Subgroups — C₂×C₅₃×#2408 #27242183

Order 13568.

8622 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#121×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₂²×#120×C₅₃	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2408	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₅₃×#394	C₂	1	2	yes	maximal, complemented
7	C₂×C₅₃×#392	C₂	1	2	yes	maximal, complemented
8	C₂×C₅₃×#390	C₂	1	2	yes	maximal, complemented
9	C₂×C₅₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅₃×#379	C₂	1	2	yes	maximal, complemented
11	C₂⁶×C₅₃	C₂²	1	2	yes	—
12	C₂³×D₄×C₅₃	—	2	16	no	—
13	C₂³×D₄×C₅₃	—	2	8	no	—
14	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
15	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
16	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂³×D₄×C₅₃	C₂²	1	8	yes	complemented
18	C₂³×D₄×C₅₃	C₂²	1	4	yes	complemented
19	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
21	C₂×#121×C₅₃	—	2	16	no	—
22	C₂×#121×C₅₃	C₂²	1	16	yes	complemented
23	C₂×#120×C₅₃	—	2	16	no	—
24	C₂×#120×C₅₃	C₂²	1	16	yes	complemented
25	C₂×C₄×D₄×C₅₃	—	2	8	no	—
26	C₂×C₄×D₄×C₅₃	C₂²	1	8	yes	complemented
27	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
28	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
29	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
30	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
31	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
32	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
33	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
34	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
35	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
36	C₅₃×#394	C₂²	1	16	yes	complemented
37	C₅₃×#392	C₂²	1	16	yes	complemented
38	C₅₃×#390	C₂²	1	16	yes	complemented
39	C₅₃×#380	C₂²	1	8	yes	complemented
40	C₅₃×#379	C₂²	1	16	yes	complemented
41	C₂×#95×C₅₃	C₂²	1	2	yes	complemented
42	C₂⁵×C₅₃	—	2	16	no	—
43	C₂⁵×C₅₃	—	2	8	no	—
44	C₂⁵×C₅₃	—	2	8	no	—
45	C₂⁵×C₅₃	—	2	8	no	—
46	C₂⁵×C₅₃	—	4	8	no	—
47	C₂⁵×C₅₃	C₂³	1	4	yes	—
48	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
49	C₂⁵×C₅₃	C₂³	1	8	yes	—
50	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
51	C₂²×D₄×C₅₃	—	2	128	no	—
52	C₂²×D₄×C₅₃	—	2	4	no	—
53	C₂²×D₄×C₅₃	—	2	16	no	—
54	C₂²×D₄×C₅₃	—	2	4	no	—
55	C₂²×D₄×C₅₃	—	2	64	no	—
56	C₂²×D₄×C₅₃	—	2	16	no	—
57	C₂²×D₄×C₅₃	—	2	16	no	—
58	C₂²×D₄×C₅₃	—	2	8	no	—
59	C₂²×D₄×C₅₃	—	2	8	no	—
60	C₂²×D₄×C₅₃	—	2	8	no	—
61	C₂²×D₄×C₅₃	—	2	32	no	—
62	C₂²×D₄×C₅₃	—	4	8	no	—
63	C₂²×D₄×C₅₃	D₄	1	16	yes	complemented
64	C₂²×D₄×C₅₃	D₄	1	8	yes	complemented
65	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
66	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
67	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
68	C₂²×D₄×C₅₃	C₂³	1	16	yes	complemented
69	C₂³×C₄×C₅₃	—	2	8	no	—
70	C₂³×C₄×C₅₃	—	2	4	no	—
71	C₂³×C₄×C₅₃	—	2	8	no	—
72	C₂³×C₄×C₅₃	D₄	1	8	yes	complemented
73	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
76	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
78	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₅₃	C₂³	1	4	yes	complemented
