Subgroups — C₄×C₅₃×#537 #27241602

Order 13568.

3186 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#537	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#41×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₅₃×#2325	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2324	C₂	1	4	yes	maximal, complemented
5	C₅₃×#2323	C₂	1	4	yes	maximal, complemented
6	C₅₃×#2316	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2311	C₂	1	2	yes	maximal, complemented
8	C₄×C₂²⋊Q₈×C₅₃	C₂	1	4	yes	maximal, complemented
9	C₄×#121×C₅₃	C₂	1	4	yes	maximal, complemented
10	C₄×#121×C₅₃	C₂	1	4	yes	maximal, complemented
11	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#537	C₄	1	16	yes	direct product factor
14	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	—
15	C₂×#121×C₅₃	C₂²	1	4	yes	—
16	C₂×#121×C₅₃	C₂²	1	4	yes	—
17	C₄×#41×C₅₃	—	2	16	no	—
18	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
24	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
25	C₂×#117×C₅₃	C₂²	1	2	yes	—
26	C₂×#117×C₅₃	C₂²	1	4	yes	—
27	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
28	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
29	C₂²×C₄²×C₅₃	C₂²	1	4	yes	—
30	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
31	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#391	C₂²	1	4	yes	—
33	C₅₃×#390	C₂²	1	4	yes	—
34	C₅₃×#390	C₂²	1	4	yes	—
35	C₅₃×#389	C₂²	1	4	yes	—
36	C₅₃×#389	C₂²	1	4	yes	—
37	C₅₃×#389	C₂²	1	4	yes	—
38	C₅₃×#388	C₂²	1	8	yes	—
39	C₅₃×#388	C₂²	1	8	yes	—
40	C₅₃×#387	C₂²	1	8	yes	—
41	C₅₃×#386	C₂²	1	8	yes	—
42	C₅₃×#386	C₂²	1	8	yes	—
43	C₅₃×#385	C₂²	1	8	yes	—
44	C₅₃×#380	C₂²	1	4	yes	—
45	C₅₃×#380	C₂²	1	4	yes	—
46	C₅₃×#380	C₂²	1	4	yes	—
47	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
48	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
49	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
50	C₄×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	—
51	C₂×#95×C₅₃	C₂²	1	2	yes	—
52	C₂×#41×C₅₃	—	2	8	no	—
53	C₂×#41×C₅₃	—	2	8	no	—
54	C₂×#41×C₅₃	C₂×C₄	1	16	yes	complemented
55	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
56	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
57	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
58	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
59	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
60	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
61	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
63	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
64	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
65	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₂²⋊Q₈×C₅₃	C₂×C₄	1	32	yes	complemented
68	#121×C₅₃	C₂×C₄	1	32	yes	complemented
69	#121×C₅₃	C₂×C₄	1	32	yes	complemented
70	C₄×Q₈×C₅₃	—	2	8	no	—
71	C₄×Q₈×C₅₃	—	2	8	no	—
72	C₄×D₄×C₅₃	—	2	8	no	—
73	C₄×D₄×C₅₃	—	2	8	no	—
74	C₄×D₄×C₅₃	—	2	16	no	—
75	C₄×D₄×C₅₃	—	2	16	no	—
76	C₄×D₄×C₅₃	—	4	8	no	—
77	#117×C₅₃	—	2	16	no	—
78	#117×C₅₃	—	2	8	no	—
79	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
80	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	complemented
81	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
85	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
86	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
88	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
90	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
91	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
92	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
93	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
