Subgroups — C₅₃×#3495 #20272432

Order 6784.

1022 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#556	C₂	1	2	yes	maximal, complemented
2	C₅₃×#554	C₂	1	2	yes	maximal, complemented
3	C₅₃×#552	C₂	1	2	yes	maximal, complemented
4	C₅₃×#552	C₂	1	4	yes	maximal, complemented
5	C₅₃×#552	C₂	1	2	yes	maximal, complemented
6	C₅₃×#551	C₂	1	4	yes	maximal, complemented
7	C₅₃×#544	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#542	C₂	1	2	yes	maximal, complemented
9	C₅₃×#541	C₂	1	2	yes	maximal, complemented
10	C₅₃×#536	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#533	C₂	1	2	yes	maximal, complemented
12	C₅₃×#529	C₂	1	2	yes	maximal, complemented
13	C₅₃×#525	C₂	1	1	yes	maximal, characteristic
14	C₂×C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#520	C₂	1	1	yes	maximal, characteristic, complemented
16	C₅₃×#519	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂×#117×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
22	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
23	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
24	#126×C₅₃	C₂²	1	4	yes	—
25	#126×C₅₃	C₂²	1	4	yes	—
26	#126×C₅₃	C₂²	1	4	yes	complemented
27	#126×C₅₃	C₂²	1	4	yes	complemented
28	#125×C₅₃	C₂²	1	2	yes	complemented
29	#125×C₅₃	C₂²	1	4	yes	complemented
30	#125×C₅₃	C₂²	1	4	yes	complemented
31	#125×C₅₃	C₂²	1	4	yes	complemented
32	#125×C₅₃	C₂²	1	2	yes	complemented
33	#124×C₅₃	C₂²	1	4	yes	—
34	#124×C₅₃	C₂²	1	2	yes	complemented
35	#123×C₅₃	C₂²	1	2	yes	—
36	#123×C₅₃	C₂²	1	4	yes	—
37	#123×C₅₃	C₂²	1	4	yes	—
38	#123×C₅₃	C₂²	1	4	yes	—
39	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
40	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
41	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
42	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
43	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
44	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	—
46	C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	—
47	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
48	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
49	C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	complemented
50	#121×C₅₃	C₂²	1	1	yes	characteristic
51	#121×C₅₃	C₂²	1	1	yes	characteristic
52	#120×C₅₃	C₂²	1	2	yes	—
53	C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
54	C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
55	C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
56	C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
57	C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
58	C₄×Q₈×C₅₃	C₂²	1	4	yes	complemented
59	C₄×D₄×C₅₃	C₂²	1	4	yes	—
60	C₄×D₄×C₅₃	C₂²	1	2	yes	—
61	C₄×D₄×C₅₃	C₂²	1	2	yes	—
62	C₄×D₄×C₅₃	C₂²	1	4	yes	—
63	C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
64	#117×C₅₃	C₂²	1	1	yes	characteristic
65	#117×C₅₃	C₂²	1	1	yes	characteristic
66	#117×C₅₃	C₂²	1	2	yes	—
67	#117×C₅₃	C₂²	1	2	yes	—
68	#117×C₅₃	C₂²	1	1	yes	characteristic
69	#117×C₅₃	C₂²	1	1	yes	characteristic
70	#117×C₅₃	C₂²	1	1	yes	characteristic
71	#117×C₅₃	C₂²	1	4	yes	—
72	#117×C₅₃	C₂²	1	4	yes	—
73	#117×C₅₃	C₂²	1	2	yes	complemented
74	#117×C₅₃	C₂²	1	4	yes	complemented
75	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
79	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
80	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
81	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
82	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
83	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
84	#41×C₅₃	—	2	4	no	—
85	C₂×Q₈×C₅₃	—	2	4	no	—
86	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
87	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
88	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
89	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
90	C₂×D₄×C₅₃	C₂³	1	2	yes	—
91	C₂×D₄×C₅₃	C₂³	1	2	yes	—
92	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
93	C₂×D₄×C₅₃	C₂³	1	2	yes	—
94	C₂²×C₄×C₅₃	—	2	4	no	—
95	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
97	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
98	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
101	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
103	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