81	#121×C₅₃	—	2	64	no	—
82	#120×C₅₃	—	2	64	no	—
83	C₄×D₄×C₅₃	—	2	32	no	—
84	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
85	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
86	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	complemented
87	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	complemented
88	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
89	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
90	C₂×C₂²⋊C₄×C₅₃	—	2	16	no	—
91	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
92	C₂×C₂²⋊C₄×C₅₃	—	2	16	no	—
93	C₂×C₂²⋊C₄×C₅₃	—	4	8	no	—
94	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
96	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	complemented
97	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	complemented
98	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	complemented
99	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	complemented
100	C₂×C₄²×C₅₃	—	2	4	no	—
101	C₂×C₄²×C₅₃	C₂³	1	4	yes	complemented
102	#95×C₅₃	C₂³	1	8	yes	complemented
103	C₂⁴×C₅₃	—	2	4	no	—
104	C₂⁴×C₅₃	—	2	4	no	—
105	C₂⁴×C₅₃	—	2	64	no	—
106	C₂⁴×C₅₃	—	2	2	no	—
107	C₂⁴×C₅₃	—	2	8	no	—
108	C₂⁴×C₅₃	—	2	4	no	—
109	C₂⁴×C₅₃	—	2	32	no	—
110	C₂⁴×C₅₃	—	2	16	no	—
111	C₂⁴×C₅₃	—	2	8	no	—
112	C₂⁴×C₅₃	—	2	8	no	—
113	C₂⁴×C₅₃	—	2	2	no	—
114	C₂⁴×C₅₃	—	2	8	no	—
115	C₂⁴×C₅₃	—	2	8	no	—
116	C₂⁴×C₅₃	—	2	4	no	—
117	C₂⁴×C₅₃	—	2	8	no	—
118	C₂⁴×C₅₃	—	2	16	no	—
119	C₂⁴×C₅₃	—	2	16	no	—
120	C₂⁴×C₅₃	—	4	16	no	—
121	C₂⁴×C₅₃	—	4	64	no	—
122	C₂⁴×C₅₃	—	4	64	no	—
123	C₂⁴×C₅₃	—	4	32	no	—
124	C₂⁴×C₅₃	—	4	32	no	—
125	C₂⁴×C₅₃	C₂×D₄	1	16	yes	—
126	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
127	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
128	C₂⁴×C₅₃	C₂×D₄	1	4	yes	complemented
129	C₂⁴×C₅₃	#41	1	4	yes	—
130	C₂⁴×C₅₃	C₂⁴	1	8	yes	—
131	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
132	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
133	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
134	C₂×D₄×C₅₃	—	2	64	no	—
135	C₂×D₄×C₅₃	—	2	256	no	—
136	C₂×D₄×C₅₃	—	2	32	no	—
137	C₂×D₄×C₅₃	—	2	8	no	—
138	C₂×D₄×C₅₃	—	2	32	no	—
139	C₂×D₄×C₅₃	—	2	8	no	—
140	C₂×D₄×C₅₃	—	2	32	no	—
141	C₂×D₄×C₅₃	—	2	32	no	—
142	C₂×D₄×C₅₃	—	2	64	no	—
143	C₂×D₄×C₅₃	—	2	16	no	—
144	C₂×D₄×C₅₃	—	2	16	no	—
145	C₂×D₄×C₅₃	—	2	16	no	—
146	C₂×D₄×C₅₃	—	4	16	no	—
147	C₂×D₄×C₅₃	—	4	64	no	—
148	C₂×D₄×C₅₃	C₂×D₄	1	32	yes	complemented
149	C₂×D₄×C₅₃	C₂×D₄	1	16	yes	complemented
150	C₂²×C₄×C₅₃	—	2	2	no	—
151	C₂²×C₄×C₅₃	—	2	4	no	—
152	C₂²×C₄×C₅₃	—	2	2	no	—
153	C₂²×C₄×C₅₃	—	2	32	no	—
154	C₂²×C₄×C₅₃	—	2	16	no	—
155	C₂²×C₄×C₅₃	—	2	2	no	—
156	C₂²×C₄×C₅₃	—	2	4	no	—
157	C₂²×C₄×C₅₃	—	2	8	no	—
158	C₂²×C₄×C₅₃	—	2	2	no	—
159	C₂²×C₄×C₅₃	—	2	8	no	—
160	C₂²×C₄×C₅₃	—	2	4	no	—
161	C₂²×C₄×C₅₃	—	2	8	no	—
162	C₂²×C₄×C₅₃	—	2	8	no	—
163	C₂²×C₄×C₅₃	—	2	16	no	—
164	C₂²×C₄×C₅₃	—	4	16	no	—
165	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
166	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
167	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
168	C₂²×C₄×C₅₃	C₂×D₄	1	16	yes	complemented