94	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
95	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
96	C₂×C₄²×C₅₃	—	2	16	no	—
97	C₂×C₄²×C₅₃	—	4	4	no	—
98	C₂×C₄²×C₅₃	D₄	1	8	yes	complemented
99	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
100	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
101	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
102	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
103	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
104	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
105	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
106	#95×C₅₃	C₂³	1	8	yes	—
107	#95×C₅₃	C₂³	1	8	yes	—
108	#95×C₅₃	C₂³	1	8	yes	—
109	#95×C₅₃	C₂³	1	8	yes	—
110	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
111	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
112	#41×C₅₃	—	2	64	no	—
113	C₂×Q₈×C₅₃	—	2	4	no	—
114	C₂×Q₈×C₅₃	—	2	4	no	—
115	C₂×Q₈×C₅₃	C₂²×C₄	1	8	yes	—
116	C₂×D₄×C₅₃	—	2	8	no	—
117	C₂×D₄×C₅₃	—	2	8	no	—
118	C₂×D₄×C₅₃	—	2	4	no	—
119	C₂×D₄×C₅₃	—	2	4	no	—
120	C₂×D₄×C₅₃	—	4	4	no	—
121	C₂×D₄×C₅₃	—	4	4	no	—
122	C₂×D₄×C₅₃	C₂²×C₄	1	16	yes	—
123	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
124	C₂×D₄×C₅₃	C₂²×C₄	1	16	yes	—
125	C₂²×C₄×C₅₃	—	2	8	no	—
126	C₂²×C₄×C₅₃	—	2	8	no	—
127	C₂²×C₄×C₅₃	—	2	2	no	—
128	C₂²×C₄×C₅₃	—	2	2	no	—
129	C₂²×C₄×C₅₃	—	2	4	no	—
130	C₂²×C₄×C₅₃	—	2	8	no	—
131	C₂²×C₄×C₅₃	—	2	8	no	—
132	C₂²×C₄×C₅₃	—	2	8	no	—
133	C₂²×C₄×C₅₃	—	2	8	no	—
134	C₂²×C₄×C₅₃	—	4	2	no	—
135	C₂²×C₄×C₅₃	—	4	2	no	—
136	C₂²×C₄×C₅₃	—	4	1	no	—
137	C₂²×C₄×C₅₃	—	4	1	no	—
138	C₂²×C₄×C₅₃	—	4	4	no	—
139	C₂²×C₄×C₅₃	—	4	4	no	—
140	C₂²×C₄×C₅₃	—	4	2	no	—
141	C₂²×C₄×C₅₃	—	4	2	no	—
142	C₂²×C₄×C₅₃	C₂²×C₄	1	16	yes	—
143	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
144	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
145	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
146	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
147	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
148	C₂²×C₄×C₅₃	#41	1	4	yes	—
149	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
150	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
151	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
154	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
156	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
158	C₄⋊C₄×C₅₃	—	2	8	no	—
159	C₄⋊C₄×C₅₃	—	2	8	no	—
160	C₄⋊C₄×C₅₃	—	2	4	no	—
161	C₄⋊C₄×C₅₃	—	2	4	no	—
162	C₄⋊C₄×C₅₃	—	2	8	no	—
163	C₄⋊C₄×C₅₃	—	4	4	no	—
164	C₄⋊C₄×C₅₃	—	4	4	no	—
165	C₄⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
166	C₄⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
167	C₂²⋊C₄×C₅₃	—	2	8	no	—
168	C₂²⋊C₄×C₅₃	—	2	8	no	—
169	C₂²⋊C₄×C₅₃	—	2	8	no	—
170	C₂²⋊C₄×C₅₃	—	4	8	no	—
171	C₂²⋊C₄×C₅₃	—	4	8	no	—
172	C₂²⋊C₄×C₅₃	C₂²×C₄	1	32	yes	—
173	C₄²×C₅₃	—	2	8	no	—
174	C₄²×C₅₃	—	2	8	no	—
175	C₄²×C₅₃	—	2	8	no	—
176	C₄²×C₅₃	—	2	8	no	—
177	C₄²×C₅₃	—	4	4	no	—
178	C₄²×C₅₃	C₂×D₄	1	8	yes	—
179	C₄²×C₅₃	C₂×D₄	1	8	yes	—
180	C₂³×C₅₃	—	2	1	no	—
181	C₂³×C₅₃	—	2	1	no	—
182	C₂³×C₅₃	—	2	1	no	—
183	C₂³×C₅₃	—	2	1	no	—
184	C₂³×C₅₃	—	2	4	no	—
185	C₂³×C₅₃	—	2	4	no	—
186	C₂³×C₅₃	—	2	4	no	—
187	C₂³×C₅₃	—	2	4	no	—
188	C₂³×C₅₃	—	2	4	no	—
189	C₂³×C₅₃	—	2	4	no	—
190	C₂³×C₅₃	C₂³×C₄	1	8	yes	—
191	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
192	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
193	C₂³×C₅₃	C₂×#41	1	2	yes	—
194	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
195	Q₈×C₅₃	—	2	16	no	—
196	D₄×C₅₃	—	2	32	no	—
197	D₄×C₅₃	—	2	16	no	—
198	D₄×C₅₃	—	4	16	no	—
