Subgroups — C₂×C₅₃×#2628 #27242119

Order 13568.

2742 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2628	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#400	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#399	C₂	1	2	yes	maximal
6	C₂×C₅₃×#395	C₂	1	2	yes	maximal, complemented
7	C₂×C₅₃×#393	C₂	1	2	yes	maximal
8	C₂×C₅₃×#389	C₂	1	2	yes	maximal, complemented
9	C₂×C₅₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅₃×#380	C₂	1	2	yes	maximal
11	C₂³×Q₈×C₅₃	C₂²	1	2	yes	—
12	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₅₃	—	2	8	no	—
14	C₂×C₄⋊Q₈×C₅₃	C₂²	1	8	yes	complemented
15	C₂×#117×C₅₃	—	2	8	no	—
16	C₂×#117×C₅₃	C₂²	1	8	yes	—
17	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
23	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
26	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#400	C₂²	1	16	yes	complemented
29	C₅₃×#399	C₂²	1	16	yes	—
30	C₅₃×#395	C₂²	1	16	yes	complemented
31	C₅₃×#393	C₂²	1	16	yes	—
32	C₅₃×#389	C₂²	1	16	yes	complemented
33	C₅₃×#382	C₂²	1	8	yes	complemented
34	C₅₃×#380	C₂²	1	16	yes	—
35	C₂×#95×C₅₃	C₂²	1	4	yes	—
36	C₂×#95×C₅₃	C₂²	1	2	yes	—
37	C₂×#95×C₅₃	C₂²	1	2	yes	—
38	C₂×#95×C₅₃	C₂²	1	4	yes	—
39	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
40	C₂²×Q₈×C₅₃	—	2	8	no	—
41	C₂²×Q₈×C₅₃	—	4	8	no	—
42	C₂²×Q₈×C₅₃	C₂³	1	8	yes	—
43	C₂³×C₄×C₅₃	—	2	4	no	—
44	C₂³×C₄×C₅₃	—	2	4	no	—
45	C₂³×C₄×C₅₃	Q₈	1	8	yes	complemented
46	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
49	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
50	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
53	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
54	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈×C₅₃	—	2	32	no	—
56	#117×C₅₃	—	2	32	no	—
57	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
58	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
59	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
60	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
61	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
62	C₂×C₄⋊C₄×C₅₃	—	4	8	no	—
63	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
64	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
65	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
66	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
67	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
68	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
69	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
70	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
71	C₂×C₂²⋊C₄×C₅₃	—	4	8	no	—
72	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
73	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
74	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
75	C₂×C₄²×C₅₃	—	2	4	no	—
76	C₂×C₄²×C₅₃	D₄	1	8	yes	—
77	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
78	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
79	#95×C₅₃	C₂³	1	16	yes	—
80	#95×C₅₃	C₂³	1	8	yes	—
81	#95×C₅₃	C₂³	1	8	yes	—
82	#95×C₅₃	C₂³	1	16	yes	—
83	C₂⁴×C₅₃	—	2	2	no	—
84	C₂⁴×C₅₃	—	2	2	no	—
85	C₂⁴×C₅₃	—	2	4	no	—
86	C₂⁴×C₅₃	—	2	4	no	—
87	C₂⁴×C₅₃	C₂×Q₈	1	4	yes	—
88	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
89	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
90	C₂×Q₈×C₅₃	—	2	16	no	—
91	C₂×Q₈×C₅₃	—	4	16	no	—
