Subgroups — C₄×H(5,𝔽₂)×D₅₃ #28290691

Order 13568.

12378 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×H(5,𝔽₂)×D₅₃	C₂	1	1	yes	maximal, characteristic
2	H(5,𝔽₂)×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#13133263	C₂	1	6	yes	maximal, complemented
4	#20287921	C₂	1	6	yes	maximal, complemented
5	C₄×#13133262	C₂	1	9	yes	maximal, complemented
6	#20287919	C₂	1	9	yes	maximal, complemented
7	#520×D₅₃	C₂	1	6	yes	maximal, complemented
8	#518×D₅₃	C₂	1	9	yes	maximal, complemented
9	C₄×#41×D₅₃	C₂	1	6	yes	maximal, complemented
10	C₂×C₄×D₄×D₅₃	C₂	1	9	yes	maximal, complemented
11	C₄×H(5,𝔽₂)×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133263	C₂²	1	6	yes	—
13	C₂×#13133262	C₂²	1	9	yes	—
14	H(5,𝔽₂)×D₅₃	C₄	1	32	yes	direct product factor
15	C₂×#41×D₅₃	C₂²	1	6	yes	—
16	C₂²×D₄×D₅₃	C₂²	1	9	yes	—
17	C₂³×C₄×D₅₃	C₂²	1	6	yes	complemented
18	#16811106	C₂²	1	6	yes	complemented
19	#16811105	C₂²	1	12	yes	complemented
20	#16811104	C₂²	1	18	yes	complemented
21	#16811103	C₂²	1	18	yes	complemented
22	#16811102	C₂²	1	36	yes	complemented
23	#16811098	C₂²	1	9	yes	complemented
24	#16811097	C₂²	1	18	yes	complemented
25	#16811096	C₂²	1	36	yes	complemented
26	#16811095	C₂²	1	36	yes	complemented
27	#16811095	C₂²	1	36	yes	complemented
28	#41×Dic₅₃	C₂²	1	6	yes	complemented
29	C₄×#11719949	C₂²	1	6	yes	complemented
30	C₄×#11719949	C₂²	1	6	yes	complemented
31	C₄×#11719945	C₂²	1	18	yes	complemented
32	#16811091	C₂²	1	18	yes	complemented
33	C₄×#11719947	C₂²	1	18	yes	complemented
34	C₄×#11719947	C₂²	1	18	yes	complemented
35	C₄×Q₈×D₅₃	C₂²	1	2	yes	complemented
36	C₄×Q₈×D₅₃	C₂²	1	6	yes	complemented
37	C₂×C₄×D₂₁₂	C₂²	1	9	yes	complemented
38	C₂×D₄×Dic₅₃	C₂²	1	9	yes	complemented
39	C₂×#13133179	C₂²	1	9	yes	complemented
40	C₂×C₄×C₂²⋊D₅₃	C₂²	1	18	yes	complemented
41	C₂×#13133177	C₂²	1	18	yes	complemented
42	C₄×D₄×D₅₃	C₂²	1	18	yes	complemented
43	C₄×D₄×D₅₃	C₂²	1	36	yes	complemented
44	C₄×D₄×D₅₃	C₂²	1	18	yes	complemented
45	C₄×D₄×D₅₃	C₂²	1	36	yes	complemented
46	C₄×D₄×D₅₃	C₂²	1	36	yes	complemented
47	#117×D₅₃	C₂²	1	18	yes	complemented
48	C₂×C₄⋊C₄×D₅₃	C₂²	1	9	yes	complemented
49	C₂×C₂²⋊C₄×D₅₃	C₂²	1	18	yes	complemented
50	C₂×C₄²×D₅₃	C₂²	1	9	yes	complemented
51	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	1	yes	characteristic
52	C₅₃×#520	C₂²	1	6	yes	complemented
53	C₅₃×#518	C₂²	1	9	yes	complemented
54	C₄×#41×C₅₃	C₂²	1	6	yes	complemented
55	C₂×C₄×D₄×C₅₃	C₂²	1	9	yes	complemented
56	C₂⁴×D₅₃	C₂³	1	6	yes	—
57	#13133263	C₂×C₄	1	96	yes	complemented
58	#13133262	C₂×C₄	1	144	yes	complemented
59	C₂×#11719949	C₂³	1	6	yes	—
60	C₂×#11719945	C₂³	1	18	yes	—
61	C₂×#11719947	C₂³	1	18	yes	—
62	#41×D₅₃	C₂×C₄	1	96	yes	complemented
63	C₂×Q₈×D₅₃	C₂³	1	2	yes	—
64	C₂²×D₂₁₂	C₂³	1	9	yes	—
65	C₂²×C₂²⋊D₅₃	C₂³	1	18	yes	—
66	C₂×D₄×D₅₃	C₂×C₄	1	144	yes	complemented