199	C₂×C₄×C₅₃	—	2	4	no	—
200	C₂×C₄×C₅₃	—	2	4	no	—
201	C₂×C₄×C₅₃	—	2	1	no	—
202	C₂×C₄×C₅₃	—	2	4	no	—
203	C₂×C₄×C₅₃	—	2	1	no	—
204	C₂×C₄×C₅₃	—	2	2	no	—
205	C₂×C₄×C₅₃	—	2	2	no	—
206	C₂×C₄×C₅₃	—	2	4	no	—
207	C₂×C₄×C₅₃	—	2	32	no	—
208	C₂×C₄×C₅₃	—	2	8	no	—
209	C₂×C₄×C₅₃	—	2	4	no	—
210	C₂×C₄×C₅₃	—	2	4	no	—
211	C₂×C₄×C₅₃	—	2	4	no	—
212	C₂×C₄×C₅₃	—	2	4	no	—
213	C₂×C₄×C₅₃	—	2	4	no	—
214	C₂×C₄×C₅₃	—	2	4	no	—
215	C₂×C₄×C₅₃	—	2	4	no	—
216	C₂×C₄×C₅₃	—	2	4	no	—
217	C₂×C₄×C₅₃	—	2	4	no	—
218	C₂×C₄×C₅₃	—	4	8	no	—
219	C₂×C₄×C₅₃	—	4	4	no	—
220	C₂×C₄×C₅₃	—	4	16	no	—
221	C₂×C₄×C₅₃	—	4	8	no	—
222	C₂×C₄×C₅₃	—	4	16	no	—
223	C₂×C₄×C₅₃	C₄×D₄	1	16	yes	complemented
224	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
225	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
226	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
227	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
228	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
229	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
230	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
231	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
232	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
233	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
234	C₄×#537	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
235	C₂²×C₅₃	—	2	2	no	—
236	C₂²×C₅₃	—	2	2	no	—
237	C₂²×C₅₃	—	2	2	no	—
238	C₂²×C₅₃	—	2	2	no	—
239	C₂²×C₅₃	—	2	2	no	—
240	C₂²×C₅₃	—	2	2	no	—
241	C₂²×C₅₃	—	2	8	no	—
242	C₂²×C₅₃	—	2	8	no	—
243	C₂²×C₅₃	—	2	8	no	—
244	C₂²×C₅₃	—	4	4	no	—
245	C₂²×C₅₃	—	4	4	no	—
246	C₂²×C₅₃	—	4	4	no	—
247	C₂²×C₅₃	—	4	4	no	—
248	C₂²×C₅₃	C₂×C₄×D₄	1	4	yes	—
249	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
250	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
251	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
252	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
253	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
254	C₂²×C₅₃	#585	1	1	yes	characteristic
255	C₂²×C₅₃	#585	1	1	yes	characteristic
256	C₄×C₅₃	—	2	8	no	—
257	C₄×C₅₃	—	2	8	no	—
258	C₄×C₅₃	—	2	2	no	—
259	C₄×C₅₃	—	2	4	no	—
260	C₄×C₅₃	—	4	4	no	—
261	C₄×C₅₃	—	4	4	no	—
262	C₄×C₅₃	—	4	4	no	—
263	C₄×C₅₃	C₂×C₄×D₄	1	8	yes	—
264	C₄×C₅₃	#537	1	4	yes	direct product factor
265	C₂×#537	C₂×C₅₃	1	1	yes	characteristic
266	C₂×C₄×#41	C₂×C₅₃	1	2	yes	complemented
267	#2325	C₂×C₅₃	1	4	yes	complemented
268	#2324	C₂×C₅₃	1	4	yes	complemented
269	#2323	C₂×C₅₃	1	4	yes	complemented
270	#2316	C₂×C₅₃	1	1	yes	characteristic, complemented
271	#2311	C₂×C₅₃	1	2	yes	complemented
272	C₄×C₂²⋊Q₈	C₂×C₅₃	1	4	yes	complemented
273	C₄×#121	C₂×C₅₃	1	4	yes	complemented
274	C₄×#121	C₂×C₅₃	1	4	yes	complemented
275	C₂×C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
276	C₂×C₅₃	—	2	4	no	—
277	C₂×C₅₃	—	4	8	no	—
278	C₂×C₅₃	C₂²×C₄×D₄	1	1	yes	characteristic
279	C₂×C₅₃	C₄×H(5,𝔽₂)	1	1	yes	characteristic
280	C₂×C₅₃	C₄×2₋¹⁺⁴	1	1	yes	characteristic
281	C₂×C₅₃	C₂×#537	1	1	yes	characteristic
282	C₂×C₅₃	#3449	1	1	yes	characteristic
283	C₂×C₅₃	#3460	1	1	yes	characteristic
284	C₂×C₅₃	#3461	1	1	yes	characteristic
285	C₂²×#41	C₂²×C₅₃	1	2	yes	—
286	#537	C₄×C₅₃	1	16	yes	direct product factor
287	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	—
288	C₂×#121	C₂²×C₅₃	1	4	yes	—
289	C₂×#121	C₂²×C₅₃	1	4	yes	—
290	C₄×#41	—	2	16	no	—
291	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
292	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
293	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
294	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
