Subgroups — C₂×#20288205 #28291054

Order 13568.

12314 conjugacy classes of subgroups.

The subgroup lattice has 584 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	#20288205	C₂	1	32	yes	maximal, direct product factor
3	C₂×Q₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16811280	C₂	1	2	yes	maximal, complemented
5	C₂×#16811278	C₂	1	3	yes	maximal, complemented
6	C₂×#16811270	C₂	1	6	yes	maximal, complemented
7	C₂×#16811258	C₂	1	3	yes	maximal, complemented
8	C₂×#16811171	C₂	1	3	yes	maximal, complemented
9	C₂×#16811133	C₂	1	6	yes	maximal, complemented
10	C₂×C₄×#11719949	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×D₄×D₅₃	C₂	1	3	yes	maximal, complemented
12	C₂×C₅₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#11719949	C₂²	1	1	yes	characteristic
14	C₂²×#11719945	C₂²	1	6	yes	complemented
15	C₂²×#11719947	C₂²	1	6	yes	complemented
16	C₂×#41×D₅₃	—	2	32	no	—
17	C₂×#41×D₅₃	C₂²	1	32	yes	complemented
18	C₂²×Q₈×D₅₃	C₂²	1	2	yes	complemented
19	C₂²×D₄×D₅₃	C₂²	1	3	yes	—
20	C₂²×D₄×D₅₃	C₂²	1	6	yes	complemented
21	C₂³×C₄×D₅₃	C₂²	1	6	yes	complemented
22	Q₈×D₂₁₂	C₂²	1	16	yes	complemented
23	#16811280	C₂²	1	32	yes	complemented
24	#16811278	C₂²	1	48	yes	complemented
25	#16811270	C₂²	1	96	yes	complemented
26	#16811258	C₂²	1	48	yes	complemented
27	#16811171	C₂²	1	48	yes	complemented
28	C₂×#13133217	C₂²	1	3	yes	complemented
29	C₂×#13133204	C₂²	1	6	yes	complemented
30	#16811133	C₂²	1	96	yes	complemented
31	C₂×#13133199	C₂²	1	12	yes	complemented
32	C₂×#13133197	C₂²	1	6	yes	complemented
33	C₂×#13133196	C₂²	1	6	yes	complemented
34	C₂×#13133188	C₂²	1	6	yes	complemented
35	C₂×#13133188	C₂²	1	6	yes	complemented
36	C₂×#13133188	C₂²	1	6	yes	complemented
37	C₂×#13133185	C₂²	1	12	yes	complemented
38	C₄×#11719949	C₂²	1	16	yes	complemented
39	C₂×Q₈×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
40	C₂×C₄×D₂₁₂	C₂²	1	3	yes	—
41	C₂×C₄×D₂₁₂	C₂²	1	3	yes	complemented
42	C₂×D₄×Dic₅₃	C₂²	1	3	yes	complemented
43	C₂×D₄×Dic₅₃	C₂²	1	6	yes	complemented
44	C₂×#13133179	C₂²	1	1	yes	characteristic
45	C₂×#13133179	C₂²	1	3	yes	complemented
46	C₂×C₄×C₂²⋊D₅₃	C₂²	1	6	yes	complemented
47	C₂×#13133177	C₂²	1	6	yes	complemented
48	C₄×D₄×D₅₃	C₂²	1	48	yes	complemented
49	C₂×#13133175	C₂²	1	3	yes	complemented
50	C₂×C₄⋊C₄×D₅₃	C₂²	1	3	yes	complemented
51	C₂×C₂²⋊C₄×D₅₃	C₂²	1	6	yes	complemented
52	C₂×C₄²×D₅₃	C₂²	1	3	yes	complemented
53	C₂²×#41×C₅₃	C₂²	1	2	yes	complemented
54	C₅₃×#550	C₂²	1	16	yes	complemented
55	C₂×#127×C₅₃	C₂²	1	3	yes	complemented
56	C₂×#121×C₅₃	C₂²	1	6	yes	complemented
57	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
58	C₂×C₄×D₄×C₅₃	C₂²	1	3	yes	complemented
59	C₂⁴×D₅₃	C₂³	1	6	yes	—
60	C₂×#11719949	D₄	1	16	yes	complemented
61	C₂×#11719949	C₂³	1	8	yes	—
62	C₂×#11719945	—	2	48	no	—
63	C₂×#11719945	C₂³	1	48	yes	complemented
64	C₂×#11719947	—	2	48	no	—