67	C₂×D₄×D₅₃	C₂³	1	18	yes	—
68	C₂×D₄×D₅₃	C₂³	1	36	yes	—
69	C₂³×Dic₅₃	C₂³	1	6	yes	complemented
70	C₂²×C₄×D₅₃	—	4	24	no	—
71	C₂²×C₄×D₅₃	C₂³	1	9	yes	—
72	C₂²×C₄×D₅₃	C₂³	1	9	yes	—
73	C₂²×C₄×D₅₃	C₂³	1	18	yes	—
74	C₂²×C₄×D₅₃	C₂³	1	9	yes	—
75	C₂²×C₄×D₅₃	C₂³	1	18	yes	complemented
76	C₂²×C₄×D₅₃	C₂³	1	18	yes	complemented
77	Q₈×Dic₅₃	C₂³	1	2	yes	complemented
78	Q₈×Dic₅₃	C₂³	1	6	yes	complemented
79	C₄×Dic₁₀₆	C₂³	1	6	yes	complemented
80	C₄×Dic₁₀₆	C₂³	1	18	yes	complemented
81	#13133182	C₂³	1	6	yes	complemented
82	#13133182	C₂³	1	18	yes	complemented
83	C₄×D₂₁₂	C₂³	1	18	yes	complemented
84	C₄×D₂₁₂	C₂³	1	36	yes	complemented
85	C₄×D₂₁₂	C₂³	1	18	yes	complemented
86	D₄×Dic₅₃	C₂³	1	18	yes	complemented
87	D₄×Dic₅₃	C₂³	1	36	yes	complemented
88	D₄×Dic₅₃	C₂³	1	18	yes	complemented
89	#13133179	C₂³	1	18	yes	complemented
90	#13133179	C₂³	1	36	yes	complemented
91	#13133179	C₂³	1	18	yes	complemented
92	C₄×C₂²⋊D₅₃	C₂³	1	36	yes	complemented
93	C₄×C₂²⋊D₅₃	C₂³	1	72	yes	complemented
94	C₄×C₂²⋊D₅₃	C₂³	1	36	yes	complemented
95	#13133177	C₂³	1	36	yes	complemented
96	#13133177	C₂³	1	72	yes	complemented
97	#13133177	C₂³	1	36	yes	complemented
98	#13133176	C₂³	1	18	yes	complemented
99	#13133175	C₂³	1	18	yes	complemented
100	#13133175	C₂³	1	18	yes	complemented
101	#13133174	C₂³	1	36	yes	complemented
102	#13133173	C₂³	1	36	yes	complemented
103	C₂×C₄⋊Dic₅₃	C₂³	1	9	yes	complemented
104	C₂×#11719919	C₂³	1	18	yes	complemented
105	C₄⋊C₄×D₅₃	C₂³	1	18	yes	—
106	C₄⋊C₄×D₅₃	C₂³	1	6	yes	—
107	C₄⋊C₄×D₅₃	C₂³	1	36	yes	complemented
108	C₂×C₂²⋊Dic₅₃	C₂³	1	18	yes	complemented
109	C₂×#11719921	C₂³	1	18	yes	complemented
110	C₂×#11719921	C₂³	1	18	yes	complemented
111	C₂²⋊C₄×D₅₃	C₂³	1	36	yes	—
112	C₂²⋊C₄×D₅₃	C₂³	1	72	yes	complemented
113	C₂×C₄×Dic₅₃	C₂³	1	9	yes	complemented
114	C₂×C₄×Dic₅₃	C₂³	1	9	yes	complemented
115	C₂×C₄×Dic₅₃	C₂³	1	9	yes	complemented
116	C₄²×D₅₃	C₂³	1	6	yes	—
117	C₄²×D₅₃	C₂³	1	18	yes	—
118	C₄²×D₅₃	C₂³	1	18	yes	complemented
119	C₄²×D₅₃	C₂³	1	18	yes	complemented
120	H(5,𝔽₂)×C₅₃	C₂×C₄	1	16	yes	complemented
121	C₂×#41×C₅₃	C₂³	1	6	yes	—
122	C₂²×D₄×C₅₃	C₂³	1	9	yes	—
123	C₂³×C₄×C₅₃	C₂³	1	6	yes	complemented
124	C₄×Q₈×C₅₃	C₂³	1	2	yes	complemented
125	C₄×Q₈×C₅₃	C₂³	1	6	yes	complemented
126	C₄×D₄×C₅₃	C₂³	1	18	yes	complemented
127	C₄×D₄×C₅₃	C₂³	1	36	yes	complemented
128	C₄×D₄×C₅₃	C₂³	1	18	yes	complemented
129	#117×C₅₃	C₂³	1	18	yes	complemented
130	C₂×C₄⋊C₄×C₅₃	C₂³	1	9	yes	complemented
131	C₂×C₂²⋊C₄×C₅₃	C₂³	1	18	yes	complemented
132	C₂×C₄²×C₅₃	C₂³	1	9	yes	complemented
133	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
134	C₂³×D₅₃	—	4	12	no	—
135	C₂³×D₅₃	—	4	12	no	—
136	C₂³×D₅₃	C₂²×C₄	1	48	yes	complemented
137	C₂³×D₅₃	C₂⁴	1	9	yes	—
138	C₂³×D₅₃	C₂⁴	1	18	yes	—
139	#41×C₅₃	C₂²×C₄	1	48	yes	complemented
140	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
