Subgroups — C₂²×#16811181 #28290807

Order 13568.

15230 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×D₅₃	C₂	1	2	yes	maximal, complemented
2	C₂×#16811181	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#13133202	C₂	1	2	yes	maximal, complemented
4	C₂²×#13133199	C₂	1	2	yes	maximal, complemented
5	C₂²×#13133191	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#13133189	C₂	1	2	yes	maximal, complemented
7	C₂²×#13133187	C₂	1	2	yes	maximal, complemented
8	C₂²×#13133185	C₂	1	2	yes	maximal, complemented
9	C₂³×#11719921	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#121×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
12	C₂⁵×D₅₃	C₂²	1	2	yes	complemented
13	C₂³×D₂₁₂	C₂²	1	2	yes	—
14	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
15	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	—
16	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
17	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
18	C₂²×D₄×D₅₃	—	2	32	no	—
19	C₂²×D₄×D₅₃	C₂²	1	48	yes	complemented
20	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
21	C₂³×C₄×D₅₃	C₂²	1	2	yes	complemented
22	#16811181	C₂²	1	256	yes	direct product factor
23	C₂×#13133202	C₂²	1	48	yes	complemented
24	C₂×#13133199	C₂²	1	48	yes	complemented
25	C₂×#13133191	C₂²	1	24	yes	complemented
26	C₂×#13133189	C₂²	1	48	yes	complemented
27	C₂×#13133187	C₂²	1	48	yes	complemented
28	C₂×#13133185	C₂²	1	48	yes	complemented
29	C₂²×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
30	C₂²×#11719919	C₂²	1	2	yes	complemented
31	C₂²×C₂²⋊Dic₅₃	C₂²	1	2	yes	complemented
32	C₂²×#11719921	C₂²	1	2	yes	complemented
33	C₂²×#11719921	C₂²	1	2	yes	complemented
34	C₂²×#11719921	C₂²	1	2	yes	complemented
35	C₂²×#11719921	C₂²	1	2	yes	complemented
36	C₂²×#11719921	C₂²	1	24	yes	complemented
37	C₂³×D₄×C₅₃	C₂²	1	2	yes	complemented
38	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
39	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
40	C₂×#121×C₅₃	C₂²	1	24	yes	complemented
41	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
42	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
43	C₂⁴×D₅₃	—	2	16	no	—
44	C₂⁴×D₅₃	—	4	4	no	—
45	C₂⁴×D₅₃	—	4	4	no	—
46	C₂⁴×D₅₃	—	4	8	no	—
47	C₂⁴×D₅₃	—	4	8	no	—
48	C₂⁴×D₅₃	D₄	1	16	yes	complemented
49	C₂⁴×D₅₃	C₂³	1	2	yes	—
50	C₂⁴×D₅₃	C₂³	1	2	yes	—
51	C₂⁴×D₅₃	C₂³	1	12	yes	—
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	—
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	—	4	8	no	—
60	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	—
61	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	complemented