65	C₂×#11719947	C₂³	1	48	yes	complemented
66	#41×D₅₃	—	2	256	no	—
67	C₂²×Dic₁₀₆	C₂³	1	6	yes	—
68	C₂×Q₈×D₅₃	—	2	16	no	—
69	C₂×Q₈×D₅₃	C₂³	1	16	yes	complemented
70	C₂²×D₂₁₂	C₂³	1	3	yes	—
71	C₂²×D₂₁₂	C₂³	1	1	yes	characteristic
72	C₂²×C₂²⋊D₅₃	C₂³	1	6	yes	—
73	C₂²×C₂²⋊D₅₃	C₂³	1	12	yes	—
74	C₂×D₄×D₅₃	—	2	48	no	—
75	C₂×D₄×D₅₃	—	2	24	no	—
76	C₂×D₄×D₅₃	C₂³	1	24	yes	—
77	C₂×D₄×D₅₃	C₂³	1	48	yes	complemented
78	C₂³×Dic₅₃	C₂³	1	6	yes	—
79	C₂²×C₄×D₅₃	—	2	48	no	—
80	C₂²×C₄×D₅₃	C₂³	1	3	yes	—
81	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
82	C₂²×C₄×D₅₃	C₂³	1	2	yes	—
83	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
84	C₂²×C₄×D₅₃	C₂³	1	3	yes	—
85	C₂²×C₄×D₅₃	C₂³	1	3	yes	—
86	C₂²×C₄×D₅₃	C₂³	1	48	yes	complemented
87	#13133217	C₂³	1	24	yes	complemented
88	#13133204	C₂³	1	48	yes	complemented
89	#13133199	C₂³	1	96	yes	complemented
90	#13133197	C₂³	1	48	yes	complemented
91	#13133196	C₂³	1	48	yes	complemented
92	#13133188	C₂³	1	48	yes	complemented
93	#13133188	C₂³	1	48	yes	complemented
94	#13133188	C₂³	1	48	yes	complemented
95	#13133185	C₂³	1	96	yes	complemented
96	Q₈×Dic₅₃	C₂³	1	8	yes	complemented
97	C₄×D₂₁₂	C₂³	1	24	yes	—
98	C₄×D₂₁₂	C₂³	1	24	yes	complemented
99	D₄×Dic₅₃	C₂³	1	24	yes	complemented
100	D₄×Dic₅₃	C₂³	1	48	yes	complemented
101	#13133179	C₂³	1	8	yes	—
102	#13133179	C₂³	1	24	yes	complemented
103	C₄×C₂²⋊D₅₃	C₂³	1	48	yes	complemented
104	#13133177	C₂³	1	48	yes	complemented
105	#13133175	C₂³	1	24	yes	complemented
106	C₂×C₄⋊Dic₅₃	C₂³	1	3	yes	—
107	C₂×C₄⋊Dic₅₃	C₂³	1	3	yes	—
108	C₂×C₄⋊Dic₅₃	C₂³	1	6	yes	—
109	C₂×#11719919	C₂³	1	6	yes	—
110	C₄⋊C₄×D₅₃	C₂³	1	24	yes	complemented
111	C₂×C₂²⋊Dic₅₃	C₂³	1	6	yes	—
112	C₂×C₂²⋊Dic₅₃	C₂³	1	12	yes	—
113	C₂×#11719921	C₂³	1	2	yes	—
114	C₂×#11719921	C₂³	1	6	yes	—
115	C₂×#11719921	C₂³	1	6	yes	—
116	C₂²⋊C₄×D₅₃	C₂³	1	48	yes	complemented
117	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
118	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
119	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
120	C₄²×D₅₃	C₂³	1	24	yes	complemented
121	C₂×#41×C₅₃	—	2	16	no	—
122	C₂×#41×C₅₃	C₂³	1	16	yes	complemented
123	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
124	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
125	C₂²×D₄×C₅₃	C₂³	1	6	yes	—
126	C₂²×D₄×C₅₃	C₂³	1	6	yes	—
127	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
128	#127×C₅₃	C₂³	1	24	yes	complemented
129	#121×C₅₃	C₂³	1	48	yes	complemented
130	C₄×Q₈×C₅₃	C₂³	1	8	yes	complemented
131	C₄×D₄×C₅₃	C₂³	1	24	yes	complemented
132	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
133	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
134	C₂×C₂²⋊C₄×C₅₃	C₂³	1	6	yes	—
135	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
136	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
137	C₂³×D₅₃	—	2	24	no	—
138	C₂³×D₅₃	—	2	24	no	—