141	C₂×D₄×C₅₃	C₂²×C₄	1	72	yes	complemented
142	C₂×D₄×C₅₃	C₂⁴	1	18	yes	—
143	C₂²×C₄×C₅₃	—	4	12	no	—
144	C₂²×C₄×C₅₃	C₂⁴	1	9	yes	—
145	C₂²×C₄×C₅₃	C₂⁴	1	9	yes	—
146	C₂²×C₄×C₅₃	C₂⁴	1	18	yes	—
147	C₂²×C₄×C₅₃	C₂⁴	1	9	yes	—
148	C₂×C₂²⋊D₅₃	C₂²×C₄	1	144	yes	complemented
149	C₂×C₂²⋊D₅₃	C₂⁴	1	36	yes	—
150	C₂²×Dic₅₃	—	4	12	no	—
151	C₂²×Dic₅₃	C₂⁴	1	9	yes	—
152	C₂²×Dic₅₃	C₂⁴	1	9	yes	—
153	C₂²×Dic₅₃	C₂⁴	1	18	yes	—
154	C₂²×Dic₅₃	C₂⁴	1	9	yes	—
155	#11719949	C₂²×C₄	1	48	yes	complemented
156	Q₈×D₅₃	C₂²×C₄	1	16	yes	complemented
157	#11719947	C₂²×C₄	1	144	yes	complemented
158	D₄×D₅₃	C₂²×C₄	1	144	yes	complemented
159	D₄×D₅₃	C₂²×C₄	1	288	yes	complemented
160	#11719945	C₂²×C₄	1	144	yes	complemented
161	C₂×D₂₁₂	C₂²×C₄	1	72	yes	complemented
162	C₂×D₂₁₂	C₂⁴	1	18	yes	—
163	C₂×C₄×D₅₃	—	2	36	no	—
164	C₂×C₄×D₅₃	—	4	36	no	—
165	C₂×C₄×D₅₃	—	4	36	no	—
166	C₂×C₄×D₅₃	—	4	36	no	—
167	C₂×C₄×D₅₃	—	4	18	no	—
168	C₂×C₄×D₅₃	C₂²×C₄	1	72	yes	complemented
169	C₂×C₄×D₅₃	C₂⁴	1	6	yes	—
170	C₂×C₄×D₅₃	C₂⁴	1	9	yes	—
171	C₂×C₄×D₅₃	C₂⁴	1	9	yes	—
172	C₂×C₄×D₅₃	C₂⁴	1	9	yes	—
173	C₂×C₄×D₅₃	C₂⁴	1	18	yes	—
174	C₂×C₄×D₅₃	C₂⁴	1	36	yes	—
175	C₂×C₄×D₅₃	C₂⁴	1	6	yes	—
176	C₂×C₄×D₅₃	C₂⁴	1	18	yes	—
177	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
178	C₂×Dic₁₀₆	C₂⁴	1	6	yes	—
179	C₄⋊C₄×C₅₃	C₂⁴	1	18	yes	—
180	C₄⋊C₄×C₅₃	C₂⁴	1	6	yes	—
181	C₂²⋊C₄×C₅₃	C₂⁴	1	36	yes	—
182	C₄²×C₅₃	C₂⁴	1	6	yes	—
183	C₄²×C₅₃	C₂⁴	1	18	yes	—
184	C₂²⋊Dic₅₃	C₂⁴	1	36	yes	—
185	#11719921	C₂⁴	1	36	yes	—
186	#11719921	C₂⁴	1	36	yes	—
187	C₄⋊Dic₅₃	C₂⁴	1	18	yes	—
188	C₄⋊Dic₅₃	C₂⁴	1	6	yes	—
189	#11719919	C₂⁴	1	36	yes	—
190	#11719919	C₂⁴	1	12	yes	—
191	C₄×Dic₅₃	C₂⁴	1	6	yes	—
192	C₄×Dic₅₃	C₂⁴	1	6	yes	—
193	C₄×Dic₅₃	C₂⁴	1	6	yes	—
194	C₄×Dic₅₃	C₂⁴	1	18	yes	—
195	C₄×Dic₅₃	C₂⁴	1	18	yes	—
196	C₄×Dic₅₃	C₂⁴	1	18	yes	—
197	C₂³×C₅₃	—	4	6	no	—
198	C₂³×C₅₃	—	4	6	no	—
199	C₂³×C₅₃	C₂³×C₄	1	24	yes	complemented
200	C₂³×C₅₃	C₂⁵	1	9	yes	—
201	C₂²×D₅₃	—	2	18	no	—
202	C₂²×D₅₃	—	2	18	no	—
203	C₂²×D₅₃	—	4	96	no	—
204	C₂²×D₅₃	—	4	18	no	—
205	C₂²×D₅₃	—	4	18	no	—
206	C₂²×D₅₃	C₂³×C₄	1	36	yes	—
207	C₂²×D₅₃	C₂³×C₄	1	72	yes	complemented
208	C₂²×D₅₃	C₂⁵	1	9	yes	—
209	C₂²×D₅₃	C₂⁵	1	1	yes	characteristic
210	Q₈×C₅₃	C₂³×C₄	1	8	yes	complemented
211	D₄×C₅₃	C₂³×C₄	1	72	yes	complemented
212	C₂×C₄×C₅₃	—	2	18	no	—
213	C₂×C₄×C₅₃	—	4	18	no	—
214	C₂×C₄×C₅₃	—	4	9	no	—
215	C₂×C₄×C₅₃	C₂³×C₄	1	36	yes	complemented
216	C₂×C₄×C₅₃	C₂⁵	1	9	yes	—
217	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
218	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
219	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
220	C₂²⋊D₅₃	C₂³×C₄	1	144	yes	complemented
221	C₂×Dic₅₃	—	2	18	no	—
222	C₂×Dic₅₃	—	4	18	no	—
223	C₂×Dic₅₃	—	4	9	no	—
224	C₂×Dic₅₃	C₂³×C₄	1	36	yes	complemented
225	C₂×Dic₅₃	C₂⁵	1	9	yes	—
226	C₂×Dic₅₃	C₂⁵	1	6	yes	—
227	C₂×Dic₅₃	C₂⁵	1	6	yes	—
