Subgroups — C₂×C₅₃×#2541 #27242435

Order 13568.

5862 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#127×C₅₃	C₂	1	2	yes	maximal, complemented
2	C₂²×#127×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#121×C₅₃	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2541	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₅₃×#392	C₂	1	4	yes	maximal, complemented
7	C₂×C₅₃×#390	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
10	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
11	C₂³×D₄×C₅₃	C₂²	1	8	yes	—
12	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
15	C₂×#127×C₅₃	—	2	16	no	—
16	C₂×#127×C₅₃	—	2	8	no	—
17	C₂×#127×C₅₃	C₂²	1	16	yes	complemented
18	C₂×#127×C₅₃	C₂²	1	8	yes	complemented
19	C₂×#121×C₅₃	—	2	32	no	—
20	C₂×#121×C₅₃	C₂²	1	32	yes	complemented
21	C₂×C₄×D₄×C₅₃	—	2	8	no	—
22	C₂×C₄×D₄×C₅₃	C₂²	1	8	yes	complemented
23	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	—
26	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
27	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
28	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#392	C₂²	1	32	yes	complemented
30	C₅₃×#390	C₂²	1	16	yes	complemented
31	C₄×C₄⋊C₄×C₅₃	C₂²	1	8	yes	complemented
32	C₂×#95×C₅₃	C₂²	1	2	yes	—
33	C₂⁵×C₅₃	C₂³	1	4	yes	—
34	C₂⁵×C₅₃	C₂³	1	2	yes	—
35	C₂²×D₄×C₅₃	—	2	32	no	—
36	C₂²×D₄×C₅₃	—	2	4	no	—
37	C₂²×D₄×C₅₃	—	2	32	no	—
38	C₂²×D₄×C₅₃	—	2	16	no	—
39	C₂²×D₄×C₅₃	—	2	8	no	—
40	C₂²×D₄×C₅₃	—	2	8	no	—
41	C₂²×D₄×C₅₃	—	2	8	no	—
42	C₂²×D₄×C₅₃	—	2	8	no	—
43	C₂²×D₄×C₅₃	—	2	4	no	—
44	C₂²×D₄×C₅₃	—	2	8	no	—
45	C₂²×D₄×C₅₃	—	4	16	no	—
46	C₂²×D₄×C₅₃	—	4	8	no	—
47	C₂²×D₄×C₅₃	D₄	1	8	yes	complemented
48	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
49	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
50	C₂²×D₄×C₅₃	C₂³	1	32	yes	—
51	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
52	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
53	C₂³×C₄×C₅₃	—	2	8	no	—
54	C₂³×C₄×C₅₃	—	2	16	no	—
55	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
56	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
57	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
58	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
61	#127×C₅₃	—	2	32	no	—
62	#127×C₅₃	—	2	64	no	—
63	#121×C₅₃	—	2	128	no	—
64	C₄×D₄×C₅₃	—	2	32	no	—
65	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
66	C₂×C₄⋊C₄×C₅₃	D₄	1	16	yes	complemented
67	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
68	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
69	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
70	C₂×C₂²⋊C₄×C₅₃	—	2	32	no	—
71	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
72	C₂×C₂²⋊C₄×C₅₃	C₂³	1	32	yes	—
73	C₂×C₄²×C₅₃	—	2	8	no	—
74	C₂×C₄²×C₅₃	D₄	1	8	yes	complemented
75	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
76	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
77	#95×C₅₃	C₂³	1	8	yes	—