169	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	complemented
170	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
171	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
173	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
174	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
176	C₄⋊C₄×C₅₃	—	2	8	no	—
177	C₄⋊C₄×C₅₃	—	2	16	no	—
178	C₂²⋊C₄×C₅₃	—	2	8	no	—
179	C₂²⋊C₄×C₅₃	—	2	16	no	—
180	C₂²⋊C₄×C₅₃	—	2	32	no	—
181	C₂²⋊C₄×C₅₃	—	2	8	no	—
182	C₂²⋊C₄×C₅₃	—	2	32	no	—
183	C₂²⋊C₄×C₅₃	—	4	16	no	—
184	C₄²×C₅₃	—	2	8	no	—
185	C₂³×C₅₃	—	2	8	no	—
186	C₂³×C₅₃	—	2	8	no	—
187	C₂³×C₅₃	—	2	16	no	—
188	C₂³×C₅₃	—	2	16	no	—
189	C₂³×C₅₃	—	2	4	no	—
190	C₂³×C₅₃	—	2	4	no	—
191	C₂³×C₅₃	—	2	64	no	—
192	C₂³×C₅₃	—	2	32	no	—
193	C₂³×C₅₃	—	2	8	no	—
194	C₂³×C₅₃	—	2	16	no	—
195	C₂³×C₅₃	—	2	32	no	—
196	C₂³×C₅₃	—	2	32	no	—
197	C₂³×C₅₃	—	2	4	no	—
198	C₂³×C₅₃	—	2	2	no	—
199	C₂³×C₅₃	—	2	8	no	—
200	C₂³×C₅₃	—	2	8	no	—
201	C₂³×C₅₃	—	2	8	no	—
202	C₂³×C₅₃	—	2	2	no	—
203	C₂³×C₅₃	—	2	8	no	—
204	C₂³×C₅₃	—	2	4	no	—
205	C₂³×C₅₃	—	2	8	no	—
206	C₂³×C₅₃	—	2	8	no	—
207	C₂³×C₅₃	—	4	64	no	—
208	C₂³×C₅₃	—	4	64	no	—
209	C₂³×C₅₃	—	4	256	no	—
210	C₂³×C₅₃	—	4	32	no	—
211	C₂³×C₅₃	—	4	4	no	—
212	C₂³×C₅₃	—	4	16	no	—
213	C₂³×C₅₃	—	4	32	no	—
214	C₂³×C₅₃	—	4	4	no	—
215	C₂³×C₅₃	—	4	16	no	—
216	C₂³×C₅₃	—	4	16	no	—
217	C₂³×C₅₃	—	4	8	no	—
218	C₂³×C₅₃	—	4	16	no	—
219	C₂³×C₅₃	#121	1	16	yes	complemented
220	C₂³×C₅₃	C₂²×D₄	1	16	yes	—
221	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
222	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
223	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
224	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
225	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
226	C₂³×C₅₃	C₂²×D₄	1	4	yes	complemented
227	C₂³×C₅₃	C₂×#41	1	4	yes	—
228	C₂³×C₅₃	C₂×#41	1	2	yes	—
229	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
230	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
231	D₄×C₅₃	—	2	64	no	—
232	D₄×C₅₃	—	2	32	no	—
233	D₄×C₅₃	—	2	64	no	—
234	D₄×C₅₃	—	4	64	no	—
235	C₂×C₄×C₅₃	—	2	8	no	—
236	C₂×C₄×C₅₃	—	2	8	no	—
237	C₂×C₄×C₅₃	—	2	4	no	—
238	C₂×C₄×C₅₃	—	2	2	no	—
239	C₂×C₄×C₅₃	—	2	2	no	—
240	C₂×C₄×C₅₃	—	2	32	no	—
241	C₂×C₄×C₅₃	—	2	4	no	—
242	C₂×C₄×C₅₃	—	2	2	no	—
243	C₂×C₄×C₅₃	—	2	8	no	—
244	C₂×C₄×C₅₃	—	2	8	no	—
245	C₂×C₄×C₅₃	—	2	2	no	—
246	C₂×C₄×C₅₃	—	2	8	no	—
247	C₂×C₄×C₅₃	—	2	4	no	—
248	C₂×C₄×C₅₃	—	4	4	no	—
249	C₂×C₄×C₅₃	—	4	4	no	—
250	C₂×C₄×C₅₃	—	4	16	no	—
251	C₂×C₄×C₅₃	—	4	16	no	—
252	C₂×C₄×C₅₃	#120	1	16	yes	complemented
253	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
254	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
255	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
256	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	complemented
257	C₂×#2408	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
258	C₂²×C₅₃	—	2	4	no	—
259	C₂²×C₅₃	—	2	32	no	—
260	C₂²×C₅₃	—	2	4	no	—
261	C₂²×C₅₃	—	2	8	no	—
262	C₂²×C₅₃	—	2	8	no	—
263	C₂²×C₅₃	—	2	16	no	—
264	C₂²×C₅₃	—	2	4	no	—
265	C₂²×C₅₃	—	2	8	no	—
266	C₂²×C₅₃	—	2	16	no	—
267	C₂²×C₅₃	—	2	16	no	—
268	C₂²×C₅₃	—	4	128	no	—