62	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	complemented
63	C₂×D₄×D₅₃	—	2	384	no	—
64	C₂×D₄×D₅₃	C₂³	1	128	yes	complemented
65	C₂³×Dic₅₃	—	4	4	no	—
66	C₂³×Dic₅₃	C₂³	1	2	yes	—
67	C₂³×Dic₅₃	C₂³	1	2	yes	—
68	C₂³×Dic₅₃	C₂³	1	12	yes	complemented
69	C₂²×C₄×D₅₃	—	2	16	no	—
70	C₂²×C₄×D₅₃	C₂³	1	24	yes	complemented
71	#13133202	C₂³	1	128	yes	complemented
72	#13133199	C₂³	1	128	yes	complemented
73	#13133191	C₂³	1	64	yes	complemented
74	#13133189	C₂³	1	128	yes	complemented
75	#13133187	C₂³	1	128	yes	complemented
76	#13133185	C₂³	1	128	yes	complemented
77	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	complemented
78	C₂×#11719919	C₂³	1	24	yes	complemented
79	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	complemented
80	C₂×#11719921	C₂³	1	24	yes	complemented
81	C₂×#11719921	C₂³	1	24	yes	complemented
82	C₂×#11719921	C₂³	1	24	yes	complemented
83	C₂×#11719921	C₂³	1	24	yes	complemented
84	C₂×#11719921	C₂³	1	64	yes	complemented
85	C₂⁵×C₅₃	C₂³	1	2	yes	—
86	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
87	C₂²×D₄×C₅₃	—	2	16	no	—
88	C₂²×D₄×C₅₃	—	4	4	no	—
89	C₂²×D₄×C₅₃	C₂³	1	24	yes	complemented
90	C₂²×D₄×C₅₃	C₂³	1	12	yes	complemented
91	C₂³×C₄×C₅₃	—	4	4	no	—
92	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
93	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
94	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
95	C₂³×C₄×C₅₃	C₂³	1	12	yes	complemented
96	#121×C₅₃	C₂³	1	64	yes	complemented
97	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	complemented
98	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	complemented
99	C₂⁴×C₅₃	—	2	4	no	—
100	C₂⁴×C₅₃	—	2	4	no	—
101	C₂⁴×C₅₃	—	2	8	no	—
102	C₂⁴×C₅₃	—	2	8	no	—
103	C₂⁴×C₅₃	—	2	8	no	—
104	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
105	C₂⁴×C₅₃	C₂⁴	1	12	yes	—
106	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
107	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
108	C₂³×D₅₃	—	2	8	no	—
109	C₂³×D₅₃	—	2	8	no	—
110	C₂³×D₅₃	—	2	8	no	—
111	C₂³×D₅₃	—	2	8	no	—
112	C₂³×D₅₃	—	2	8	no	—
113	C₂³×D₅₃	—	2	96	no	—
114	C₂³×D₅₃	—	4	64	no	—
115	C₂³×D₅₃	—	4	24	no	—
116	C₂³×D₅₃	—	4	24	no	—
117	C₂³×D₅₃	—	4	48	no	—
118	C₂³×D₅₃	—	4	48	no	—
119	C₂³×D₅₃	—	4	128	no	—
120	C₂³×D₅₃	C₂×D₄	1	16	yes	—
121	C₂³×D₅₃	C₂×D₄	1	96	yes	complemented
122	C₂³×D₅₃	C₂⁴	1	12	yes	—
123	C₂³×D₅₃	C₂⁴	1	12	yes	—
124	C₂³×D₅₃	C₂⁴	1	16	yes	—
125	C₂³×D₅₃	C₂⁴	1	32	yes	complemented
126	C₂×D₄×C₅₃	—	2	96	no	—
127	C₂×D₄×C₅₃	—	4	24	no	—
128	C₂×D₄×C₅₃	C₂⁴	1	32	yes	complemented
129	C₂×D₄×C₅₃	C₂⁴	1	16	yes	complemented
130	C₂²×C₄×C₅₃	—	2	8	no	—
131	C₂²×C₄×C₅₃	—	2	4	no	—
