Subgroups — C₄×D₄²×C₅₃ #27241668

Order 13568.

4804 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄²×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×D₄×C₅₃	C₂	1	4	yes	maximal, complemented
3	C₅₃×#2366	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2361	C₂	1	4	yes	maximal, complemented
5	C₅₃×#2358	C₂	1	4	yes	maximal, complemented
6	D₄×C₄⋊C₄×C₅₃	C₂	1	2	yes	maximal, complemented
7	D₄×C₂²⋊C₄×C₅₃	C₂	1	4	yes	maximal, complemented
8	C₄×#127×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#121×C₅₃	C₂	1	4	yes	maximal, complemented
10	C₄×#120×C₅₃	C₂	1	4	yes	maximal, complemented
11	C₄²×D₄×C₅₃	C₂	1	2	yes	maximal, complemented
12	C₂³×D₄×C₅₃	C₂²	1	4	yes	—
13	C₂⁴×C₄×C₅₃	C₂²	1	4	yes	complemented
14	D₄²×C₅₃	C₄	1	16	yes	direct product factor
15	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#121×C₅₃	C₂²	1	4	yes	—
17	C₂×#120×C₅₃	C₂²	1	4	yes	—
18	C₂×C₄×D₄×C₅₃	—	2	32	no	—
19	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₅₃	C₂²	1	8	yes	complemented
24	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
25	C₂×C₄×D₄×C₅₃	C₂²	1	8	yes	complemented
26	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
27	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
28	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	complemented
29	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	complemented
30	C₂²×C₄²×C₅₃	C₂²	1	4	yes	complemented
31	C₅₃×#390	C₂²	1	4	yes	complemented
32	C₅₃×#390	C₂²	1	4	yes	complemented
33	C₅₃×#390	C₂²	1	2	yes	complemented
34	C₅₃×#389	C₂²	1	4	yes	complemented
35	C₅₃×#388	C₂²	1	8	yes	complemented
36	C₅₃×#386	C₂²	1	8	yes	complemented
37	C₅₃×#386	C₂²	1	4	yes	complemented
38	C₅₃×#386	C₂²	1	8	yes	complemented
39	C₅₃×#385	C₂²	1	8	yes	complemented
40	C₅₃×#385	C₂²	1	4	yes	complemented
41	C₅₃×#384	C₂²	1	1	yes	characteristic, complemented
42	C₅₃×#380	C₂²	1	4	yes	complemented
43	C₅₃×#379	C₂²	1	4	yes	complemented
44	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
45	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
46	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
47	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
48	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
49	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
50	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
51	C₄³×C₅₃	C₂²	1	1	yes	characteristic, complemented
52	C₂⁵×C₅₃	C₂³	1	4	yes	—
53	C₂²×D₄×C₅₃	—	2	16	no	—
54	C₂²×D₄×C₅₃	—	2	16	no	—
55	C₂²×D₄×C₅₃	C₂×C₄	1	32	yes	complemented
56	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
57	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
58	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
59	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
60	C₂³×C₄×C₅₃	—	2	32	no	—
61	C₂³×C₄×C₅₃	—	4	8	no	—
62	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
63	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
64	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
65	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
66	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