104	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
105	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
106	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
109	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
110	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
111	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
112	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
113	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
114	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
115	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
116	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
117	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
118	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
119	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
120	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
121	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
122	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
123	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
124	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
125	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
126	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
127	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
128	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
129	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
130	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
131	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
132	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
133	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
134	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
135	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
136	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
137	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
138	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
139	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
140	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
141	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
142	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
143	C₄²×C₅₃	C₂³	1	2	yes	—
144	C₄²×C₅₃	C₂³	1	2	yes	—
145	C₄²×C₅₃	C₂³	1	2	yes	—
146	C₄²×C₅₃	C₂³	1	2	yes	—
147	C₄²×C₅₃	C₂³	1	4	yes	—
148	C₄²×C₅₃	C₂³	1	4	yes	—
149	C₄²×C₅₃	C₂³	1	4	yes	—
150	C₄²×C₅₃	C₂³	1	2	yes	—
151	C₂³×C₅₃	—	2	2	no	—
152	C₂³×C₅₃	—	2	2	no	—
153	C₂³×C₅₃	—	4	1	no	—
154	C₂³×C₅₃	C₂⁴	1	2	yes	—
155	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
156	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
157	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
158	Q₈×C₅₃	—	2	4	no	—
159	Q₈×C₅₃	—	2	4	no	—
160	Q₈×C₅₃	—	2	2	no	—
161	Q₈×C₅₃	—	4	4	no	—
162	D₄×C₅₃	—	2	4	no	—
163	D₄×C₅₃	—	2	4	no	—
164	D₄×C₅₃	—	2	2	no	—
165	D₄×C₅₃	—	4	2	no	—
166	C₂×C₄×C₅₃	—	2	2	no	—
167	C₂×C₄×C₅₃	—	2	4	no	—
168	C₂×C₄×C₅₃	—	2	4	no	—
169	C₂×C₄×C₅₃	—	2	2	no	—
170	C₂×C₄×C₅₃	—	2	2	no	—
171	C₂×C₄×C₅₃	—	2	2	no	—
172	C₂×C₄×C₅₃	—	2	2	no	—
173	C₂×C₄×C₅₃	—	2	4	no	—
174	C₂×C₄×C₅₃	—	4	4	no	—
175	C₂×C₄×C₅₃	—	4	2	no	—
176	C₂×C₄×C₅₃	—	4	2	no	—
177	C₂×C₄×C₅₃	#41	1	4	yes	—
178	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
179	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
180	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
181	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
182	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
183	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
184	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
185	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
186	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
187	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
188	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
189	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
190	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
191	C₂²×C₅₃	—	2	1	no	—
192	C₂²×C₅₃	—	2	1	no	—
193	C₂²×C₅₃	—	2	1	no	—
194	C₂²×C₅₃	—	2	1	no	—
195	C₂²×C₅₃	—	2	1	no	—
196	C₂²×C₅₃	—	2	2	no	—
197	C₂²×C₅₃	—	2	2	no	—
198	C₂²×C₅₃	—	2	1	no	—
199	C₂²×C₅₃	—	2	2	no	—
200	C₂²×C₅₃	—	2	1	no	—
201	C₂²×C₅₃	—	4	4	no	—
202	C₂²×C₅₃	C₂×#41	1	2	yes	—