132	C₂²×C₄×C₅₃	—	2	4	no	—
133	C₂²×C₄×C₅₃	—	4	24	no	—
134	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
135	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
136	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
137	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	complemented
138	C₂×C₂²⋊D₅₃	—	2	96	no	—
139	C₂×C₂²⋊D₅₃	—	2	96	no	—
140	C₂×C₂²⋊D₅₃	—	4	48	no	—
141	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
142	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	complemented
143	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	complemented
144	C₂²×Dic₅₃	—	2	8	no	—
145	C₂²×Dic₅₃	—	2	8	no	—
146	C₂²×Dic₅₃	—	4	24	no	—
147	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
148	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
149	C₂²×Dic₅₃	C₂⁴	1	16	yes	complemented
150	D₄×D₅₃	—	2	512	no	—
151	C₂×D₂₁₂	—	2	96	no	—
152	C₂×D₂₁₂	—	4	24	no	—
153	C₂×D₂₁₂	C₂⁴	1	32	yes	—
154	C₂×D₂₁₂	C₂⁴	1	16	yes	complemented
155	C₂×C₄×D₅₃	—	2	96	no	—
156	C₂×C₄×D₅₃	C₂⁴	1	32	yes	complemented
157	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	complemented
158	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
159	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	complemented
160	#11719921	C₂⁴	1	32	yes	complemented
161	#11719921	C₂⁴	1	32	yes	complemented
162	#11719921	C₂⁴	1	32	yes	complemented
163	#11719921	C₂⁴	1	32	yes	complemented
164	C₄⋊Dic₅₃	C₂⁴	1	16	yes	complemented
165	#11719919	C₂⁴	1	32	yes	complemented
166	C₂³×C₅₃	—	2	32	no	—
167	C₂³×C₅₃	—	2	12	no	—
168	C₂³×C₅₃	—	2	12	no	—
169	C₂³×C₅₃	—	2	24	no	—
170	C₂³×C₅₃	—	2	24	no	—
171	C₂³×C₅₃	—	2	24	no	—
172	C₂³×C₅₃	—	4	32	no	—
173	C₂³×C₅₃	C₂²×D₄	1	24	yes	—
174	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
175	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
176	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
177	C₂³×C₅₃	C₂⁵	1	8	yes	—
178	C₂³×C₅₃	C₂⁵	1	4	yes	—
179	C₂³×C₅₃	C₂⁵	1	3	yes	—
180	C₂²×D₅₃	—	2	64	no	—
181	C₂²×D₅₃	—	2	24	no	—
182	C₂²×D₅₃	—	2	24	no	—
183	C₂²×D₅₃	—	2	24	no	—
184	C₂²×D₅₃	—	2	24	no	—
185	C₂²×D₅₃	—	2	24	no	—
186	C₂²×D₅₃	—	2	64	no	—
187	C₂²×D₅₃	—	4	192	no	—
188	C₂²×D₅₃	—	4	16	no	—
189	C₂²×D₅₃	—	4	16	no	—
190	C₂²×D₅₃	—	4	32	no	—
191	C₂²×D₅₃	—	4	32	no	—
192	C₂²×D₅₃	—	4	384	no	—
193	C₂²×D₅₃	—	4	32	no	—
194	C₂²×D₅₃	C₂²×D₄	1	48	yes	—
195	C₂²×D₅₃	C₂²×D₄	1	64	yes	complemented
196	C₂²×D₅₃	C₂⁵	1	8	yes	—
197	C₂²×D₅₃	C₂⁵	1	8	yes	—
198	D₄×C₅₃	—	2	64	no	—
199	D₄×C₅₃	—	4	16	no	—
200	C₂×C₄×C₅₃	—	2	24	no	—
201	C₂×C₄×C₅₃	—	2	12	no	—
202	C₂×C₄×C₅₃	—	2	12	no	—
203	C₂×C₄×C₅₃	—	4	16	no	—
