³×C₄	—	106	2	no	—
344	C₂³×C₄	—	106	2	no	—
345	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
346	#137	—	53	128	no	—
347	#137	—	53	128	no	—
348	AGL(1,ℤ/8)	—	53	128	no	—
349	AGL(1,ℤ/8)	—	53	128	no	—
350	#135	—	53	128	no	—
351	#135	—	53	64	no	—
352	#135	C₂²×D₅₃	1	64	yes	complemented
353	C₂×Q₁₆	—	53	24	no	—
354	C₂×Q₁₆	—	53	12	no	—
355	C₂×Q₁₆	C₂²×D₅₃	1	12	yes	complemented
356	C₂×SD₁₆	—	53	24	no	—
357	C₂×SD₁₆	—	53	24	no	—
358	C₂×SD₁₆	—	53	24	no	—
359	C₂×SD₁₆	C₂²×D₅₃	1	24	yes	complemented
360	C₂×D₈	—	53	24	no	—
361	C₂×D₈	—	53	12	no	—
362	C₂×D₈	C₂²×D₅₃	1	12	yes	complemented
363	C₂×M₁₆	—	53	24	no	—
364	C₂×M₁₆	—	53	24	no	—
365	C₂×M₁₆	—	53	64	no	—
366	C₂²×C₈	—	53	2	no	—
367	C₂²×C₈	—	53	12	no	—
368	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
369	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
370	C₂²×C₈	C₂²×D₅₃	1	12	yes	complemented
371	C₂⁴	—	2	2	no	—
372	C₂⁴	—	53	2	no	—
373	C₂⁴	—	53	12	no	—
374	C₂⁴	—	106	24	no	—
375	C₂⁴	—	106	2	no	—
376	C₂⁴	—	212	4	no	—
377	C₂⁴	—	212	16	no	—
378	C₂⁴	D₄×D₅₃	1	1	yes	characteristic
379	#41	—	2	48	no	—
380	#41	—	53	32	no	—
381	#41	—	53	32	no	—
382	#41	—	53	32	no	—
383	#41	—	106	128	no	—
384	#41	—	106	128	no	—
385	#41	—	106	48	no	—
386	#41	—	106	48	no	—
387	#41	—	106	48	no	—
388	#41	C₂³×D₅₃	1	32	yes	complemented
389	C₂×Q₈	—	2	6	no	—
390	C₂×Q₈	—	53	32	no	—
391	C₂×Q₈	—	53	12	no	—
392	C₂×Q₈	—	106	24	no	—
393	C₂×Q₈	—	106	48	no	—
394	C₂×Q₈	—	106	6	no	—
395	C₂×Q₈	C₂³×D₅₃	1	12	yes	—
396	C₂×D₄	—	2	24	no	—
397	C₂×D₄	—	2	6	no	—
398	C₂×D₄	—	53	32	no	—
399	C₂×D₄	—	53	12	no	—
400	C₂×D₄	—	106	24	no	—
401	C₂×D₄	—	106	48	no	—
402	C₂×D₄	—	106	24	no	—
403	C₂×D₄	—	106	128	no	—
404	C₂×D₄	—	106	24	no	—
405	C₂×D₄	—	106	24	no	—
406	C₂×D₄	—	106	6	no	—
407	C₂×D₄	C₂³×D₅₃	1	12	yes	—
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	—	2	24	no	—
410	C₂²×C₄	—	53	2	no	—
411	C₂²×C₄	—	53	12	no	—
412	C₂²×C₄	—	53	24	no	—
413	C₂²×C₄	—	53	12	no	—
414	C₂²×C₄	—	53	48	no	—
415	C₂²×C₄	—	53	12	no	—
416	C₂²×C₄	—	53	16	no	—
417	C₂²×C₄	—	106	24	no	—
418	C₂²×C₄	—	106	128	no	—
419	C₂²×C₄	—	106	24	no	—
420	C₂²×C₄	—	106	24	no	—
421	C₂²×C₄	—	106	2	no	—