78	C₂⁴×C₅₃	—	2	4	no	—
79	C₂⁴×C₅₃	—	2	8	no	—
80	C₂⁴×C₅₃	—	2	4	no	—
81	C₂⁴×C₅₃	—	2	8	no	—
82	C₂⁴×C₅₃	—	2	16	no	—
83	C₂⁴×C₅₃	—	2	8	no	—
84	C₂⁴×C₅₃	—	2	8	no	—
85	C₂⁴×C₅₃	—	2	8	no	—
86	C₂⁴×C₅₃	—	2	16	no	—
87	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
88	C₂⁴×C₅₃	C₂⁴	1	8	yes	—
89	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
90	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
91	C₂×D₄×C₅₃	—	2	64	no	—
92	C₂×D₄×C₅₃	—	2	32	no	—
93	C₂×D₄×C₅₃	—	2	8	no	—
94	C₂×D₄×C₅₃	—	2	64	no	—
95	C₂×D₄×C₅₃	—	2	32	no	—
96	C₂×D₄×C₅₃	—	2	16	no	—
97	C₂×D₄×C₅₃	—	2	16	no	—
98	C₂×D₄×C₅₃	—	2	16	no	—
99	C₂×D₄×C₅₃	—	2	16	no	—
100	C₂×D₄×C₅₃	—	2	8	no	—
101	C₂×D₄×C₅₃	—	2	16	no	—
102	C₂×D₄×C₅₃	—	4	32	no	—
103	C₂×D₄×C₅₃	—	4	16	no	—
104	C₂×D₄×C₅₃	—	4	128	no	—
105	C₂×D₄×C₅₃	—	4	32	no	—
106	C₂×D₄×C₅₃	—	4	32	no	—
107	C₂×D₄×C₅₃	—	4	32	no	—
108	C₂×D₄×C₅₃	C₂×D₄	1	16	yes	complemented
109	C₂²×C₄×C₅₃	—	2	8	no	—
110	C₂²×C₄×C₅₃	—	2	4	no	—
111	C₂²×C₄×C₅₃	—	2	8	no	—
112	C₂²×C₄×C₅₃	—	2	8	no	—
113	C₂²×C₄×C₅₃	—	2	4	no	—
114	C₂²×C₄×C₅₃	—	2	16	no	—
115	C₂²×C₄×C₅₃	—	2	32	no	—
116	C₂²×C₄×C₅₃	—	4	32	no	—
117	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
118	C₂²×C₄×C₅₃	C₂×D₄	1	16	yes	—
119	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
120	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
121	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
122	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
123	C₂²×C₄×C₅₃	#41	1	4	yes	—
124	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
125	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
126	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
127	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
128	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
129	C₄⋊C₄×C₅₃	—	2	8	no	—
130	C₄⋊C₄×C₅₃	C₂×D₄	1	32	yes	complemented
131	C₂²⋊C₄×C₅₃	—	2	16	no	—
132	C₂²⋊C₄×C₅₃	—	2	64	no	—
133	C₄²×C₅₃	—	2	16	no	—
134	C₄²×C₅₃	C₂×D₄	1	16	yes	complemented
135	C₂³×C₅₃	—	2	16	no	—
136	C₂³×C₅₃	—	2	32	no	—
137	C₂³×C₅₃	—	2	8	no	—
138	C₂³×C₅₃	—	2	4	no	—
139	C₂³×C₅₃	—	2	4	no	—
140	C₂³×C₅₃	—	2	8	no	—
141	C₂³×C₅₃	—	2	8	no	—
142	C₂³×C₅₃	—	2	8	no	—
143	C₂³×C₅₃	—	2	16	no	—
144	C₂³×C₅₃	—	2	8	no	—
145	C₂³×C₅₃	—	2	16	no	—
146	C₂³×C₅₃	—	4	8	no	—
147	C₂³×C₅₃	—	4	8	no	—
148	C₂³×C₅₃	—	4	16	no	—
149	C₂³×C₅₃	—	4	16	no	—
150	C₂³×C₅₃	—	4	16	no	—
151	C₂³×C₅₃	—	4	16	no	—
152	C₂³×C₅₃	—	4	32	no	—
153	C₂³×C₅₃	—	4	32	no	—
154	C₂³×C₅₃	C₂²×D₄	1	8	yes	—
155	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
156	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
157	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
158	C₂³×C₅₃	C₂×#41	1	2	yes	—
159	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
160	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
161	D₄×C₅₃	—	2	32	no	—
162	D₄×C₅₃	—	4	128	no	—
163	D₄×C₅₃	—	4	32	no	—
164	D₄×C₅₃	—	4	32	no	—
165	D₄×C₅₃	—	4	32	no	—