203	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
204	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
205	C₄×C₅₃	—	2	1	no	—
206	C₄×C₅₃	—	2	1	no	—
207	C₄×C₅₃	—	2	2	no	—
208	C₄×C₅₃	—	2	4	no	—
209	C₄×C₅₃	—	2	2	no	—
210	C₄×C₅₃	—	2	2	no	—
211	C₄×C₅₃	—	2	2	no	—
212	C₄×C₅₃	—	2	2	no	—
213	C₄×C₅₃	—	2	4	no	—
214	C₄×C₅₃	—	2	2	no	—
215	C₄×C₅₃	—	2	2	no	—
216	C₄×C₅₃	C₂×#41	1	2	yes	—
217	C₄×C₅₃	C₂×#41	1	2	yes	—
218	#3495	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
219	C₂×C₅₃	—	2	2	no	—
220	C₂×C₅₃	—	2	2	no	—
221	C₂×C₅₃	—	4	2	no	—
222	C₂×C₅₃	—	4	1	no	—
223	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
224	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
225	C₂×C₅₃	#585	1	1	yes	characteristic
226	#556	C₂×C₅₃	1	2	yes	complemented
227	#554	C₂×C₅₃	1	2	yes	complemented
228	#552	C₂×C₅₃	1	2	yes	complemented
229	#552	C₂×C₅₃	1	4	yes	complemented
230	#552	C₂×C₅₃	1	2	yes	complemented
231	#551	C₂×C₅₃	1	4	yes	complemented
232	#544	C₂×C₅₃	1	1	yes	characteristic, complemented
233	#542	C₂×C₅₃	1	2	yes	complemented
234	#541	C₂×C₅₃	1	2	yes	complemented
235	#536	C₂×C₅₃	1	1	yes	characteristic, complemented
236	#533	C₂×C₅₃	1	2	yes	complemented
237	#529	C₂×C₅₃	1	2	yes	complemented
238	#525	C₂×C₅₃	1	1	yes	characteristic
239	C₂×C₂²⋊Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
240	#520	C₂×C₅₃	1	1	yes	characteristic, complemented
241	#519	C₂×C₅₃	1	1	yes	characteristic, complemented
242	C₂×#117	C₂×C₅₃	1	1	yes	characteristic, complemented
243	C₅₃	#3495	1	1	yes	characteristic, direct product factor, Sylow
244	C₂×#41	C₂²×C₅₃	1	1	yes	characteristic
245	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
246	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
247	C₄⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
248	C₄⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
249	C₄⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
250	#126	C₂²×C₅₃	1	4	yes	—
251	#126	C₂²×C₅₃	1	4	yes	—
252	#126	C₂²×C₅₃	1	4	yes	complemented
253	#126	C₂²×C₅₃	1	4	yes	complemented
254	#125	C₂²×C₅₃	1	2	yes	complemented
255	#125	C₂²×C₅₃	1	4	yes	complemented
256	#125	C₂²×C₅₃	1	4	yes	complemented
257	#125	C₂²×C₅₃	1	4	yes	complemented
258	#125	C₂²×C₅₃	1	2	yes	complemented
259	#124	C₂²×C₅₃	1	4	yes	—
260	#124	C₂²×C₅₃	1	2	yes	complemented
261	#123	C₂²×C₅₃	1	2	yes	—
262	#123	C₂²×C₅₃	1	4	yes	—
263	#123	C₂²×C₅₃	1	4	yes	—
264	#123	C₂²×C₅₃	1	4	yes	—
265	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
266	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
267	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
268	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
269	C₂²⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
270	C₂²⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
271	C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	—
272	C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	—
273	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
274	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
275	C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	complemented
276	#121	C₂²×C₅₃	1	1	yes	characteristic
277	#121	C₂²×C₅₃	1	1	yes	characteristic
278	#120	C₂²×C₅₃	1	2	yes	—
279	C₄×Q₈	C₂²×C₅₃	1	2	yes	complemented
280	C₄×Q₈	C₂²×C₅₃	1	2	yes	complemented
281	C₄×Q₈	C₂²×C₅₃	1	2	yes	complemented
282	C₄×Q₈	C₂²×C₅₃	1	2	yes	complemented
283	C₄×Q₈	C₂²×C₅₃	1	2	yes	complemented
284	C₄×Q₈	C₂²×C₅₃	1	4	yes	complemented
285	C₄×D₄	C₂²×C₅₃	1	4	yes	—
286	C₄×D₄	C₂²×C₅₃	1	2	yes	—
287	C₄×D₄	C₂²×C₅₃	1	2	yes	—
288	C₄×D₄	C₂²×C₅₃	1	4	yes	—
289	C₄×D₄	C₂²×C₅₃	1	2	yes	complemented
290	#117	C₂²×C₅₃	1	1	yes	characteristic
291	#117	C₂²×C₅₃	1	1	yes	characteristic
292	#117	C₂²×C₅₃	1	2	yes	—
293	#117	C₂²×C₅₃	1	1	yes	characteristic
294	#117	C₂²×C₅₃	1	2	yes	—
295	#117	C₂²×C₅₃	1	1	yes	characteristic
296	#117	C₂²×C₅₃	1	1	yes	characteristic
297	#117	C₂²×C₅₃	1	4	yes	—
298	#117	C₂²×C₅₃	1	4	yes	—
299	#117	C₂²×C₅₃	1	2	yes	complemented
300	#117	C₂²×C₅₃	1	4	yes	complemented
301	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
302	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
304	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
305	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
306	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