422	C₂²×C₄	—	106	24	no	—
423	C₂²×C₄	D₄×D₅₃	1	6	yes	—
424	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
425	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
426	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
427	Q₁₆	—	53	32	no	—
428	Q₁₆	—	53	16	no	—
429	Q₁₆	C₂³×D₅₃	1	16	yes	complemented
430	SD₁₆	—	53	32	no	—
431	SD₁₆	—	53	32	no	—
432	SD₁₆	—	53	32	no	—
433	SD₁₆	C₂³×D₅₃	1	32	yes	complemented
434	D₈	—	53	32	no	—
435	D₈	—	53	16	no	—
436	D₈	C₂³×D₅₃	1	16	yes	complemented
437	M₁₆	—	53	32	no	—
438	M₁₆	—	53	32	no	—
439	C₂×C₈	—	53	12	no	—
440	C₂×C₈	—	53	16	no	—
441	C₂×C₈	C₂³×D₅₃	1	6	yes	—
442	C₂×C₈	C₂³×D₅₃	1	6	yes	—
443	C₂×C₈	C₂³×D₅₃	1	16	yes	complemented
444	C₂³	—	2	4	no	—
445	C₂³	—	2	12	no	—
446	C₂³	—	4	8	no	—
447	C₂³	—	53	12	no	—
448	C₂³	—	53	16	no	—
449	C₂³	—	106	32	no	—
450	C₂³	—	106	8	no	—
451	C₂³	—	106	12	no	—
452	C₂³	—	212	24	no	—
453	C₂³	—	212	96	no	—
454	C₂³	—	212	8	no	—
455	C₂³	C₂×D₄×D₅₃	1	6	yes	—
456	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
457	Q₈	—	2	12	no	—
458	Q₈	—	53	8	no	—
459	Q₈	—	106	32	no	—
460	Q₈	—	106	12	no	—
461	Q₈	C₂⁴×D₅₃	1	8	yes	—
462	D₄	—	2	32	no	—
463	D₄	—	2	12	no	—
464	D₄	—	53	8	no	—
465	D₄	—	106	32	no	—
466	D₄	—	106	32	no	—
467	D₄	—	106	32	no	—
468	D₄	—	106	32	no	—
469	D₄	—	106	12	no	—
470	D₄	C₂⁴×D₅₃	1	8	yes	—
471	C₂×C₄	—	2	12	no	—
472	C₂×C₄	—	2	32	no	—
473	C₂×C₄	—	53	12	no	—
474	C₂×C₄	—	53	16	no	—
475	C₂×C₄	—	53	32	no	—
476	C₂×C₄	—	53	24	no	—
477	C₂×C₄	—	53	8	no	—
478	C₂×C₄	—	106	32	no	—
479	C₂×C₄	—	106	32	no	—
480	C₂×C₄	—	106	32	no	—
481	C₂×C₄	—	106	12	no	—
482	C₂×C₄	—	106	32	no	—
483	C₂×C₄	C₂×D₄×D₅₃	1	6	yes	—
484	C₂×C₄	C₂×D₄×D₅₃	1	3	yes	—
485	C₂×C₄	C₂×D₄×D₅₃	1	12	yes	—
486	C₂×C₄	C₂⁴×D₅₃	1	3	yes	—
487	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
488	C₈	—	53	8	no	—
489	C₈	C₂⁴×D₅₃	1	4	yes	—
490	C₈	C₂⁴×D₅₃	1	4	yes	—
491	C₂²	—	2	12	no	—
492	C₂²	—	2	8	no	—
493	C₂²	—	4	24	no	—
494	C₂²	—	53	8	no	—
495	C₂²	—	106	24	no	—
496	C₂²	—	106	8	no	—
497	C₂²	—	212	16	no	—
498	C₂²	—	212	64	no	—
499	C₂²	—	212	24	no	—
500	C₂²	#16811420	1	4	yes	direct product factor
501	C₂²	C₂²×D₄×D₅₃	1	4	yes	—
502	C₂²	C₂²×D₄×D₅₃	1	3	yes	—
