Subgroups — C₂²×#16811255 #28290999

Order 13568.

20038 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₂²⋊D₅₃	C₂	1	2	yes	maximal, complemented
2	C₂³×D₄×D₅₃	C₂	1	2	yes	maximal, complemented
3	C₂×#16811255	C₂	1	48	yes	maximal, direct product factor
4	C₂²×#13133213	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#13133190	C₂	1	2	yes	maximal, complemented
6	C₂²×#13133189	C₂	1	2	yes	maximal, complemented
7	C₂²×#13133187	C₂	1	2	yes	maximal, complemented
8	C₂²×#13133185	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#13133177	C₂	1	2	yes	maximal, complemented
10	C₂²×#120×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic, complemented
12	C₂⁵×D₅₃	C₂²	1	2	yes	complemented
13	C₂³×D₂₁₂	C₂²	1	2	yes	complemented
14	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
15	C₂³×C₂²⋊D₅₃	—	2	32	no	—
16	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	—
17	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
18	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
19	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
20	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
21	C₂³×C₂²⋊D₅₃	C₂²	1	48	yes	complemented
22	C₂²×D₄×D₅₃	—	2	32	no	—
23	C₂²×D₄×D₅₃	C₂²	1	48	yes	complemented
24	C₂⁴×Dic₅₃	C₂²	1	2	yes	complemented
25	C₂³×C₄×D₅₃	C₂²	1	2	yes	complemented
26	#16811255	C₂²	1	256	yes	direct product factor
27	C₂×#13133213	C₂²	1	24	yes	complemented
28	C₂×#13133190	C₂²	1	48	yes	complemented
29	C₂×#13133189	C₂²	1	48	yes	complemented
30	C₂×#13133187	C₂²	1	48	yes	complemented
31	C₂×#13133185	C₂²	1	24	yes	complemented
32	C₂×#13133177	C₂²	1	48	yes	complemented
33	C₂²×#11719919	C₂²	1	2	yes	complemented
34	C₂²×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
35	C₂²×#11719921	C₂²	1	2	yes	complemented
36	C₂²×#11719921	C₂²	1	2	yes	complemented
37	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
38	C₂⁶×C₅₃	C₂²	1	1	yes	characteristic, complemented
39	C₂³×D₄×C₅₃	C₂²	1	2	yes	complemented
40	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
41	C₂×#120×C₅₃	C₂²	1	24	yes	complemented
42	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
43	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
44	C₂⁴×D₅₃	—	2	16	no	—
45	C₂⁴×D₅₃	—	2	16	no	—
46	C₂⁴×D₅₃	—	2	16	no	—
47	C₂⁴×D₅₃	—	4	4	no	—
48	C₂⁴×D₅₃	—	4	8	no	—
49	C₂⁴×D₅₃	C₂³	1	2	yes	—
50	C₂⁴×D₅₃	C₂³	1	2	yes	—
51	C₂⁴×D₅₃	C₂³	1	12	yes	complemented
52	C₂⁴×D₅₃	C₂³	1	24	yes	complemented
53	C₂²×D₂₁₂	—	2	16	no	—
54	C₂²×D₂₁₂	—	4	4	no	—
55	C₂²×D₂₁₂	C₂³	1	24	yes	complemented
56	C₂²×D₂₁₂	C₂³	1	12	yes	complemented
57	C₂²×C₂²⋊D₅₃	—	2	16	no	—
58	C₂²×C₂²⋊D₅₃	—	2	16	no	—
59	C₂²×C₂²⋊D₅₃	—	2	384	no	—
60	C₂²×C₂²⋊D₅₃	—	2	16	no	—
61	C₂²×C₂²⋊D₅₃	—	2	16	no	—
62	C₂²×C₂²⋊D₅₃	D₄	1	32	yes	complemented