228	C₂×Dic₅₃	C₂⁵	1	1	yes	characteristic
229	D₂₁₂	C₂³×C₄	1	72	yes	complemented
230	C₄×D₅₃	—	2	18	no	—
231	C₄×D₅₃	—	2	12	no	—
232	C₄×D₅₃	—	2	18	no	—
233	C₄×D₅₃	—	2	18	no	—
234	C₄×D₅₃	—	4	36	no	—
235	C₄×D₅₃	—	4	18	no	—
236	C₄×D₅₃	C₂³×C₄	1	24	yes	—
237	C₄×D₅₃	C₂³×C₄	1	72	yes	complemented
238	C₄×D₅₃	H(5,𝔽₂)	1	4	yes	direct product factor
239	Dic₁₀₆	C₂³×C₄	1	24	yes	complemented
240	C₂×C₄×H(5,𝔽₂)	—	53	1	no	maximal, Sylow
241	C₂²×C₅₃	—	2	9	no	—
242	C₂²×C₅₃	—	2	9	no	—
243	C₂²×C₅₃	—	4	24	no	—
244	C₂²×C₅₃	C₂⁴×C₄	1	18	yes	—
245	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
246	C₂×D₅₃	—	2	9	no	—
247	C₂×D₅₃	—	2	9	no	—
248	C₂×D₅₃	—	2	72	no	—
249	C₂×D₅₃	—	4	72	no	—
250	C₂×D₅₃	C₂⁴×C₄	1	18	yes	—
251	C₂×D₅₃	C₂⁴×C₄	1	2	yes	—
252	C₂×D₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
253	C₂×D₅₃	#585	1	2	yes	—
254	C₄×C₅₃	—	2	9	no	—
255	C₄×C₅₃	—	2	6	no	—
256	C₄×C₅₃	C₂⁴×C₄	1	12	yes	—
257	C₄×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	complemented
258	Dic₅₃	—	2	9	no	—
259	Dic₅₃	—	2	6	no	—
260	Dic₅₃	C₂⁴×C₄	1	12	yes	—
261	Dic₅₃	C₂×H(5,𝔽₂)	1	2	yes	complemented
262	C₂²×H(5,𝔽₂)	—	53	1	no	—
263	C₄×H(5,𝔽₂)	—	53	1	no	—
264	C₄×H(5,𝔽₂)	—	53	6	no	—
265	C₄×H(5,𝔽₂)	—	53	9	no	—
266	C₄×H(5,𝔽₂)	—	53	6	no	—
267	C₄×H(5,𝔽₂)	—	53	9	no	—
268	C₄×H(5,𝔽₂)	D₅₃	1	1	yes	characteristic, direct product factor
269	C₂×#520	—	53	6	no	—
270	C₂×#518	—	53	9	no	—
271	C₂×C₄×#41	—	53	6	no	—
272	C₂²×C₄×D₄	—	53	9	no	—
273	C₂×C₅₃	—	2	18	no	—
274	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
275	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
276	C₂×C₅₃	C₂×#585	1	1	yes	characteristic
277	D₅₃	—	2	18	no	—
278	D₅₃	C₄×H(5,𝔽₂)	1	4	yes	direct product factor
279	C₂×H(5,𝔽₂)	—	53	32	no	—
280	C₂×H(5,𝔽₂)	—	53	6	no	—
281	C₂×H(5,𝔽₂)	—	53	9	no	—
282	C₂×H(5,𝔽₂)	C₂×D₅₃	1	1	yes	characteristic
283	C₂²×#41	—	53	6	no	—
284	C₂³×D₄	—	53	9	no	—
285	C₂⁴×C₄	—	53	6	no	—
286	#520	—	53	6	no	—
287	#520	—	53	12	no	—
288	#520	—	53	36	no	—
289	#520	—	53	18	no	—
290	#520	—	53	18	no	—
291	#520	C₂×D₅₃	1	6	yes	complemented
292	#518	—	53	9	no	—
293	#518	—	53	36	no	—
294	#518	—	53	36	no	—
295	#518	—	53	18	no	—
296	#518	—	53	36	no	—
297	#518	C₂×D₅₃	1	9	yes	complemented
298	C₄×#41	—	53	6	no	—
299	C₄×#41	—	53	6	no	—
300	C₄×#41	—	53	6	no	—
301	C₄×#41	—	53	18	no	—
302	C₄×#41	—	53	18	no	—
303	C₄×#41	—	53	18	no	—
304	C₄×#41	—	53	18	no	—
305	C₄×#41	C₂×D₅₃	1	6	yes	complemented
306	C₂×C₄×Q₈	—	53	2	no	—
307	C₂×C₄×Q₈	—	53	6	no	—
308	C₂×C₄×D₄	—	53	9	no	—
309	C₂×C₄×D₄	—	53	18	no	—
310	C₂×C₄×D₄	—	53	18	no	—
311	C₂×C₄×D₄	—	53	36	no	—
312	C₂×C₄×D₄	—	53	36	no	—
313	C₂×C₄×D₄	—	53	36	no	—
314	C₂×C₄×D₄	—	53	18	no	—
315	C₂×C₄×D₄	—	53	9	no	—
316	C₂×C₄×D₄	—	53	18	no	—
317	C₂×C₄×D₄	—	53	9	no	—
318	C₂×C₄×D₄	C₂×D₅₃	1	9	yes	complemented
319	C₂×#117	—	53	18	no	—
320	C₂²×C₄⋊C₄	—	53	9	no	—
