Subgroups — C₂×C₅₃×#2326 #27241579

Order 13568.

3566 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#124×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2326	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₅₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×#390	C₂	1	1	yes	maximal, characteristic
7	C₂×C₅₃×#388	C₂	1	4	yes	maximal, complemented
8	C₂×C₅₃×#385	C₂	1	4	yes	maximal, complemented
9	C₂×C₅₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
14	C₂×#124×C₅₃	C₄	1	16	yes	complemented
15	C₂×#124×C₅₃	C₂²	1	8	yes	—
16	C₂×C₄×Q₈×C₅₃	—	2	8	no	—
17	C₂×C₄×Q₈×C₅₃	C₂²	1	8	yes	complemented
18	C₂×C₄×D₄×C₅₃	—	2	8	no	—
19	C₂×C₄×D₄×C₅₃	C₂²	1	8	yes	—
20	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
24	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
26	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
27	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
29	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
30	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#391	C₂²	1	8	yes	complemented
32	C₅₃×#390	C₂²	1	8	yes	—
33	C₅₃×#388	C₂²	1	32	yes	complemented
34	C₅₃×#385	C₂²	1	32	yes	complemented
35	C₅₃×#384	C₂²	1	8	yes	complemented
36	C₅₃×#376	C₂²	1	8	yes	complemented
37	C₂×#95×C₅₃	C₂²	1	2	yes	—
38	C₂×#95×C₅₃	C₂²	1	2	yes	—
39	C₂×#95×C₅₃	C₂²	1	4	yes	—
40	C₂⁵×C₅₃	C₂³	1	2	yes	—
41	C₂²×Q₈×C₅₃	—	2	4	no	—
42	C₂²×Q₈×C₅₃	—	2	4	no	—
43	C₂²×Q₈×C₅₃	C₂×C₄	1	8	yes	complemented
44	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
45	C₂²×D₄×C₅₃	—	2	4	no	—
46	C₂²×D₄×C₅₃	—	2	4	no	—
47	C₂²×D₄×C₅₃	C₂×C₄	1	8	yes	—
48	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
49	C₂³×C₄×C₅₃	—	2	8	no	—
50	C₂³×C₄×C₅₃	—	4	8	no	—
51	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
52	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
53	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
54	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
55	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
56	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
59	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
60	#124×C₅₃	C₂×C₄	1	64	yes	complemented
61	C₄×Q₈×C₅₃	—	2	32	no	—
62	C₄×D₄×C₅₃	—	2	32	no	—
63	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
64	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
65	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
66	C₂×C₄⋊C₄×C₅₃	—	4	8	no	—
67	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
68	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
69	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
70	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
71	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
72	C₂×C₂²⋊C₄×C₅₃	—	4	8	no	—
73	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	32	yes	complemented
74	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
75	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
76	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
77	C₂×C₄²×C₅₃	—	2	8	no	—
78	C₂×C₄²×C₅₃	—	2	4	no	—