63	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	—
64	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	complemented
65	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	complemented
66	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	complemented
67	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	complemented
68	C₂²×C₂²⋊D₅₃	C₂³	1	128	yes	complemented
69	C₂×D₄×D₅₃	—	2	384	no	—
70	C₂×D₄×D₅₃	C₂³	1	128	yes	complemented
71	C₂³×Dic₅₃	—	2	16	no	—
72	C₂³×Dic₅₃	C₂³	1	2	yes	—
73	C₂³×Dic₅₃	C₂³	1	2	yes	—
74	C₂³×Dic₅₃	C₂³	1	24	yes	complemented
75	C₂²×C₄×D₅₃	—	2	16	no	—
76	C₂²×C₄×D₅₃	C₂³	1	24	yes	complemented
77	#13133213	C₂³	1	64	yes	complemented
78	#13133190	C₂³	1	128	yes	complemented
79	#13133189	C₂³	1	128	yes	complemented
80	#13133187	C₂³	1	128	yes	complemented
81	#13133185	C₂³	1	64	yes	complemented
82	#13133177	C₂³	1	128	yes	complemented
83	C₂×#11719919	C₂³	1	24	yes	complemented
84	C₂×C₂²⋊Dic₅₃	C₂³	1	12	yes	complemented
85	C₂×#11719921	C₂³	1	24	yes	complemented
86	C₂×#11719921	C₂³	1	24	yes	complemented
87	C₂×C₄×Dic₅₃	C₂³	1	12	yes	complemented
88	C₂⁵×C₅₃	—	2	16	no	—
89	C₂⁵×C₅₃	—	4	4	no	—
90	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
91	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
92	C₂⁵×C₅₃	C₂³	1	2	yes	—
93	C₂⁵×C₅₃	C₂³	1	12	yes	complemented
94	C₂²×D₄×C₅₃	—	2	16	no	—
95	C₂²×D₄×C₅₃	—	4	4	no	—
96	C₂²×D₄×C₅₃	C₂³	1	24	yes	complemented
97	C₂²×D₄×C₅₃	C₂³	1	12	yes	complemented
98	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
99	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
100	#120×C₅₃	C₂³	1	64	yes	complemented
101	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	complemented
102	C₂×C₂²⋊C₄×C₅₃	C₂³	1	12	yes	complemented
103	C₂⁴×C₅₃	—	2	8	no	—
104	C₂⁴×C₅₃	—	2	8	no	—
105	C₂⁴×C₅₃	—	2	8	no	—
106	C₂⁴×C₅₃	—	2	8	no	—
107	C₂⁴×C₅₃	—	2	96	no	—
108	C₂⁴×C₅₃	—	2	4	no	—
109	C₂⁴×C₅₃	—	2	4	no	—
110	C₂⁴×C₅₃	—	4	24	no	—
111	C₂⁴×C₅₃	—	4	64	no	—
112	C₂⁴×C₅₃	C₂×D₄	1	16	yes	complemented
113	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
114	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
115	C₂⁴×C₅₃	C₂⁴	1	12	yes	—
116	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
117	C₂⁴×C₅₃	C₂⁴	1	16	yes	complemented
118	C₂³×D₅₃	—	2	8	no	—
119	C₂³×D₅₃	—	2	8	no	—
120	C₂³×D₅₃	—	2	8	no	—
121	C₂³×D₅₃	—	2	8	no	—
122	C₂³×D₅₃	—	2	96	no	—
123	C₂³×D₅₃	—	2	8	no	—
124	C₂³×D₅₃	—	2	96	no	—
125	C₂³×D₅₃	—	2	96	no	—
126	C₂³×D₅₃	—	4	24	no	—
127	C₂³×D₅₃	—	4	48	no	—
128	C₂³×D₅₃	—	4	64	no	—
129	C₂³×D₅₃	—	4	128	no	—
130	C₂³×D₅₃	C₂×D₄	1	16	yes	complemented
131	C₂³×D₅₃	C₂⁴	1	12	yes	—
132	C₂³×D₅₃	C₂⁴	1	12	yes	—
133	C₂³×D₅₃	C₂⁴	1	16	yes	complemented
134	C₂³×D₅₃	C₂⁴	1	32	yes	complemented