139	C₂³×D₅₃	—	4	12	no	—
140	C₂³×D₅₃	C₂⁴	1	24	yes	—
141	C₂³×D₅₃	C₂⁴	1	3	yes	—
142	C₂³×D₅₃	C₂⁴	1	2	yes	—
143	#41×C₅₃	—	2	64	no	—
144	C₂×Q₈×C₅₃	C₂×D₄	1	8	yes	complemented
145	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
146	C₂×D₄×C₅₃	—	2	24	no	—
147	C₂×D₄×C₅₃	—	2	12	no	—
148	C₂×D₄×C₅₃	—	4	12	no	—
149	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
150	C₂×D₄×C₅₃	C₂⁴	1	24	yes	—
151	C₂×D₄×C₅₃	C₂⁴	1	24	yes	—
152	C₂²×C₄×C₅₃	—	2	24	no	—
153	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
154	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
156	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
157	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
158	C₂×C₂²⋊D₅₃	—	2	48	no	—
159	C₂×C₂²⋊D₅₃	—	2	24	no	—
160	C₂×C₂²⋊D₅₃	C₂⁴	1	24	yes	—
161	C₂×C₂²⋊D₅₃	C₂⁴	1	48	yes	—
162	C₂²×Dic₅₃	—	2	24	no	—
163	C₂²×Dic₅₃	C₂⁴	1	1	yes	characteristic
164	C₂²×Dic₅₃	C₂⁴	1	3	yes	—
165	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
166	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
167	C₂²×Dic₅₃	C₂⁴	1	1	yes	characteristic
168	#11719949	C₂×D₄	1	64	yes	complemented
169	Q₈×D₅₃	—	2	64	no	—
170	#11719947	—	2	192	no	—
171	D₄×D₅₃	—	2	192	no	—
172	D₄×D₅₃	—	2	96	no	—
173	#11719945	—	2	192	no	—
174	C₂×D₂₁₂	C₂×D₄	1	24	yes	complemented
175	C₂×D₂₁₂	#41	1	8	yes	complemented
176	C₂×D₂₁₂	C₂⁴	1	12	yes	—
177	C₂×D₂₁₂	C₂⁴	1	4	yes	—
178	C₂×C₄×D₅₃	—	2	24	no	—
179	C₂×C₄×D₅₃	—	2	24	no	—
180	C₂×C₄×D₅₃	—	2	12	no	—
181	C₂×C₄×D₅₃	—	2	8	no	—
182	C₂×C₄×D₅₃	—	2	192	no	—
183	C₂×C₄×D₅₃	—	4	6	no	—
184	C₂×C₄×D₅₃	C₂×D₄	1	24	yes	complemented
185	C₂×C₄×D₅₃	C₂⁴	1	12	yes	—
186	C₂×C₄×D₅₃	C₂⁴	1	24	yes	—
187	C₂×C₄×D₅₃	C₂⁴	1	8	yes	—
188	C₂×C₄×D₅₃	C₂⁴	1	24	yes	—
189	C₂×C₄×D₅₃	C₂⁴	1	12	yes	—
190	C₂×C₄×D₅₃	C₂⁴	1	12	yes	—
191	C₂×Dic₁₀₆	—	2	24	no	—
192	C₂×Dic₁₀₆	C₂⁴	1	24	yes	—
193	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
194	C₄⋊C₄×C₅₃	C₂⁴	1	12	yes	—
195	C₂²⋊C₄×C₅₃	C₂⁴	1	24	yes	—
196	C₄²×C₅₃	C₂⁴	1	12	yes	—
197	C₂²⋊Dic₅₃	C₂⁴	1	24	yes	—
198	C₂²⋊Dic₅₃	C₂⁴	1	48	yes	—
199	#11719921	C₂⁴	1	8	yes	—
200	#11719921	C₂⁴	1	24	yes	—
201	#11719921	C₂⁴	1	24	yes	—
202	C₄⋊Dic₅₃	C₂⁴	1	12	yes	—
203	C₄⋊Dic₅₃	C₂⁴	1	12	yes	—
204	C₄⋊Dic₅₃	C₂⁴	1	24	yes	—
205	#11719919	C₂⁴	1	24	yes	—
206	C₄×Dic₅₃	C₂⁴	1	4	yes	—
207	C₄×Dic₅₃	C₂⁴	1	12	yes	—
208	C₄×Dic₅₃	C₂⁴	1	12	yes	—
209	C₂³×C₅₃	—	2	12	no	—
210	C₂³×C₅₃	—	2	12	no	—
211	C₂³×C₅₃	—	2	12	no	—
212	C₂³×C₅₃	C₂⁵	1	12	yes	—
213	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
214	C₂²×D₅₃	—	2	4	no	—
215	C₂²×D₅₃	—	2	6	no	—
216	C₂²×D₅₃	—	2	6	no	—
217	C₂²×D₅₃	—	2	4	no	—
218	C₂²×D₅₃	—	2	48	no	—
219	C₂²×D₅₃	—	2	48	no	—
220	C₂²×D₅₃	—	4	24	no	—
221	C₂²×D₅₃	—	4	96	no	—
222	C₂²×D₅₃	C₂²×D₄	1	12	yes	—
223	C₂²×D₅₃	C₂×#41	1	8	yes	complemented
224	C₂²×D₅₃	C₂⁵	1	6	yes	—
225	C₂²×D₅₃	C₂⁵	1	4	yes	—
226	Q₈×C₅₃	C₂²×D₄	1	16	yes	complemented
227	D₄×C₅₃	—	2	48	no	—