204	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
205	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
206	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
207	C₂²⋊D₅₃	—	2	64	no	—
208	C₂²⋊D₅₃	—	2	64	no	—
209	C₂²⋊D₅₃	—	4	32	no	—
210	C₂×Dic₅₃	—	2	24	no	—
211	C₂×Dic₅₃	—	2	24	no	—
212	C₂×Dic₅₃	—	4	16	no	—
213	C₂×Dic₅₃	C₂⁵	1	8	yes	—
214	C₂×Dic₅₃	C₂⁵	1	8	yes	—
215	D₂₁₂	—	2	64	no	—
216	D₂₁₂	—	4	16	no	—
217	C₄×D₅₃	—	2	64	no	—
218	C₂²×#534	—	53	1	no	maximal, Sylow
219	C₂²×C₅₃	—	2	4	no	—
220	C₂²×C₅₃	—	2	4	no	—
221	C₂²×C₅₃	—	2	48	no	—
222	C₂²×C₅₃	—	2	8	no	—
223	C₂²×C₅₃	—	2	8	no	—
224	C₂²×C₅₃	—	2	8	no	—
225	C₂²×C₅₃	—	4	48	no	—
226	C₂²×C₅₃	#534	1	16	yes	complemented
227	C₂²×C₅₃	C₂³×D₄	1	8	yes	—
228	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
229	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
230	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
231	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
232	C₂×D₅₃	—	2	8	no	—
233	C₂×D₅₃	—	2	8	no	—
234	C₂×D₅₃	—	2	96	no	—
235	C₂×D₅₃	—	2	8	no	—
236	C₂×D₅₃	—	2	8	no	—
237	C₂×D₅₃	—	2	8	no	—
238	C₂×D₅₃	—	4	64	no	—
239	C₂×D₅₃	—	4	128	no	—
240	C₂×D₅₃	—	4	48	no	—
241	C₂×D₅₃	C₂³×D₄	1	16	yes	—
242	C₄×C₅₃	—	2	8	no	—
243	C₄×C₅₃	—	2	4	no	—
244	C₄×C₅₃	—	2	4	no	—
245	Dic₅₃	—	2	8	no	—
246	Dic₅₃	—	2	8	no	—
247	C₂⁴×D₄	—	53	2	no	—
248	C₂×#534	—	53	48	no	—
249	C₂²×#123	—	53	2	no	—
250	C₂²×#123	—	53	2	no	—
251	C₂²×#121	—	53	2	no	—
252	C₂²×#121	—	53	1	no	—
253	C₂²×#121	D₅₃	1	1	yes	characteristic, complemented
254	C₂²×#120	—	53	2	no	—
255	C₂²×#120	—	53	2	no	—
256	C₂³×C₂²⋊C₄	—	53	1	no	—
257	C₂×C₅₃	—	2	8	no	—
258	C₂×C₅₃	—	4	8	no	—
259	C₂×C₅₃	C₂×#534	1	12	yes	complemented
260	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
261	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
262	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
263	D₅₃	—	2	16	no	—
264	D₅₃	—	4	8	no	—
265	C₂⁶	—	53	1	no	—
266	C₂⁶	—	53	2	no	—
267	C₂³×D₄	—	53	2	no	—
268	C₂³×D₄	—	53	2	no	—
269	C₂³×D₄	—	53	2	no	—
270	C₂³×D₄	—	53	48	no	—
271	C₂³×D₄	—	53	2	no	—
272	C₂³×D₄	—	53	1	no	—
273	C₂³×D₄	—	106	32	no	—
274	C₂³×D₄	C₂×D₅₃	1	2	yes	complemented
275	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic, complemented
276	C₂⁴×C₄	—	53	1	no	—
277	C₂⁴×C₄	—	53	2	no	—
278	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
279	#534	—	53	256	no	—
280	C₂×#123	—	53	48	no	—
281	C₂×#123	—	53	48	no	—
282	C₂×#121	—	53	48	no	—
283	C₂×#121	—	53	24	no	—
284	C₂×#121	C₂×D₅₃	1	24	yes	complemented
285	C₂×#120	—	53	48	no	—
286	C₂×#120	—	53	48	no	—
287	C₂²×C₄⋊C₄	—	53	2	no	—
288	C₂²×C₄⋊C₄	—	53	1	no	—