135	C₂×D₄×C₅₃	—	2	96	no	—
136	C₂×D₄×C₅₃	—	4	24	no	—
137	C₂×D₄×C₅₃	C₂⁴	1	32	yes	complemented
138	C₂×D₄×C₅₃	C₂⁴	1	16	yes	complemented
139	C₂²×C₄×C₅₃	—	2	8	no	—
140	C₂²×C₄×C₅₃	—	2	4	no	—
141	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
142	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
143	C₂×C₂²⋊D₅₃	—	2	96	no	—
144	C₂×C₂²⋊D₅₃	—	2	96	no	—
145	C₂×C₂²⋊D₅₃	—	2	512	no	—
146	C₂×C₂²⋊D₅₃	—	2	96	no	—
147	C₂×C₂²⋊D₅₃	—	2	96	no	—
148	C₂×C₂²⋊D₅₃	C₂×D₄	1	192	yes	complemented
149	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
150	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	complemented
151	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	complemented
152	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	complemented
153	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	complemented
154	C₂²×Dic₅₃	—	2	8	no	—
155	C₂²×Dic₅₃	—	2	96	no	—
156	C₂²×Dic₅₃	C₂×D₄	1	16	yes	complemented
157	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
158	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
159	C₂²×Dic₅₃	C₂⁴	1	32	yes	complemented
160	D₄×D₅₃	—	2	512	no	—
161	C₂×D₂₁₂	—	2	96	no	—
162	C₂×D₂₁₂	—	4	24	no	—
163	C₂×D₂₁₂	C₂⁴	1	32	yes	complemented
164	C₂×D₂₁₂	C₂⁴	1	16	yes	complemented
165	C₂×C₄×D₅₃	—	2	96	no	—
166	C₂×C₄×D₅₃	C₂⁴	1	32	yes	complemented
167	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
168	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	complemented
169	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	complemented
170	#11719921	C₂⁴	1	32	yes	complemented
171	#11719921	C₂⁴	1	32	yes	complemented
172	#11719919	C₂⁴	1	32	yes	complemented
173	C₄×Dic₅₃	C₂⁴	1	16	yes	complemented
174	C₂³×C₅₃	—	2	64	no	—
175	C₂³×C₅₃	—	2	24	no	—
176	C₂³×C₅₃	—	2	24	no	—
177	C₂³×C₅₃	—	2	24	no	—
178	C₂³×C₅₃	—	2	24	no	—
179	C₂³×C₅₃	—	2	64	no	—
180	C₂³×C₅₃	—	2	12	no	—
181	C₂³×C₅₃	—	2	12	no	—
182	C₂³×C₅₃	—	4	16	no	—
183	C₂³×C₅₃	—	4	192	no	—
184	C₂³×C₅₃	—	4	16	no	—
185	C₂³×C₅₃	—	4	16	no	—
186	C₂³×C₅₃	C₂²×D₄	1	8	yes	—
187	C₂³×C₅₃	C₂²×D₄	1	48	yes	complemented
188	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
189	C₂³×C₅₃	C₂⁵	1	4	yes	—
190	C₂³×C₅₃	C₂⁵	1	4	yes	—
191	C₂³×C₅₃	C₂⁵	1	8	yes	—
192	C₂³×C₅₃	C₂⁵	1	3	yes	—
193	C₂²×D₅₃	—	2	64	no	—
194	C₂²×D₅₃	—	2	24	no	—
195	C₂²×D₅₃	—	2	24	no	—
196	C₂²×D₅₃	—	2	24	no	—
197	C₂²×D₅₃	—	2	24	no	—
198	C₂²×D₅₃	—	2	64	no	—
199	C₂²×D₅₃	—	2	24	no	—
200	C₂²×D₅₃	—	2	64	no	—
201	C₂²×D₅₃	—	2	64	no	—
202	C₂²×D₅₃	—	4	16	no	—
203	C₂²×D₅₃	—	4	32	no	—
204	C₂²×D₅₃	—	4	192	no	—
205	C₂²×D₅₃	—	4	384	no	—
206	C₂²×D₅₃	—	4	32	no	—
207	C₂²×D₅₃	C₂²×D₄	1	48	yes	complemented
208	C₂²×D₅₃	C₂⁵	1	8	yes	—
209	C₂²×D₅₃	C₂⁵	1	8	yes	—
210	D₄×C₅₃	—	2	64	no	—
211	D₄×C₅₃	—	4	16	no	—
212	C₂×C₄×C₅₃	—	2	24	no	—