228	D₄×C₅₃	—	2	24	no	—
229	D₄×C₅₃	—	4	24	no	—
230	C₂×C₄×C₅₃	—	2	4	no	—
231	C₂×C₄×C₅₃	—	2	6	no	—
232	C₂×C₄×C₅₃	—	2	48	no	—
233	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
234	C₂×C₄×C₅₃	C₂×#41	1	4	yes	complemented
235	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
236	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
237	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
238	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
239	C₂²⋊D₅₃	—	2	96	no	—
240	C₂²⋊D₅₃	—	2	48	no	—
241	C₂×Dic₅₃	—	2	12	no	—
242	C₂×Dic₅₃	—	2	6	no	—
243	C₂×Dic₅₃	—	2	2	no	—
244	C₂×Dic₅₃	—	2	48	no	—
245	C₂×Dic₅₃	C₂²×D₄	1	4	yes	—
246	C₂×Dic₅₃	C₂⁵	1	12	yes	—
247	C₂×Dic₅₃	C₂⁵	1	2	yes	—
248	C₂×Dic₅₃	C₂⁵	1	6	yes	—
249	C₂×Dic₅₃	C₂⁵	1	2	yes	—
250	D₂₁₂	C₂²×D₄	1	48	yes	complemented
251	D₂₁₂	C₂×#41	1	16	yes	complemented
252	C₄×D₅₃	—	2	48	no	—
253	C₄×D₅₃	—	2	48	no	—
254	C₄×D₅₃	—	2	24	no	—
255	C₄×D₅₃	—	2	16	no	—
256	C₄×D₅₃	—	4	12	no	—
257	C₄×D₅₃	C₂²×D₄	1	48	yes	complemented
258	Dic₁₀₆	—	2	48	no	—
259	C₂×D₄×#41	—	53	1	no	maximal, Sylow
260	C₂²×C₅₃	—	2	12	no	—
261	C₂²×C₅₃	—	2	12	no	—
262	C₂²×C₅₃	—	2	12	no	—
263	C₂²×C₅₃	—	4	24	no	—
264	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
265	C₂²×C₅₃	C₂²×#41	1	2	yes	—
266	C₂²×C₅₃	#585	1	2	yes	—
267	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
268	C₂×D₅₃	—	2	4	no	—
269	C₂×D₅₃	—	2	6	no	—
270	C₂×D₅₃	—	2	6	no	—
271	C₂×D₅₃	—	2	4	no	—
272	C₂×D₅₃	—	2	16	no	—
273	C₂×D₅₃	—	2	24	no	—
274	C₂×D₅₃	—	4	96	no	—
275	C₂×D₅₃	C₂³×D₄	1	12	yes	—
276	C₂×D₅₃	C₂²×#41	1	8	yes	complemented
277	C₄×C₅₃	—	2	4	no	—
278	C₄×C₅₃	—	2	6	no	—
279	C₄×C₅₃	C₂³×D₄	1	12	yes	—
280	C₄×C₅₃	C₂²×#41	1	4	yes	complemented
281	Dic₅₃	—	2	12	no	—
282	Dic₅₃	—	2	6	no	—
283	Dic₅₃	—	2	2	no	—
284	Dic₅₃	C₂³×D₄	1	4	yes	—
285	C₂³×#41	—	53	2	no	—
286	D₄×#41	—	53	32	no	—
287	C₂×#550	D₅₃	1	1	yes	characteristic, complemented
288	C₂×D₄×Q₈	—	53	1	no	—
289	C₂×#548	—	53	2	no	—
290	C₂×#547	—	53	3	no	—
291	C₂×#546	—	53	6	no	—
292	C₂×D₄²	—	53	3	no	—
293	C₂×#532	—	53	3	no	—
294	C₂×#525	—	53	6	no	—
295	C₂×C₄×#41	—	53	1	no	—
296	C₂²×C₄×D₄	—	53	3	no	—
297	C₂×C₅₃	—	4	12	no	—
298	C₂×C₅₃	D₄×#41	1	4	yes	complemented
299	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
300	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
301	C₂×C₅₃	C₂×#585	1	1	yes	characteristic
302	D₅₃	—	2	8	no	—
303	D₅₃	—	2	12	no	—
304	C₂²×#41	—	53	32	no	—
305	C₂²×#41	—	53	1	no	—
306	C₂²×#41	—	53	6	no	—
307	C₂²×#41	—	53	6	no	—
308	C₂²×#41	—	106	32	no	—
309	C₂²×#41	C₂×D₅₃	1	2	yes	complemented
310	C₂³×Q₈	—	53	2	no	—
311	C₂³×D₄	—	53	6	no	—
312	C₂³×D₄	—	53	3	no	—
313	C₂⁴×C₄	—	53	6	no	—
314	#550	C₂×D₅₃	1	16	yes	complemented
315	D₄×Q₈	—	53	16	no	—
316	#548	—	53	32	no	—
317	#547	—	53	48	no	—
318	#546	—	53	96	no	—
319	D₄²	—	53	48	no	—
320	#532	—	53	48	no	—
321	C₂×C₄⋊Q₈	—	53	3	no	—