289	C₂²×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
290	C₂²×C₂²⋊C₄	—	53	2	no	—
291	C₂²×C₂²⋊C₄	—	53	2	no	—
292	C₂²×C₂²⋊C₄	—	53	2	no	—
293	C₂²×C₂²⋊C₄	—	53	2	no	—
294	C₂²×C₂²⋊C₄	—	53	2	no	—
295	C₂²×C₂²⋊C₄	—	53	24	no	—
296	C₂²×C₂²⋊C₄	C₂×D₅₃	1	2	yes	complemented
297	C₅₃	C₂²×#534	1	1	yes	characteristic, complemented, Sylow
298	C₂⁵	—	53	2	no	—
299	C₂⁵	—	53	16	no	—
300	C₂⁵	—	53	12	no	—
301	C₂⁵	—	53	2	no	—
302	C₂⁵	—	53	24	no	—
303	C₂⁵	—	106	16	no	—
304	C₂⁵	—	212	4	no	—
305	C₂⁵	—	212	4	no	—
306	C₂⁵	—	212	8	no	—
307	C₂⁵	—	212	8	no	—
308	C₂⁵	C₂²×D₅₃	1	2	yes	—
309	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
310	C₂²×D₄	—	2	16	no	—
311	C₂²×D₄	—	4	4	no	—
312	C₂²×D₄	—	53	24	no	—
313	C₂²×D₄	—	53	24	no	—
314	C₂²×D₄	—	53	24	no	—
315	C₂²×D₄	—	53	128	no	—
316	C₂²×D₄	—	53	24	no	—
317	C₂²×D₄	—	53	12	no	—
318	C₂²×D₄	—	106	16	no	—
319	C₂²×D₄	—	106	384	no	—
320	C₂²×D₄	—	106	16	no	—
321	C₂²×D₄	—	106	16	no	—
322	C₂²×D₄	—	212	4	no	—
323	C₂²×D₄	—	212	8	no	—
324	C₂²×D₄	C₂²×D₅₃	1	24	yes	complemented
325	C₂²×D₄	C₂²×D₅₃	1	12	yes	complemented
326	C₂³×C₄	—	4	4	no	—
327	C₂³×C₄	—	53	2	no	—
328	C₂³×C₄	—	53	2	no	—
329	C₂³×C₄	—	53	12	no	—
330	C₂³×C₄	—	53	24	no	—
331	C₂³×C₄	—	106	16	no	—
332	C₂³×C₄	—	212	4	no	—
333	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
334	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
335	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
336	C₂³×C₄	C₂²×D₅₃	1	12	yes	complemented
337	#123	—	53	128	no	—
338	#123	—	53	128	no	—
339	#121	—	53	128	no	—
340	#121	—	53	64	no	—
341	#121	C₂²×D₅₃	1	64	yes	complemented
342	#120	—	53	128	no	—
343	#120	—	53	128	no	—
344	C₂×C₄⋊C₄	—	53	24	no	—
345	C₂×C₄⋊C₄	—	53	12	no	—
346	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	complemented
347	C₂×C₂²⋊C₄	—	53	24	no	—
348	C₂×C₂²⋊C₄	—	53	24	no	—
349	C₂×C₂²⋊C₄	—	53	24	no	—
350	C₂×C₂²⋊C₄	—	53	24	no	—
351	C₂×C₂²⋊C₄	—	53	24	no	—
352	C₂×C₂²⋊C₄	—	53	64	no	—
353	C₂×C₂²⋊C₄	C₂²×D₅₃	1	24	yes	complemented
354	C₂⁴	—	2	4	no	—
355	C₂⁴	—	2	4	no	—
356	C₂⁴	—	2	8	no	—
357	C₂⁴	—	2	8	no	—
358	C₂⁴	—	2	8	no	—
359	C₂⁴	—	53	16	no	—
360	C₂⁴	—	53	12	no	—
361	C₂⁴	—	53	96	no	—
362	C₂⁴	—	53	16	no	—
363	C₂⁴	—	53	12	no	—
364	C₂⁴	—	53	32	no	—
365	C₂⁴	—	106	96	no	—
366	C₂⁴	—	106	8	no	—
367	C₂⁴	—	106	8	no	—
368	C₂⁴	—	106	8	no	—
369	C₂⁴	—	106	8	no	—
370	C₂⁴	—	106	8	no	—
371	C₂⁴	—	212	64	no	—
372	C₂⁴	—	212	24	no	—
373	C₂⁴	—	212	24	no	—
374	C₂⁴	—	212	48	no	—
375	C₂⁴	—	212	48	no	—
376	C₂⁴	—	212	128	no	—
377	C₂⁴	D₄×D₅₃	1	8	yes	—
378	C₂⁴	C₂³×D₅₃	1	12	yes	—
379	C₂⁴	C₂³×D₅₃	1	6	yes	—
380	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