213	C₂×C₄×C₅₃	—	2	12	no	—
214	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
215	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
216	C₂²⋊D₅₃	—	2	64	no	—
217	C₂²⋊D₅₃	—	2	64	no	—
218	C₂²⋊D₅₃	—	2	64	no	—
219	C₂²⋊D₅₃	—	2	64	no	—
220	C₂²⋊D₅₃	C₂²×D₄	1	128	yes	complemented
221	C₂×Dic₅₃	—	2	24	no	—
222	C₂×Dic₅₃	—	2	64	no	—
223	C₂×Dic₅₃	C₂²×D₄	1	48	yes	complemented
224	C₂×Dic₅₃	C₂⁵	1	8	yes	—
225	C₂×Dic₅₃	C₂⁵	1	8	yes	—
226	D₂₁₂	—	2	64	no	—
227	D₂₁₂	—	4	16	no	—
228	C₄×D₅₃	—	2	64	no	—
229	C₂²×D₄²	—	53	1	no	maximal, Sylow
230	C₂²×C₅₃	—	2	8	no	—
231	C₂²×C₅₃	—	2	8	no	—
232	C₂²×C₅₃	—	2	96	no	—
233	C₂²×C₅₃	—	2	8	no	—
234	C₂²×C₅₃	—	2	8	no	—
235	C₂²×C₅₃	—	2	4	no	—
236	C₂²×C₅₃	—	2	4	no	—
237	C₂²×C₅₃	—	4	64	no	—
238	C₂²×C₅₃	—	4	24	no	—
239	C₂²×C₅₃	—	4	24	no	—
240	C₂²×C₅₃	D₄²	1	16	yes	complemented
241	C₂²×C₅₃	C₂³×D₄	1	12	yes	—
242	C₂²×C₅₃	C₂³×D₄	1	16	yes	complemented
243	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
244	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
245	C₂×D₅₃	—	2	8	no	—
246	C₂×D₅₃	—	2	8	no	—
247	C₂×D₅₃	—	2	96	no	—
248	C₂×D₅₃	—	2	8	no	—
249	C₂×D₅₃	—	2	8	no	—
250	C₂×D₅₃	—	2	8	no	—
251	C₂×D₅₃	—	4	64	no	—
252	C₂×D₅₃	—	4	128	no	—
253	C₂×D₅₃	—	4	48	no	—
254	C₂×D₅₃	C₂³×D₄	1	16	yes	complemented
255	C₄×C₅₃	—	2	8	no	—
256	C₄×C₅₃	—	2	4	no	—
257	Dic₅₃	—	2	8	no	—
258	Dic₅₃	C₂³×D₄	1	16	yes	complemented
259	C₂⁴×D₄	—	53	2	no	—
260	C₂⁴×D₄	—	53	2	no	—
261	C₂×D₄²	—	53	48	no	—
262	C₂²×#127	—	53	1	no	—
263	C₂²×#121	—	53	2	no	—
264	C₂²×#121	—	53	2	no	—
265	C₂²×#120	—	53	1	no	—
266	C₂²×#120	—	53	2	no	—
267	C₂²×#120	D₅₃	1	1	yes	characteristic, complemented
268	C₂²×C₄×D₄	—	53	2	no	—
269	C₂×C₅₃	—	2	16	no	—
270	C₂×C₅₃	—	4	4	no	—
271	C₂×C₅₃	—	4	4	no	—
272	C₂×C₅₃	C₂×D₄²	1	12	yes	complemented
273	C₂×C₅₃	C₂⁴×D₄	1	2	yes	—
274	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
275	D₅₃	—	2	16	no	—
276	D₅₃	—	4	8	no	—
277	C₂⁶	—	53	1	no	—
278	C₂⁶	—	53	2	no	—
279	C₂⁶	C₂×D₅₃	1	1	yes	characteristic, complemented
280	C₂³×D₄	—	53	2	no	—
281	C₂³×D₄	—	53	2	no	—
282	C₂³×D₄	—	53	2	no	—
283	C₂³×D₄	—	53	2	no	—
284	C₂³×D₄	—	53	2	no	—
285	C₂³×D₄	—	53	48	no	—
286	C₂³×D₄	—	53	48	no	—
287	C₂³×D₄	—	53	2	no	—
288	C₂³×D₄	—	53	1	no	—
289	C₂³×D₄	—	106	32	no	—
290	C₂³×D₄	—	106	32	no	—
291	C₂³×D₄	C₂×D₅₃	1	2	yes	complemented
292	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic, complemented
293	C₂⁴×C₄	—	53	2	no	—
294	C₂⁴×C₄	—	53	2	no	—
295	D₄²	—	53	256	no	—
296	C₂×#127	—	53	24	no	—
297	C₂×#121	—	53	48	no	—
298	C₂×#121	—	53	48	no	—
299	C₂×#120	—	53	24	no	—
300	C₂×#120	—	53	48	no	—
301	C₂×#120	C₂×D₅₃	1	24	yes	complemented
302	C₂×C₄×D₄	—	53	48	no	—
303	C₂²×C₄⋊C₄	—	53	2	no	—
304	C₂²×C₂²⋊C₄	—	53	1	no	—
305	C₂²×C₂²⋊C₄	—	53	2	no	—
306	C₂²×C₂²⋊C₄	—	53	2	no	—