503	C₄	—	2	8	no	—
504	C₄	—	53	8	no	—
505	C₄	—	106	8	no	—
506	C₄	C₂²×D₄×D₅₃	1	4	yes	—
507	C₄	C₂²×D₄×D₅₃	1	3	yes	—
508	C₄	C₂⁵×D₅₃	1	1	yes	characteristic
509	C₂	—	2	4	no	—
510	C₂	—	4	8	no	—
511	C₂	—	106	8	no	—
512	C₂	—	212	8	no	—
513	C₂	C₂×#16811420	1	6	yes	direct product factor
514	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator, cosocle	C₄×C₅₃	C₂⁶
Frattini	C₄	C₂⁵×D₅₃
Fitting	C₂²×#135×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×D₄
hypercenter	C₂²×#135	D₅₃
hypocenter	C₅₃	C₂²×#575
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#135

Complements

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

N	H ≅ G/N	type
1. C₂²×#41×D₅₃	C₂	semidirect
2. C₂×#16811420	C₂	direct (N × H)
3. C₂²×#13133249	C₂	semidirect
4. C₂²×#13133246	C₂	semidirect
5. C₂²×#13133244	C₂	semidirect
6. C₂²×#13133241	C₂	semidirect
7. C₂²×#13133239	C₂	semidirect
8. C₂²×#13133237	C₂	semidirect
9. C₂³×#11719912	C₂	semidirect
10. C₂²×#135×C₅₃	C₂	semidirect
13. C₂²×#11719945	C₂²	semidirect
17. C₂×#41×D₅₃	C₂²	semidirect
18. C₂²×Q₈×D₅₃	C₂²	semidirect
20. C₂²×D₄×D₅₃	C₂²	semidirect
23. #16811420	C₂²	direct (N × H)
24. C₂×#13133249	C₂²	semidirect
25. C₂×#13133246	C₂²	semidirect
26. C₂×#13133244	C₂²	semidirect
27. C₂×#13133241	C₂²	semidirect
28. C₂×#13133239	C₂²	semidirect
29. C₂×#13133237	C₂²	semidirect
30. C₂²×Dic₂₁₂	C₂²	semidirect
31. C₂²×#11719925	C₂²	semidirect
32. C₂²×#11719913	C₂²	semidirect
33. C₂²×#11719923	C₂²	semidirect
34. C₂²×#11719924	C₂²	semidirect
35. C₂²×D₄₂₄	C₂²	semidirect
36. C₂²×#11719922	C₂²	semidirect
37. C₂²×#11719912	C₂²	semidirect
38. C₂²×#11719912	C₂²	semidirect
39. C₂²×#11719912	C₂²	semidirect
40. C₂³×#88322	C₂²	semidirect
41. C₂²×#41×C₅₃	C₂²	semidirect
42. C₂×#135×C₅₃	C₂²	semidirect
43. C₂²×Q₁₆×C₅₃	C₂²	semidirect
44. C₂²×SD₁₆×C₅₃	C₂²	semidirect
45. C₂²×D₈×C₅₃	C₂²	semidirect
46. C₂³×C₈×C₅₃	C₂²	semidirect
53. C₂×#11719945	C₂³	semidirect
58. #41×D₅₃	C₂³	semidirect
62. C₂×Q₈×D₅₃	C₂³	semidirect
70. C₂×D₄×D₅₃	C₂³	semidirect
81. #13133249	C₂³	semidirect
82. #13133246	C₂³	semidirect
83. #13133244	C₂³	semidirect
84. #13133241	C₂³	semidirect
85. #13133239	C₂³	semidirect