79	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	complemented
80	C₂×C₄²×C₅₃	D₄	1	8	yes	—
81	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
82	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
83	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
84	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
85	#95×C₅₃	C₂³	1	8	yes	—
86	#95×C₅₃	C₂³	1	8	yes	—
87	#95×C₅₃	C₂³	1	16	yes	—
88	C₂⁴×C₅₃	—	2	4	no	—
89	C₂⁴×C₅₃	—	2	4	no	—
90	C₂⁴×C₅₃	—	2	8	no	—
91	C₂⁴×C₅₃	—	2	8	no	—
92	C₂⁴×C₅₃	C₂²×C₄	1	8	yes	—
93	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
94	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
95	C₂×Q₈×C₅₃	—	2	32	no	—
96	C₂×Q₈×C₅₃	—	2	8	no	—
97	C₂×Q₈×C₅₃	—	2	8	no	—
98	C₂×Q₈×C₅₃	C₂²×C₄	1	16	yes	complemented
99	C₂×D₄×C₅₃	—	2	32	no	—
100	C₂×D₄×C₅₃	—	2	8	no	—
101	C₂×D₄×C₅₃	—	2	8	no	—
102	C₂×D₄×C₅₃	C₂²×C₄	1	16	yes	—
103	C₂²×C₄×C₅₃	—	2	4	no	—
104	C₂²×C₄×C₅₃	—	2	4	no	—
105	C₂²×C₄×C₅₃	—	2	4	no	—
106	C₂²×C₄×C₅₃	—	2	4	no	—
107	C₂²×C₄×C₅₃	—	2	4	no	—
108	C₂²×C₄×C₅₃	—	2	4	no	—
109	C₂²×C₄×C₅₃	—	2	4	no	—
110	C₂²×C₄×C₅₃	—	2	4	no	—
111	C₂²×C₄×C₅₃	—	2	4	no	—
112	C₂²×C₄×C₅₃	—	2	4	no	—
113	C₂²×C₄×C₅₃	—	2	8	no	—
114	C₂²×C₄×C₅₃	—	2	8	no	—
115	C₂²×C₄×C₅₃	—	2	4	no	—
116	C₂²×C₄×C₅₃	—	2	16	no	—
117	C₂²×C₄×C₅₃	—	4	16	no	—
118	C₂²×C₄×C₅₃	—	4	32	no	—
119	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
120	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
121	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
122	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
123	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
124	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
125	C₂²×C₄×C₅₃	#41	1	4	yes	—
126	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
127	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
128	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
129	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
131	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
132	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
133	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
134	C₄⋊C₄×C₅₃	—	2	8	no	—
135	C₄⋊C₄×C₅₃	—	2	16	no	—
136	C₄⋊C₄×C₅₃	—	2	8	no	—
137	C₄⋊C₄×C₅₃	—	4	16	no	—
138	C₂²⋊C₄×C₅₃	—	2	16	no	—
139	C₂²⋊C₄×C₅₃	—	4	16	no	—
140	C₂²⋊C₄×C₅₃	C₂²×C₄	1	64	yes	complemented
141	C₄²×C₅₃	—	2	16	no	—
142	C₄²×C₅₃	—	2	8	no	—
143	C₄²×C₅₃	C₂²×C₄	1	16	yes	complemented
144	C₄²×C₅₃	C₂×D₄	1	16	yes	—
145	C₂³×C₅₃	—	2	16	no	—
146	C₂³×C₅₃	—	2	32	no	—
147	C₂³×C₅₃	—	2	4	no	—
148	C₂³×C₅₃	—	2	4	no	—
149	C₂³×C₅₃	—	2	8	no	—
150	C₂³×C₅₃	—	2	8	no	—
151	C₂³×C₅₃	—	4	8	no	—
152	C₂³×C₅₃	—	4	8	no	—
153	C₂³×C₅₃	—	4	16	no	—
154	C₂³×C₅₃	C₂³×C₄	1	8	yes	—
155	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
156	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
157	C₂³×C₅₃	C₂×#41	1	2	yes	—
158	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
159	C₂³×C₅₃	2₋¹⁺⁴	1	4	yes	—
160	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
161	Q₈×C₅₃	—	2	32	no	—
162	D₄×C₅₃	—	2	32	no	—
163	C₂×C₄×C₅₃	—	2	16	no	—
164	C₂×C₄×C₅₃	—	2	16	no	—