322	C₂×#127	C₂×D₅₃	1	3	yes	complemented
323	C₂×#124	—	53	6	no	—
324	#525	—	53	96	no	—
325	C₂×#123	—	53	12	no	—
326	C₂×C₂²⋊Q₈	—	53	6	no	—
327	C₂×C₂²⋊Q₈	—	53	6	no	—
328	C₂×#121	—	53	6	no	—
329	C₂×#121	—	53	6	no	—
330	C₂×#121	—	53	6	no	—
331	C₂×#121	C₂×D₅₃	1	6	yes	complemented
332	C₂×#120	—	53	12	no	—
333	C₄×#41	—	53	16	no	—
334	C₂×C₄×Q₈	—	53	1	no	—
335	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
336	C₂×C₄×D₄	—	53	3	no	—
337	C₂×C₄×D₄	—	53	6	no	—
338	C₂×C₄×D₄	—	53	6	no	—
339	C₂×C₄×D₄	—	53	48	no	—
340	C₂×C₄×D₄	—	53	1	no	—
341	C₂×C₄×D₄	—	53	3	no	—
342	C₂×C₄×D₄	—	53	6	no	—
343	C₂×C₄×D₄	—	53	3	no	—
344	C₂×C₄×D₄	—	53	3	no	—
345	C₂×C₄×D₄	C₂×D₅₃	1	3	yes	complemented
346	C₂×#117	—	53	3	no	—
347	C₂²×C₄⋊C₄	—	53	3	no	—
348	C₂²×C₂²⋊C₄	—	53	6	no	—
349	C₂²×C₄²	—	53	3	no	—
350	C₅₃	C₂×D₄×#41	1	1	yes	characteristic, complemented, Sylow
351	C₂⁵	—	53	6	no	—
352	C₂×#41	—	2	16	no	—
353	C₂×#41	—	53	16	no	—
354	C₂×#41	—	53	8	no	—
355	C₂×#41	—	53	48	no	—
356	C₂×#41	—	53	48	no	—
357	C₂×#41	—	106	48	no	—
358	C₂×#41	—	106	48	no	—
359	C₂×#41	—	106	256	no	—
360	C₂×#41	C₂²×D₅₃	1	16	yes	complemented
361	C₂²×Q₈	—	53	16	no	—
362	C₂²×Q₈	—	53	6	no	—
363	C₂²×Q₈	—	106	16	no	—
364	C₂²×Q₈	C₂²×D₅₃	1	1	yes	characteristic
365	C₂²×D₄	—	53	6	no	—
366	C₂²×D₄	—	53	12	no	—
367	C₂²×D₄	—	53	24	no	—
368	C₂²×D₄	—	53	48	no	—
369	C₂²×D₄	—	53	1	no	—
370	C₂²×D₄	—	53	3	no	—
371	C₂²×D₄	—	106	48	no	—
372	C₂²×D₄	—	106	24	no	—
373	C₂²×D₄	C₂²×D₅₃	1	3	yes	—
374	C₂²×D₄	C₂²×D₅₃	1	6	yes	—
375	C₂²×D₄	C₂²×D₅₃	1	6	yes	—
376	C₂³×C₄	—	53	3	no	—
377	C₂³×C₄	—	53	6	no	—
378	C₂³×C₄	—	53	6	no	—
379	C₂³×C₄	—	53	3	no	—
380	C₂³×C₄	—	53	2	no	—
381	C₂³×C₄	—	53	6	no	—
382	C₂³×C₄	—	53	48	no	—
383	C₂³×C₄	—	53	3	no	—
384	C₂³×C₄	—	106	48	no	—
385	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
386	C₄⋊Q₈	—	53	24	no	—
387	#127	C₂²×D₅₃	1	24	yes	complemented
388	#124	—	53	48	no	—
389	#123	—	53	96	no	—
390	C₂²⋊Q₈	—	53	48	no	—
391	C₂²⋊Q₈	—	53	48	no	—
392	#121	—	53	48	no	—
393	#121	—	53	48	no	—
394	#121	—	53	48	no	—
395	#121	C₂²×D₅₃	1	48	yes	complemented
396	#120	—	53	96	no	—
397	C₄×Q₈	—	53	8	no	—
398	C₄×Q₈	C₂²×D₅₃	1	8	yes	complemented
399	C₄×D₄	—	53	24	no	—
400	C₄×D₄	—	53	48	no	—
401	C₄×D₄	—	53	48	no	—
402	C₄×D₄	—	53	8	no	—
403	C₄×D₄	—	53	24	no	—
404	C₄×D₄	—	53	48	no	—
405	C₄×D₄	—	53	24	no	—
406	C₄×D₄	—	53	24	no	—
407	C₄×D₄	C₂²×D₅₃	1	24	yes	complemented
408	#117	—	53	24	no	—
409	C₂×C₄⋊C₄	—	53	3	no	—
410	C₂×C₄⋊C₄	—	53	6	no	—
411	C₂×C₄⋊C₄	—	53	3	no	—
412	C₂×C₄⋊C₄	—	53	6	no	—
413	C₂×C₄⋊C₄	—	53	24	no	—
414	C₂×C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
415	C₂×C₄⋊C₄	C₂²×D₅₃	1	3	yes	—
416	C₂×C₂²⋊C₄	—	53	6	no	—
417	C₂×C₂²⋊C₄	—	53	12	no	—
418	C₂×C₂²⋊C₄	—	53	2	no	—
419	C₂×C₂²⋊C₄	—	53	6	no	—
420	C₂×C₂²⋊C₄	—	53	6	no	—
421	C₂×C₂²⋊C₄	—	53	48	no	—
422	C₂×C₂²⋊C₄	C₂²×D₅₃	1	6	yes	—
423	C₂×C₄²	—	53	3	no	—
424	C₂×C₄²	—	53	1	no	—