92	C₂×Q₈×C₅₃	—	4	32	no	—
93	C₂²×C₄×C₅₃	—	2	2	no	—
94	C₂²×C₄×C₅₃	—	2	2	no	—
95	C₂²×C₄×C₅₃	—	2	4	no	—
96	C₂²×C₄×C₅₃	—	2	4	no	—
97	C₂²×C₄×C₅₃	—	2	32	no	—
98	C₂²×C₄×C₅₃	—	2	8	no	—
99	C₂²×C₄×C₅₃	—	2	8	no	—
100	C₂²×C₄×C₅₃	—	2	2	no	—
101	C₂²×C₄×C₅₃	—	2	2	no	—
102	C₂²×C₄×C₅₃	—	2	8	no	—
103	C₂²×C₄×C₅₃	—	2	8	no	—
104	C₂²×C₄×C₅₃	—	2	8	no	—
105	C₂²×C₄×C₅₃	—	2	8	no	—
106	C₂²×C₄×C₅₃	—	2	4	no	—
107	C₂²×C₄×C₅₃	—	2	8	no	—
108	C₂²×C₄×C₅₃	—	2	4	no	—
109	C₂²×C₄×C₅₃	—	2	8	no	—
110	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
111	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
112	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
113	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
114	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
115	C₂²×C₄×C₅₃	C₂×Q₈	1	16	yes	complemented
116	C₂²×C₄×C₅₃	#41	1	4	yes	—
117	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
118	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
119	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
120	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
121	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
122	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
123	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
124	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
125	C₄⋊C₄×C₅₃	—	2	8	no	—
126	C₄⋊C₄×C₅₃	—	2	8	no	—
127	C₄⋊C₄×C₅₃	—	2	16	no	—
128	C₄⋊C₄×C₅₃	—	2	16	no	—
129	C₄⋊C₄×C₅₃	—	2	16	no	—
130	C₄⋊C₄×C₅₃	—	4	16	no	—
131	C₂²⋊C₄×C₅₃	—	2	8	no	—
132	C₂²⋊C₄×C₅₃	—	2	8	no	—
133	C₂²⋊C₄×C₅₃	—	4	16	no	—
134	C₄²×C₅₃	—	2	8	no	—
135	C₄²×C₅₃	C₂×D₄	1	16	yes	—
136	C₂³×C₅₃	—	2	8	no	—
137	C₂³×C₅₃	—	2	16	no	—
138	C₂³×C₅₃	—	2	2	no	—
139	C₂³×C₅₃	—	2	2	no	—
140	C₂³×C₅₃	—	2	4	no	—
141	C₂³×C₅₃	—	2	4	no	—
142	C₂³×C₅₃	—	4	4	no	—
143	C₂³×C₅₃	—	4	4	no	—
144	C₂³×C₅₃	—	4	8	no	—
145	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
146	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
147	C₂³×C₅₃	C₂²×Q₈	1	2	yes	—
148	C₂³×C₅₃	C₂×#41	1	2	yes	—
149	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
150	C₂³×C₅₃	2₋¹⁺⁴	1	4	yes	—
151	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
152	Q₈×C₅₃	—	4	32	no	—
153	C₂×C₄×C₅₃	—	2	8	no	—
154	C₂×C₄×C₅₃	—	2	8	no	—
155	C₂×C₄×C₅₃	—	2	8	no	—
156	C₂×C₄×C₅₃	—	2	2	no	—
157	C₂×C₄×C₅₃	—	2	2	no	—
158	C₂×C₄×C₅₃	—	2	4	no	—
159	C₂×C₄×C₅₃	—	2	4	no	—
160	C₂×C₄×C₅₃	—	2	32	no	—
161	C₂×C₄×C₅₃	—	2	8	no	—
162	C₂×C₄×C₅₃	—	2	2	no	—
163	C₂×C₄×C₅₃	—	2	2	no	—
164	C₂×C₄×C₅₃	—	2	8	no	—
165	C₂×C₄×C₅₃	—	2	8	no	—
166	C₂×C₄×C₅₃	—	2	8	no	—
167	C₂×C₄×C₅₃	—	2	8	no	—
168	C₂×C₄×C₅₃	—	2	4	no	—
169	C₂×C₄×C₅₃	—	2	4	no	—
170	C₂×C₄×C₅₃	—	2	8	no	—
171	C₂×C₄×C₅₃	—	4	4	no	—
172	C₂×C₄×C₅₃	—	4	16	no	—
173	C₂×C₄×C₅₃	—	4	4	no	—
174	C₂×C₄×C₅₃	—	4	16	no	—
175	C₂×C₄×C₅₃	C₂²⋊Q₈	1	16	yes	—
176	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
177	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
178	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
179	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
180	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
181	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
182	C₂×#2628	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂²×C₅₃	—	2	4	no	—
184	C₂²×C₅₃	—	2	8	no	—
185	C₂²×C₅₃	—	4	16	no	—
186	C₂²×C₅₃	—	4	2	no	—
187	C₂²×C₅₃	—	4	2	no	—