67	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
68	#127×C₅₃	C₂×C₄	1	8	yes	complemented
69	#121×C₅₃	C₂×C₄	1	32	yes	complemented
70	#120×C₅₃	C₂×C₄	1	32	yes	complemented
71	C₄×D₄×C₅₃	—	2	16	no	—
72	C₄×D₄×C₅₃	—	2	32	no	—
73	C₄×D₄×C₅₃	—	2	16	no	—
74	C₄×D₄×C₅₃	—	2	16	no	—
75	C₄×D₄×C₅₃	—	2	8	no	—
76	C₄×D₄×C₅₃	—	2	32	no	—
77	C₄×D₄×C₅₃	—	4	4	no	—
78	C₄×D₄×C₅₃	C₂×C₄	1	16	yes	complemented
79	C₄×D₄×C₅₃	D₄	1	16	yes	direct product factor
80	C₂×C₄⋊C₄×C₅₃	—	2	16	no	—
81	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
83	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
85	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
87	C₂×C₂²⋊C₄×C₅₃	—	2	32	no	—
88	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
89	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
90	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
91	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
92	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
93	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
96	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
97	C₂×C₄²×C₅₃	—	2	16	no	—
98	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
99	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
100	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
101	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
104	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
106	#95×C₅₃	C₂³	1	4	yes	—
107	#95×C₅₃	C₂³	1	4	yes	—
108	#95×C₅₃	C₂³	1	4	yes	—
109	#95×C₅₃	C₂³	1	8	yes	—
110	C₂⁴×C₅₃	—	2	16	no	—
111	C₂⁴×C₅₃	—	2	16	no	—
112	C₂⁴×C₅₃	—	4	4	no	—
113	C₂⁴×C₅₃	—	4	4	no	—
114	C₂⁴×C₅₃	C₂²×C₄	1	16	yes	complemented
115	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
116	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
117	C₂×D₄×C₅₃	—	2	16	no	—
118	C₂×D₄×C₅₃	—	2	16	no	—
119	C₂×D₄×C₅₃	—	2	128	no	—
120	C₂×D₄×C₅₃	—	2	8	no	—
121	C₂×D₄×C₅₃	—	2	8	no	—
122	C₂×D₄×C₅₃	—	4	2	no	—
123	C₂×D₄×C₅₃	—	4	2	no	—
124	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
125	C₂×D₄×C₅₃	C₂²×C₄	1	32	yes	complemented
126	C₂×D₄×C₅₃	C₂²×C₄	1	16	yes	complemented
127	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	complemented
128	C₂×D₄×C₅₃	C₂×D₄	1	8	yes	—
129	C₂×D₄×C₅₃	#41	1	8	yes	—
130	C₂²×C₄×C₅₃	—	2	8	no	—
131	C₂²×C₄×C₅₃	—	2	8	no	—
132	C₂²×C₄×C₅₃	—	2	8	no	—
133	C₂²×C₄×C₅₃	—	2	8	no	—
134	C₂²×C₄×C₅₃	—	2	16	no	—
135	C₂²×C₄×C₅₃	—	2	8	no	—
136	C₂²×C₄×C₅₃	—	2	8	no	—
137	C₂²×C₄×C₅₃	—	2	16	no	—
138	C₂²×C₄×C₅₃	—	2	16	no	—
139	C₂²×C₄×C₅₃	—	2	16	no	—
140	C₂²×C₄×C₅₃	—	2	8	no	—
141	C₂²×C₄×C₅₃	—	4	4	no	—
142	C₂²×C₄×C₅₃	—	4	8	no	—
143	C₂²×C₄×C₅₃	—	4	4	no	—
144	C₂²×C₄×C₅₃	—	4	64	no	—
145	C₂²×C₄×C₅₃	C₂²×C₄	1	16	yes	complemented
146	C₂²×C₄×C₅₃	C₂×D₄	1	16	yes	complemented
147	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
148	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
149	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
150	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
151	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
153	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