166	C₂×C₄×C₅₃	—	2	8	no	—
167	C₂×C₄×C₅₃	—	2	32	no	—
168	C₂×C₄×C₅₃	—	2	8	no	—
169	C₂×C₄×C₅₃	—	2	4	no	—
170	C₂×C₄×C₅₃	—	2	8	no	—
171	C₂×C₄×C₅₃	—	2	8	no	—
172	C₂×C₄×C₅₃	—	2	4	no	—
173	C₂×C₄×C₅₃	—	4	8	no	—
174	C₂×C₄×C₅₃	—	4	32	no	—
175	C₂×C₄×C₅₃	#121	1	16	yes	complemented
176	C₂×C₄×C₅₃	#127	1	16	yes	complemented
177	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
178	C₂×C₄×C₅₃	C₂²×D₄	1	16	yes	—
179	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
180	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
181	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
182	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
183	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
184	C₂×#2541	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
185	C₂²×C₅₃	—	2	8	no	—
186	C₂²×C₅₃	—	2	16	no	—
187	C₂²×C₅₃	—	4	4	no	—
188	C₂²×C₅₃	—	4	4	no	—
189	C₂²×C₅₃	—	4	8	no	—
190	C₂²×C₅₃	—	4	32	no	—
191	C₂²×C₅₃	—	4	8	no	—
192	C₂²×C₅₃	—	4	8	no	—
193	C₂²×C₅₃	—	4	8	no	—
194	C₂²×C₅₃	—	4	16	no	—
195	C₂²×C₅₃	—	4	16	no	—
196	C₂²×C₅₃	—	8	32	no	—
197	C₂²×C₅₃	C₂×#121	1	4	yes	—
198	C₂²×C₅₃	C₂×#127	1	4	yes	—
199	C₂²×C₅₃	#535	1	4	yes	—
200	C₂²×C₅₃	D₄²	1	8	yes	—
201	C₂²×C₅₃	#550	1	8	yes	—
202	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
203	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
204	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
205	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
206	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
207	C₄×C₅₃	—	2	4	no	—
208	C₄×C₅₃	—	2	16	no	—
209	C₄×C₅₃	—	4	4	no	—
210	C₄×C₅₃	C₂×#121	1	8	yes	complemented
211	C₄×C₅₃	C₂×#127	1	8	yes	complemented
212	C₂²×#127	C₂×C₅₃	1	2	yes	complemented
213	C₂²×#127	C₂×C₅₃	1	1	yes	characteristic, complemented
214	C₂²×#121	C₂×C₅₃	1	4	yes	complemented
215	C₂²×C₄×D₄	C₂×C₅₃	1	1	yes	characteristic, complemented
216	#2541	C₂×C₅₃	1	16	yes	direct product factor
217	C₂×#392	C₂×C₅₃	1	4	yes	complemented
218	C₂×#390	C₂×C₅₃	1	2	yes	complemented
219	C₂×C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
220	C₂×C₅₃	—	4	8	no	—
221	C₂×C₅₃	—	8	8	no	—
222	C₂×C₅₃	#2541	1	8	yes	direct product factor
223	C₂×C₅₃	C₂²×#121	1	1	yes	characteristic
224	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
225	C₂×C₅₃	C₂×#535	1	1	yes	characteristic
226	C₂×C₅₃	C₂×D₄²	1	2	yes	—
227	C₂×C₅₃	C₂×#550	1	2	yes	—
228	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
229	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
230	C₂³×D₄	C₂²×C₅₃	1	8	yes	—
231	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
232	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
233	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	—
234	C₂×#127	—	2	16	no	—
235	C₂×#127	—	2	8	no	—
236	C₂×#127	C₂²×C₅₃	1	16	yes	complemented
237	C₂×#127	C₂²×C₅₃	1	8	yes	complemented
238	C₂×#121	—	2	32	no	—
239	C₂×#121	C₂²×C₅₃	1	32	yes	complemented
240	C₂×C₄×D₄	—	2	8	no	—