321	C₂²×C₂²⋊C₄	—	53	18	no	—
322	C₂²×C₄²	—	53	9	no	—
323	C₅₃	C₂×C₄×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
324	C₂⁵	—	53	6	no	—
325	H(5,𝔽₂)	—	53	96	no	—
326	H(5,𝔽₂)	—	53	144	no	—
327	H(5,𝔽₂)	C₄×D₅₃	1	16	yes	direct product factor
328	C₂×#41	—	53	96	no	—
329	C₂×#41	—	53	6	no	—
330	C₂×#41	—	53	18	no	—
331	C₂×#41	—	53	18	no	—
332	C₂×#41	C₂²×D₅₃	1	6	yes	—
333	C₂²×Q₈	—	53	2	no	—
334	C₂²×D₄	—	53	18	no	—
335	C₂²×D₄	—	53	144	no	—
336	C₂²×D₄	—	53	18	no	—
337	C₂²×D₄	—	53	36	no	—
338	C₂²×D₄	—	53	9	no	—
339	C₂²×D₄	C₂²×D₅₃	1	9	yes	—
340	C₂³×C₄	—	53	9	no	—
341	C₂³×C₄	—	53	6	no	—
342	C₂³×C₄	—	53	9	no	—
343	C₂³×C₄	—	53	9	no	—
344	C₂³×C₄	—	53	18	no	—
345	C₂³×C₄	—	53	18	no	—
346	C₂³×C₄	—	53	18	no	—
347	C₂³×C₄	—	212	24	no	—
348	C₂³×C₄	C₂²×D₅₃	1	6	yes	complemented
349	C₄×Q₈	—	53	2	no	—
350	C₄×Q₈	—	53	6	no	—
351	C₄×Q₈	—	53	6	no	—
352	C₄×Q₈	—	53	18	no	—
353	C₄×Q₈	—	53	6	no	—
354	C₄×Q₈	—	53	18	no	—
355	C₄×Q₈	C₂²×D₅₃	1	2	yes	complemented
356	C₄×Q₈	C₂²×D₅₃	1	6	yes	complemented
357	C₄×D₄	—	53	18	no	—
358	C₄×D₄	—	53	36	no	—
359	C₄×D₄	—	53	18	no	—
360	C₄×D₄	—	53	36	no	—
361	C₄×D₄	—	53	72	no	—
362	C₄×D₄	—	53	36	no	—
363	C₄×D₄	—	53	18	no	—
364	C₄×D₄	—	53	36	no	—
365	C₄×D₄	—	53	18	no	—
366	C₄×D₄	—	53	36	no	—
367	C₄×D₄	—	53	72	no	—
368	C₄×D₄	—	53	36	no	—
369	C₄×D₄	—	53	18	no	—
370	C₄×D₄	—	53	36	no	—
371	C₄×D₄	—	53	18	no	—
372	C₄×D₄	C₂²×D₅₃	1	18	yes	complemented
373	C₄×D₄	C₂²×D₅₃	1	36	yes	complemented
374	C₄×D₄	C₂²×D₅₃	1	18	yes	complemented
375	#117	—	53	18	no	—
376	#117	—	53	18	no	—
377	#117	—	53	18	no	—
378	#117	—	53	36	no	—
379	#117	—	53	36	no	—
380	#117	C₂²×D₅₃	1	18	yes	complemented
381	C₂×C₄⋊C₄	—	53	18	no	—
382	C₂×C₄⋊C₄	—	53	9	no	—
383	C₂×C₄⋊C₄	—	53	18	no	—
384	C₂×C₄⋊C₄	—	53	36	no	—
385	C₂×C₄⋊C₄	—	53	6	no	—
386	C₂×C₄⋊C₄	C₂²×D₅₃	1	9	yes	complemented
387	C₂×C₂²⋊C₄	—	53	18	no	—
388	C₂×C₂²⋊C₄	—	53	18	no	—
389	C₂×C₂²⋊C₄	—	53	18	no	—
390	C₂×C₂²⋊C₄	—	53	36	no	—
391	C₂×C₂²⋊C₄	—	53	72	no	—
392	C₂×C₂²⋊C₄	C₂²×D₅₃	1	18	yes	complemented
393	C₂×C₄²	—	53	9	no	—
394	C₂×C₄²	—	53	9	no	—
395	C₂×C₄²	—	53	9	no	—
396	C₂×C₄²	—	53	18	no	—
397	C₂×C₄²	—	53	6	no	—
398	C₂×C₄²	—	53	18	no	—
399	C₂×C₄²	—	53	18	no	—
400	C₂×C₄²	C₂²×D₅₃	1	9	yes	complemented
401	C₂⁴	—	53	9	no	—
402	C₂⁴	—	53	48	no	—
403	C₂⁴	—	53	18	no	—
404	C₂⁴	—	212	12	no	—
405	C₂⁴	—	212	12	no	—
406	C₂⁴	C₂³×D₅₃	1	6	yes	—
407	#41	—	53	48	no	—
408	#41	—	53	144	no	—
409	#41	—	53	144	no	—
410	#41	C₂×C₄×D₅₃	1	48	yes	complemented
411	C₂×Q₈	—	53	16	no	—
412	C₂×Q₈	—	53	6	no	—
413	C₂×Q₈	C₂³×D₅₃	1	2	yes	—
414	C₂×D₄	—	53	144	no	—
415	C₂×D₄	—	53	36	no	—
416	C₂×D₄	—	53	144	no	—
417	C₂×D₄	—	53	288	no	—
418	C₂×D₄	—	53	72	no	—
419	C₂×D₄	—	53	18	no	—
420	C₂×D₄	C₂×C₄×D₅₃	1	72	yes	complemented
421	C₂×D₄	C₂³×D₅₃	1	18	yes	—
422	C₂²×C₄	—	4	12	no	—
423	C₂²×C₄	—	53	9	no	—
424	C₂²×C₄	—	53	72	no	—
425	C₂²×C₄	—	53	6	no	—