295	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
296	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
297	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
298	C₂×#117	C₂²×C₅₃	1	2	yes	—
299	C₂×#117	C₂²×C₅₃	1	4	yes	—
300	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
301	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
302	C₂²×C₄²	C₂²×C₅₃	1	4	yes	—
303	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
304	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
305	#391	C₂²×C₅₃	1	4	yes	—
306	#390	C₂²×C₅₃	1	4	yes	—
307	#390	C₂²×C₅₃	1	4	yes	—
308	#389	C₂²×C₅₃	1	4	yes	—
309	#389	C₂²×C₅₃	1	4	yes	—
310	#389	C₂²×C₅₃	1	4	yes	—
311	#388	C₂²×C₅₃	1	8	yes	—
312	#388	C₂²×C₅₃	1	8	yes	—
313	#387	C₂²×C₅₃	1	8	yes	—
314	#386	C₂²×C₅₃	1	8	yes	—
315	#386	C₂²×C₅₃	1	8	yes	—
316	#385	C₂²×C₅₃	1	8	yes	—
317	#380	C₂²×C₅₃	1	4	yes	—
318	#380	C₂²×C₅₃	1	4	yes	—
319	#380	C₂²×C₅₃	1	4	yes	—
320	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
321	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
322	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
323	C₄×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	—
324	C₂×#95	C₂²×C₅₃	1	2	yes	—
325	C₅₃	C₄×#537	1	1	yes	characteristic, direct product factor, Sylow
326	C₂×#41	—	2	8	no	—
327	C₂×#41	—	2	8	no	—
328	C₂×#41	C₂×C₄×C₅₃	1	16	yes	complemented
329	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
330	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
331	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
332	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
333	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
334	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
335	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
336	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
337	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
338	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
339	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
340	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
341	C₂²⋊Q₈	C₂×C₄×C₅₃	1	32	yes	complemented
342	#121	C₂×C₄×C₅₃	1	32	yes	complemented
343	#121	C₂×C₄×C₅₃	1	32	yes	complemented
344	C₄×Q₈	—	2	8	no	—
345	C₄×Q₈	—	2	8	no	—
346	C₄×D₄	—	2	16	no	—
347	C₄×D₄	—	2	16	no	—
348	C₄×D₄	—	2	8	no	—
349	C₄×D₄	—	2	8	no	—
350	C₄×D₄	—	4	8	no	—
351	#117	—	2	8	no	—
352	#117	—	2	16	no	—
353	C₂×C₄⋊C₄	—	2	4	no	—
354	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	8	yes	complemented
355	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
356	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
357	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
358	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
359	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
360	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
361	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
362	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
363	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
364	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
365	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
366	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
367	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
368	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
369	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
370	C₂×C₄²	—	2	16	no	—
371	C₂×C₄²	—	4	4	no	—
372	C₂×C₄²	D₄×C₅₃	1	8	yes	complemented
373	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
374	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