307	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
308	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
309	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
310	#41	—	2	4	no	—
311	C₂×Q₈	—	2	4	no	—
312	C₂×Q₈	C₂³×C₅₃	1	2	yes	—
313	C₂×Q₈	C₂³×C₅₃	1	2	yes	—
314	C₂×Q₈	C₂³×C₅₃	1	1	yes	characteristic
315	C₂×Q₈	C₂³×C₅₃	1	4	yes	—
316	C₂×D₄	C₂³×C₅₃	1	2	yes	—
317	C₂×D₄	C₂³×C₅₃	1	2	yes	—
318	C₂×D₄	C₂³×C₅₃	1	1	yes	characteristic
319	C₂×D₄	C₂³×C₅₃	1	2	yes	—
320	C₂²×C₄	—	2	4	no	—
321	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
322	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
323	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
324	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
325	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
326	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
327	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
328	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
329	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
330	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
331	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
332	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
333	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
334	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
335	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
336	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
337	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
338	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
339	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
340	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
341	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
342	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
343	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
344	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
345	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
346	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
347	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
348	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
349	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
350	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
351	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
352	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
353	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
354	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
355	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
356	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
357	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
358	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
359	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
360	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
361	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
362	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
363	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
364	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
365	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
366	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
367	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
368	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
369	C₄²	C₂³×C₅₃	1	2	yes	—
370	C₄²	C₂³×C₅₃	1	2	yes	—
371	C₄²	C₂³×C₅₃	1	2	yes	—
372	C₄²	C₂³×C₅₃	1	2	yes	—
373	C₄²	C₂³×C₅₃	1	4	yes	—
374	C₄²	C₂³×C₅₃	1	4	yes	—
375	C₄²	C₂³×C₅₃	1	4	yes	—
376	C₄²	C₂³×C₅₃	1	2	yes	—
377	C₂³	—	2	2	no	—
378	C₂³	—	2	2	no	—
379	C₂³	—	4	1	no	—
380	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
381	C₂³	C₂⁴×C₅₃	1	2	yes	—
382	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
383	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
384	Q₈	—	2	4	no	—
385	Q₈	—	2	4	no	—
386	Q₈	—	2	2	no	—
387	Q₈	—	4	4	no	—
388	D₄	—	2	4	no	—
389	D₄	—	2	4	no	—
390	D₄	—	2	2	no	—
391	D₄	—	4	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	4	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	4	4	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	#41×C₅₃	1	4	yes	—
404	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
405	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