307	C₂²×C₂²⋊C₄	C₂×D₅₃	1	2	yes	complemented
308	C₂²×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
309	C₂²×C₄²	—	53	1	no	—
310	C₅₃	C₂²×D₄²	1	1	yes	characteristic, complemented, Sylow
311	C₂⁵	—	2	16	no	—
312	C₂⁵	—	4	4	no	—
313	C₂⁵	—	53	2	no	—
314	C₂⁵	—	53	2	no	—
315	C₂⁵	—	53	12	no	—
316	C₂⁵	—	53	24	no	—
317	C₂⁵	—	106	16	no	—
318	C₂⁵	—	106	16	no	—
319	C₂⁵	—	106	16	no	—
320	C₂⁵	—	212	8	no	—
321	C₂⁵	—	212	4	no	—
322	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
323	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
324	C₂⁵	C₂²×D₅₃	1	2	yes	—
325	C₂⁵	C₂²×D₅₃	1	12	yes	complemented
326	C₂²×D₄	—	2	16	no	—
327	C₂²×D₄	—	4	4	no	—
328	C₂²×D₄	—	53	32	no	—
329	C₂²×D₄	—	53	24	no	—
330	C₂²×D₄	—	53	24	no	—
331	C₂²×D₄	—	53	24	no	—
332	C₂²×D₄	—	53	24	no	—
333	C₂²×D₄	—	53	24	no	—
334	C₂²×D₄	—	53	128	no	—
335	C₂²×D₄	—	53	128	no	—
336	C₂²×D₄	—	53	24	no	—
337	C₂²×D₄	—	53	12	no	—
338	C₂²×D₄	—	106	16	no	—
339	C₂²×D₄	—	106	16	no	—
340	C₂²×D₄	—	106	16	no	—
341	C₂²×D₄	—	106	384	no	—
342	C₂²×D₄	—	106	384	no	—
343	C₂²×D₄	—	106	16	no	—
344	C₂²×D₄	—	106	16	no	—
345	C₂²×D₄	—	212	4	no	—
346	C₂²×D₄	C₂²×D₅₃	1	24	yes	complemented
347	C₂²×D₄	C₂²×D₅₃	1	12	yes	complemented
348	C₂³×C₄	—	53	2	no	—
349	C₂³×C₄	—	53	24	no	—
350	C₂³×C₄	—	53	2	no	—
351	C₂³×C₄	—	53	24	no	—
352	C₂³×C₄	—	106	16	no	—
353	C₂³×C₄	—	106	16	no	—
354	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
355	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
356	#127	—	53	64	no	—
357	#121	—	53	128	no	—
358	#121	—	53	128	no	—
359	#120	—	53	64	no	—
360	#120	—	53	128	no	—
361	#120	C₂²×D₅₃	1	64	yes	complemented
362	C₄×D₄	—	53	128	no	—
363	C₂×C₄⋊C₄	—	53	24	no	—
364	C₂×C₂²⋊C₄	—	53	12	no	—
365	C₂×C₂²⋊C₄	—	53	24	no	—
366	C₂×C₂²⋊C₄	—	53	24	no	—
367	C₂×C₂²⋊C₄	C₂²×D₅₃	1	24	yes	complemented
368	C₂×C₂²⋊C₄	C₂²×D₅₃	1	12	yes	complemented
369	C₂×C₄²	—	53	12	no	—
370	C₂⁴	—	2	8	no	—
371	C₂⁴	—	2	8	no	—
372	C₂⁴	—	2	8	no	—
373	C₂⁴	—	2	8	no	—
374	C₂⁴	—	2	96	no	—
375	C₂⁴	—	2	4	no	—
376	C₂⁴	—	2	4	no	—
377	C₂⁴	—	4	24	no	—
378	C₂⁴	—	4	64	no	—
379	C₂⁴	—	53	16	no	—
380	C₂⁴	—	53	12	no	—
381	C₂⁴	—	53	12	no	—
382	C₂⁴	—	53	16	no	—
383	C₂⁴	—	53	32	no	—
384	C₂⁴	—	106	96	no	—
385	C₂⁴	—	106	96	no	—
386	C₂⁴	—	106	8	no	—
387	C₂⁴	—	106	8	no	—
388	C₂⁴	—	106	8	no	—
389	C₂⁴	—	106	8	no	—
390	C₂⁴	—	106	96	no	—
391	C₂⁴	—	106	8	no	—
392	C₂⁴	—	212	48	no	—
393	C₂⁴	—	212	24	no	—
394	C₂⁴	—	212	128	no	—
395	C₂⁴	—	212	64	no	—
396	C₂⁴	D₄×D₅₃	1	16	yes	complemented
397	C₂⁴	C₂³×D₅₃	1	6	yes	—
398	C₂⁴	C₂³×D₅₃	1	6	yes	—
399	C₂⁴	C₂³×D₅₃	1	12	yes	—
400	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
401	C₂⁴	C₂³×D₅₃	1	16	yes	complemented
402	C₂×D₄	—	2	96	no	—
403	C₂×D₄	—	4	24	no	—
404	C₂×D₄	—	53	192	no	—
405	C₂×D₄	—	53	32	no	—