188	C₂²×C₅₃	—	4	4	no	—
189	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	4	yes	—
190	C₂²×C₅₃	#535	1	4	yes	—
191	C₂²×C₅₃	#536	1	4	yes	—
192	C₂²×C₅₃	#542	1	8	yes	—
193	C₂²×C₅₃	#544	1	8	yes	—
194	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
195	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
196	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
197	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
198	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	2	yes	—
199	C₄×C₅₃	—	2	4	no	—
200	C₄×C₅₃	—	2	4	no	—
201	C₄×C₅₃	—	2	4	no	—
202	C₄×C₅₃	—	4	2	no	—
203	C₄×C₅₃	—	4	8	no	—
204	C₄×C₅₃	—	4	2	no	—
205	C₄×C₅₃	—	4	8	no	—
206	C₄×C₅₃	C₂×C₂²⋊Q₈	1	8	yes	—
207	C₂²×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
208	C₂²×#117	C₂×C₅₃	1	1	yes	characteristic
209	#2628	C₂×C₅₃	1	16	yes	direct product factor
210	C₂×#400	C₂×C₅₃	1	2	yes	complemented
211	C₂×#399	C₂×C₅₃	1	2	yes	—
212	C₂×#395	C₂×C₅₃	1	2	yes	complemented
213	C₂×#393	C₂×C₅₃	1	2	yes	—
214	C₂×#389	C₂×C₅₃	1	2	yes	complemented
215	C₂×#382	C₂×C₅₃	1	1	yes	characteristic, complemented
216	C₂×#380	C₂×C₅₃	1	2	yes	—
217	C₂×C₅₃	—	4	4	no	—
218	C₂×C₅₃	#2628	1	8	yes	direct product factor
219	C₂×C₅₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
220	C₂×C₅₃	C₂×#535	1	1	yes	characteristic
221	C₂×C₅₃	C₂×#536	1	1	yes	characteristic
222	C₂×C₅₃	C₂×#542	1	2	yes	—
223	C₂×C₅₃	C₂×#544	1	2	yes	—
224	C₂³×Q₈	C₂²×C₅₃	1	2	yes	—
225	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
226	C₂×C₄⋊Q₈	—	2	8	no	—
227	C₂×C₄⋊Q₈	C₂²×C₅₃	1	8	yes	complemented
228	C₂×#117	—	2	8	no	—
229	C₂×#117	C₂²×C₅₃	1	8	yes	—
230	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
231	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
232	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
233	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
234	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
235	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
236	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
237	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
238	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
239	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
240	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
241	#400	C₂²×C₅₃	1	16	yes	complemented
242	#399	C₂²×C₅₃	1	16	yes	—
243	#395	C₂²×C₅₃	1	16	yes	complemented
244	#393	C₂²×C₅₃	1	16	yes	—
245	#389	C₂²×C₅₃	1	16	yes	complemented
246	#382	C₂²×C₅₃	1	8	yes	complemented
247	#380	C₂²×C₅₃	1	16	yes	—
248	C₂×#95	C₂²×C₅₃	1	2	yes	—
249	C₂×#95	C₂²×C₅₃	1	4	yes	—
250	C₂×#95	C₂²×C₅₃	1	4	yes	—
251	C₂×#95	C₂²×C₅₃	1	2	yes	—
252	C₅₃	C₂×#2628	1	1	yes	characteristic, direct product factor, Sylow
253	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
254	C₂²×Q₈	—	2	8	no	—
255	C₂²×Q₈	—	4	8	no	—
256	C₂²×Q₈	C₂³×C₅₃	1	8	yes	—
257	C₂³×C₄	—	2	4	no	—
258	C₂³×C₄	—	2	4	no	—
259	C₂³×C₄	Q₈×C₅₃	1	8	yes	complemented
260	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
261	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
262	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
263	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
264	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
265	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