241	C₂×C₄×D₄	C₂²×C₅₃	1	8	yes	complemented
242	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
243	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
244	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
245	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	—
246	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
247	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
248	#392	C₂²×C₅₃	1	32	yes	complemented
249	#390	C₂²×C₅₃	1	16	yes	complemented
250	C₄×C₄⋊C₄	C₂²×C₅₃	1	8	yes	complemented
251	C₂×#95	C₂²×C₅₃	1	2	yes	—
252	C₅₃	C₂×#2541	1	1	yes	characteristic, direct product factor, Sylow
253	C₂⁵	C₂³×C₅₃	1	4	yes	—
254	C₂⁵	C₂³×C₅₃	1	2	yes	—
255	C₂²×D₄	—	2	32	no	—
256	C₂²×D₄	—	2	4	no	—
257	C₂²×D₄	—	2	32	no	—
258	C₂²×D₄	—	2	16	no	—
259	C₂²×D₄	—	2	8	no	—
260	C₂²×D₄	—	2	8	no	—
261	C₂²×D₄	—	2	8	no	—
262	C₂²×D₄	—	2	8	no	—
263	C₂²×D₄	—	2	4	no	—
264	C₂²×D₄	—	2	8	no	—
265	C₂²×D₄	—	4	16	no	—
266	C₂²×D₄	—	4	8	no	—
267	C₂²×D₄	D₄×C₅₃	1	8	yes	complemented
268	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
269	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
270	C₂²×D₄	C₂³×C₅₃	1	32	yes	—
271	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
272	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
273	C₂³×C₄	—	2	8	no	—
274	C₂³×C₄	—	2	16	no	—
275	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
276	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
277	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
278	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
279	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
280	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
281	#127	—	2	64	no	—
282	#127	—	2	32	no	—
283	#121	—	2	128	no	—
284	C₄×D₄	—	2	32	no	—
285	C₂×C₄⋊C₄	—	2	4	no	—
286	C₂×C₄⋊C₄	D₄×C₅₃	1	16	yes	complemented
287	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
288	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
289	C₂×C₂²⋊C₄	—	2	8	no	—
290	C₂×C₂²⋊C₄	—	2	32	no	—
291	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
292	C₂×C₂²⋊C₄	C₂³×C₅₃	1	32	yes	—
293	C₂×C₄²	—	2	8	no	—
294	C₂×C₄²	D₄×C₅₃	1	8	yes	complemented
295	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
296	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
297	#95	C₂³×C₅₃	1	8	yes	—
298	C₂⁴	—	2	4	no	—
299	C₂⁴	—	2	8	no	—
300	C₂⁴	—	2	4	no	—
301	C₂⁴	—	2	8	no	—
302	C₂⁴	—	2	16	no	—
303	C₂⁴	—	2	8	no	—
304	C₂⁴	—	2	8	no	—
305	C₂⁴	—	2	8	no	—
306	C₂⁴	—	2	16	no	—
307	C₂⁴	C₂×D₄×C₅₃	1	8	yes	—
308	C₂⁴	C₂⁴×C₅₃	1	8	yes	—
309	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
310	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
311	C₂×D₄	—	2	64	no	—
312	C₂×D₄	—	2	32	no	—
313	C₂×D₄	—	2	8	no	—
314	C₂×D₄	—	2	64	no	—
315	C₂×D₄	—	2	32	no	—
316	C₂×D₄	—	2	16	no	—
317	C₂×D₄	—	2	16	no	—
318	C₂×D₄	—	2	16	no	—
319	C₂×D₄	—	2	16	no	—
320	C₂×D₄	—	2	8	no	—
321	C₂×D₄	—	2	16	no	—