154	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
156	C₄⋊C₄×C₅₃	—	2	16	no	—
157	C₄⋊C₄×C₅₃	—	2	8	no	—
158	C₄⋊C₄×C₅₃	—	2	4	no	—
159	C₄⋊C₄×C₅₃	—	4	2	no	—
160	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	complemented
161	C₄⋊C₄×C₅₃	C₂×D₄	1	8	yes	complemented
162	C₂²⋊C₄×C₅₃	—	2	16	no	—
163	C₂²⋊C₄×C₅₃	—	2	8	no	—
164	C₂²⋊C₄×C₅₃	—	2	16	no	—
165	C₂²⋊C₄×C₅₃	—	2	16	no	—
166	C₂²⋊C₄×C₅₃	—	4	4	no	—
167	C₂²⋊C₄×C₅₃	—	4	4	no	—
168	C₂²⋊C₄×C₅₃	C₂²×C₄	1	16	yes	complemented
169	C₂²⋊C₄×C₅₃	C₂²×C₄	1	16	yes	complemented
170	C₂²⋊C₄×C₅₃	C₂×D₄	1	16	yes	complemented
171	C₄²×C₅₃	—	2	8	no	—
172	C₄²×C₅₃	—	2	8	no	—
173	C₄²×C₅₃	—	2	8	no	—
174	C₄²×C₅₃	—	2	4	no	—
175	C₄²×C₅₃	—	2	2	no	—
176	C₄²×C₅₃	—	4	1	no	—
177	C₄²×C₅₃	C₂²×C₄	1	4	yes	complemented
178	C₄²×C₅₃	C₂×D₄	1	8	yes	complemented
179	C₂³×C₅₃	—	2	4	no	—
180	C₂³×C₅₃	—	2	4	no	—
181	C₂³×C₅₃	—	2	4	no	—
182	C₂³×C₅₃	—	2	4	no	—
183	C₂³×C₅₃	—	2	64	no	—
184	C₂³×C₅₃	—	2	8	no	—
185	C₂³×C₅₃	—	2	8	no	—
186	C₂³×C₅₃	—	2	4	no	—
187	C₂³×C₅₃	—	2	4	no	—
188	C₂³×C₅₃	—	4	16	no	—
189	C₂³×C₅₃	—	4	16	no	—
190	C₂³×C₅₃	—	4	32	no	—
191	C₂³×C₅₃	—	4	16	no	—
192	C₂³×C₅₃	C₂³×C₄	1	8	yes	—
193	C₂³×C₅₃	C₂³×C₄	1	8	yes	—
194	C₂³×C₅₃	C₂²×D₄	1	8	yes	—
195	C₂³×C₅₃	C₂×#41	1	8	yes	—
196	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
197	D₄×C₅₃	—	2	64	no	—
198	D₄×C₅₃	—	2	32	no	—
199	D₄×C₅₃	—	4	8	no	—
200	D₄×C₅₃	C₄×D₄	1	32	yes	direct product factor
201	C₂×C₄×C₅₃	—	2	4	no	—
202	C₂×C₄×C₅₃	—	2	4	no	—
203	C₂×C₄×C₅₃	—	2	2	no	—
204	C₂×C₄×C₅₃	—	2	4	no	—
205	C₂×C₄×C₅₃	—	2	4	no	—
206	C₂×C₄×C₅₃	—	2	1	no	—
207	C₂×C₄×C₅₃	—	2	2	no	—
208	C₂×C₄×C₅₃	—	2	1	no	—
209	C₂×C₄×C₅₃	—	2	32	no	—
210	C₂×C₄×C₅₃	—	2	32	no	—
211	C₂×C₄×C₅₃	—	2	8	no	—
212	C₂×C₄×C₅₃	—	2	4	no	—
213	C₂×C₄×C₅₃	—	2	4	no	—
214	C₂×C₄×C₅₃	—	2	2	no	—
215	C₂×C₄×C₅₃	—	2	4	no	—
216	C₂×C₄×C₅₃	—	2	1	no	—
217	C₂×C₄×C₅₃	—	2	4	no	—
218	C₂×C₄×C₅₃	—	4	16	no	—
219	C₂×C₄×C₅₃	—	4	16	no	—
220	C₂×C₄×C₅₃	—	4	32	no	—
221	C₂×C₄×C₅₃	—	4	16	no	—
222	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
223	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
224	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
225	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
226	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
227	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
228	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
229	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
230	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
231	C₄×D₄²	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
232	C₂²×C₅₃	—	2	8	no	—
233	C₂²×C₅₃	—	2	8	no	—
234	C₂²×C₅₃	—	2	8	no	—
235	C₂²×C₅₃	—	2	8	no	—
236	C₂²×C₅₃	—	2	8	no	—
237	C₂²×C₅₃	—	2	8	no	—
238	C₂²×C₅₃	—	2	16	no	—
239	C₂²×C₅₃	—	2	8	no	—
240	C₂²×C₅₃	—	4	64	no	—
241	C₂²×C₅₃	—	4	4	no	—
242	C₂²×C₅₃	—	4	8	no	—
243	C₂²×C₅₃	—	4	4	no	—
244	C₂²×C₅₃	C₂×C₄×D₄	1	16	yes	complemented
245	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
246	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
247	C₂²×C₅₃	C₂²×#41	1	2	yes	—
248	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