425	C₂×C₄²	—	53	3	no	—
426	C₂×C₄²	—	53	24	no	—
427	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
428	C₂⁴	—	53	2	no	—
429	C₂⁴	—	53	3	no	—
430	C₂⁴	—	53	24	no	—
431	C₂⁴	—	106	24	no	—
432	C₂⁴	—	106	24	no	—
433	C₂⁴	—	212	12	no	—
434	C₂⁴	C₂³×D₅₃	1	6	yes	—
435	#41	—	2	64	no	—
436	#41	—	53	64	no	—
437	#41	—	106	192	no	—
438	#41	—	106	192	no	—
439	C₂×Q₈	—	53	24	no	—
440	C₂×Q₈	—	106	64	no	—
441	C₂×Q₈	—	106	24	no	—
442	C₂×Q₈	D₄×D₅₃	1	8	yes	complemented
443	C₂×Q₈	C₂³×D₅₃	1	4	yes	—
444	C₂×D₄	—	2	24	no	—
445	C₂×D₄	—	2	12	no	—
446	C₂×D₄	—	4	12	no	—
447	C₂×D₄	—	53	24	no	—
448	C₂×D₄	—	53	48	no	—
449	C₂×D₄	—	53	24	no	—
450	C₂×D₄	—	53	8	no	—
451	C₂×D₄	—	53	4	no	—
452	C₂×D₄	—	53	12	no	—
453	C₂×D₄	—	106	96	no	—
454	C₂×D₄	—	106	192	no	—
455	C₂×D₄	—	106	24	no	—
456	C₂×D₄	—	106	48	no	—
457	C₂×D₄	C₂³×D₅₃	1	12	yes	—
458	C₂×D₄	C₂³×D₅₃	1	24	yes	—
459	C₂×D₄	C₂³×D₅₃	1	24	yes	—
460	C₂²×C₄	—	2	24	no	—
461	C₂²×C₄	—	53	1	no	—
462	C₂²×C₄	—	53	24	no	—
463	C₂²×C₄	—	53	12	no	—
464	C₂²×C₄	—	53	24	no	—
465	C₂²×C₄	—	53	1	no	—
466	C₂²×C₄	—	53	3	no	—
467	C₂²×C₄	—	53	6	no	—
468	C₂²×C₄	—	53	12	no	—
469	C₂²×C₄	—	53	24	no	—
470	C₂²×C₄	—	53	8	no	—
471	C₂²×C₄	—	53	24	no	—
472	C₂²×C₄	—	53	12	no	—
473	C₂²×C₄	—	106	8	no	—
474	C₂²×C₄	—	106	192	no	—
475	C₂²×C₄	—	106	24	no	—
476	C₂²×C₄	—	106	24	no	—
477	C₂²×C₄	—	106	24	no	—
478	C₂²×C₄	—	106	12	no	—
479	C₂²×C₄	—	212	6	no	—
480	C₂²×C₄	C₂³×D₅₃	1	24	yes	—
481	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
482	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
483	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
484	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
485	C₄⋊C₄	—	53	12	no	—
486	C₄⋊C₄	—	53	24	no	—
487	C₄⋊C₄	—	53	12	no	—
488	C₄⋊C₄	—	53	24	no	—
489	C₄⋊C₄	C₂³×D₅₃	1	4	yes	—
490	C₄⋊C₄	C₂³×D₅₃	1	12	yes	—
491	C₂²⋊C₄	—	53	24	no	—
492	C₂²⋊C₄	—	53	48	no	—
493	C₂²⋊C₄	—	53	8	no	—
494	C₂²⋊C₄	—	53	24	no	—
495	C₂²⋊C₄	—	53	24	no	—
496	C₂²⋊C₄	C₂³×D₅₃	1	24	yes	—
497	C₄²	—	53	4	no	—
498	C₄²	—	53	12	no	—
499	C₄²	—	53	12	no	—
500	C₄²	C₂³×D₅₃	1	12	yes	—
501	C₂³	—	2	12	no	—
502	C₂³	—	2	12	no	—
503	C₂³	—	2	12	no	—
504	C₂³	—	53	12	no	—
505	C₂³	—	53	8	no	—
506	C₂³	—	53	4	no	—
507	C₂³	—	53	6	no	—
508	C₂³	—	106	48	no	—
509	C₂³	—	106	48	no	—
510	C₂³	—	106	4	no	—
511	C₂³	—	106	6	no	—
512	C₂³	—	106	6	no	—
513	C₂³	—	106	4	no	—
514	C₂³	—	212	24	no	—
515	C₂³	—	212	96	no	—
516	C₂³	C₂⁴×D₅₃	1	12	yes	—
517	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
518	Q₈	—	106	48	no	—
519	Q₈	C₂×D₄×D₅₃	1	16	yes	complemented
520	D₄	—	2	48	no	—
521	D₄	—	2	24	no	—
522	D₄	—	4	24	no	—
523	D₄	—	53	48	no	—
524	D₄	—	53	16	no	—
525	D₄	—	106	48	no	—
526	D₄	—	106	96	no	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	2	6	no	—
529	C₂×C₄	—	2	48	no	—
530	C₂×C₄	—	53	4	no	—
531	C₂×C₄	—	53	2	no	—
532	C₂×C₄	—	53	48	no	—
533	C₂×C₄	—	53	2	no	—