165	C₂×C₄×C₅₃	—	2	8	no	—
166	C₂×C₄×C₅₃	—	2	8	no	—
167	C₂×C₄×C₅₃	—	2	4	no	—
168	C₂×C₄×C₅₃	—	2	4	no	—
169	C₂×C₄×C₅₃	—	2	4	no	—
170	C₂×C₄×C₅₃	—	2	4	no	—
171	C₂×C₄×C₅₃	—	2	4	no	—
172	C₂×C₄×C₅₃	—	2	4	no	—
173	C₂×C₄×C₅₃	—	2	4	no	—
174	C₂×C₄×C₅₃	—	2	4	no	—
175	C₂×C₄×C₅₃	—	2	4	no	—
176	C₂×C₄×C₅₃	—	2	8	no	—
177	C₂×C₄×C₅₃	—	2	4	no	—
178	C₂×C₄×C₅₃	—	2	8	no	—
179	C₂×C₄×C₅₃	—	2	4	no	—
180	C₂×C₄×C₅₃	—	4	32	no	—
181	C₂×C₄×C₅₃	—	4	8	no	—
182	C₂×C₄×C₅₃	—	4	8	no	—
183	C₂×C₄×C₅₃	—	4	8	no	—
184	C₂×C₄×C₅₃	C₄×D₄	1	16	yes	—
185	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
186	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
187	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
188	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
189	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
190	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
191	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
192	C₂×#2326	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂²×C₅₃	—	2	8	no	—
194	C₂²×C₅₃	—	2	16	no	—
195	C₂²×C₅₃	—	4	32	no	—
196	C₂²×C₅₃	—	4	4	no	—
197	C₂²×C₅₃	—	4	4	no	—
198	C₂²×C₅₃	—	4	8	no	—
199	C₂²×C₅₃	C₂×C₄×D₄	1	4	yes	—
200	C₂²×C₅₃	#520	1	8	yes	—
201	C₂²×C₅₃	#535	1	4	yes	—
202	C₂²×C₅₃	#536	1	4	yes	—
203	C₂²×C₅₃	#542	1	8	yes	—
204	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
205	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
206	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
207	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
208	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	2	yes	—
209	C₄×C₅₃	—	2	8	no	—
210	C₄×C₅₃	—	2	8	no	—
211	C₄×C₅₃	—	2	4	no	—
212	C₄×C₅₃	—	2	4	no	—
213	C₄×C₅₃	—	4	4	no	—
214	C₄×C₅₃	—	4	4	no	—
215	C₄×C₅₃	—	4	4	no	—
216	C₄×C₅₃	C₂×C₄×D₄	1	8	yes	—
217	C₂²×#124	C₂×C₅₃	1	1	yes	characteristic
218	C₂²×C₄×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
219	C₂²×C₄×D₄	C₂×C₅₃	1	1	yes	characteristic
220	#2326	C₂×C₅₃	1	16	yes	direct product factor
221	C₂×#391	C₂×C₅₃	1	1	yes	characteristic, complemented
222	C₂×#390	C₂×C₅₃	1	1	yes	characteristic
223	C₂×#388	C₂×C₅₃	1	4	yes	complemented
224	C₂×#385	C₂×C₅₃	1	4	yes	complemented
225	C₂×#384	C₂×C₅₃	1	1	yes	characteristic, complemented
226	C₂×#376	C₂×C₅₃	1	1	yes	characteristic, complemented
227	C₂×C₅₃	—	4	8	no	—
228	C₂×C₅₃	#2326	1	8	yes	direct product factor
229	C₂×C₅₃	C₂²×C₄×D₄	1	1	yes	characteristic
230	C₂×C₅₃	C₂×#520	1	2	yes	—
231	C₂×C₅₃	C₂×#535	1	1	yes	characteristic
232	C₂×C₅₃	C₂×#536	1	1	yes	characteristic
233	C₂×C₅₃	C₂×#542	1	2	yes	—
234	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
235	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
236	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	—
237	C₂×#124	C₄×C₅₃	1	16	yes	complemented
238	C₂×#124	C₂²×C₅₃	1	8	yes	—
239	C₂×C₄×Q₈	—	2	8	no	—
240	C₂×C₄×Q₈	C₂²×C₅₃	1	8	yes	complemented
241	C₂×C₄×D₄	—	2	8	no	—
242	C₂×C₄×D₄	C₂²×C₅₃	1	8	yes	—
243	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
244	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