322	C₂×D₄	—	4	32	no	—
323	C₂×D₄	—	4	16	no	—
324	C₂×D₄	—	4	128	no	—
325	C₂×D₄	—	4	32	no	—
326	C₂×D₄	—	4	32	no	—
327	C₂×D₄	—	4	32	no	—
328	C₂×D₄	C₂×D₄×C₅₃	1	16	yes	complemented
329	C₂²×C₄	—	2	8	no	—
330	C₂²×C₄	—	2	4	no	—
331	C₂²×C₄	—	2	8	no	—
332	C₂²×C₄	—	2	8	no	—
333	C₂²×C₄	—	2	4	no	—
334	C₂²×C₄	—	2	16	no	—
335	C₂²×C₄	—	2	32	no	—
336	C₂²×C₄	—	4	32	no	—
337	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
338	C₂²×C₄	C₂×D₄×C₅₃	1	16	yes	—
339	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
340	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
341	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
342	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
343	C₂²×C₄	#41×C₅₃	1	4	yes	—
344	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
345	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
346	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
347	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
348	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
349	C₄⋊C₄	—	2	8	no	—
350	C₄⋊C₄	C₂×D₄×C₅₃	1	32	yes	complemented
351	C₂²⋊C₄	—	2	16	no	—
352	C₂²⋊C₄	—	2	64	no	—
353	C₄²	—	2	16	no	—
354	C₄²	C₂×D₄×C₅₃	1	16	yes	complemented
355	C₂³	—	2	16	no	—
356	C₂³	—	2	32	no	—
357	C₂³	—	2	4	no	—
358	C₂³	—	2	8	no	—
359	C₂³	—	2	4	no	—
360	C₂³	—	2	8	no	—
361	C₂³	—	2	16	no	—
362	C₂³	—	2	8	no	—
363	C₂³	—	2	8	no	—
364	C₂³	—	2	8	no	—
365	C₂³	—	2	16	no	—
366	C₂³	—	4	8	no	—
367	C₂³	—	4	8	no	—
368	C₂³	—	4	16	no	—
369	C₂³	—	4	16	no	—
370	C₂³	—	4	16	no	—
371	C₂³	—	4	16	no	—
372	C₂³	—	4	32	no	—
373	C₂³	—	4	32	no	—
374	C₂³	C₂²×D₄×C₅₃	1	8	yes	—
375	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
376	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
377	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
378	C₂³	C₂×#41×C₅₃	1	2	yes	—
379	C₂³	H(5,𝔽₂)×C₅₃	1	4	yes	—
380	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
381	D₄	—	2	32	no	—
382	D₄	—	4	128	no	—
383	D₄	—	4	32	no	—
384	D₄	—	4	32	no	—
385	D₄	—	4	32	no	—
386	C₂×C₄	—	2	8	no	—
387	C₂×C₄	—	2	32	no	—
388	C₂×C₄	—	2	8	no	—
389	C₂×C₄	—	2	8	no	—
390	C₂×C₄	—	2	4	no	—
391	C₂×C₄	—	2	8	no	—
392	C₂×C₄	—	2	4	no	—
393	C₂×C₄	—	4	8	no	—
394	C₂×C₄	—	4	32	no	—
395	C₂×C₄	#121×C₅₃	1	16	yes	complemented
396	C₂×C₄	#127×C₅₃	1	16	yes	complemented
397	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
398	C₂×C₄	C₂²×D₄×C₅₃	1	16	yes	—
399	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
400	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
401	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
402	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
403	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
404	C₂²	—	2	16	no	—
405	C₂²	—	2	8	no	—
406	C₂²	—	4	8	no	—
407	C₂²	—	4	32	no	—
408	C₂²	—	4	4	no	—
409	C₂²	—	4	4	no	—
410	C₂²	—	4	16	no	—
411	C₂²	—	4	16	no	—
412	C₂²	—	4	8	no	—
413	C₂²	—	4	8	no	—
414	C₂²	—	4	8	no	—
415	C₂²	—	8	32	no	—