375	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
376	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
377	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
378	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
379	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
380	#95	C₂³×C₅₃	1	8	yes	—
381	#95	C₂³×C₅₃	1	8	yes	—
382	#95	C₂³×C₅₃	1	8	yes	—
383	#95	C₂³×C₅₃	1	8	yes	—
384	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
385	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
386	#41	—	2	64	no	—
387	C₂×Q₈	—	2	4	no	—
388	C₂×Q₈	—	2	4	no	—
389	C₂×Q₈	C₂²×C₄×C₅₃	1	8	yes	—
390	C₂×D₄	—	2	8	no	—
391	C₂×D₄	—	2	8	no	—
392	C₂×D₄	—	2	4	no	—
393	C₂×D₄	—	2	4	no	—
394	C₂×D₄	—	4	4	no	—
395	C₂×D₄	—	4	4	no	—
396	C₂×D₄	C₂²×C₄×C₅₃	1	16	yes	—
397	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
398	C₂×D₄	C₂²×C₄×C₅₃	1	16	yes	—
399	C₂²×C₄	—	2	8	no	—
400	C₂²×C₄	—	2	8	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	4	no	—
404	C₂²×C₄	—	2	8	no	—
405	C₂²×C₄	—	2	8	no	—
406	C₂²×C₄	—	2	8	no	—
407	C₂²×C₄	—	2	8	no	—
408	C₂²×C₄	—	4	2	no	—
409	C₂²×C₄	—	4	2	no	—
410	C₂²×C₄	—	4	2	no	—
411	C₂²×C₄	—	4	1	no	—
412	C₂²×C₄	—	4	1	no	—
413	C₂²×C₄	—	4	4	no	—
414	C₂²×C₄	—	4	4	no	—
415	C₂²×C₄	—	4	2	no	—
416	C₂²×C₄	C₂²×C₄×C₅₃	1	16	yes	—
417	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
418	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
419	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
420	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
421	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
422	C₂²×C₄	#41×C₅₃	1	4	yes	—
423	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
424	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
425	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
426	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
427	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
428	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
429	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
430	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
431	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
432	C₄⋊C₄	—	2	8	no	—
433	C₄⋊C₄	—	2	8	no	—
434	C₄⋊C₄	—	2	4	no	—
435	C₄⋊C₄	—	2	4	no	—
436	C₄⋊C₄	—	2	8	no	—
437	C₄⋊C₄	—	4	4	no	—
438	C₄⋊C₄	—	4	4	no	—
439	C₄⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
440	C₄⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
441	C₂²⋊C₄	—	2	8	no	—
442	C₂²⋊C₄	—	2	8	no	—
443	C₂²⋊C₄	—	2	8	no	—
444	C₂²⋊C₄	—	4	8	no	—
445	C₂²⋊C₄	—	4	8	no	—
446	C₂²⋊C₄	C₂²×C₄×C₅₃	1	32	yes	—
447	C₄²	—	2	8	no	—
448	C₄²	—	2	8	no	—
449	C₄²	—	2	8	no	—
450	C₄²	—	2	8	no	—
451	C₄²	—	4	4	no	—
452	C₄²	C₂×D₄×C₅₃	1	8	yes	—
453	C₄²	C₂×D₄×C₅₃	1	8	yes	—
454	C₂³	—	2	1	no	—
455	C₂³	—	2	1	no	—
456	C₂³	—	2	1	no	—
457	C₂³	—	2	1	no	—
458	C₂³	—	2	4	no	—
459	C₂³	—	2	4	no	—
460	C₂³	—	2	4	no	—
461	C₂³	—	2	4	no	—
462	C₂³	—	2	4	no	—
463	C₂³	—	2	4	no	—
464	C₂³	C₂³×C₄×C₅₃	1	8	yes	—
465	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
466	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
467	C₂³	C₂×#41×C₅₃	1	2	yes	—
468	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
469	Q₈	—	2	16	no	—
470	D₄	—	2	32	no	—
471	D₄	—	2	16	no	—
472	D₄	—	4	16	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	1	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	32	no	—
482	C₂×C₄	—	2	8	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	4	no	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	4	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	4	8	no	—
493	C₂×C₄	—	4	4	no	—