269	C₂²×C₅₃	—	4	2	no	—
270	C₂²×C₅₃	—	4	16	no	—
271	C₂²×C₅₃	—	4	64	no	—
272	C₂²×C₅₃	—	4	8	no	—
273	C₂²×C₅₃	—	4	2	no	—
274	C₂²×C₅₃	—	4	8	no	—
275	C₂²×C₅₃	—	4	8	no	—
276	C₂²×C₅₃	—	4	4	no	—
277	C₂²×C₅₃	—	4	8	no	—
278	C₂²×C₅₃	C₂×#120	1	4	yes	—
279	C₂²×C₅₃	C₂×#121	1	4	yes	—
280	C₂²×C₅₃	C₂×#121	1	8	yes	complemented
281	C₂²×C₅₃	#535	1	4	yes	—
282	C₂²×C₅₃	D₄²	1	8	yes	—
283	C₂²×C₅₃	#546	1	8	yes	—
284	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
285	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
286	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
287	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
288	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
289	C₄×C₅₃	—	2	4	no	—
290	C₄×C₅₃	—	2	4	no	—
291	C₄×C₅₃	—	4	2	no	—
292	C₄×C₅₃	—	4	2	no	—
293	C₄×C₅₃	—	4	8	no	—
294	C₄×C₅₃	C₂×#120	1	8	yes	complemented
295	C₂⁴×D₄	C₂×C₅₃	1	1	yes	characteristic, complemented
296	C₂²×#121	C₂×C₅₃	1	2	yes	complemented
297	C₂²×#120	C₂×C₅₃	1	2	yes	complemented
298	C₂²×C₄×D₄	C₂×C₅₃	1	1	yes	characteristic, complemented
299	#2408	C₂×C₅₃	1	16	yes	direct product factor
300	C₂×#394	C₂×C₅₃	1	2	yes	complemented
301	C₂×#392	C₂×C₅₃	1	2	yes	complemented
302	C₂×#390	C₂×C₅₃	1	2	yes	complemented
303	C₂×#380	C₂×C₅₃	1	1	yes	characteristic, complemented
304	C₂×#379	C₂×C₅₃	1	2	yes	complemented
305	C₂×C₅₃	—	2	8	no	—
306	C₂×C₅₃	—	4	4	no	—
307	C₂×C₅₃	—	4	16	no	—
308	C₂×C₅₃	#2408	1	8	yes	direct product factor
309	C₂×C₅₃	C₂²×#120	1	1	yes	characteristic
310	C₂×C₅₃	C₂²×#121	1	1	yes	characteristic
311	C₂×C₅₃	C₂×#535	1	1	yes	characteristic
312	C₂×C₅₃	C₂×D₄²	1	2	yes	—
313	C₂×C₅₃	C₂×#546	1	2	yes	—
314	C₂⁶	C₂²×C₅₃	1	2	yes	—
315	C₂³×D₄	—	2	16	no	—
316	C₂³×D₄	—	2	8	no	—
317	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
318	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
319	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
320	C₂³×D₄	C₂²×C₅₃	1	8	yes	complemented
321	C₂³×D₄	C₂²×C₅₃	1	4	yes	complemented
322	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
323	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
324	C₂×#121	—	2	16	no	—
325	C₂×#121	C₂²×C₅₃	1	16	yes	complemented
326	C₂×#120	—	2	16	no	—
327	C₂×#120	C₂²×C₅₃	1	16	yes	complemented
328	C₂×C₄×D₄	—	2	8	no	—
329	C₂×C₄×D₄	C₂²×C₅₃	1	8	yes	complemented
330	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
331	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
332	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
333	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
334	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
335	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
336	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
337	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
338	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic, complemented
339	#394	C₂²×C₅₃	1	16	yes	complemented
340	#392	C₂²×C₅₃	1	16	yes	complemented
341	#390	C₂²×C₅₃	1	16	yes	complemented
342	#380	C₂²×C₅₃	1	8	yes	complemented
343	#379	C₂²×C₅₃	1	16	yes	complemented
344	C₂×#95	C₂²×C₅₃	1	2	yes	complemented
345	C₅₃	C₂×#2408	1	1	yes	characteristic, direct product factor, Sylow
