Subgroups — C₅₃×#3561 #20272498

Order 6784.

1032 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#563	C₂	1	2	yes	maximal, complemented
2	C₅₃×#562	C₂	1	2	yes	maximal, complemented
3	C₅₃×#556	C₂	1	2	yes	maximal, complemented
4	C₅₃×#551	C₂	1	4	yes	maximal, complemented
5	D₄×Q₈×C₅₃	C₂	1	2	yes	maximal, complemented
6	C₅₃×#548	C₂	1	2	yes	maximal, complemented
7	C₅₃×#546	C₂	1	2	yes	maximal, complemented
8	C₅₃×#543	C₂	1	2	yes	maximal, complemented
9	C₅₃×#542	C₂	1	2	yes	maximal, complemented
10	C₅₃×#542	C₂	1	4	yes	maximal, complemented
11	C₅₃×#538	C₂	1	2	yes	maximal, complemented
12	C₅₃×#536	C₂	1	2	yes	maximal, complemented
13	C₅₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅₃×#529	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#124×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41×C₅₃	C₂²	1	2	yes	—
17	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×Q₈×C₅₃	C₂²	1	2	yes	complemented
19	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
21	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
22	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
23	C₄⋊Q₈×C₅₃	C₂²	1	4	yes	complemented
24	#126×C₅₃	C₂²	1	2	yes	complemented
25	#126×C₅₃	C₂²	1	4	yes	complemented
26	#126×C₅₃	C₂²	1	4	yes	complemented
27	#126×C₅₃	C₂²	1	4	yes	complemented
28	#125×C₅₃	C₂²	1	1	yes	characteristic, complemented
29	#125×C₅₃	C₂²	1	2	yes	complemented
30	#124×C₅₃	C₂²	1	2	yes	—
31	#124×C₅₃	C₂²	1	2	yes	—
32	#124×C₅₃	C₂²	1	2	yes	—
33	#124×C₅₃	C₂²	1	2	yes	—
34	#124×C₅₃	C₂²	1	2	yes	—
35	#124×C₅₃	C₂²	1	1	yes	characteristic, complemented
36	#124×C₅₃	C₂²	1	4	yes	complemented
37	#124×C₅₃	C₂²	1	4	yes	complemented
38	#123×C₅₃	C₂²	1	2	yes	—
39	#123×C₅₃	C₂²	1	4	yes	—
40	#123×C₅₃	C₂²	1	4	yes	complemented
41	#123×C₅₃	C₂²	1	4	yes	complemented
42	#123×C₅₃	C₂²	1	4	yes	complemented
43	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
44	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
45	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
46	C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	—
47	C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	complemented
48	C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	complemented
49	C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	complemented
50	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
51	C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	complemented
52	C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	complemented
53	C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	complemented
54	#121×C₅₃	C₂²	1	1	yes	characteristic
55	#121×C₅₃	C₂²	1	2	yes	—
56	#121×C₅₃	C₂²	1	4	yes	—
57	#121×C₅₃	C₂²	1	4	yes	complemented
58	#120×C₅₃	C₂²	1	4	yes	—
59	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
60	C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
61	C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
62	C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
63	C₄×D₄×C₅₃	C₂²	1	2	yes	—
64	C₄×D₄×C₅₃	C₂²	1	4	yes	—
65	C₄×D₄×C₅₃	C₂²	1	2	yes	—
66	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
67	C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
68	#117×C₅₃	C₂²	1	2	yes	—
69	#117×C₅₃	C₂²	1	2	yes	complemented
70	#117×C₅₃	C₂²	1	4	yes	complemented
71	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
72	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
73	C₂×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
74	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
75	C₂⁴×C₅₃	C₂³	1	2	yes	—
76	#41×C₅₃	—	4	4	no	—
77	C₂×Q₈×C₅₃	—	4	2	no	—
78	C₂×Q₈×C₅₃	—	4	4	no	—
79	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
80	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
81	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
83	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
84	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
85	C₂×D₄×C₅₃	—	4	2	no	—
86	C₂×D₄×C₅₃	C₂³	1	2	yes	—
87	C₂×D₄×C₅₃	C₂³	1	2	yes	—
88	C₂×D₄×C₅₃	C₂³	1	4	yes	—
89	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