249	C₂²×C₅₃	#585	1	1	yes	characteristic
250	C₄×C₅₃	—	2	8	no	—
251	C₄×C₅₃	—	2	2	no	—
252	C₄×C₅₃	—	2	8	no	—
253	C₄×C₅₃	—	2	4	no	—
254	C₄×C₅₃	—	4	4	no	—
255	C₄×C₅₃	—	4	4	no	—
256	C₄×C₅₃	—	4	1	no	—
257	C₄×C₅₃	C₂×C₄×D₄	1	8	yes	complemented
258	C₄×C₅₃	D₄²	1	4	yes	direct product factor
259	C₂×D₄²	C₂×C₅₃	1	1	yes	characteristic
260	C₂²×C₄×D₄	C₂×C₅₃	1	4	yes	complemented
261	#2366	C₂×C₅₃	1	1	yes	characteristic, complemented
262	#2361	C₂×C₅₃	1	4	yes	complemented
263	#2358	C₂×C₅₃	1	4	yes	complemented
264	D₄×C₄⋊C₄	C₂×C₅₃	1	2	yes	complemented
265	D₄×C₂²⋊C₄	C₂×C₅₃	1	4	yes	complemented
266	C₄×#127	C₂×C₅₃	1	1	yes	characteristic, complemented
267	C₄×#121	C₂×C₅₃	1	4	yes	complemented
268	C₄×#120	C₂×C₅₃	1	4	yes	complemented
269	C₄²×D₄	C₂×C₅₃	1	2	yes	complemented
270	C₂×C₅₃	—	2	16	no	—
271	C₂×C₅₃	—	4	8	no	—
272	C₂×C₅₃	C₂²×C₄×D₄	1	2	yes	—
273	C₂×C₅₃	C₄×H(5,𝔽₂)	1	1	yes	characteristic
274	C₂×C₅₃	C₂×D₄²	1	1	yes	characteristic
275	C₂×C₅₃	D₄×#41	1	2	yes	—
276	C₂×C₅₃	#3481	1	1	yes	characteristic
277	C₂³×D₄	C₂²×C₅₃	1	4	yes	—
278	C₂⁴×C₄	C₂²×C₅₃	1	4	yes	complemented
279	D₄²	C₄×C₅₃	1	16	yes	direct product factor
280	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
281	C₂×#121	C₂²×C₅₃	1	4	yes	—
282	C₂×#120	C₂²×C₅₃	1	4	yes	—
283	C₂×C₄×D₄	—	2	32	no	—
284	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
285	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
286	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
287	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
288	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	complemented
289	C₂×C₄×D₄	C₂²×C₅₃	1	8	yes	complemented
290	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	complemented
291	C₂×C₄×D₄	C₂²×C₅₃	1	8	yes	complemented
292	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	complemented
293	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	complemented
294	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	complemented
295	C₂²×C₄²	C₂²×C₅₃	1	4	yes	complemented
296	#390	C₂²×C₅₃	1	4	yes	complemented
297	#390	C₂²×C₅₃	1	2	yes	complemented
298	#390	C₂²×C₅₃	1	4	yes	complemented
299	#389	C₂²×C₅₃	1	4	yes	complemented
300	#388	C₂²×C₅₃	1	8	yes	complemented
301	#386	C₂²×C₅₃	1	8	yes	complemented
302	#386	C₂²×C₅₃	1	8	yes	complemented
303	#386	C₂²×C₅₃	1	4	yes	complemented
304	#385	C₂²×C₅₃	1	8	yes	complemented
305	#385	C₂²×C₅₃	1	4	yes	complemented
306	#384	C₂²×C₅₃	1	1	yes	characteristic, complemented
307	#380	C₂²×C₅₃	1	4	yes	complemented
308	#379	C₂²×C₅₃	1	4	yes	complemented
309	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
310	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
311	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
312	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
313	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
314	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
315	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
316	C₄³	C₂²×C₅₃	1	1	yes	characteristic, complemented
317	C₅₃	C₄×D₄²	1	1	yes	characteristic, direct product factor, Sylow
318	C₂⁵	C₂³×C₅₃	1	4	yes	—