406	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
407	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
408	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
409	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
410	C₂×C₄	C₂⁴×C₅₃	1	4	yes	—
411	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
412	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
413	C₂×C₄	C₂⁴×C₅₃	1	4	yes	—
414	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
415	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
416	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
417	C₂²	—	2	1	no	—
418	C₂²	—	2	1	no	—
419	C₂²	—	2	1	no	—
420	C₂²	—	2	1	no	—
421	C₂²	—	2	1	no	—
422	C₂²	—	2	2	no	—
423	C₂²	—	2	2	no	—
424	C₂²	—	2	1	no	—
425	C₂²	—	2	2	no	—
426	C₂²	—	2	1	no	—
427	C₂²	—	4	4	no	—
428	C₂²	C₂×#41×C₅₃	1	2	yes	—
429	C₂²	2₋¹⁺⁴×C₅₃	1	2	yes	—
430	C₂²	C₂⁵×C₅₃	1	1	yes	characteristic
431	C₄	—	2	1	no	—
432	C₄	—	2	1	no	—
433	C₄	—	2	2	no	—
434	C₄	—	2	4	no	—
435	C₄	—	2	2	no	—
436	C₄	—	2	2	no	—
437	C₄	—	2	2	no	—
438	C₄	—	2	2	no	—
439	C₄	—	2	4	no	—
440	C₄	—	2	2	no	—
441	C₄	—	2	2	no	—
442	C₄	C₂×#41×C₅₃	1	2	yes	—
443	C₄	C₂×#41×C₅₃	1	2	yes	—
444	C₂	—	2	2	no	—
445	C₂	—	2	2	no	—
446	C₂	—	4	1	no	—
447	C₂	—	4	2	no	—
448	C₂	C₂²×#41×C₅₃	1	1	yes	characteristic
449	C₂	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
450	C₂	C₅₃×#585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₅₃
prime radical	C₅₃×#525	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#556	C₂	semidirect
2. C₅₃×#554	C₂	semidirect
3. C₅₃×#552	C₂	semidirect
4. C₅₃×#552	C₂	semidirect
5. C₅₃×#552	C₂	semidirect
6. C₅₃×#551	C₂	semidirect
7. C₅₃×#544	C₂	semidirect
8. C₅₃×#542	C₂	semidirect
9. C₅₃×#541	C₂	semidirect
10. C₅₃×#536	C₂	semidirect
11. C₅₃×#533	C₂	semidirect
12. C₅₃×#529	C₂	semidirect
14. C₂×C₂²⋊Q₈×C₅₃	C₂	semidirect
15. C₅₃×#520	C₂	semidirect
16. C₅₃×#519	C₂	semidirect
17. C₂×#117×C₅₃	C₂	semidirect
21. C₄⋊Q₈×C₅₃	C₂²	semidirect
22. C₄⋊Q₈×C₅₃	C₂²	semidirect
23. C₄⋊Q₈×C₅₃	C₂²	semidirect
26. #126×C₅₃	C₂²	semidirect
27. #126×C₅₃	C₂²	semidirect
28. #125×C₅₃	C₂²	semidirect
29. #125×C₅₃	C₂²	semidirect
30. #125×C₅₃	C₂²	semidirect
31. #125×C₅₃	C₂²	semidirect
32. #125×C₅₃	C₂²	semidirect
34. #124×C₅₃	C₂²	semidirect
47. C₂²⋊Q₈×C₅₃	C₂²	semidirect
48. C₂²⋊Q₈×C₅₃	C₂²	semidirect
49. C₂²⋊Q₈×C₅₃	C₂²	semidirect
53. C₄×Q₈×C₅₃	C₂²	semidirect
54. C₄×Q₈×C₅₃	C₂²	semidirect
55. C₄×Q₈×C₅₃	C₂²	semidirect
56. C₄×Q₈×C₅₃	C₂²	semidirect
57. C₄×Q₈×C₅₃	C₂²	semidirect
58. C₄×Q₈×C₅₃	C₂²	semidirect
63. C₄×D₄×C₅₃	C₂²	semidirect
73. #117×C₅₃	C₂²	semidirect
74. #117×C₅₃	C₂²	semidirect
79. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
218. #3495	C₅₃	direct (N × H)
226. #556	C₂×C₅₃	semidirect
227. #554	C₂×C₅₃	semidirect
228. #552	C₂×C₅₃	semidirect
229. #552	C₂×C₅₃	semidirect
230. #552	C₂×C₅₃	semidirect
231. #551	C₂×C₅₃	semidirect
232. #544	C₂×C₅₃	semidirect
233. #542	C₂×C₅₃	semidirect
234. #541	C₂×C₅₃	semidirect
235. #536	C₂×C₅₃	semidirect
236. #533	C₂×C₅₃	semidirect
237. #529	C₂×C₅₃	semidirect
239. C₂×C₂²⋊Q₈	C₂×C₅₃	semidirect
240. #520	C₂×C₅₃	semidirect
241. #519	C₂×C₅₃	semidirect
242. C₂×#117	C₂×C₅₃	semidirect
243. C₅₃	#3495	direct (N × H)
247. C₄⋊Q₈	C₂²×C₅₃	semidirect
248. C₄⋊Q₈	C₂²×C₅₃	semidirect
249. C₄⋊Q₈	C₂²×C₅₃	semidirect
252. #126	C₂²×C₅₃	semidirect
253. #126	C₂²×C₅₃	semidirect
254. #125	C₂²×C₅₃	semidirect
255. #125	C₂²×C₅₃	semidirect
256. #125	C₂²×C₅₃	semidirect
257. #125	C₂²×C₅₃	semidirect
258. #125	C₂²×C₅₃	semidirect
260. #124	C₂²×C₅₃	semidirect
273. C₂²⋊Q₈	C₂²×C₅₃	semidirect
274. C₂²⋊Q₈	C₂²×C₅₃	semidirect
275. C₂²⋊Q₈	C₂²×C₅₃	semidirect
279. C₄×Q₈	C₂²×C₅₃	semidirect
280. C₄×Q₈	C₂²×C₅₃	semidirect
281. C₄×Q₈	C₂²×C₅₃	semidirect
282. C₄×Q₈	C₂²×C₅₃	semidirect
283. C₄×Q₈	C₂²×C₅₃	semidirect
284. C₄×Q₈	C₂²×C₅₃	semidirect
289. C₄×D₄	C₂²×C₅₃	semidirect
299. #117	C₂²×C₅₃	semidirect
300. #117	C₂²×C₅₃	semidirect
305. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#556	2
C₅₃×#554	2
C₅₃×#552	2
C₅₃×#551	2
C₅₃×#544	2
C₅₃×#542	2
C₅₃×#541	2
C₅₃×#536	2
C₅₃×#533	2
C₅₃×#529	2
C₅₃×#525	2
C₂×C₂²⋊Q₈×C₅₃	2
C₅₃×#520	2
C₅₃×#519	2
C₂×#117×C₅₃	2
#3495	53

Subgroups — C53×#3495 #20272432

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#3495 #20272432