406	C₂×D₄	—	53	32	no	—
407	C₂×D₄	—	53	32	no	—
408	C₂×D₄	—	53	32	no	—
409	C₂×D₄	—	53	32	no	—
410	C₂×D₄	—	53	32	no	—
411	C₂×D₄	—	53	16	no	—
412	C₂×D₄	—	106	96	no	—
413	C₂×D₄	—	106	96	no	—
414	C₂×D₄	—	106	96	no	—
415	C₂×D₄	—	106	512	no	—
416	C₂×D₄	—	106	512	no	—
417	C₂×D₄	—	106	96	no	—
418	C₂×D₄	—	106	96	no	—
419	C₂×D₄	—	212	24	no	—
420	C₂×D₄	C₂³×D₅₃	1	32	yes	complemented
421	C₂×D₄	C₂³×D₅₃	1	16	yes	complemented
422	C₂²×C₄	—	2	8	no	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	53	16	no	—
425	C₂²×C₄	—	53	12	no	—
426	C₂²×C₄	—	53	12	no	—
427	C₂²×C₄	—	53	32	no	—
428	C₂²×C₄	—	53	32	no	—
429	C₂²×C₄	—	106	96	no	—
430	C₂²×C₄	—	106	96	no	—
431	C₂²×C₄	—	106	8	no	—
432	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
433	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
434	C₄⋊C₄	—	53	32	no	—
435	C₂²⋊C₄	—	53	16	no	—
436	C₂²⋊C₄	—	53	32	no	—
437	C₂²⋊C₄	—	53	32	no	—
438	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	complemented
439	C₂²⋊C₄	C₂³×D₅₃	1	16	yes	complemented
440	C₄²	—	53	16	no	—
441	C₂³	—	2	24	no	—
442	C₂³	—	2	24	no	—
443	C₂³	—	2	64	no	—
444	C₂³	—	2	24	no	—
445	C₂³	—	2	24	no	—
446	C₂³	—	2	64	no	—
447	C₂³	—	2	12	no	—
448	C₂³	—	2	12	no	—
449	C₂³	—	4	16	no	—
450	C₂³	—	4	192	no	—
451	C₂³	—	4	16	no	—
452	C₂³	—	4	16	no	—
453	C₂³	—	53	48	no	—
454	C₂³	—	53	8	no	—
455	C₂³	—	53	8	no	—
456	C₂³	—	106	64	no	—
457	C₂³	—	106	64	no	—
458	C₂³	—	106	24	no	—
459	C₂³	—	106	24	no	—
460	C₂³	—	106	64	no	—
461	C₂³	—	106	24	no	—
462	C₂³	—	106	24	no	—
463	C₂³	—	106	64	no	—
464	C₂³	—	106	24	no	—
465	C₂³	—	212	32	no	—
466	C₂³	—	212	16	no	—
467	C₂³	—	212	384	no	—
468	C₂³	—	212	192	no	—
469	C₂³	—	212	32	no	—
470	C₂³	C₂×D₄×D₅₃	1	8	yes	—
471	C₂³	C₂×D₄×D₅₃	1	48	yes	complemented
472	C₂³	#13133262	1	4	yes	—
473	C₂³	C₂⁴×D₅₃	1	4	yes	—
474	C₂³	C₂⁴×D₅₃	1	4	yes	—
475	C₂³	C₂⁴×D₅₃	1	8	yes	—
476	C₂³	C₂⁴×D₅₃	1	3	yes	—
477	D₄	—	2	64	no	—
478	D₄	—	4	16	no	—
479	D₄	—	53	128	no	—
480	D₄	—	106	64	no	—
481	D₄	—	106	64	no	—
482	D₄	—	106	64	no	—
483	D₄	—	106	64	no	—
484	D₄	—	106	64	no	—
485	D₄	—	212	16	no	—
486	C₂×C₄	—	2	24	no	—
487	C₂×C₄	—	2	12	no	—
488	C₂×C₄	—	53	48	no	—
489	C₂×C₄	—	53	8	no	—
490	C₂×C₄	—	53	8	no	—
491	C₂×C₄	—	106	64	no	—
492	C₂×C₄	—	106	64	no	—
493	C₂×C₄	—	106	24	no	—
494	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
495	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
496	C₂²	—	2	96	no	—
497	C₂²	—	2	8	no	—
498	C₂²	—	2	8	no	—
499	C₂²	—	2	8	no	—
500	C₂²	—	2	8	no	—
501	C₂²	—	2	4	no	—
502	C₂²	—	2	4	no	—
503	C₂²	—	4	64	no	—
504	C₂²	—	4	24	no	—
505	C₂²	—	4	24	no	—
506	C₂²	—	53	16	no	—
507	C₂²	—	106	96	no	—
508	C₂²	—	106	8	no	—
509	C₂²	—	106	8	no	—
510	C₂²	—	106	8	no	—
511	C₂²	—	106	8	no	—
512	C₂²	—	106	8	no	—