266	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
267	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
268	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
269	C₄⋊Q₈	—	2	32	no	—
270	#117	—	2	32	no	—
271	C₂×C₄⋊C₄	—	2	4	no	—
272	C₂×C₄⋊C₄	—	2	4	no	—
273	C₂×C₄⋊C₄	—	2	8	no	—
274	C₂×C₄⋊C₄	—	2	8	no	—
275	C₂×C₄⋊C₄	—	2	8	no	—
276	C₂×C₄⋊C₄	—	4	8	no	—
277	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
278	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
279	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
280	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
281	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
282	C₂×C₄⋊C₄	C₂³×C₅₃	1	16	yes	—
283	C₂×C₂²⋊C₄	—	2	4	no	—
284	C₂×C₂²⋊C₄	—	2	4	no	—
285	C₂×C₂²⋊C₄	—	4	8	no	—
286	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
287	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
288	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
289	C₂×C₄²	—	2	4	no	—
290	C₂×C₄²	D₄×C₅₃	1	8	yes	—
291	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
292	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
293	#95	C₂³×C₅₃	1	16	yes	—
294	#95	C₂³×C₅₃	1	8	yes	—
295	#95	C₂³×C₅₃	1	8	yes	—
296	#95	C₂³×C₅₃	1	16	yes	—
297	C₂⁴	—	2	2	no	—
298	C₂⁴	—	2	2	no	—
299	C₂⁴	—	2	4	no	—
300	C₂⁴	—	2	4	no	—
301	C₂⁴	C₂×Q₈×C₅₃	1	4	yes	—
302	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
303	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
304	C₂×Q₈	—	2	16	no	—
305	C₂×Q₈	—	4	16	no	—
306	C₂×Q₈	—	4	32	no	—
307	C₂²×C₄	—	2	2	no	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	4	no	—
310	C₂²×C₄	—	2	4	no	—
311	C₂²×C₄	—	2	32	no	—
312	C₂²×C₄	—	2	8	no	—
313	C₂²×C₄	—	2	8	no	—
314	C₂²×C₄	—	2	2	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	2	8	no	—
317	C₂²×C₄	—	2	8	no	—
318	C₂²×C₄	—	2	8	no	—
319	C₂²×C₄	—	2	8	no	—
320	C₂²×C₄	—	2	4	no	—
321	C₂²×C₄	—	2	8	no	—
322	C₂²×C₄	—	2	4	no	—
323	C₂²×C₄	—	2	8	no	—
324	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
325	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
326	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
327	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
328	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
329	C₂²×C₄	C₂×Q₈×C₅₃	1	16	yes	complemented
330	C₂²×C₄	#41×C₅₃	1	4	yes	—
331	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
332	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
333	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
334	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
335	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
336	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
337	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
338	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
339	C₄⋊C₄	—	2	8	no	—
340	C₄⋊C₄	—	2	8	no	—
341	C₄⋊C₄	—	2	16	no	—
342	C₄⋊C₄	—	2	16	no	—
343	C₄⋊C₄	—	2	16	no	—
344	C₄⋊C₄	—	4	16	no	—
345	C₂²⋊C₄	—	2	8	no	—
346	C₂²⋊C₄	—	2	8	no	—
347	C₂²⋊C₄	—	4	16	no	—
348	C₄²	—	2	8	no	—
349	C₄²	C₂×D₄×C₅₃	1	16	yes	—
350	C₂³	—	2	8	no	—
351	C₂³	—	2	16	no	—
352	C₂³	—	2	2	no	—
353	C₂³	—	2	2	no	—
354	C₂³	—	2	4	no	—
355	C₂³	—	2	4	no	—
356	C₂³	—	4	4	no	—
357	C₂³	—	4	4	no	—
358	C₂³	—	4	8	no	—
359	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
360	C₂³	C₂²×Q₈×C₅₃	1	4	yes	—
361	C₂³	C₂²×Q₈×C₅₃	1	2	yes	—