86. #13133237	C₂³	semidirect
87. C₂×Dic₂₁₂	C₂³	semidirect
88. C₂×#11719925	C₂³	semidirect
89. C₂×#11719913	C₂³	semidirect
90. C₂×#11719923	C₂³	semidirect
91. C₂×#11719924	C₂³	semidirect
92. C₂×D₄₂₄	C₂³	semidirect
93. C₂×#11719922	C₂³	semidirect
94. C₂×#11719912	C₂³	semidirect
95. C₂×#11719912	C₂³	semidirect
96. C₂×#11719912	C₂³	semidirect
98. C₂²×#88322	C₂³	semidirect
100. C₂×#41×C₅₃	C₂³	semidirect
106. #135×C₅₃	C₂³	semidirect
107. C₂×Q₁₆×C₅₃	C₂³	semidirect
108. C₂×SD₁₆×C₅₃	C₂³	semidirect
109. C₂×D₈×C₅₃	C₂³	semidirect
112. C₂²×C₈×C₅₃	C₂³	semidirect
122. #41×C₅₃	C₂⁴	semidirect
144. Q₈×D₅₃	C₂⁴	semidirect
150. D₄×D₅₃	C₂⁴	semidirect
153. #11719945	C₂⁴	semidirect
169. Q₁₆×C₅₃	C₂⁴	semidirect
170. SD₁₆×C₅₃	C₂⁴	semidirect
171. D₈×C₅₃	C₂⁴	semidirect
174. C₂×C₈×C₅₃	C₂⁴	semidirect
175. #11719925	C₂⁴	semidirect
176. #11719924	C₂⁴	semidirect
177. #11719923	C₂⁴	semidirect
178. #11719922	C₂⁴	semidirect
180. C₂×#88322	C₂⁴	semidirect
181. Dic₂₁₂	C₂⁴	semidirect
182. D₄₂₄	C₂⁴	semidirect
183. #11719913	C₂⁴	semidirect
184. #11719912	C₂⁴	semidirect
185. #11719912	C₂⁴	semidirect
237. C₂²×C₅₃	#575	semidirect
260. C₂²×#135	D₅₃	semidirect
264. C₂×C₅₃	C₂×#575	semidirect
275. C₂²×#41	C₂×D₅₃	semidirect
288. C₂×#135	C₂×D₅₃	semidirect
291. C₂²×Q₁₆	C₂×D₅₃	semidirect
295. C₂²×SD₁₆	C₂×D₅₃	semidirect
298. C₂²×D₈	C₂×D₅₃	semidirect
303. C₂³×C₈	C₂×D₅₃	semidirect
304. C₅₃	C₂²×#575	semidirect
318. C₂×#41	C₂²×D₅₃	semidirect
352. #135	C₂²×D₅₃	semidirect
355. C₂×Q₁₆	C₂²×D₅₃	semidirect
359. C₂×SD₁₆	C₂²×D₅₃	semidirect
362. C₂×D₈	C₂²×D₅₃	semidirect
370. C₂²×C₈	C₂²×D₅₃	semidirect
388. #41	C₂³×D₅₃	semidirect
429. Q₁₆	C₂³×D₅₃	semidirect
433. SD₁₆	C₂³×D₅₃	semidirect
436. D₈	C₂³×D₅₃	semidirect
443. C₂×C₈	C₂³×D₅₃	semidirect
500. C₂²	#16811420	direct (N × H)
513. C₂	C₂×#16811420	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×#41×D₅₃	2
C₂×#16811420	2
C₂²×#13133249	2
C₂²×#13133246	2
C₂²×#13133244	2
C₂²×#13133241	2
C₂²×#13133239	2
C₂²×#13133237	2
C₂³×#11719912	2
C₂²×#135×C₅₃	2
C₂²×#575	53

Subgroups — C22×#16811420 #28297895

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811420 #28297895