346	C₂⁵	—	2	16	no	—
347	C₂⁵	—	2	8	no	—
348	C₂⁵	—	2	8	no	—
349	C₂⁵	—	2	8	no	—
350	C₂⁵	—	4	8	no	—
351	C₂⁵	C₂³×C₅₃	1	4	yes	—
352	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
353	C₂⁵	C₂³×C₅₃	1	8	yes	—
354	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
355	C₂²×D₄	—	2	128	no	—
356	C₂²×D₄	—	2	4	no	—
357	C₂²×D₄	—	2	16	no	—
358	C₂²×D₄	—	2	4	no	—
359	C₂²×D₄	—	2	64	no	—
360	C₂²×D₄	—	2	16	no	—
361	C₂²×D₄	—	2	16	no	—
362	C₂²×D₄	—	2	8	no	—
363	C₂²×D₄	—	2	8	no	—
364	C₂²×D₄	—	2	8	no	—
365	C₂²×D₄	—	2	32	no	—
366	C₂²×D₄	—	4	8	no	—
367	C₂²×D₄	D₄×C₅₃	1	16	yes	complemented
368	C₂²×D₄	D₄×C₅₃	1	8	yes	complemented
369	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
370	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
371	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
372	C₂²×D₄	C₂³×C₅₃	1	16	yes	complemented
373	C₂³×C₄	—	2	8	no	—
374	C₂³×C₄	—	2	4	no	—
375	C₂³×C₄	—	2	8	no	—
376	C₂³×C₄	D₄×C₅₃	1	8	yes	complemented
377	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
378	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
379	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
380	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
381	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
382	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
383	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
384	C₂³×C₄	C₂³×C₅₃	1	4	yes	complemented
385	#121	—	2	64	no	—
386	#120	—	2	64	no	—
387	C₄×D₄	—	2	32	no	—
388	C₂×C₄⋊C₄	—	2	4	no	—
389	C₂×C₄⋊C₄	—	2	8	no	—
390	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	complemented
391	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	complemented
392	C₂×C₂²⋊C₄	—	2	4	no	—
393	C₂×C₂²⋊C₄	—	2	8	no	—
394	C₂×C₂²⋊C₄	—	2	16	no	—
395	C₂×C₂²⋊C₄	—	2	4	no	—
396	C₂×C₂²⋊C₄	—	2	16	no	—
397	C₂×C₂²⋊C₄	—	4	8	no	—
398	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
399	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
400	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	complemented
401	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	complemented
402	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	complemented
403	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	complemented
404	C₂×C₄²	—	2	4	no	—
405	C₂×C₄²	C₂³×C₅₃	1	4	yes	complemented
406	#95	C₂³×C₅₃	1	8	yes	complemented
407	C₂⁴	—	2	4	no	—
408	C₂⁴	—	2	4	no	—
409	C₂⁴	—	2	64	no	—
410	C₂⁴	—	2	2	no	—
411	C₂⁴	—	2	8	no	—
412	C₂⁴	—	2	4	no	—
413	C₂⁴	—	2	32	no	—
414	C₂⁴	—	2	16	no	—
415	C₂⁴	—	2	8	no	—
416	C₂⁴	—	2	8	no	—
417	C₂⁴	—	2	2	no	—
418	C₂⁴	—	2	8	no	—
419	C₂⁴	—	2	8	no	—
420	C₂⁴	—	2	4	no	—
421	C₂⁴	—	2	8	no	—
422	C₂⁴	—	2	16	no	—
423	C₂⁴	—	2	16	no	—
424	C₂⁴	—	4	16	no	—
425	C₂⁴	—	4	64	no	—
426	C₂⁴	—	4	64	no	—
427	C₂⁴	—	4	32	no	—
428	C₂⁴	—	4	32	no	—
429	C₂⁴	C₂×D₄×C₅₃	1	16	yes	—
430	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
431	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
432	C₂⁴	C₂×D₄×C₅₃	1	4	yes	complemented
433	C₂⁴	#41×C₅₃	1	4	yes	—
434	C₂⁴	C₂⁴×C₅₃	1	8	yes	—
435	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