381	C₂×D₄	—	2	96	no	—
382	C₂×D₄	—	4	24	no	—
383	C₂×D₄	—	53	32	no	—
384	C₂×D₄	—	53	32	no	—
385	C₂×D₄	—	53	32	no	—
386	C₂×D₄	—	53	32	no	—
387	C₂×D₄	—	53	16	no	—
388	C₂×D₄	—	106	96	no	—
389	C₂×D₄	—	106	512	no	—
390	C₂×D₄	—	106	96	no	—
391	C₂×D₄	—	106	96	no	—
392	C₂×D₄	—	212	24	no	—
393	C₂×D₄	—	212	48	no	—
394	C₂×D₄	C₂³×D₅₃	1	32	yes	complemented
395	C₂×D₄	C₂³×D₅₃	1	16	yes	complemented
396	C₂²×C₄	—	2	8	no	—
397	C₂²×C₄	—	2	4	no	—
398	C₂²×C₄	—	2	4	no	—
399	C₂²×C₄	—	4	24	no	—
400	C₂²×C₄	—	53	12	no	—
401	C₂²×C₄	—	53	12	no	—
402	C₂²×C₄	—	53	16	no	—
403	C₂²×C₄	—	53	32	no	—
404	C₂²×C₄	—	106	96	no	—
405	C₂²×C₄	—	106	8	no	—
406	C₂²×C₄	—	106	8	no	—
407	C₂²×C₄	—	212	24	no	—
408	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
409	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
410	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
411	C₂²×C₄	C₂³×D₅₃	1	16	yes	complemented
412	C₄⋊C₄	—	53	32	no	—
413	C₄⋊C₄	—	53	16	no	—
414	C₄⋊C₄	C₂³×D₅₃	1	16	yes	complemented
415	C₂²⋊C₄	—	53	32	no	—
416	C₂²⋊C₄	—	53	32	no	—
417	C₂²⋊C₄	—	53	32	no	—
418	C₂²⋊C₄	—	53	32	no	—
419	C₂²⋊C₄	—	53	32	no	—
420	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	complemented
421	C₂³	—	2	32	no	—
422	C₂³	—	2	12	no	—
423	C₂³	—	2	12	no	—
424	C₂³	—	2	24	no	—
425	C₂³	—	2	24	no	—
426	C₂³	—	2	24	no	—
427	C₂³	—	4	32	no	—
428	C₂³	—	53	48	no	—
429	C₂³	—	53	8	no	—
430	C₂³	—	53	64	no	—
431	C₂³	—	53	8	no	—
432	C₂³	—	106	64	no	—
433	C₂³	—	106	64	no	—
434	C₂³	—	106	24	no	—
435	C₂³	—	106	24	no	—
436	C₂³	—	106	24	no	—
437	C₂³	—	106	24	no	—
438	C₂³	—	106	24	no	—
439	C₂³	—	212	192	no	—
440	C₂³	—	212	16	no	—
441	C₂³	—	212	16	no	—
442	C₂³	—	212	32	no	—
443	C₂³	—	212	32	no	—
444	C₂³	—	212	384	no	—
445	C₂³	—	212	32	no	—
446	C₂³	C₂×D₄×D₅₃	1	24	yes	—
447	C₂³	C₂×D₄×D₅₃	1	4	yes	—
448	C₂³	#13133262	1	4	yes	—
449	C₂³	#13133263	1	4	yes	—
450	C₂³	C₂⁴×D₅₃	1	8	yes	—
451	C₂³	C₂⁴×D₅₃	1	4	yes	—
452	C₂³	C₂⁴×D₅₃	1	3	yes	—
453	D₄	—	2	64	no	—
454	D₄	—	4	16	no	—
455	D₄	—	106	64	no	—
456	D₄	—	106	64	no	—
457	D₄	—	106	64	no	—
458	D₄	—	212	16	no	—
459	D₄	—	212	32	no	—
460	C₂×C₄	—	2	24	no	—
461	C₂×C₄	—	2	12	no	—
462	C₂×C₄	—	2	12	no	—
463	C₂×C₄	—	4	16	no	—
464	C₂×C₄	—	53	8	no	—
465	C₂×C₄	—	53	8	no	—
466	C₂×C₄	—	106	64	no	—
467	C₂×C₄	—	106	24	no	—
468	C₂×C₄	—	106	24	no	—
469	C₂×C₄	—	212	16	no	—
470	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
471	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
472	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
473	C₂²	—	2	48	no	—
474	C₂²	—	2	4	no	—
475	C₂²	—	2	4	no	—
476	C₂²	—	2	8	no	—
477	C₂²	—	2	8	no	—
478	C₂²	—	2	8	no	—