319	C₂²×D₄	—	2	16	no	—
320	C₂²×D₄	—	2	16	no	—
321	C₂²×D₄	C₂×C₄×C₅₃	1	32	yes	complemented
322	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
323	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
324	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
325	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
326	C₂³×C₄	—	2	32	no	—
327	C₂³×C₄	—	4	8	no	—
328	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
329	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
330	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
331	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
332	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
333	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
334	#127	C₂×C₄×C₅₃	1	8	yes	complemented
335	#121	C₂×C₄×C₅₃	1	32	yes	complemented
336	#120	C₂×C₄×C₅₃	1	32	yes	complemented
337	C₄×D₄	—	2	16	no	—
338	C₄×D₄	—	2	8	no	—
339	C₄×D₄	—	2	32	no	—
340	C₄×D₄	—	2	16	no	—
341	C₄×D₄	—	2	32	no	—
342	C₄×D₄	—	2	16	no	—
343	C₄×D₄	—	4	4	no	—
344	C₄×D₄	C₂×C₄×C₅₃	1	16	yes	complemented
345	C₄×D₄	D₄×C₅₃	1	16	yes	direct product factor
346	C₂×C₄⋊C₄	—	2	16	no	—
347	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
348	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
349	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
350	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
351	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
352	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
353	C₂×C₂²⋊C₄	—	2	32	no	—
354	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
355	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
356	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
357	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
358	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
359	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
360	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
361	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
362	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
363	C₂×C₄²	—	2	16	no	—
364	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
365	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
366	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
367	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
368	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
369	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
370	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
371	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
372	#95	C₂³×C₅₃	1	4	yes	—
373	#95	C₂³×C₅₃	1	4	yes	—
374	#95	C₂³×C₅₃	1	4	yes	—
375	#95	C₂³×C₅₃	1	8	yes	—
376	C₂⁴	—	2	16	no	—
377	C₂⁴	—	2	16	no	—
378	C₂⁴	—	4	4	no	—
379	C₂⁴	—	4	4	no	—
380	C₂⁴	C₂²×C₄×C₅₃	1	16	yes	complemented
381	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
382	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
383	C₂×D₄	—	2	16	no	—
384	C₂×D₄	—	2	16	no	—
385	C₂×D₄	—	2	128	no	—
386	C₂×D₄	—	2	8	no	—
387	C₂×D₄	—	2	8	no	—
388	C₂×D₄	—	4	2	no	—
389	C₂×D₄	—	4	2	no	—
390	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