534	C₂×C₄	—	53	6	no	—
535	C₂×C₄	—	53	12	no	—
536	C₂×C₄	—	106	16	no	—
537	C₂×C₄	—	106	48	no	—
538	C₂×C₄	—	106	48	no	—
539	C₂×C₄	—	106	12	no	—
540	C₂×C₄	—	106	6	no	—
541	C₂×C₄	—	106	2	no	—
542	C₂×C₄	—	106	48	no	—
543	C₂×C₄	—	106	24	no	—
544	C₂×C₄	—	212	12	no	—
545	C₂×C₄	C₂×D₄×D₅₃	1	12	yes	—
546	C₂×C₄	#41×D₅₃	1	4	yes	complemented
547	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
548	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
549	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
550	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
551	C₂²	—	2	12	no	—
552	C₂²	—	2	12	no	—
553	C₂²	—	2	12	no	—
554	C₂²	—	4	24	no	—
555	C₂²	—	53	12	no	—
556	C₂²	—	53	8	no	—
557	C₂²	—	106	16	no	—
558	C₂²	—	106	24	no	—
559	C₂²	—	106	4	no	—
560	C₂²	—	106	6	no	—
561	C₂²	—	106	6	no	—
562	C₂²	—	106	4	no	—
563	C₂²	—	212	96	no	—
564	C₂²	C₂²×D₄×D₅₃	1	2	yes	—
565	C₂²	C₂×#41×D₅₃	1	2	yes	—
566	C₂²	#16811458	1	2	yes	—
567	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
568	C₄	—	2	4	no	—
569	C₄	—	2	6	no	—
570	C₄	—	53	4	no	—
571	C₄	—	106	12	no	—
572	C₄	—	106	6	no	—
573	C₄	—	106	2	no	—
574	C₄	C₂²×D₄×D₅₃	1	12	yes	—
575	C₄	C₂×#41×D₅₃	1	4	yes	complemented
576	C₂	—	4	12	no	—
577	C₂	—	106	8	no	—
578	C₂	—	106	12	no	—
579	C₂	#20288205	1	4	yes	direct product factor
580	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
581	C₂	C₂²×#41×D₅₃	1	1	yes	characteristic
582	C₂	C₂×#16811458	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂×C₅₃×#550	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂×#550	D₅₃
hypocenter	C₅₃	C₂×D₄×#41
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#550

Complements

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

N	H ≅ G/N	type
1. C₂²×#41×D₅₃	C₂	semidirect
2. #20288205	C₂	direct (N × H)
3. C₂×Q₈×D₂₁₂	C₂	semidirect
4. C₂×#16811280	C₂	semidirect
5. C₂×#16811278	C₂	semidirect
6. C₂×#16811270	C₂	semidirect
7. C₂×#16811258	C₂	semidirect
8. C₂×#16811171	C₂	semidirect
9. C₂×#16811133	C₂	semidirect
10. C₂×C₄×#11719949	C₂	semidirect
11. C₂×C₄×D₄×D₅₃	C₂	semidirect
12. C₂×C₅₃×#550	C₂	semidirect
14. C₂²×#11719945	C₂²	semidirect
15. C₂²×#11719947	C₂²	semidirect
17. C₂×#41×D₅₃	C₂²	semidirect
18. C₂²×Q₈×D₅₃	C₂²	semidirect
20. C₂²×D₄×D₅₃	C₂²	semidirect
21. C₂³×C₄×D₅₃	C₂²	semidirect
22. Q₈×D₂₁₂	C₂²	semidirect
23. #16811280	C₂²	semidirect
24. #16811278	C₂²	semidirect
25. #16811270	C₂²	semidirect
26. #16811258	C₂²	semidirect
27. #16811171	C₂²	semidirect
28. C₂×#13133217	C₂²	semidirect
29. C₂×#13133204	C₂²	semidirect
30. #16811133	C₂²	semidirect
31. C₂×#13133199	C₂²	semidirect
32. C₂×#13133197	C₂²	semidirect
33. C₂×#13133196	C₂²	semidirect
34. C₂×#13133188	C₂²	semidirect
35. C₂×#13133188	C₂²	semidirect
36. C₂×#13133188	C₂²	semidirect
37. C₂×#13133185	C₂²	semidirect
38. C₄×#11719949	C₂²	semidirect
39. C₂×Q₈×Dic₅₃	C₂²	semidirect