90	C₂×D₄×C₅₃	C₂³	1	4	yes	—
91	C₂×D₄×C₅₃	C₂³	1	2	yes	—
92	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
95	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
97	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
98	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
99	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
100	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
101	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
102	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
103	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
104	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
105	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
106	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
107	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
108	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
109	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
110	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
111	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
112	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
113	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
114	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
115	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
116	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
117	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
118	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
119	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
120	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
121	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
122	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
123	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
124	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
125	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
126	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
127	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
128	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
129	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
130	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
131	C₄²×C₅₃	C₂³	1	1	yes	characteristic
132	C₄²×C₅₃	C₂³	1	1	yes	characteristic
133	C₄²×C₅₃	C₂³	1	1	yes	characteristic
134	C₄²×C₅₃	C₂³	1	1	yes	characteristic
135	C₄²×C₅₃	C₂³	1	4	yes	—
136	C₄²×C₅₃	C₂³	1	2	yes	—
137	C₄²×C₅₃	C₂³	1	4	yes	—
138	C₂³×C₅₃	—	2	4	no	—
139	C₂³×C₅₃	—	4	2	no	—
140	C₂³×C₅₃	—	4	2	no	—
141	C₂³×C₅₃	C₂⁴	1	2	yes	—
142	C₂³×C₅₃	C₂⁴	1	4	yes	—
143	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
144	Q₈×C₅₃	—	2	4	no	—
145	Q₈×C₅₃	—	2	4	no	—
146	Q₈×C₅₃	—	4	4	no	—
147	Q₈×C₅₃	—	4	2	no	—
148	Q₈×C₅₃	—	4	1	no	—
149	Q₈×C₅₃	—	4	2	no	—
150	D₄×C₅₃	—	2	4	no	—
151	D₄×C₅₃	—	2	4	no	—
152	D₄×C₅₃	—	4	2	no	—
153	D₄×C₅₃	—	4	4	no	—
154	D₄×C₅₃	—	4	1	no	—
155	D₄×C₅₃	—	4	4	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	—	4	4	no	—
160	C₂×C₄×C₅₃	—	4	1	no	—
161	C₂×C₄×C₅₃	—	4	4	no	—
162	C₂×C₄×C₅₃	—	4	2	no	—
163	C₂×C₄×C₅₃	—	4	4	no	—
164	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
165	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
166	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
167	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
168	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
169	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
170	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
171	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
172	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
175	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
176	C₂²×C₅₃	—	2	1	no	—
177	C₂²×C₅₃	—	2	1	no	—
178	C₂²×C₅₃	—	2	2	no	—
179	C₂²×C₅₃	—	2	4	no	—
180	C₂²×C₅₃	—	2	2	no	—
181	C₂²×C₅₃	—	2	4	no	—
182	C₂²×C₅₃	—	2	4	no	—
183	C₂²×C₅₃	—	2	2	no	—
184	C₂²×C₅₃	—	8	4	no	—
185	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
186	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
187	C₄×C₅₃	—	2	2	no	—
188	C₄×C₅₃	—	2	1	no	—
189	C₄×C₅₃	—	2	4	no	—