436	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
437	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
438	C₂×D₄	—	2	64	no	—
439	C₂×D₄	—	2	256	no	—
440	C₂×D₄	—	2	32	no	—
441	C₂×D₄	—	2	8	no	—
442	C₂×D₄	—	2	32	no	—
443	C₂×D₄	—	2	8	no	—
444	C₂×D₄	—	2	32	no	—
445	C₂×D₄	—	2	32	no	—
446	C₂×D₄	—	2	64	no	—
447	C₂×D₄	—	2	16	no	—
448	C₂×D₄	—	2	16	no	—
449	C₂×D₄	—	2	16	no	—
450	C₂×D₄	—	4	16	no	—
451	C₂×D₄	—	4	64	no	—
452	C₂×D₄	C₂×D₄×C₅₃	1	32	yes	complemented
453	C₂×D₄	C₂×D₄×C₅₃	1	16	yes	complemented
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	2	4	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	32	no	—
458	C₂²×C₄	—	2	16	no	—
459	C₂²×C₄	—	2	2	no	—
460	C₂²×C₄	—	2	4	no	—
461	C₂²×C₄	—	2	8	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	8	no	—
464	C₂²×C₄	—	2	4	no	—
465	C₂²×C₄	—	2	8	no	—
466	C₂²×C₄	—	2	8	no	—
467	C₂²×C₄	—	2	16	no	—
468	C₂²×C₄	—	4	16	no	—
469	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
470	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
471	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
472	C₂²×C₄	C₂×D₄×C₅₃	1	16	yes	complemented
473	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	complemented
474	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
475	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
476	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
477	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
478	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
479	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
480	C₄⋊C₄	—	2	8	no	—
481	C₄⋊C₄	—	2	16	no	—
482	C₂²⋊C₄	—	2	8	no	—
483	C₂²⋊C₄	—	2	16	no	—
484	C₂²⋊C₄	—	2	32	no	—
485	C₂²⋊C₄	—	2	8	no	—
486	C₂²⋊C₄	—	2	32	no	—
487	C₂²⋊C₄	—	4	16	no	—
488	C₄²	—	2	8	no	—
489	C₂³	—	2	8	no	—
490	C₂³	—	2	8	no	—
491	C₂³	—	2	16	no	—
492	C₂³	—	2	16	no	—
493	C₂³	—	2	4	no	—
494	C₂³	—	2	4	no	—
495	C₂³	—	2	64	no	—
496	C₂³	—	2	32	no	—
497	C₂³	—	2	8	no	—
498	C₂³	—	2	16	no	—
499	C₂³	—	2	32	no	—
500	C₂³	—	2	32	no	—
501	C₂³	—	2	2	no	—
502	C₂³	—	2	8	no	—
503	C₂³	—	2	4	no	—
504	C₂³	—	2	8	no	—
505	C₂³	—	2	8	no	—
506	C₂³	—	2	2	no	—
507	C₂³	—	2	8	no	—
508	C₂³	—	2	8	no	—
509	C₂³	—	2	4	no	—
510	C₂³	—	2	8	no	—
511	C₂³	—	4	64	no	—
512	C₂³	—	4	64	no	—
513	C₂³	—	4	256	no	—
514	C₂³	—	4	32	no	—
515	C₂³	—	4	4	no	—
516	C₂³	—	4	16	no	—
517	C₂³	—	4	32	no	—
518	C₂³	—	4	4	no	—
519	C₂³	—	4	16	no	—
520	C₂³	—	4	16	no	—
521	C₂³	—	4	8	no	—
522	C₂³	—	4	16	no	—
523	C₂³	#121×C₅₃	1	16	yes	complemented
524	C₂³	C₂²×D₄×C₅₃	1	16	yes	—
525	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
526	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
527	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
528	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
529	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
530	C₂³	C₂²×D₄×C₅₃	1	4	yes	complemented
531	C₂³	C₂×#41×C₅₃	1	4	yes	—
532	C₂³	C₂×#41×C₅₃	1	2	yes	—
533	C₂³	H(5,𝔽₂)×C₅₃	1	4	yes	—
534	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
535	D₄	—	2	64	no	—
536	D₄	—	2	32	no	—
537	D₄	—	2	64	no	—
538	D₄	—	4	64	no	—
539	C₂×C₄	—	2	8	no	—