479	C₂²	—	4	48	no	—
480	C₂²	—	53	16	no	—
481	C₂²	—	106	96	no	—
482	C₂²	—	106	8	no	—
483	C₂²	—	106	8	no	—
484	C₂²	—	106	8	no	—
485	C₂²	—	106	8	no	—
486	C₂²	—	106	8	no	—
487	C₂²	—	212	64	no	—
488	C₂²	—	212	128	no	—
489	C₂²	—	212	48	no	—
490	C₂²	#16811181	1	16	yes	direct product factor
491	C₂²	C₂²×D₄×D₅₃	1	8	yes	—
492	C₂²	C₂²×D₄×D₅₃	1	6	yes	—
493	C₂²	C₂×#13133262	1	6	yes	—
494	C₂²	C₂×#13133263	1	6	yes	—
495	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
496	C₄	—	2	8	no	—
497	C₄	—	2	4	no	—
498	C₄	—	2	4	no	—
499	C₄	—	106	8	no	—
500	C₄	—	106	8	no	—
501	C₂	—	2	8	no	—
502	C₂	—	4	8	no	—
503	C₂	—	106	16	no	—
504	C₂	—	212	8	no	—
505	C₂	C₂×#16811181	1	12	yes	direct product factor
506	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
507	C₂	C₂²×#13133262	1	1	yes	characteristic
508	C₂	C₂²×#13133263	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×#121×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂²×#121	D₅₃
hypocenter	C₅₃	C₂²×#534
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#121

Complements

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

N	H ≅ G/N	type
1. C₂³×D₄×D₅₃	C₂	semidirect
2. C₂×#16811181	C₂	direct (N × H)
3. C₂²×#13133202	C₂	semidirect
4. C₂²×#13133199	C₂	semidirect
5. C₂²×#13133191	C₂	semidirect
6. C₂²×#13133189	C₂	semidirect
7. C₂²×#13133187	C₂	semidirect
8. C₂²×#13133185	C₂	semidirect
9. C₂³×#11719921	C₂	semidirect
10. C₂²×#121×C₅₃	C₂	semidirect
12. C₂⁵×D₅₃	C₂²	semidirect
14. C₂³×D₂₁₂	C₂²	semidirect
16. C₂³×C₂²⋊D₅₃	C₂²	semidirect
17. C₂³×C₂²⋊D₅₃	C₂²	semidirect
19. C₂²×D₄×D₅₃	C₂²	semidirect
20. C₂⁴×Dic₅₃	C₂²	semidirect
21. C₂³×C₄×D₅₃	C₂²	semidirect
22. #16811181	C₂²	direct (N × H)
23. C₂×#13133202	C₂²	semidirect
24. C₂×#13133199	C₂²	semidirect
25. C₂×#13133191	C₂²	semidirect
26. C₂×#13133189	C₂²	semidirect
27. C₂×#13133187	C₂²	semidirect
28. C₂×#13133185	C₂²	semidirect
29. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
30. C₂²×#11719919	C₂²	semidirect
31. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
32. C₂²×#11719921	C₂²	semidirect
33. C₂²×#11719921	C₂²	semidirect
34. C₂²×#11719921	C₂²	semidirect
35. C₂²×#11719921	C₂²	semidirect
36. C₂²×#11719921	C₂²	semidirect
37. C₂³×D₄×C₅₃	C₂²	semidirect
38. C₂³×D₄×C₅₃	C₂²	semidirect
39. C₂⁴×C₄×C₅₃	C₂²	semidirect
40. C₂×#121×C₅₃	C₂²	semidirect
41. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
42. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
48. C₂⁴×D₅₃	D₄	semidirect
52. C₂⁴×D₅₃	C₂³	semidirect
56. C₂²×D₂₁₂	C₂³	semidirect
61. C₂²×C₂²⋊D₅₃	C₂³	semidirect
62. C₂²×C₂²⋊D₅₃	C₂³	semidirect