245	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
246	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
247	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
248	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
249	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
250	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
251	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
252	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
253	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
254	#391	C₂²×C₅₃	1	8	yes	complemented
255	#390	C₂²×C₅₃	1	8	yes	—
256	#388	C₂²×C₅₃	1	32	yes	complemented
257	#385	C₂²×C₅₃	1	32	yes	complemented
258	#384	C₂²×C₅₃	1	8	yes	complemented
259	#376	C₂²×C₅₃	1	8	yes	complemented
260	C₂×#95	C₂²×C₅₃	1	2	yes	—
261	C₂×#95	C₂²×C₅₃	1	2	yes	—
262	C₂×#95	C₂²×C₅₃	1	4	yes	—
263	C₅₃	C₂×#2326	1	1	yes	characteristic, direct product factor, Sylow
264	C₂⁵	C₂³×C₅₃	1	2	yes	—
265	C₂²×Q₈	—	2	4	no	—
266	C₂²×Q₈	—	2	4	no	—
267	C₂²×Q₈	C₂×C₄×C₅₃	1	8	yes	complemented
268	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
269	C₂²×D₄	—	2	4	no	—
270	C₂²×D₄	—	2	4	no	—
271	C₂²×D₄	C₂×C₄×C₅₃	1	8	yes	—
272	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
273	C₂³×C₄	—	2	8	no	—
274	C₂³×C₄	—	4	8	no	—
275	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
276	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
277	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
278	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
279	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
280	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
281	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
282	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
283	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
284	#124	C₂×C₄×C₅₃	1	64	yes	complemented
285	C₄×Q₈	—	2	32	no	—
286	C₄×D₄	—	2	32	no	—
287	C₂×C₄⋊C₄	—	2	4	no	—
288	C₂×C₄⋊C₄	—	2	8	no	—
289	C₂×C₄⋊C₄	—	2	4	no	—
290	C₂×C₄⋊C₄	—	4	8	no	—
291	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
292	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
293	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
294	C₂×C₄⋊C₄	C₂³×C₅₃	1	16	yes	—
295	C₂×C₂²⋊C₄	—	2	8	no	—
296	C₂×C₂²⋊C₄	—	4	8	no	—
297	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	32	yes	complemented
298	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
299	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
300	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
301	C₂×C₄²	—	2	8	no	—
302	C₂×C₄²	—	2	4	no	—
303	C₂×C₄²	C₂×C₄×C₅₃	1	8	yes	complemented
304	C₂×C₄²	D₄×C₅₃	1	8	yes	—
305	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
306	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
307	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
308	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
309	#95	C₂³×C₅₃	1	8	yes	—
310	#95	C₂³×C₅₃	1	8	yes	—
311	#95	C₂³×C₅₃	1	16	yes	—
312	C₂⁴	—	2	4	no	—
313	C₂⁴	—	2	4	no	—
314	C₂⁴	—	2	8	no	—
315	C₂⁴	—	2	8	no	—
316	C₂⁴	C₂²×C₄×C₅₃	1	8	yes	—