391	C₂×D₄	C₂²×C₄×C₅₃	1	32	yes	complemented
392	C₂×D₄	C₂²×C₄×C₅₃	1	16	yes	complemented
393	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	complemented
394	C₂×D₄	C₂×D₄×C₅₃	1	8	yes	—
395	C₂×D₄	#41×C₅₃	1	8	yes	—
396	C₂²×C₄	—	2	8	no	—
397	C₂²×C₄	—	2	8	no	—
398	C₂²×C₄	—	2	8	no	—
399	C₂²×C₄	—	2	8	no	—
400	C₂²×C₄	—	2	16	no	—
401	C₂²×C₄	—	2	8	no	—
402	C₂²×C₄	—	2	8	no	—
403	C₂²×C₄	—	2	16	no	—
404	C₂²×C₄	—	2	16	no	—
405	C₂²×C₄	—	2	16	no	—
406	C₂²×C₄	—	2	8	no	—
407	C₂²×C₄	—	4	4	no	—
408	C₂²×C₄	—	4	8	no	—
409	C₂²×C₄	—	4	4	no	—
410	C₂²×C₄	—	4	64	no	—
411	C₂²×C₄	C₂²×C₄×C₅₃	1	16	yes	complemented
412	C₂²×C₄	C₂×D₄×C₅₃	1	16	yes	complemented
413	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
414	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
415	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
416	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
417	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
418	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
419	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
420	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
421	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
422	C₄⋊C₄	—	2	16	no	—
423	C₄⋊C₄	—	2	8	no	—
424	C₄⋊C₄	—	2	4	no	—
425	C₄⋊C₄	—	4	2	no	—
426	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	complemented
427	C₄⋊C₄	C₂×D₄×C₅₃	1	8	yes	complemented
428	C₂²⋊C₄	—	2	16	no	—
429	C₂²⋊C₄	—	2	8	no	—
430	C₂²⋊C₄	—	2	16	no	—
431	C₂²⋊C₄	—	2	16	no	—
432	C₂²⋊C₄	—	4	4	no	—
433	C₂²⋊C₄	—	4	4	no	—
434	C₂²⋊C₄	C₂²×C₄×C₅₃	1	16	yes	complemented
435	C₂²⋊C₄	C₂²×C₄×C₅₃	1	16	yes	complemented
436	C₂²⋊C₄	C₂×D₄×C₅₃	1	16	yes	complemented
437	C₄²	—	2	8	no	—
438	C₄²	—	2	8	no	—
439	C₄²	—	2	8	no	—
440	C₄²	—	2	4	no	—
441	C₄²	—	2	2	no	—
442	C₄²	—	4	1	no	—
443	C₄²	C₂²×C₄×C₅₃	1	4	yes	complemented
444	C₄²	C₂×D₄×C₅₃	1	8	yes	complemented
445	C₂³	—	2	4	no	—
446	C₂³	—	2	4	no	—
447	C₂³	—	2	4	no	—
448	C₂³	—	2	4	no	—
449	C₂³	—	2	64	no	—
450	C₂³	—	2	8	no	—
451	C₂³	—	2	8	no	—
452	C₂³	—	2	4	no	—
453	C₂³	—	2	4	no	—
454	C₂³	—	4	16	no	—
455	C₂³	—	4	16	no	—
456	C₂³	—	4	32	no	—
457	C₂³	—	4	16	no	—
458	C₂³	C₂³×C₄×C₅₃	1	8	yes	—
459	C₂³	C₂³×C₄×C₅₃	1	8	yes	—
460	C₂³	C₂²×D₄×C₅₃	1	8	yes	—
461	C₂³	C₂×#41×C₅₃	1	8	yes	—
462	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
463	D₄	—	2	64	no	—
464	D₄	—	2	32	no	—
465	D₄	—	4	8	no	—
466	D₄	C₄×D₄×C₅₃	1	32	yes	direct product factor
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	32	no	—
476	C₂×C₄	—	2	32	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	8	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	1	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	4	16	no	—
485	C₂×C₄	—	4	16	no	—
486	C₂×C₄	—	4	32	no	—
487	C₂×C₄	—	4	16	no	—
488	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
489	C₂×C₄	C₂³×C₄×C₅₃	1	8	yes	—
490	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
491	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
492	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
493	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