362	C₂³	C₂×#41×C₅₃	1	2	yes	—
363	C₂³	H(5,𝔽₂)×C₅₃	1	4	yes	—
364	C₂³	2₋¹⁺⁴×C₅₃	1	4	yes	—
365	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
366	Q₈	—	4	32	no	—
367	C₂×C₄	—	2	8	no	—
368	C₂×C₄	—	2	8	no	—
369	C₂×C₄	—	2	8	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	2	32	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	8	no	—
378	C₂×C₄	—	2	8	no	—
379	C₂×C₄	—	2	8	no	—
380	C₂×C₄	—	2	8	no	—
381	C₂×C₄	—	2	4	no	—
382	C₂×C₄	—	2	8	no	—
383	C₂×C₄	—	2	4	no	—
384	C₂×C₄	—	2	8	no	—
385	C₂×C₄	—	4	4	no	—
386	C₂×C₄	—	4	16	no	—
387	C₂×C₄	—	4	4	no	—
388	C₂×C₄	—	4	16	no	—
389	C₂×C₄	C₂²⋊Q₈×C₅₃	1	16	yes	—
390	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
391	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
392	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
393	C₂×C₄	C₂²×Q₈×C₅₃	1	4	yes	—
394	C₂×C₄	C₂²×Q₈×C₅₃	1	4	yes	—
395	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
396	C₂²	—	2	8	no	—
397	C₂²	—	2	4	no	—
398	C₂²	—	4	2	no	—
399	C₂²	—	4	4	no	—
400	C₂²	—	4	16	no	—
401	C₂²	—	4	2	no	—
402	C₂²	C₂×C₂²⋊Q₈×C₅₃	1	4	yes	—
403	C₂²	C₅₃×#535	1	4	yes	—
404	C₂²	C₅₃×#536	1	4	yes	—
405	C₂²	C₅₃×#542	1	8	yes	—
406	C₂²	C₅₃×#544	1	8	yes	—
407	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
408	C₂²	C₂³×Q₈×C₅₃	1	1	yes	characteristic
409	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
410	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
411	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	2	yes	—
412	C₄	—	2	4	no	—
413	C₄	—	2	4	no	—
414	C₄	—	2	4	no	—
415	C₄	—	4	2	no	—
416	C₄	—	4	8	no	—
417	C₄	—	4	2	no	—
418	C₄	—	4	8	no	—
419	C₄	C₂×C₂²⋊Q₈×C₅₃	1	8	yes	—
420	C₂	—	4	4	no	—
421	C₂	C₅₃×#2628	1	8	yes	direct product factor
422	C₂	C₂²×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
423	C₂	C₂×C₅₃×#535	1	1	yes	characteristic
424	C₂	C₂×C₅₃×#536	1	1	yes	characteristic
425	C₂	C₂×C₅₃×#542	1	2	yes	—
426	C₂	C₂×C₅₃×#544	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂⁵×C₅₃	C₂³
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. C₂²×C₄⋊Q₈×C₅₃	C₂	semidirect
3. C₅₃×#2628	C₂	direct (N × H)
4. C₂×C₅₃×#400	C₂	semidirect
6. C₂×C₅₃×#395	C₂	semidirect
8. C₂×C₅₃×#389	C₂	semidirect
9. C₂×C₅₃×#382	C₂	semidirect
14. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
28. C₅₃×#400	C₂²	semidirect
30. C₅₃×#395	C₂²	semidirect
32. C₅₃×#389	C₂²	semidirect
33. C₅₃×#382	C₂²	semidirect
45. C₂³×C₄×C₅₃	Q₈	semidirect
115. C₂²×C₄×C₅₃	C₂×Q₈	semidirect
182. C₂×#2628	C₅₃	direct (N × H)
207. C₂²×C₄⋊Q₈	C₂×C₅₃	semidirect
209. #2628	C₂×C₅₃	direct (N × H)
210. C₂×#400	C₂×C₅₃	semidirect
212. C₂×#395	C₂×C₅₃	semidirect
214. C₂×#389	C₂×C₅₃	semidirect
215. C₂×#382	C₂×C₅₃	semidirect
218. C₂×C₅₃	#2628	direct (N × H)
227. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
241. #400	C₂²×C₅₃	semidirect
243. #395	C₂²×C₅₃	semidirect
245. #389	C₂²×C₅₃	semidirect
246. #382	C₂²×C₅₃	semidirect
252. C₅₃	C₂×#2628	direct (N × H)
259. C₂³×C₄	Q₈×C₅₃	semidirect
329. C₂²×C₄	C₂×Q₈×C₅₃	semidirect
421. C₂	C₅₃×#2628	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×C₄⋊Q₈×C₅₃	2
C₂²×#117×C₅₃	2
C₅₃×#2628	2
C₂×C₅₃×#400	2
C₂×C₅₃×#399	2
C₂×C₅₃×#395	2
C₂×C₅₃×#393	2
C₂×C₅₃×#389	2
C₂×C₅₃×#382	2
C₂×C₅₃×#380	2
C₂×#2628	53

Subgroups — C2×C53×#2628 #27242119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2628 #27242119