317	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
318	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
319	C₂×Q₈	—	2	32	no	—
320	C₂×Q₈	—	2	8	no	—
321	C₂×Q₈	—	2	8	no	—
322	C₂×Q₈	C₂²×C₄×C₅₃	1	16	yes	complemented
323	C₂×D₄	—	2	32	no	—
324	C₂×D₄	—	2	8	no	—
325	C₂×D₄	—	2	8	no	—
326	C₂×D₄	C₂²×C₄×C₅₃	1	16	yes	—
327	C₂²×C₄	—	2	4	no	—
328	C₂²×C₄	—	2	4	no	—
329	C₂²×C₄	—	2	4	no	—
330	C₂²×C₄	—	2	4	no	—
331	C₂²×C₄	—	2	4	no	—
332	C₂²×C₄	—	2	4	no	—
333	C₂²×C₄	—	2	4	no	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	—	2	4	no	—
336	C₂²×C₄	—	2	4	no	—
337	C₂²×C₄	—	2	8	no	—
338	C₂²×C₄	—	2	8	no	—
339	C₂²×C₄	—	2	4	no	—
340	C₂²×C₄	—	2	16	no	—
341	C₂²×C₄	—	4	16	no	—
342	C₂²×C₄	—	4	32	no	—
343	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
344	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
345	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
346	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
347	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
348	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
349	C₂²×C₄	#41×C₅₃	1	4	yes	—
350	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
351	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
352	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
353	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
354	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
355	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
356	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
357	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
358	C₄⋊C₄	—	2	8	no	—
359	C₄⋊C₄	—	2	16	no	—
360	C₄⋊C₄	—	2	8	no	—
361	C₄⋊C₄	—	4	16	no	—
362	C₂²⋊C₄	—	2	16	no	—
363	C₂²⋊C₄	—	4	16	no	—
364	C₂²⋊C₄	C₂²×C₄×C₅₃	1	64	yes	complemented
365	C₄²	—	2	16	no	—
366	C₄²	—	2	8	no	—
367	C₄²	C₂²×C₄×C₅₃	1	16	yes	complemented
368	C₄²	C₂×D₄×C₅₃	1	16	yes	—
369	C₂³	—	2	16	no	—
370	C₂³	—	2	32	no	—
371	C₂³	—	2	4	no	—
372	C₂³	—	2	4	no	—
373	C₂³	—	2	8	no	—
374	C₂³	—	2	8	no	—
375	C₂³	—	4	8	no	—
376	C₂³	—	4	8	no	—
377	C₂³	—	4	16	no	—
378	C₂³	C₂³×C₄×C₅₃	1	8	yes	—
379	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
380	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
381	C₂³	C₂×#41×C₅₃	1	2	yes	—
382	C₂³	H(5,𝔽₂)×C₅₃	1	4	yes	—
383	C₂³	2₋¹⁺⁴×C₅₃	1	4	yes	—
384	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
385	Q₈	—	2	32	no	—
386	D₄	—	2	32	no	—
387	C₂×C₄	—	2	16	no	—
388	C₂×C₄	—	2	16	no	—
389	C₂×C₄	—	2	8	no	—
390	C₂×C₄	—	2	8	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	2	4	no	—
393	C₂×C₄	—	2	4	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	4	no	—
396	C₂×C₄	—	2	4	no	—
397	C₂×C₄	—	2	4	no	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	2	4	no	—
401	C₂×C₄	—	2	8	no	—
402	C₂×C₄	—	2	8	no	—
403	C₂×C₄	—	2	4	no	—
404	C₂×C₄	—	4	32	no	—
405	C₂×C₄	—	4	8	no	—
406	C₂×C₄	—	4	8	no	—
407	C₂×C₄	—	4	8	no	—
408	C₂×C₄	C₄×D₄×C₅₃	1	16	yes	—
409	C₂×C₄	C₂³×C₄×C₅₃	1	8	yes	—
410	C₂×C₄	C₂³×C₄×C₅₃	1	8	yes	—
411	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
412	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