416	C₂²	C₂×#121×C₅₃	1	4	yes	—
417	C₂²	C₂×#127×C₅₃	1	4	yes	—
418	C₂²	C₅₃×#535	1	4	yes	—
419	C₂²	D₄²×C₅₃	1	8	yes	—
420	C₂²	C₅₃×#550	1	8	yes	—
421	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
422	C₂²	C₂³×D₄×C₅₃	1	2	yes	—
423	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
424	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
425	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
426	C₄	—	2	16	no	—
427	C₄	—	2	4	no	—
428	C₄	—	4	4	no	—
429	C₄	C₂×#121×C₅₃	1	8	yes	complemented
430	C₄	C₂×#127×C₅₃	1	8	yes	complemented
431	C₂	—	4	8	no	—
432	C₂	—	8	8	no	—
433	C₂	C₅₃×#2541	1	8	yes	direct product factor
434	C₂	C₂²×#121×C₅₃	1	1	yes	characteristic
435	C₂	C₂²×#127×C₅₃	1	1	yes	characteristic
436	C₂	C₂×C₅₃×#535	1	1	yes	characteristic
437	C₂	C₂×D₄²×C₅₃	1	2	yes	—
438	C₂	C₂×C₅₃×#550	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. C₂²×#127×C₅₃	C₂	semidirect
2. C₂²×#127×C₅₃	C₂	semidirect
3. C₂²×#121×C₅₃	C₂	semidirect
4. C₂²×C₄×D₄×C₅₃	C₂	semidirect
5. C₅₃×#2541	C₂	direct (N × H)
6. C₂×C₅₃×#392	C₂	semidirect
7. C₂×C₅₃×#390	C₂	semidirect
8. C₂×C₄×C₄⋊C₄×C₅₃	C₂	semidirect
17. C₂×#127×C₅₃	C₂²	semidirect
18. C₂×#127×C₅₃	C₂²	semidirect
20. C₂×#121×C₅₃	C₂²	semidirect
22. C₂×C₄×D₄×C₅₃	C₂²	semidirect
29. C₅₃×#392	C₂²	semidirect
30. C₅₃×#390	C₂²	semidirect
31. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
47. C₂²×D₄×C₅₃	D₄	semidirect
66. C₂×C₄⋊C₄×C₅₃	D₄	semidirect
74. C₂×C₄²×C₅₃	D₄	semidirect
108. C₂×D₄×C₅₃	C₂×D₄	semidirect
130. C₄⋊C₄×C₅₃	C₂×D₄	semidirect
134. C₄²×C₅₃	C₂×D₄	semidirect
175. C₂×C₄×C₅₃	#121	semidirect
176. C₂×C₄×C₅₃	#127	semidirect
184. C₂×#2541	C₅₃	direct (N × H)
210. C₄×C₅₃	C₂×#121	semidirect
211. C₄×C₅₃	C₂×#127	semidirect
212. C₂²×#127	C₂×C₅₃	semidirect
213. C₂²×#127	C₂×C₅₃	semidirect
214. C₂²×#121	C₂×C₅₃	semidirect
215. C₂²×C₄×D₄	C₂×C₅₃	semidirect
216. #2541	C₂×C₅₃	direct (N × H)
217. C₂×#392	C₂×C₅₃	semidirect
218. C₂×#390	C₂×C₅₃	semidirect
219. C₂×C₄×C₄⋊C₄	C₂×C₅₃	semidirect
222. C₂×C₅₃	#2541	direct (N × H)
236. C₂×#127	C₂²×C₅₃	semidirect
237. C₂×#127	C₂²×C₅₃	semidirect
239. C₂×#121	C₂²×C₅₃	semidirect
241. C₂×C₄×D₄	C₂²×C₅₃	semidirect
248. #392	C₂²×C₅₃	semidirect
249. #390	C₂²×C₅₃	semidirect
250. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
252. C₅₃	C₂×#2541	direct (N × H)
267. C₂²×D₄	D₄×C₅₃	semidirect
286. C₂×C₄⋊C₄	D₄×C₅₃	semidirect
294. C₂×C₄²	D₄×C₅₃	semidirect
328. C₂×D₄	C₂×D₄×C₅₃	semidirect
350. C₄⋊C₄	C₂×D₄×C₅₃	semidirect
354. C₄²	C₂×D₄×C₅₃	semidirect
395. C₂×C₄	#121×C₅₃	semidirect
396. C₂×C₄	#127×C₅₃	semidirect
429. C₄	C₂×#121×C₅₃	semidirect
430. C₄	C₂×#127×C₅₃	semidirect
433. C₂	C₅₃×#2541	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×#127×C₅₃	2
C₂²×#121×C₅₃	2
C₂²×C₄×D₄×C₅₃	2
C₅₃×#2541	2
C₂×C₅₃×#392	2
C₂×C₅₃×#390	2
C₂×C₄×C₄⋊C₄×C₅₃	2
C₂×#2541	53

Subgroups — C2×C53×#2541 #27242435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2541 #27242435