41. C₂×C₄×D₂₁₂	C₂²	semidirect
42. C₂×D₄×Dic₅₃	C₂²	semidirect
43. C₂×D₄×Dic₅₃	C₂²	semidirect
45. C₂×#13133179	C₂²	semidirect
46. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
47. C₂×#13133177	C₂²	semidirect
48. C₄×D₄×D₅₃	C₂²	semidirect
49. C₂×#13133175	C₂²	semidirect
50. C₂×C₄⋊C₄×D₅₃	C₂²	semidirect
51. C₂×C₂²⋊C₄×D₅₃	C₂²	semidirect
52. C₂×C₄²×D₅₃	C₂²	semidirect
53. C₂²×#41×C₅₃	C₂²	semidirect
54. C₅₃×#550	C₂²	semidirect
55. C₂×#127×C₅₃	C₂²	semidirect
56. C₂×#121×C₅₃	C₂²	semidirect
57. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
58. C₂×C₄×D₄×C₅₃	C₂²	semidirect
60. C₂×#11719949	D₄	semidirect
63. C₂×#11719945	C₂³	semidirect
65. C₂×#11719947	C₂³	semidirect
69. C₂×Q₈×D₅₃	C₂³	semidirect
77. C₂×D₄×D₅₃	C₂³	semidirect
86. C₂²×C₄×D₅₃	C₂³	semidirect
87. #13133217	C₂³	semidirect
88. #13133204	C₂³	semidirect
89. #13133199	C₂³	semidirect
90. #13133197	C₂³	semidirect
91. #13133196	C₂³	semidirect
92. #13133188	C₂³	semidirect
93. #13133188	C₂³	semidirect
94. #13133188	C₂³	semidirect
95. #13133185	C₂³	semidirect
96. Q₈×Dic₅₃	C₂³	semidirect
98. C₄×D₂₁₂	C₂³	semidirect
99. D₄×Dic₅₃	C₂³	semidirect
100. D₄×Dic₅₃	C₂³	semidirect
102. #13133179	C₂³	semidirect
103. C₄×C₂²⋊D₅₃	C₂³	semidirect
104. #13133177	C₂³	semidirect
105. #13133175	C₂³	semidirect
110. C₄⋊C₄×D₅₃	C₂³	semidirect
116. C₂²⋊C₄×D₅₃	C₂³	semidirect
120. C₄²×D₅₃	C₂³	semidirect
122. C₂×#41×C₅₃	C₂³	semidirect
128. #127×C₅₃	C₂³	semidirect
129. #121×C₅₃	C₂³	semidirect
130. C₄×Q₈×C₅₃	C₂³	semidirect
131. C₄×D₄×C₅₃	C₂³	semidirect
144. C₂×Q₈×C₅₃	C₂×D₄	semidirect
168. #11719949	C₂×D₄	semidirect
174. C₂×D₂₁₂	C₂×D₄	semidirect
175. C₂×D₂₁₂	#41	semidirect
184. C₂×C₄×D₅₃	C₂×D₄	semidirect
223. C₂²×D₅₃	C₂×#41	semidirect
226. Q₈×C₅₃	C₂²×D₄	semidirect
234. C₂×C₄×C₅₃	C₂×#41	semidirect
250. D₂₁₂	C₂²×D₄	semidirect
251. D₂₁₂	C₂×#41	semidirect
257. C₄×D₅₃	C₂²×D₄	semidirect
276. C₂×D₅₃	C₂²×#41	semidirect
280. C₄×C₅₃	C₂²×#41	semidirect
287. C₂×#550	D₅₃	semidirect
298. C₂×C₅₃	D₄×#41	semidirect
309. C₂²×#41	C₂×D₅₃	semidirect
314. #550	C₂×D₅₃	semidirect
322. C₂×#127	C₂×D₅₃	semidirect
331. C₂×#121	C₂×D₅₃	semidirect
335. C₂×C₄×Q₈	C₂×D₅₃	semidirect
345. C₂×C₄×D₄	C₂×D₅₃	semidirect
350. C₅₃	C₂×D₄×#41	semidirect
360. C₂×#41	C₂²×D₅₃	semidirect
387. #127	C₂²×D₅₃	semidirect
395. #121	C₂²×D₅₃	semidirect
398. C₄×Q₈	C₂²×D₅₃	semidirect
407. C₄×D₄	C₂²×D₅₃	semidirect
442. C₂×Q₈	D₄×D₅₃	semidirect
519. Q₈	C₂×D₄×D₅₃	semidirect
546. C₂×C₄	#41×D₅₃	semidirect
575. C₄	C₂×#41×D₅₃	semidirect
579. C₂	#20288205	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
#20288205	2
C₂×Q₈×D₂₁₂	2
C₂×#16811280	2
C₂×#16811278	2
C₂×#16811270	2
C₂×#16811258	2
C₂×#16811171	2
C₂×#16811133	2
C₂×C₄×#11719949	2
C₂×C₄×D₄×D₅₃	2
C₂×C₅₃×#550	2
C₂×D₄×#41	53

Subgroups — C2×#20288205 #28291054

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20288205 #28291054