190	C₄×C₅₃	—	2	4	no	—
191	C₄×C₅₃	—	2	1	no	—
192	C₄×C₅₃	—	2	1	no	—
193	C₄×C₅₃	—	2	2	no	—
194	C₄×C₅₃	—	2	2	no	—
195	C₄×C₅₃	—	2	4	no	—
196	C₄×C₅₃	—	2	1	no	—
197	C₄×C₅₃	—	2	1	no	—
198	C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
199	#3561	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂×C₅₃	—	2	2	no	—
201	C₂×C₅₃	—	4	2	no	—
202	C₂×C₅₃	—	4	4	no	—
203	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
204	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
205	C₂×C₅₃	#585	1	1	yes	characteristic
206	#563	C₂×C₅₃	1	2	yes	complemented
207	#562	C₂×C₅₃	1	2	yes	complemented
208	#556	C₂×C₅₃	1	2	yes	complemented
209	#551	C₂×C₅₃	1	4	yes	complemented
210	D₄×Q₈	C₂×C₅₃	1	2	yes	complemented
211	#548	C₂×C₅₃	1	2	yes	complemented
212	#546	C₂×C₅₃	1	2	yes	complemented
213	#543	C₂×C₅₃	1	2	yes	complemented
214	#542	C₂×C₅₃	1	2	yes	complemented
215	#542	C₂×C₅₃	1	4	yes	complemented
216	#538	C₂×C₅₃	1	2	yes	complemented
217	#536	C₂×C₅₃	1	2	yes	complemented
218	#533	C₂×C₅₃	1	1	yes	characteristic, complemented
219	#529	C₂×C₅₃	1	1	yes	characteristic, complemented
220	C₂×#124	C₂×C₅₃	1	1	yes	characteristic, complemented
221	C₅₃	#3561	1	1	yes	characteristic, direct product factor, Sylow
222	C₂×#41	C₂²×C₅₃	1	2	yes	—
223	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
224	C₂²×Q₈	C₂²×C₅₃	1	2	yes	complemented
225	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
226	C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
227	C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
228	C₄⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
229	C₄⋊Q₈	C₂²×C₅₃	1	4	yes	complemented
230	#126	C₂²×C₅₃	1	2	yes	complemented
231	#126	C₂²×C₅₃	1	4	yes	complemented
232	#126	C₂²×C₅₃	1	4	yes	complemented
233	#126	C₂²×C₅₃	1	4	yes	complemented
234	#125	C₂²×C₅₃	1	1	yes	characteristic, complemented
235	#125	C₂²×C₅₃	1	2	yes	complemented
236	#124	C₂²×C₅₃	1	2	yes	—
237	#124	C₂²×C₅₃	1	2	yes	—
238	#124	C₂²×C₅₃	1	2	yes	—
239	#124	C₂²×C₅₃	1	2	yes	—
240	#124	C₂²×C₅₃	1	2	yes	—
241	#124	C₂²×C₅₃	1	1	yes	characteristic, complemented
242	#124	C₂²×C₅₃	1	4	yes	complemented
243	#124	C₂²×C₅₃	1	4	yes	complemented
244	#123	C₂²×C₅₃	1	2	yes	—
245	#123	C₂²×C₅₃	1	4	yes	—
246	#123	C₂²×C₅₃	1	4	yes	complemented
247	#123	C₂²×C₅₃	1	4	yes	complemented
248	#123	C₂²×C₅₃	1	4	yes	complemented
249	C₂²⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
250	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
251	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
252	C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	—
253	C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	complemented
254	C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	complemented
255	C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	complemented
256	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
257	C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	complemented
258	C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	complemented
259	C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	complemented
260	#121	C₂²×C₅₃	1	1	yes	characteristic
261	#121	C₂²×C₅₃	1	2	yes	—
262	#121	C₂²×C₅₃	1	4	yes	—
263	#121	C₂²×C₅₃	1	4	yes	complemented
264	#120	C₂²×C₅₃	1	4	yes	—
265	C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
266	C₄×Q₈	C₂²×C₅₃	1	2	yes	complemented
267	C₄×Q₈	C₂²×C₅₃	1	2	yes	complemented
268	C₄×Q₈	C₂²×C₅₃	1	2	yes	complemented
269	C₄×D₄	C₂²×C₅₃	1	2	yes	—
270	C₄×D₄	C₂²×C₅₃	1	4	yes	—
271	C₄×D₄	C₂²×C₅₃	1	2	yes	—
272	C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
273	C₄×D₄	C₂²×C₅₃	1	4	yes	complemented
274	#117	C₂²×C₅₃	1	2	yes	—
275	#117	C₂²×C₅₃	1	2	yes	complemented
276	#117	C₂²×C₅₃	1	4	yes	complemented
277	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
278	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
279	C₂×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
280	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
281	C₂⁴	C₂³×C₅₃	1	2	yes	—
282	#41	—	4	4	no	—
283	C₂×Q₈	—	4	2	no	—
284	C₂×Q₈	—	4	4	no	—