494	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
495	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
496	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
497	C₂²	—	2	8	no	—
498	C₂²	—	2	8	no	—
499	C₂²	—	2	8	no	—
500	C₂²	—	2	8	no	—
501	C₂²	—	2	8	no	—
502	C₂²	—	2	8	no	—
503	C₂²	—	2	16	no	—
504	C₂²	—	2	8	no	—
505	C₂²	—	4	64	no	—
506	C₂²	—	4	4	no	—
507	C₂²	—	4	8	no	—
508	C₂²	—	4	4	no	—
509	C₂²	C₂×C₄×D₄×C₅₃	1	16	yes	complemented
510	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
511	C₂²	C₂³×D₄×C₅₃	1	2	yes	—
512	C₂²	C₂²×#41×C₅₃	1	2	yes	—
513	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
514	C₂²	C₅₃×#585	1	1	yes	characteristic
515	C₄	—	2	8	no	—
516	C₄	—	2	4	no	—
517	C₄	—	2	8	no	—
518	C₄	—	2	2	no	—
519	C₄	—	4	4	no	—
520	C₄	—	4	4	no	—
521	C₄	—	4	1	no	—
522	C₄	C₂×C₄×D₄×C₅₃	1	8	yes	complemented
523	C₄	D₄²×C₅₃	1	4	yes	direct product factor
524	C₂	—	2	16	no	—
525	C₂	—	4	8	no	—
526	C₂	C₂²×C₄×D₄×C₅₃	1	2	yes	—
527	C₂	C₄×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
528	C₂	C₂×D₄²×C₅₃	1	1	yes	characteristic
529	C₂	D₄×#41×C₅₃	1	2	yes	—
530	C₂	C₅₃×#3481	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁵×C₅₃
socle	C₂³×C₅₃	C₂⁵
prime radical	C₂×D₄²×C₅₃	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×D₄×C₅₃	C₂	semidirect
3. C₅₃×#2366	C₂	semidirect
4. C₅₃×#2361	C₂	semidirect
5. C₅₃×#2358	C₂	semidirect
6. D₄×C₄⋊C₄×C₅₃	C₂	semidirect
7. D₄×C₂²⋊C₄×C₅₃	C₂	semidirect
8. C₄×#127×C₅₃	C₂	semidirect
9. C₄×#121×C₅₃	C₂	semidirect
10. C₄×#120×C₅₃	C₂	semidirect
11. C₄²×D₄×C₅₃	C₂	semidirect
13. C₂⁴×C₄×C₅₃	C₂²	semidirect
14. D₄²×C₅₃	C₄	direct (N × H)
23. C₂×C₄×D₄×C₅₃	C₂²	semidirect
24. C₂×C₄×D₄×C₅₃	C₂²	semidirect
25. C₂×C₄×D₄×C₅₃	C₂²	semidirect
26. C₂×C₄×D₄×C₅₃	C₂²	semidirect
27. C₂×C₄×D₄×C₅₃	C₂²	semidirect
28. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
29. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
30. C₂²×C₄²×C₅₃	C₂²	semidirect
31. C₅₃×#390	C₂²	semidirect
32. C₅₃×#390	C₂²	semidirect
33. C₅₃×#390	C₂²	semidirect
34. C₅₃×#389	C₂²	semidirect
35. C₅₃×#388	C₂²	semidirect
36. C₅₃×#386	C₂²	semidirect
37. C₅₃×#386	C₂²	semidirect
38. C₅₃×#386	C₂²	semidirect
39. C₅₃×#385	C₂²	semidirect
40. C₅₃×#385	C₂²	semidirect
41. C₅₃×#384	C₂²	semidirect
42. C₅₃×#380	C₂²	semidirect
43. C₅₃×#379	C₂²	semidirect
44. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
45. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
46. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
47. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
48. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
49. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
50. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
51. C₄³×C₅₃	C₂²	semidirect
55. C₂²×D₄×C₅₃	C₂×C₄	semidirect
68. #127×C₅₃	C₂×C₄	semidirect
69. #121×C₅₃	C₂×C₄	semidirect
70. #120×C₅₃	C₂×C₄	semidirect
78. C₄×D₄×C₅₃	C₂×C₄	semidirect
79. C₄×D₄×C₅₃	D₄	direct (N × H)
114. C₂⁴×C₅₃	C₂²×C₄	semidirect
125. C₂×D₄×C₅₃	C₂²×C₄	semidirect
126. C₂×D₄×C₅₃	C₂²×C₄	semidirect
127. C₂×D₄×C₅₃	C₂²×C₄	semidirect
145. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