540	C₂×C₄	—	2	8	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	32	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	8	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	8	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	2	8	no	—
552	C₂×C₄	—	4	4	no	—
553	C₂×C₄	—	4	4	no	—
554	C₂×C₄	—	4	16	no	—
555	C₂×C₄	—	4	16	no	—
556	C₂×C₄	#120×C₅₃	1	16	yes	complemented
557	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
558	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
559	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
560	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	complemented
561	C₂²	—	2	8	no	—
562	C₂²	—	2	8	no	—
563	C₂²	—	2	4	no	—
564	C₂²	—	2	32	no	—
565	C₂²	—	2	4	no	—
566	C₂²	—	2	8	no	—
567	C₂²	—	2	16	no	—
568	C₂²	—	2	16	no	—
569	C₂²	—	2	16	no	—
570	C₂²	—	2	4	no	—
571	C₂²	—	4	128	no	—
572	C₂²	—	4	8	no	—
573	C₂²	—	4	8	no	—
574	C₂²	—	4	4	no	—
575	C₂²	—	4	2	no	—
576	C₂²	—	4	64	no	—
577	C₂²	—	4	16	no	—
578	C₂²	—	4	8	no	—
579	C₂²	—	4	8	no	—
580	C₂²	—	4	2	no	—
581	C₂²	C₂×#120×C₅₃	1	4	yes	—
582	C₂²	C₂×#121×C₅₃	1	4	yes	—
583	C₂²	C₂×#121×C₅₃	1	8	yes	complemented
584	C₂²	C₅₃×#535	1	4	yes	—
585	C₂²	D₄²×C₅₃	1	8	yes	—
586	C₂²	C₅₃×#546	1	8	yes	—
587	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
588	C₂²	C₂³×D₄×C₅₃	1	2	yes	—
589	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
590	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
591	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
592	C₄	—	2	4	no	—
593	C₄	—	2	4	no	—
594	C₄	—	4	8	no	—
595	C₄	—	4	2	no	—
596	C₄	—	4	2	no	—
597	C₄	C₂×#120×C₅₃	1	8	yes	complemented
598	C₂	—	2	8	no	—
599	C₂	—	4	16	no	—
600	C₂	—	4	4	no	—
601	C₂	C₅₃×#2408	1	8	yes	direct product factor
602	C₂	C₂²×#120×C₅₃	1	1	yes	characteristic
603	C₂	C₂²×#121×C₅₃	1	1	yes	characteristic
604	C₂	C₂×C₅₃×#535	1	1	yes	characteristic
605	C₂	C₂×D₄²×C₅₃	1	2	yes	—
606	C₂	C₂×C₅₃×#546	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₄×C₅₃	C₂	semidirect
2. C₂²×#121×C₅₃	C₂	semidirect
3. C₂²×#120×C₅₃	C₂	semidirect
4. C₂²×C₄×D₄×C₅₃	C₂	semidirect
5. C₅₃×#2408	C₂	direct (N × H)
6. C₂×C₅₃×#394	C₂	semidirect
7. C₂×C₅₃×#392	C₂	semidirect
8. C₂×C₅₃×#390	C₂	semidirect
9. C₂×C₅₃×#380	C₂	semidirect
10. C₂×C₅₃×#379	C₂	semidirect
17. C₂³×D₄×C₅₃	C₂²	semidirect
18. C₂³×D₄×C₅₃	C₂²	semidirect
20. C₂⁴×C₄×C₅₃	C₂²	semidirect
22. C₂×#121×C₅₃	C₂²	semidirect
24. C₂×#120×C₅₃	C₂²	semidirect
26. C₂×C₄×D₄×C₅₃	C₂²	semidirect
27. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
28. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
31. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
32. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
33. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
34. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
35. C₂²×C₄²×C₅₃	C₂²	semidirect
36. C₅₃×#394	C₂²	semidirect
37. C₅₃×#392	C₂²	semidirect
38. C₅₃×#390	C₂²	semidirect
39. C₅₃×#380	C₂²	semidirect
40. C₅₃×#379	C₂²	semidirect
41. C₂×#95×C₅₃	C₂²	semidirect
63. C₂²×D₄×C₅₃	D₄	semidirect
64. C₂²×D₄×C₅₃	D₄	semidirect
68. C₂²×D₄×C₅₃	C₂³	semidirect
72. C₂³×C₄×C₅₃	D₄	semidirect