413	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
414	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
415	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
416	C₂²	—	2	8	no	—
417	C₂²	—	2	16	no	—
418	C₂²	—	4	4	no	—
419	C₂²	—	4	8	no	—
420	C₂²	—	4	32	no	—
421	C₂²	—	4	4	no	—
422	C₂²	C₂×C₄×D₄×C₅₃	1	4	yes	—
423	C₂²	C₅₃×#520	1	8	yes	—
424	C₂²	C₅₃×#535	1	4	yes	—
425	C₂²	C₅₃×#536	1	4	yes	—
426	C₂²	C₅₃×#542	1	8	yes	—
427	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
428	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
429	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
430	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
431	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	2	yes	—
432	C₄	—	2	4	no	—
433	C₄	—	2	8	no	—
434	C₄	—	2	8	no	—
435	C₄	—	2	4	no	—
436	C₄	—	4	4	no	—
437	C₄	—	4	4	no	—
438	C₄	—	4	4	no	—
439	C₄	C₂×C₄×D₄×C₅₃	1	8	yes	—
440	C₂	—	4	8	no	—
441	C₂	C₅₃×#2326	1	8	yes	direct product factor
442	C₂	C₂²×C₄×D₄×C₅₃	1	1	yes	characteristic
443	C₂	C₂×C₅₃×#520	1	2	yes	—
444	C₂	C₂×C₅₃×#535	1	1	yes	characteristic
445	C₂	C₂×C₅₃×#536	1	1	yes	characteristic
446	C₂	C₂×C₅₃×#542	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂³×D₄×C₅₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×Q₈×C₅₃	C₂	semidirect
4. C₅₃×#2326	C₂	direct (N × H)
5. C₂×C₅₃×#391	C₂	semidirect
7. C₂×C₅₃×#388	C₂	semidirect
8. C₂×C₅₃×#385	C₂	semidirect
9. C₂×C₅₃×#384	C₂	semidirect
10. C₂×C₅₃×#376	C₂	semidirect
14. C₂×#124×C₅₃	C₄	semidirect
17. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
31. C₅₃×#391	C₂²	semidirect
33. C₅₃×#388	C₂²	semidirect
34. C₅₃×#385	C₂²	semidirect
35. C₅₃×#384	C₂²	semidirect
36. C₅₃×#376	C₂²	semidirect
43. C₂²×Q₈×C₅₃	C₂×C₄	semidirect
60. #124×C₅₃	C₂×C₄	semidirect
73. C₂×C₂²⋊C₄×C₅₃	C₂×C₄	semidirect
79. C₂×C₄²×C₅₃	C₂×C₄	semidirect
98. C₂×Q₈×C₅₃	C₂²×C₄	semidirect
140. C₂²⋊C₄×C₅₃	C₂²×C₄	semidirect
143. C₄²×C₅₃	C₂²×C₄	semidirect
192. C₂×#2326	C₅₃	direct (N × H)
218. C₂²×C₄×Q₈	C₂×C₅₃	semidirect
220. #2326	C₂×C₅₃	direct (N × H)
221. C₂×#391	C₂×C₅₃	semidirect
223. C₂×#388	C₂×C₅₃	semidirect
224. C₂×#385	C₂×C₅₃	semidirect
225. C₂×#384	C₂×C₅₃	semidirect
226. C₂×#376	C₂×C₅₃	semidirect
228. C₂×C₅₃	#2326	direct (N × H)
237. C₂×#124	C₄×C₅₃	semidirect
240. C₂×C₄×Q₈	C₂²×C₅₃	semidirect
254. #391	C₂²×C₅₃	semidirect
256. #388	C₂²×C₅₃	semidirect
257. #385	C₂²×C₅₃	semidirect
258. #384	C₂²×C₅₃	semidirect
259. #376	C₂²×C₅₃	semidirect
263. C₅₃	C₂×#2326	direct (N × H)
267. C₂²×Q₈	C₂×C₄×C₅₃	semidirect
284. #124	C₂×C₄×C₅₃	semidirect
297. C₂×C₂²⋊C₄	C₂×C₄×C₅₃	semidirect
303. C₂×C₄²	C₂×C₄×C₅₃	semidirect
322. C₂×Q₈	C₂²×C₄×C₅₃	semidirect
364. C₂²⋊C₄	C₂²×C₄×C₅₃	semidirect
367. C₄²	C₂²×C₄×C₅₃	semidirect
441. C₂	C₅₃×#2326	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×#124×C₅₃	2
C₂²×C₄×Q₈×C₅₃	2
C₂²×C₄×D₄×C₅₃	2
C₅₃×#2326	2
C₂×C₅₃×#391	2
C₂×C₅₃×#390	2
C₂×C₅₃×#388	2
C₂×C₅₃×#385	2
C₂×C₅₃×#384	2
C₂×C₅₃×#376	2
C₂×#2326	53

Subgroups — C2×C53×#2326 #27241579

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2326 #27241579