494	C₂×C₄	—	4	16	no	—
495	C₂×C₄	—	4	8	no	—
496	C₂×C₄	—	4	16	no	—
497	C₂×C₄	C₄×D₄×C₅₃	1	16	yes	complemented
498	C₂×C₄	C₂³×C₄×C₅₃	1	8	yes	—
499	C₂×C₄	C₂³×C₄×C₅₃	1	8	yes	—
500	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
501	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
502	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
503	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
504	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
505	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
506	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
507	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
508	C₂²	—	2	2	no	—
509	C₂²	—	2	2	no	—
510	C₂²	—	2	2	no	—
511	C₂²	—	2	2	no	—
512	C₂²	—	2	2	no	—
513	C₂²	—	2	2	no	—
514	C₂²	—	2	8	no	—
515	C₂²	—	2	8	no	—
516	C₂²	—	2	8	no	—
517	C₂²	—	4	4	no	—
518	C₂²	—	4	4	no	—
519	C₂²	—	4	4	no	—
520	C₂²	—	4	4	no	—
521	C₂²	C₂×C₄×D₄×C₅₃	1	4	yes	—
522	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
523	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
524	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
525	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
526	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
527	C₂²	C₅₃×#585	1	1	yes	characteristic
528	C₂²	C₅₃×#585	1	1	yes	characteristic
529	C₄	—	2	2	no	—
530	C₄	—	2	4	no	—
531	C₄	—	2	8	no	—
532	C₄	—	2	8	no	—
533	C₄	—	4	4	no	—
534	C₄	—	4	4	no	—
535	C₄	—	4	4	no	—
536	C₄	C₂×C₄×D₄×C₅₃	1	8	yes	—
537	C₄	C₅₃×#537	1	4	yes	direct product factor
538	C₂	—	2	4	no	—
539	C₂	—	4	8	no	—
540	C₂	C₂²×C₄×D₄×C₅₃	1	1	yes	characteristic
541	C₂	C₄×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
542	C₂	C₄×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
543	C₂	C₂×C₅₃×#537	1	1	yes	characteristic
544	C₂	C₅₃×#3449	1	1	yes	characteristic
545	C₂	C₅₃×#3460	1	1	yes	characteristic
546	C₂	C₅₃×#3461	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁵×C₅₃
socle	C₂³×C₅₃	C₂⁵
prime radical	C₂×C₅₃×#537	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#41×C₅₃	C₂	semidirect
3. C₅₃×#2325	C₂	semidirect
4. C₅₃×#2324	C₂	semidirect
5. C₅₃×#2323	C₂	semidirect
6. C₅₃×#2316	C₂	semidirect
7. C₅₃×#2311	C₂	semidirect
8. C₄×C₂²⋊Q₈×C₅₃	C₂	semidirect
9. C₄×#121×C₅₃	C₂	semidirect
10. C₄×#121×C₅₃	C₂	semidirect
11. C₂×C₄×C₄⋊C₄×C₅₃	C₂	semidirect
13. C₅₃×#537	C₄	direct (N × H)
54. C₂×#41×C₅₃	C₂×C₄	semidirect
67. C₂²⋊Q₈×C₅₃	C₂×C₄	semidirect
68. #121×C₅₃	C₂×C₄	semidirect
69. #121×C₅₃	C₂×C₄	semidirect
80. C₂×C₄⋊C₄×C₅₃	C₂×C₄	semidirect
98. C₂×C₄²×C₅₃	D₄	semidirect
223. C₂×C₄×C₅₃	C₄×D₄	semidirect
234. C₄×#537	C₅₃	direct (N × H)
264. C₄×C₅₃	#537	direct (N × H)
266. C₂×C₄×#41	C₂×C₅₃	semidirect
267. #2325	C₂×C₅₃	semidirect
268. #2324	C₂×C₅₃	semidirect
269. #2323	C₂×C₅₃	semidirect
270. #2316	C₂×C₅₃	semidirect
271. #2311	C₂×C₅₃	semidirect
272. C₄×C₂²⋊Q₈	C₂×C₅₃	semidirect
273. C₄×#121	C₂×C₅₃	semidirect
274. C₄×#121	C₂×C₅₃	semidirect
275. C₂×C₄×C₄⋊C₄	C₂×C₅₃	semidirect
286. #537	C₄×C₅₃	direct (N × H)
325. C₅₃	C₄×#537	direct (N × H)
328. C₂×#41	C₂×C₄×C₅₃	semidirect
341. C₂²⋊Q₈	C₂×C₄×C₅₃	semidirect
342. #121	C₂×C₄×C₅₃	semidirect
343. #121	C₂×C₄×C₅₃	semidirect
354. C₂×C₄⋊C₄	C₂×C₄×C₅₃	semidirect
372. C₂×C₄²	D₄×C₅₃	semidirect
497. C₂×C₄	C₄×D₄×C₅₃	semidirect
537. C₄	C₅₃×#537	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#537	2
C₂×C₄×#41×C₅₃	2
C₅₃×#2325	2
C₅₃×#2324	2
C₅₃×#2323	2
C₅₃×#2316	2
C₅₃×#2311	2
C₄×C₂²⋊Q₈×C₅₃	2
C₄×#121×C₅₃	2
C₂×C₄×C₄⋊C₄×C₅₃	2
C₄×#537	53

Subgroups — C4×C53×#537 #27241602

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅₃×#537 #27241602