426	C₂²×C₄	—	53	18	no	—
427	C₂²×C₄	—	53	1	no	—
428	C₂²×C₄	—	53	9	no	—
429	C₂²×C₄	—	53	9	no	—
430	C₂²×C₄	—	53	18	no	—
431	C₂²×C₄	—	53	6	no	—
432	C₂²×C₄	—	53	9	no	—
433	C₂²×C₄	—	53	9	no	—
434	C₂²×C₄	—	53	9	no	—
435	C₂²×C₄	—	53	18	no	—
436	C₂²×C₄	—	53	36	no	—
437	C₂²×C₄	—	106	36	no	—
438	C₂²×C₄	—	212	36	no	—
439	C₂²×C₄	—	212	36	no	—
440	C₂²×C₄	—	212	18	no	—
441	C₂²×C₄	—	212	36	no	—
442	C₂²×C₄	—	212	12	no	—
443	C₂²×C₄	C₂³×D₅₃	1	9	yes	—
444	C₂²×C₄	C₂³×D₅₃	1	9	yes	—
445	C₂²×C₄	C₂³×D₅₃	1	18	yes	—
446	C₂²×C₄	C₂³×D₅₃	1	9	yes	—
447	C₄⋊C₄	—	53	36	no	—
448	C₄⋊C₄	—	53	18	no	—
449	C₄⋊C₄	—	53	12	no	—
450	C₄⋊C₄	—	53	6	no	—
451	C₄⋊C₄	C₂³×D₅₃	1	18	yes	—
452	C₄⋊C₄	C₂³×D₅₃	1	6	yes	—
453	C₂²⋊C₄	—	53	36	no	—
454	C₂²⋊C₄	—	53	36	no	—
455	C₂²⋊C₄	—	53	36	no	—
456	C₂²⋊C₄	C₂³×D₅₃	1	36	yes	—
457	C₄²	—	53	6	no	—
458	C₄²	—	53	6	no	—
459	C₄²	—	53	18	no	—
460	C₄²	—	53	18	no	—
461	C₄²	—	53	18	no	—
462	C₄²	—	53	6	no	—
463	C₄²	C₂³×D₅₃	1	18	yes	—
464	C₄²	C₂³×D₅₃	1	6	yes	—
465	C₂³	—	4	6	no	—
466	C₂³	—	4	6	no	—
467	C₂³	—	53	1	no	—
468	C₂³	—	53	36	no	—
469	C₂³	—	53	9	no	—
470	C₂³	—	53	72	no	—
471	C₂³	—	106	18	no	—
472	C₂³	—	106	18	no	—
473	C₂³	—	212	18	no	—
474	C₂³	—	212	18	no	—
475	C₂³	—	212	96	no	—
476	C₂³	C₂²×C₄×D₅₃	1	24	yes	complemented
477	C₂³	C₂⁴×D₅₃	1	9	yes	—
478	Q₈	—	53	24	no	—
479	Q₈	C₂²×C₄×D₅₃	1	8	yes	complemented
480	D₄	—	53	144	no	—
481	D₄	—	53	72	no	—
482	D₄	C₂²×C₄×D₅₃	1	72	yes	complemented
483	C₂×C₄	—	2	18	no	—
484	C₂×C₄	—	4	18	no	—
485	C₂×C₄	—	4	9	no	—
486	C₂×C₄	—	53	36	no	—
487	C₂×C₄	—	53	1	no	—
488	C₂×C₄	—	53	6	no	—
489	C₂×C₄	—	53	24	no	—
490	C₂×C₄	—	53	72	no	—
491	C₂×C₄	—	53	4	no	—
492	C₂×C₄	—	53	6	no	—
493	C₂×C₄	—	53	9	no	—
494	C₂×C₄	—	106	18	no	—
495	C₂×C₄	—	106	18	no	—
496	C₂×C₄	—	106	12	no	—
497	C₂×C₄	—	106	18	no	—
498	C₂×C₄	—	106	18	no	—
499	C₂×C₄	—	212	18	no	—
500	C₂×C₄	—	212	36	no	—
501	C₂×C₄	—	212	9	no	—
502	C₂×C₄	—	212	18	no	—
503	C₂×C₄	C₂²×C₄×D₅₃	1	36	yes	complemented
504	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
505	C₂×C₄	C₂⁴×D₅₃	1	9	yes	—
506	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
507	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
508	C₂²	—	2	9	no	—
509	C₂²	—	2	9	no	—
510	C₂²	—	4	24	no	—
511	C₂²	—	53	2	no	—
512	C₂²	—	53	2	no	—
513	C₂²	—	53	2	no	—
514	C₂²	—	53	18	no	—
515	C₂²	—	106	72	no	—
516	C₂²	—	106	9	no	—
517	C₂²	—	106	9	no	—
518	C₂²	—	212	72	no	—
519	C₂²	C₂³×C₄×D₅₃	1	18	yes	—
520	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
521	C₄	—	2	9	no	—
522	C₄	—	2	6	no	—
523	C₄	—	53	12	no	—
524	C₄	—	53	2	no	—
525	C₄	—	106	9	no	—
526	C₄	—	106	6	no	—
527	C₄	C₂³×C₄×D₅₃	1	12	yes	—
528	C₄	H(5,𝔽₂)×D₅₃	1	2	yes	direct product factor
529	C₂	—	2	18	no	—
530	C₂	—	53	4	no	—
531	C₂	—	106	18	no	—
532	C₂	C₂⁴×C₄×D₅₃	1	1	yes	characteristic
533	C₂	C₂×H(5,𝔽₂)×D₅₃	1	1	yes	characteristic