285	C₂×Q₈	C₂³×C₅₃	1	2	yes	—
286	C₂×Q₈	C₂³×C₅₃	1	2	yes	—
287	C₂×Q₈	C₂³×C₅₃	1	1	yes	characteristic
288	C₂×Q₈	C₂³×C₅₃	1	2	yes	—
289	C₂×Q₈	C₂³×C₅₃	1	4	yes	—
290	C₂×Q₈	C₂³×C₅₃	1	2	yes	—
291	C₂×D₄	—	4	2	no	—
292	C₂×D₄	C₂³×C₅₃	1	2	yes	—
293	C₂×D₄	C₂³×C₅₃	1	2	yes	—
294	C₂×D₄	C₂³×C₅₃	1	4	yes	—
295	C₂×D₄	C₂³×C₅₃	1	1	yes	characteristic
296	C₂×D₄	C₂³×C₅₃	1	4	yes	—
297	C₂×D₄	C₂³×C₅₃	1	2	yes	—
298	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
299	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
300	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
301	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
302	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
303	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
304	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
305	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
306	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
307	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
308	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
309	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
310	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
311	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
312	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
313	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
314	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
315	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
316	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
317	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
318	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
319	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
320	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
321	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
322	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
323	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
324	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
325	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
326	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
327	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
328	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
329	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
330	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
331	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
332	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
333	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
334	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
335	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
336	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
337	C₄²	C₂³×C₅₃	1	1	yes	characteristic
338	C₄²	C₂³×C₅₃	1	1	yes	characteristic
339	C₄²	C₂³×C₅₃	1	1	yes	characteristic
340	C₄²	C₂³×C₅₃	1	1	yes	characteristic
341	C₄²	C₂³×C₅₃	1	4	yes	—
342	C₄²	C₂³×C₅₃	1	2	yes	—
343	C₄²	C₂³×C₅₃	1	4	yes	—
344	C₂³	—	2	4	no	—
345	C₂³	—	4	2	no	—
346	C₂³	—	4	2	no	—
347	C₂³	C₂⁴×C₅₃	1	4	yes	—
348	C₂³	C₂⁴×C₅₃	1	2	yes	—
349	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
350	Q₈	—	2	4	no	—
351	Q₈	—	2	4	no	—
352	Q₈	—	4	4	no	—
353	Q₈	—	4	2	no	—
354	Q₈	—	4	1	no	—
355	Q₈	—	4	2	no	—
356	D₄	—	2	4	no	—
357	D₄	—	2	4	no	—
358	D₄	—	4	2	no	—
359	D₄	—	4	4	no	—
360	D₄	—	4	1	no	—
361	D₄	—	4	4	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	4	no	—
365	C₂×C₄	—	4	4	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	4	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	4	no	—
370	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
371	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
372	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
373	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
374	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
375	C₂×C₄	C₂⁴×C₅₃	1	4	yes	—
376	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
377	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
378	C₂×C₄	C₂⁴×C₅₃	1	4	yes	—
379	C₂×C₄	C₂⁴×C₅₃	1	4	yes	—