146. C₂²×C₄×C₅₃	C₂×D₄	semidirect
160. C₄⋊C₄×C₅₃	C₂²×C₄	semidirect
161. C₄⋊C₄×C₅₃	C₂×D₄	semidirect
168. C₂²⋊C₄×C₅₃	C₂²×C₄	semidirect
169. C₂²⋊C₄×C₅₃	C₂²×C₄	semidirect
170. C₂²⋊C₄×C₅₃	C₂×D₄	semidirect
177. C₄²×C₅₃	C₂²×C₄	semidirect
178. C₄²×C₅₃	C₂×D₄	semidirect
200. D₄×C₅₃	C₄×D₄	direct (N × H)
231. C₄×D₄²	C₅₃	direct (N × H)
244. C₂²×C₅₃	C₂×C₄×D₄	semidirect
257. C₄×C₅₃	C₂×C₄×D₄	semidirect
258. C₄×C₅₃	D₄²	direct (N × H)
260. C₂²×C₄×D₄	C₂×C₅₃	semidirect
261. #2366	C₂×C₅₃	semidirect
262. #2361	C₂×C₅₃	semidirect
263. #2358	C₂×C₅₃	semidirect
264. D₄×C₄⋊C₄	C₂×C₅₃	semidirect
265. D₄×C₂²⋊C₄	C₂×C₅₃	semidirect
266. C₄×#127	C₂×C₅₃	semidirect
267. C₄×#121	C₂×C₅₃	semidirect
268. C₄×#120	C₂×C₅₃	semidirect
269. C₄²×D₄	C₂×C₅₃	semidirect
278. C₂⁴×C₄	C₂²×C₅₃	semidirect
279. D₄²	C₄×C₅₃	direct (N × H)
288. C₂×C₄×D₄	C₂²×C₅₃	semidirect
289. C₂×C₄×D₄	C₂²×C₅₃	semidirect
290. C₂×C₄×D₄	C₂²×C₅₃	semidirect
291. C₂×C₄×D₄	C₂²×C₅₃	semidirect
292. C₂×C₄×D₄	C₂²×C₅₃	semidirect
293. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
294. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
295. C₂²×C₄²	C₂²×C₅₃	semidirect
296. #390	C₂²×C₅₃	semidirect
297. #390	C₂²×C₅₃	semidirect
298. #390	C₂²×C₅₃	semidirect
299. #389	C₂²×C₅₃	semidirect
300. #388	C₂²×C₅₃	semidirect
301. #386	C₂²×C₅₃	semidirect
302. #386	C₂²×C₅₃	semidirect
303. #386	C₂²×C₅₃	semidirect
304. #385	C₂²×C₅₃	semidirect
305. #385	C₂²×C₅₃	semidirect
306. #384	C₂²×C₅₃	semidirect
307. #380	C₂²×C₅₃	semidirect
308. #379	C₂²×C₅₃	semidirect
309. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
310. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
311. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
312. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
313. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
314. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
315. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
316. C₄³	C₂²×C₅₃	semidirect
317. C₅₃	C₄×D₄²	direct (N × H)
321. C₂²×D₄	C₂×C₄×C₅₃	semidirect
334. #127	C₂×C₄×C₅₃	semidirect
335. #121	C₂×C₄×C₅₃	semidirect
336. #120	C₂×C₄×C₅₃	semidirect
344. C₄×D₄	C₂×C₄×C₅₃	semidirect
345. C₄×D₄	D₄×C₅₃	direct (N × H)
380. C₂⁴	C₂²×C₄×C₅₃	semidirect
391. C₂×D₄	C₂²×C₄×C₅₃	semidirect
392. C₂×D₄	C₂²×C₄×C₅₃	semidirect
393. C₂×D₄	C₂²×C₄×C₅₃	semidirect
411. C₂²×C₄	C₂²×C₄×C₅₃	semidirect
412. C₂²×C₄	C₂×D₄×C₅₃	semidirect
426. C₄⋊C₄	C₂²×C₄×C₅₃	semidirect
427. C₄⋊C₄	C₂×D₄×C₅₃	semidirect
434. C₂²⋊C₄	C₂²×C₄×C₅₃	semidirect
435. C₂²⋊C₄	C₂²×C₄×C₅₃	semidirect
436. C₂²⋊C₄	C₂×D₄×C₅₃	semidirect
443. C₄²	C₂²×C₄×C₅₃	semidirect
444. C₄²	C₂×D₄×C₅₃	semidirect
466. D₄	C₄×D₄×C₅₃	direct (N × H)
509. C₂²	C₂×C₄×D₄×C₅₃	semidirect
522. C₄	C₂×C₄×D₄×C₅₃	semidirect
523. C₄	D₄²×C₅₃	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×D₄²×C₅₃	2
C₂²×C₄×D₄×C₅₃	2
C₅₃×#2366	2
C₅₃×#2361	2
C₅₃×#2358	2
D₄×C₄⋊C₄×C₅₃	2
D₄×C₂²⋊C₄×C₅₃	2
C₄×#127×C₅₃	2
C₄×#121×C₅₃	2
C₄×#120×C₅₃	2
C₄²×D₄×C₅₃	2
C₄×D₄²	53

Subgroups — C4×D42×C53 #27241668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₄²×C₅₃ #27241668