534	C₂	#585×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂⁴×D₅₃
commutator	C₂×C₅₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₄×H(5,𝔽₂)×C₅₃	C₂
socle, cosocle	C₂²×C₅₃	C₂⁶
prime radical	C₂×H(5,𝔽₂)×D₅₃	C₂
hypercenter	C₄×H(5,𝔽₂)	D₅₃
hypocenter	C₅₃	C₂×C₄×H(5,𝔽₂)
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
2. H(5,𝔽₂)×Dic₅₃	C₂	semidirect
3. C₄×#13133263	C₂	semidirect
4. #20287921	C₂	semidirect
5. C₄×#13133262	C₂	semidirect
6. #20287919	C₂	semidirect
7. #520×D₅₃	C₂	semidirect
8. #518×D₅₃	C₂	semidirect
9. C₄×#41×D₅₃	C₂	semidirect
10. C₂×C₄×D₄×D₅₃	C₂	semidirect
11. C₄×H(5,𝔽₂)×C₅₃	C₂	semidirect
14. H(5,𝔽₂)×D₅₃	C₄	direct (N × H)
17. C₂³×C₄×D₅₃	C₂²	semidirect
18. #16811106	C₂²	semidirect
19. #16811105	C₂²	semidirect
20. #16811104	C₂²	semidirect
21. #16811103	C₂²	semidirect
22. #16811102	C₂²	semidirect
23. #16811098	C₂²	semidirect
24. #16811097	C₂²	semidirect
25. #16811096	C₂²	semidirect
26. #16811095	C₂²	semidirect
27. #16811095	C₂²	semidirect
28. #41×Dic₅₃	C₂²	semidirect
29. C₄×#11719949	C₂²	semidirect
30. C₄×#11719949	C₂²	semidirect
31. C₄×#11719945	C₂²	semidirect
32. #16811091	C₂²	semidirect
33. C₄×#11719947	C₂²	semidirect
34. C₄×#11719947	C₂²	semidirect
35. C₄×Q₈×D₅₃	C₂²	semidirect
36. C₄×Q₈×D₅₃	C₂²	semidirect
37. C₂×C₄×D₂₁₂	C₂²	semidirect
38. C₂×D₄×Dic₅₃	C₂²	semidirect
39. C₂×#13133179	C₂²	semidirect
40. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
41. C₂×#13133177	C₂²	semidirect
42. C₄×D₄×D₅₃	C₂²	semidirect
43. C₄×D₄×D₅₃	C₂²	semidirect
44. C₄×D₄×D₅₃	C₂²	semidirect
45. C₄×D₄×D₅₃	C₂²	semidirect
46. C₄×D₄×D₅₃	C₂²	semidirect
47. #117×D₅₃	C₂²	semidirect
48. C₂×C₄⋊C₄×D₅₃	C₂²	semidirect
49. C₂×C₂²⋊C₄×D₅₃	C₂²	semidirect
50. C₂×C₄²×D₅₃	C₂²	semidirect
52. C₅₃×#520	C₂²	semidirect
53. C₅₃×#518	C₂²	semidirect
54. C₄×#41×C₅₃	C₂²	semidirect
55. C₂×C₄×D₄×C₅₃	C₂²	semidirect
57. #13133263	C₂×C₄	semidirect
58. #13133262	C₂×C₄	semidirect
62. #41×D₅₃	C₂×C₄	semidirect
66. C₂×D₄×D₅₃	C₂×C₄	semidirect
69. C₂³×Dic₅₃	C₂³	semidirect
75. C₂²×C₄×D₅₃	C₂³	semidirect
76. C₂²×C₄×D₅₃	C₂³	semidirect
77. Q₈×Dic₅₃	C₂³	semidirect
78. Q₈×Dic₅₃	C₂³	semidirect
79. C₄×Dic₁₀₆	C₂³	semidirect
80. C₄×Dic₁₀₆	C₂³	semidirect
81. #13133182	C₂³	semidirect
82. #13133182	C₂³	semidirect
83. C₄×D₂₁₂	C₂³	semidirect
84. C₄×D₂₁₂	C₂³	semidirect
85. C₄×D₂₁₂	C₂³	semidirect
86. D₄×Dic₅₃	C₂³	semidirect
87. D₄×Dic₅₃	C₂³	semidirect
88. D₄×Dic₅₃	C₂³	semidirect
89. #13133179	C₂³	semidirect
90. #13133179	C₂³	semidirect
91. #13133179	C₂³	semidirect
92. C₄×C₂²⋊D₅₃	C₂³	semidirect
93. C₄×C₂²⋊D₅₃	C₂³	semidirect
94. C₄×C₂²⋊D₅₃	C₂³	semidirect
95. #13133177	C₂³	semidirect
96. #13133177	C₂³	semidirect
97. #13133177	C₂³	semidirect
98. #13133176	C₂³	semidirect
99. #13133175	C₂³	semidirect
100. #13133175	C₂³	semidirect
101. #13133174	C₂³	semidirect