64. C₂×D₄×D₅₃	C₂³	semidirect
68. C₂³×Dic₅₃	C₂³	semidirect
70. C₂²×C₄×D₅₃	C₂³	semidirect
71. #13133202	C₂³	semidirect
72. #13133199	C₂³	semidirect
73. #13133191	C₂³	semidirect
74. #13133189	C₂³	semidirect
75. #13133187	C₂³	semidirect
76. #13133185	C₂³	semidirect
77. C₂×C₄⋊Dic₅₃	C₂³	semidirect
78. C₂×#11719919	C₂³	semidirect
79. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
80. C₂×#11719921	C₂³	semidirect
81. C₂×#11719921	C₂³	semidirect
82. C₂×#11719921	C₂³	semidirect
83. C₂×#11719921	C₂³	semidirect
84. C₂×#11719921	C₂³	semidirect
89. C₂²×D₄×C₅₃	C₂³	semidirect
90. C₂²×D₄×C₅₃	C₂³	semidirect
95. C₂³×C₄×C₅₃	C₂³	semidirect
96. #121×C₅₃	C₂³	semidirect
97. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
98. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
121. C₂³×D₅₃	C₂×D₄	semidirect
125. C₂³×D₅₃	C₂⁴	semidirect
128. C₂×D₄×C₅₃	C₂⁴	semidirect
129. C₂×D₄×C₅₃	C₂⁴	semidirect
137. C₂²×C₄×C₅₃	C₂⁴	semidirect
142. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
143. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
149. C₂²×Dic₅₃	C₂⁴	semidirect
154. C₂×D₂₁₂	C₂⁴	semidirect
156. C₂×C₄×D₅₃	C₂⁴	semidirect
157. C₄⋊C₄×C₅₃	C₂⁴	semidirect
158. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
159. C₂²⋊Dic₅₃	C₂⁴	semidirect
160. #11719921	C₂⁴	semidirect
161. #11719921	C₂⁴	semidirect
162. #11719921	C₂⁴	semidirect
163. #11719921	C₂⁴	semidirect
164. C₄⋊Dic₅₃	C₂⁴	semidirect
165. #11719919	C₂⁴	semidirect
195. C₂²×D₅₃	C₂²×D₄	semidirect
226. C₂²×C₅₃	#534	semidirect
253. C₂²×#121	D₅₃	semidirect
259. C₂×C₅₃	C₂×#534	semidirect
274. C₂³×D₄	C₂×D₅₃	semidirect
275. C₂³×D₄	C₂×D₅₃	semidirect
278. C₂⁴×C₄	C₂×D₅₃	semidirect
284. C₂×#121	C₂×D₅₃	semidirect
289. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
296. C₂²×C₂²⋊C₄	C₂×D₅₃	semidirect
297. C₅₃	C₂²×#534	semidirect
324. C₂²×D₄	C₂²×D₅₃	semidirect
325. C₂²×D₄	C₂²×D₅₃	semidirect
336. C₂³×C₄	C₂²×D₅₃	semidirect
341. #121	C₂²×D₅₃	semidirect
346. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
353. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
394. C₂×D₄	C₂³×D₅₃	semidirect
395. C₂×D₄	C₂³×D₅₃	semidirect
411. C₂²×C₄	C₂³×D₅₃	semidirect
414. C₄⋊C₄	C₂³×D₅₃	semidirect
420. C₂²⋊C₄	C₂³×D₅₃	semidirect
490. C₂²	#16811181	direct (N × H)
505. C₂	C₂×#16811181	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₄×D₅₃	2
C₂×#16811181	2
C₂²×#13133202	2
C₂²×#13133199	2
C₂²×#13133191	2
C₂²×#13133189	2
C₂²×#13133187	2
C₂²×#13133185	2
C₂³×#11719921	2
C₂²×#121×C₅₃	2
C₂²×#534	53

Subgroups — C22×#16811181 #28290807

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811181 #28290807