513	C₂²	—	212	128	no	—
514	C₂²	—	212	64	no	—
515	C₂²	—	212	48	no	—
516	C₂²	#16811255	1	16	yes	direct product factor
517	C₂²	C₂²×D₄×D₅₃	1	12	yes	—
518	C₂²	C₂²×D₄×D₅₃	1	16	yes	complemented
519	C₂²	C₂×#13133262	1	6	yes	—
520	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
521	C₄	—	2	8	no	—
522	C₄	—	2	4	no	—
523	C₄	—	53	16	no	—
524	C₄	—	106	8	no	—
525	C₂	—	2	16	no	—
526	C₂	—	4	4	no	—
527	C₂	—	4	4	no	—
528	C₂	—	106	16	no	—
529	C₂	—	212	8	no	—
530	C₂	C₂×#16811255	1	12	yes	direct product factor
531	C₂	C₂³×D₄×D₅₃	1	2	yes	—
532	C₂	C₂²×#13133262	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×#120×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂²×#120	D₅₃
hypocenter	C₅₃	C₂²×D₄²
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#120

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₂²⋊D₅₃	C₂	semidirect
2. C₂³×D₄×D₅₃	C₂	semidirect
3. C₂×#16811255	C₂	direct (N × H)
4. C₂²×#13133213	C₂	semidirect
5. C₂²×#13133190	C₂	semidirect
6. C₂²×#13133189	C₂	semidirect
7. C₂²×#13133187	C₂	semidirect
8. C₂²×#13133185	C₂	semidirect
9. C₂²×#13133177	C₂	semidirect
10. C₂²×#120×C₅₃	C₂	semidirect
11. C₂⁵×D₅₃	C₂²	semidirect
12. C₂⁵×D₅₃	C₂²	semidirect
13. C₂³×D₂₁₂	C₂²	semidirect
14. C₂³×D₂₁₂	C₂²	semidirect
17. C₂³×C₂²⋊D₅₃	C₂²	semidirect
18. C₂³×C₂²⋊D₅₃	C₂²	semidirect
19. C₂³×C₂²⋊D₅₃	C₂²	semidirect
20. C₂³×C₂²⋊D₅₃	C₂²	semidirect
21. C₂³×C₂²⋊D₅₃	C₂²	semidirect
23. C₂²×D₄×D₅₃	C₂²	semidirect
24. C₂⁴×Dic₅₃	C₂²	semidirect
25. C₂³×C₄×D₅₃	C₂²	semidirect
26. #16811255	C₂²	direct (N × H)
27. C₂×#13133213	C₂²	semidirect
28. C₂×#13133190	C₂²	semidirect
29. C₂×#13133189	C₂²	semidirect
30. C₂×#13133187	C₂²	semidirect
31. C₂×#13133185	C₂²	semidirect
32. C₂×#13133177	C₂²	semidirect
33. C₂²×#11719919	C₂²	semidirect
34. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
35. C₂²×#11719921	C₂²	semidirect
36. C₂²×#11719921	C₂²	semidirect
37. C₂²×C₄×Dic₅₃	C₂²	semidirect
38. C₂⁶×C₅₃	C₂²	semidirect
39. C₂³×D₄×C₅₃	C₂²	semidirect
40. C₂³×D₄×C₅₃	C₂²	semidirect
41. C₂×#120×C₅₃	C₂²	semidirect
42. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
43. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
51. C₂⁴×D₅₃	C₂³	semidirect
52. C₂⁴×D₅₃	C₂³	semidirect
55. C₂²×D₂₁₂	C₂³	semidirect
56. C₂²×D₂₁₂	C₂³	semidirect
62. C₂²×C₂²⋊D₅₃	D₄	semidirect
64. C₂²×C₂²⋊D₅₃	C₂³	semidirect
65. C₂²×C₂²⋊D₅₃	C₂³	semidirect
66. C₂²×C₂²⋊D₅₃	C₂³	semidirect
67. C₂²×C₂²⋊D₅₃	C₂³	semidirect
68. C₂²×C₂²⋊D₅₃	C₂³	semidirect
70. C₂×D₄×D₅₃	C₂³	semidirect
74. C₂³×Dic₅₃	C₂³	semidirect
76. C₂²×C₄×D₅₃	C₂³	semidirect
77. #13133213	C₂³	semidirect
78. #13133190	C₂³	semidirect
79. #13133189	C₂³	semidirect
80. #13133187	C₂³	semidirect
81. #13133185	C₂³	semidirect
82. #13133177	C₂³	semidirect