102. #13133173	C₂³	semidirect
103. C₂×C₄⋊Dic₅₃	C₂³	semidirect
104. C₂×#11719919	C₂³	semidirect
107. C₄⋊C₄×D₅₃	C₂³	semidirect
108. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
109. C₂×#11719921	C₂³	semidirect
110. C₂×#11719921	C₂³	semidirect
112. C₂²⋊C₄×D₅₃	C₂³	semidirect
113. C₂×C₄×Dic₅₃	C₂³	semidirect
114. C₂×C₄×Dic₅₃	C₂³	semidirect
115. C₂×C₄×Dic₅₃	C₂³	semidirect
118. C₄²×D₅₃	C₂³	semidirect
119. C₄²×D₅₃	C₂³	semidirect
120. H(5,𝔽₂)×C₅₃	C₂×C₄	semidirect
123. C₂³×C₄×C₅₃	C₂³	semidirect
124. C₄×Q₈×C₅₃	C₂³	semidirect
125. C₄×Q₈×C₅₃	C₂³	semidirect
126. C₄×D₄×C₅₃	C₂³	semidirect
127. C₄×D₄×C₅₃	C₂³	semidirect
128. C₄×D₄×C₅₃	C₂³	semidirect
129. #117×C₅₃	C₂³	semidirect
130. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
131. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
132. C₂×C₄²×C₅₃	C₂³	semidirect
136. C₂³×D₅₃	C₂²×C₄	semidirect
139. #41×C₅₃	C₂²×C₄	semidirect
141. C₂×D₄×C₅₃	C₂²×C₄	semidirect
148. C₂×C₂²⋊D₅₃	C₂²×C₄	semidirect
155. #11719949	C₂²×C₄	semidirect
156. Q₈×D₅₃	C₂²×C₄	semidirect
157. #11719947	C₂²×C₄	semidirect
158. D₄×D₅₃	C₂²×C₄	semidirect
159. D₄×D₅₃	C₂²×C₄	semidirect
160. #11719945	C₂²×C₄	semidirect
161. C₂×D₂₁₂	C₂²×C₄	semidirect
168. C₂×C₄×D₅₃	C₂²×C₄	semidirect
199. C₂³×C₅₃	C₂³×C₄	semidirect
207. C₂²×D₅₃	C₂³×C₄	semidirect
210. Q₈×C₅₃	C₂³×C₄	semidirect
211. D₄×C₅₃	C₂³×C₄	semidirect
215. C₂×C₄×C₅₃	C₂³×C₄	semidirect
220. C₂²⋊D₅₃	C₂³×C₄	semidirect
224. C₂×Dic₅₃	C₂³×C₄	semidirect
229. D₂₁₂	C₂³×C₄	semidirect
237. C₄×D₅₃	C₂³×C₄	semidirect
238. C₄×D₅₃	H(5,𝔽₂)	direct (N × H)
239. Dic₁₀₆	C₂³×C₄	semidirect
257. C₄×C₅₃	C₂×H(5,𝔽₂)	semidirect
261. Dic₅₃	C₂×H(5,𝔽₂)	semidirect
268. C₄×H(5,𝔽₂)	D₅₃	direct (N × H)
278. D₅₃	C₄×H(5,𝔽₂)	direct (N × H)
291. #520	C₂×D₅₃	semidirect
297. #518	C₂×D₅₃	semidirect
305. C₄×#41	C₂×D₅₃	semidirect
318. C₂×C₄×D₄	C₂×D₅₃	semidirect
323. C₅₃	C₂×C₄×H(5,𝔽₂)	semidirect
327. H(5,𝔽₂)	C₄×D₅₃	direct (N × H)
348. C₂³×C₄	C₂²×D₅₃	semidirect
355. C₄×Q₈	C₂²×D₅₃	semidirect
356. C₄×Q₈	C₂²×D₅₃	semidirect
372. C₄×D₄	C₂²×D₅₃	semidirect
373. C₄×D₄	C₂²×D₅₃	semidirect
374. C₄×D₄	C₂²×D₅₃	semidirect
380. #117	C₂²×D₅₃	semidirect
386. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
392. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
400. C₂×C₄²	C₂²×D₅₃	semidirect
410. #41	C₂×C₄×D₅₃	semidirect
420. C₂×D₄	C₂×C₄×D₅₃	semidirect
476. C₂³	C₂²×C₄×D₅₃	semidirect
479. Q₈	C₂²×C₄×D₅₃	semidirect
482. D₄	C₂²×C₄×D₅₃	semidirect
503. C₂×C₄	C₂²×C₄×D₅₃	semidirect
528. C₄	H(5,𝔽₂)×D₅₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×H(5,𝔽₂)×D₅₃	2
H(5,𝔽₂)×Dic₅₃	2
C₄×#13133263	2
#20287921	2
C₄×#13133262	2
#20287919	2
#520×D₅₃	2
#518×D₅₃	2
C₄×#41×D₅₃	2
C₂×C₄×D₄×D₅₃	2
C₄×H(5,𝔽₂)×C₅₃	2
C₂×C₄×H(5,𝔽₂)	53

Subgroups — C4×H(5,𝔽2)×D53 #28290691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×H(5,𝔽₂)×D₅₃ #28290691