80. C₂³×C₄×C₅₃	C₂³	semidirect
86. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
87. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
96. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
97. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
98. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
99. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
101. C₂×C₄²×C₅₃	C₂³	semidirect
102. #95×C₅₃	C₂³	semidirect
128. C₂⁴×C₅₃	C₂×D₄	semidirect
148. C₂×D₄×C₅₃	C₂×D₄	semidirect
149. C₂×D₄×C₅₃	C₂×D₄	semidirect
168. C₂²×C₄×C₅₃	C₂×D₄	semidirect
169. C₂²×C₄×C₅₃	C₂×D₄	semidirect
219. C₂³×C₅₃	#121	semidirect
226. C₂³×C₅₃	C₂²×D₄	semidirect
252. C₂×C₄×C₅₃	#120	semidirect
256. C₂×C₄×C₅₃	C₂²×D₄	semidirect
257. C₂×#2408	C₅₃	direct (N × H)
280. C₂²×C₅₃	C₂×#121	semidirect
294. C₄×C₅₃	C₂×#120	semidirect
295. C₂⁴×D₄	C₂×C₅₃	semidirect
296. C₂²×#121	C₂×C₅₃	semidirect
297. C₂²×#120	C₂×C₅₃	semidirect
298. C₂²×C₄×D₄	C₂×C₅₃	semidirect
299. #2408	C₂×C₅₃	direct (N × H)
300. C₂×#394	C₂×C₅₃	semidirect
301. C₂×#392	C₂×C₅₃	semidirect
302. C₂×#390	C₂×C₅₃	semidirect
303. C₂×#380	C₂×C₅₃	semidirect
304. C₂×#379	C₂×C₅₃	semidirect
308. C₂×C₅₃	#2408	direct (N × H)
320. C₂³×D₄	C₂²×C₅₃	semidirect
321. C₂³×D₄	C₂²×C₅₃	semidirect
323. C₂⁴×C₄	C₂²×C₅₃	semidirect
325. C₂×#121	C₂²×C₅₃	semidirect
327. C₂×#120	C₂²×C₅₃	semidirect
329. C₂×C₄×D₄	C₂²×C₅₃	semidirect
330. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
331. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
334. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
335. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
336. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
337. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
338. C₂²×C₄²	C₂²×C₅₃	semidirect
339. #394	C₂²×C₅₃	semidirect
340. #392	C₂²×C₅₃	semidirect
341. #390	C₂²×C₅₃	semidirect
342. #380	C₂²×C₅₃	semidirect
343. #379	C₂²×C₅₃	semidirect
344. C₂×#95	C₂²×C₅₃	semidirect
345. C₅₃	C₂×#2408	direct (N × H)
367. C₂²×D₄	D₄×C₅₃	semidirect
368. C₂²×D₄	D₄×C₅₃	semidirect
372. C₂²×D₄	C₂³×C₅₃	semidirect
376. C₂³×C₄	D₄×C₅₃	semidirect
384. C₂³×C₄	C₂³×C₅₃	semidirect
390. C₂×C₄⋊C₄	C₂³×C₅₃	semidirect
391. C₂×C₄⋊C₄	C₂³×C₅₃	semidirect
400. C₂×C₂²⋊C₄	C₂³×C₅₃	semidirect
401. C₂×C₂²⋊C₄	C₂³×C₅₃	semidirect
402. C₂×C₂²⋊C₄	C₂³×C₅₃	semidirect
403. C₂×C₂²⋊C₄	C₂³×C₅₃	semidirect
405. C₂×C₄²	C₂³×C₅₃	semidirect
406. #95	C₂³×C₅₃	semidirect
432. C₂⁴	C₂×D₄×C₅₃	semidirect
452. C₂×D₄	C₂×D₄×C₅₃	semidirect
453. C₂×D₄	C₂×D₄×C₅₃	semidirect
472. C₂²×C₄	C₂×D₄×C₅₃	semidirect
473. C₂²×C₄	C₂×D₄×C₅₃	semidirect
523. C₂³	#121×C₅₃	semidirect
530. C₂³	C₂²×D₄×C₅₃	semidirect
556. C₂×C₄	#120×C₅₃	semidirect
560. C₂×C₄	C₂²×D₄×C₅₃	semidirect
583. C₂²	C₂×#121×C₅₃	semidirect
597. C₄	C₂×#120×C₅₃	semidirect
601. C₂	C₅₃×#2408	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₄×C₅₃	2
C₂²×#121×C₅₃	2
C₂²×#120×C₅₃	2
C₂²×C₄×D₄×C₅₃	2
C₅₃×#2408	2
C₂×C₅₃×#394	2
C₂×C₅₃×#392	2
C₂×C₅₃×#390	2
C₂×C₅₃×#380	2
C₂×C₅₃×#379	2
C₂×#2408	53

Subgroups — C2×C53×#2408 #27242183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2408 #27242183