83. C₂×#11719919	C₂³	semidirect
84. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
85. C₂×#11719921	C₂³	semidirect
86. C₂×#11719921	C₂³	semidirect
87. C₂×C₄×Dic₅₃	C₂³	semidirect
93. C₂⁵×C₅₃	C₂³	semidirect
96. C₂²×D₄×C₅₃	C₂³	semidirect
97. C₂²×D₄×C₅₃	C₂³	semidirect
100. #120×C₅₃	C₂³	semidirect
101. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
102. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
112. C₂⁴×C₅₃	C₂×D₄	semidirect
117. C₂⁴×C₅₃	C₂⁴	semidirect
130. C₂³×D₅₃	C₂×D₄	semidirect
133. C₂³×D₅₃	C₂⁴	semidirect
134. C₂³×D₅₃	C₂⁴	semidirect
137. C₂×D₄×C₅₃	C₂⁴	semidirect
138. C₂×D₄×C₅₃	C₂⁴	semidirect
148. C₂×C₂²⋊D₅₃	C₂×D₄	semidirect
150. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
151. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
152. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
153. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
156. C₂²×Dic₅₃	C₂×D₄	semidirect
159. C₂²×Dic₅₃	C₂⁴	semidirect
163. C₂×D₂₁₂	C₂⁴	semidirect
164. C₂×D₂₁₂	C₂⁴	semidirect
166. C₂×C₄×D₅₃	C₂⁴	semidirect
167. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
168. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
169. C₂²⋊Dic₅₃	C₂⁴	semidirect
170. #11719921	C₂⁴	semidirect
171. #11719921	C₂⁴	semidirect
172. #11719919	C₂⁴	semidirect
173. C₄×Dic₅₃	C₂⁴	semidirect
187. C₂³×C₅₃	C₂²×D₄	semidirect
207. C₂²×D₅₃	C₂²×D₄	semidirect
220. C₂²⋊D₅₃	C₂²×D₄	semidirect
223. C₂×Dic₅₃	C₂²×D₄	semidirect
240. C₂²×C₅₃	D₄²	semidirect
242. C₂²×C₅₃	C₂³×D₄	semidirect
254. C₂×D₅₃	C₂³×D₄	semidirect
258. Dic₅₃	C₂³×D₄	semidirect
267. C₂²×#120	D₅₃	semidirect
272. C₂×C₅₃	C₂×D₄²	semidirect
279. C₂⁶	C₂×D₅₃	semidirect
291. C₂³×D₄	C₂×D₅₃	semidirect
292. C₂³×D₄	C₂×D₅₃	semidirect
301. C₂×#120	C₂×D₅₃	semidirect
307. C₂²×C₂²⋊C₄	C₂×D₅₃	semidirect
308. C₂²×C₂²⋊C₄	C₂×D₅₃	semidirect
310. C₅₃	C₂²×D₄²	semidirect
325. C₂⁵	C₂²×D₅₃	semidirect
346. C₂²×D₄	C₂²×D₅₃	semidirect
347. C₂²×D₄	C₂²×D₅₃	semidirect
361. #120	C₂²×D₅₃	semidirect
367. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
368. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
396. C₂⁴	D₄×D₅₃	semidirect
401. C₂⁴	C₂³×D₅₃	semidirect
420. C₂×D₄	C₂³×D₅₃	semidirect
421. C₂×D₄	C₂³×D₅₃	semidirect
438. C₂²⋊C₄	C₂³×D₅₃	semidirect
439. C₂²⋊C₄	C₂³×D₅₃	semidirect
471. C₂³	C₂×D₄×D₅₃	semidirect
516. C₂²	#16811255	direct (N × H)
518. C₂²	C₂²×D₄×D₅₃	semidirect
530. C₂	C₂×#16811255	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₂²⋊D₅₃	2
C₂³×D₄×D₅₃	2
C₂×#16811255	2
C₂²×#13133213	2
C₂²×#13133190	2
C₂²×#13133189	2
C₂²×#13133187	2
C₂²×#13133185	2
C₂²×#13133177	2
C₂²×#120×C₅₃	2
C₂²×D₄²	53

Subgroups — C22×#16811255 #28290999

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811255 #28290999