380	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
381	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	2	no	—
385	C₂²	—	2	4	no	—
386	C₂²	—	2	2	no	—
387	C₂²	—	2	4	no	—
388	C₂²	—	2	4	no	—
389	C₂²	—	2	2	no	—
390	C₂²	—	8	4	no	—
391	C₂²	2₋¹⁺⁴×C₅₃	1	2	yes	—
392	C₂²	C₂⁵×C₅₃	1	1	yes	characteristic
393	C₄	—	2	2	no	—
394	C₄	—	2	1	no	—
395	C₄	—	2	4	no	—
396	C₄	—	2	4	no	—
397	C₄	—	2	1	no	—
398	C₄	—	2	1	no	—
399	C₄	—	2	2	no	—
400	C₄	—	2	2	no	—
401	C₄	—	2	4	no	—
402	C₄	—	2	1	no	—
403	C₄	—	2	1	no	—
404	C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
405	C₂	—	2	2	no	—
406	C₂	—	4	4	no	—
407	C₂	—	4	2	no	—
408	C₂	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
409	C₂	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
410	C₂	C₅₃×#585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₅₃
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×#563	C₂	semidirect
2. C₅₃×#562	C₂	semidirect
3. C₅₃×#556	C₂	semidirect
4. C₅₃×#551	C₂	semidirect
5. D₄×Q₈×C₅₃	C₂	semidirect
6. C₅₃×#548	C₂	semidirect
7. C₅₃×#546	C₂	semidirect
8. C₅₃×#543	C₂	semidirect
9. C₅₃×#542	C₂	semidirect
10. C₅₃×#542	C₂	semidirect
11. C₅₃×#538	C₂	semidirect
12. C₅₃×#536	C₂	semidirect
13. C₅₃×#533	C₂	semidirect
14. C₅₃×#529	C₂	semidirect
15. C₂×#124×C₅₃	C₂	semidirect
18. C₂²×Q₈×C₅₃	C₂²	semidirect
20. C₄⋊Q₈×C₅₃	C₂²	semidirect
21. C₄⋊Q₈×C₅₃	C₂²	semidirect
22. C₄⋊Q₈×C₅₃	C₂²	semidirect
23. C₄⋊Q₈×C₅₃	C₂²	semidirect
24. #126×C₅₃	C₂²	semidirect
25. #126×C₅₃	C₂²	semidirect
26. #126×C₅₃	C₂²	semidirect
27. #126×C₅₃	C₂²	semidirect
28. #125×C₅₃	C₂²	semidirect
29. #125×C₅₃	C₂²	semidirect
35. #124×C₅₃	C₂²	semidirect
36. #124×C₅₃	C₂²	semidirect
37. #124×C₅₃	C₂²	semidirect
40. #123×C₅₃	C₂²	semidirect
41. #123×C₅₃	C₂²	semidirect
42. #123×C₅₃	C₂²	semidirect
47. C₂²⋊Q₈×C₅₃	C₂²	semidirect
48. C₂²⋊Q₈×C₅₃	C₂²	semidirect
49. C₂²⋊Q₈×C₅₃	C₂²	semidirect
50. C₂²⋊Q₈×C₅₃	C₂²	semidirect
51. C₂²⋊Q₈×C₅₃	C₂²	semidirect
52. C₂²⋊Q₈×C₅₃	C₂²	semidirect
53. C₂²⋊Q₈×C₅₃	C₂²	semidirect
57. #121×C₅₃	C₂²	semidirect
59. C₄×Q₈×C₅₃	C₂²	semidirect
60. C₄×Q₈×C₅₃	C₂²	semidirect
61. C₄×Q₈×C₅₃	C₂²	semidirect
62. C₄×Q₈×C₅₃	C₂²	semidirect
66. C₄×D₄×C₅₃	C₂²	semidirect
67. C₄×D₄×C₅₃	C₂²	semidirect
69. #117×C₅₃	C₂²	semidirect
70. #117×C₅₃	C₂²	semidirect
199. #3561	C₅₃	direct (N × H)
206. #563	C₂×C₅₃	semidirect
207. #562	C₂×C₅₃	semidirect
208. #556	C₂×C₅₃	semidirect
209. #551	C₂×C₅₃	semidirect
210. D₄×Q₈	C₂×C₅₃	semidirect
211. #548	C₂×C₅₃	semidirect
212. #546	C₂×C₅₃	semidirect
213. #543	C₂×C₅₃	semidirect
214. #542	C₂×C₅₃	semidirect
215. #542	C₂×C₅₃	semidirect
216. #538	C₂×C₅₃	semidirect
217. #536	C₂×C₅₃	semidirect
218. #533	C₂×C₅₃	semidirect
219. #529	C₂×C₅₃	semidirect
220. C₂×#124	C₂×C₅₃	semidirect
221. C₅₃	#3561	direct (N × H)
224. C₂²×Q₈	C₂²×C₅₃	semidirect
226. C₄⋊Q₈	C₂²×C₅₃	semidirect
227. C₄⋊Q₈	C₂²×C₅₃	semidirect
228. C₄⋊Q₈	C₂²×C₅₃	semidirect
229. C₄⋊Q₈	C₂²×C₅₃	semidirect
230. #126	C₂²×C₅₃	semidirect
231. #126	C₂²×C₅₃	semidirect
232. #126	C₂²×C₅₃	semidirect
233. #126	C₂²×C₅₃	semidirect
234. #125	C₂²×C₅₃	semidirect
235. #125	C₂²×C₅₃	semidirect
241. #124	C₂²×C₅₃	semidirect
242. #124	C₂²×C₅₃	semidirect
243. #124	C₂²×C₅₃	semidirect
246. #123	C₂²×C₅₃	semidirect
247. #123	C₂²×C₅₃	semidirect
248. #123	C₂²×C₅₃	semidirect
253. C₂²⋊Q₈	C₂²×C₅₃	semidirect
254. C₂²⋊Q₈	C₂²×C₅₃	semidirect
255. C₂²⋊Q₈	C₂²×C₅₃	semidirect
256. C₂²⋊Q₈	C₂²×C₅₃	semidirect
257. C₂²⋊Q₈	C₂²×C₅₃	semidirect
258. C₂²⋊Q₈	C₂²×C₅₃	semidirect
259. C₂²⋊Q₈	C₂²×C₅₃	semidirect
263. #121	C₂²×C₅₃	semidirect
265. C₄×Q₈	C₂²×C₅₃	semidirect
266. C₄×Q₈	C₂²×C₅₃	semidirect
267. C₄×Q₈	C₂²×C₅₃	semidirect
268. C₄×Q₈	C₂²×C₅₃	semidirect
272. C₄×D₄	C₂²×C₅₃	semidirect
273. C₄×D₄	C₂²×C₅₃	semidirect
275. #117	C₂²×C₅₃	semidirect
276. #117	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×#563	2
C₅₃×#562	2
C₅₃×#556	2
C₅₃×#551	2
D₄×Q₈×C₅₃	2
C₅₃×#548	2
C₅₃×#546	2
C₅₃×#543	2
C₅₃×#542	2
C₅₃×#538	2
C₅₃×#536	2
C₅₃×#533	2
C₅₃×#529	2
C₂×#124×C₅₃	2
#3561	53

Subgroups — C53×#3561 #20272498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#3561 #20272498