Subgroups — C₂²×#16811257 #28291001

Order 13568.

20038 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₄×D₅₃	C₂	1	2	yes	maximal, complemented
3	C₂³×D₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16811257	C₂	1	48	yes	maximal, direct product factor
5	C₂²×#13133213	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#13133192	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#13133190	C₂	1	2	yes	maximal, complemented
8	C₂²×#13133187	C₂	1	2	yes	maximal, complemented
9	C₂²×#13133185	C₂	1	2	yes	maximal, complemented
10	C₂²×#13133179	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×C₄×C₂²⋊D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#121×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂⁵×D₅₃	C₂²	1	2	yes	complemented
14	C₂⁵×D₅₃	C₂²	1	2	yes	complemented
15	C₂³×D₂₁₂	—	2	16	no	—
16	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic
17	C₂³×D₂₁₂	C₂²	1	2	yes	complemented
18	C₂³×D₂₁₂	C₂²	1	2	yes	complemented
19	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
20	C₂³×D₂₁₂	C₂²	1	24	yes	complemented
21	C₂³×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
22	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
23	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
24	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
25	C₂²×D₄×D₅₃	—	2	16	no	—
26	C₂²×D₄×D₅₃	—	2	32	no	—
27	C₂²×D₄×D₅₃	C₂²	1	48	yes	complemented
28	C₂²×D₄×D₅₃	C₂²	1	24	yes	complemented
29	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic, complemented
30	C₂³×C₄×D₅₃	C₂²	1	2	yes	complemented
31	#16811257	C₂²	1	256	yes	direct product factor
32	C₂×#13133213	C₂²	1	24	yes	complemented
33	C₂×#13133192	C₂²	1	24	yes	complemented
34	C₂×#13133190	C₂²	1	48	yes	complemented
35	C₂×#13133187	C₂²	1	48	yes	complemented
36	C₂×#13133185	C₂²	1	48	yes	complemented
37	C₂×#13133179	C₂²	1	24	yes	complemented
38	C₂×C₄×C₂²⋊D₅₃	C₂²	1	24	yes	complemented
39	C₂²×#11719919	C₂²	1	1	yes	characteristic, complemented
40	C₂²×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
41	C₂²×#11719921	C₂²	1	1	yes	characteristic, complemented
42	C₂²×#11719921	C₂²	1	2	yes	complemented
43	C₂²×#11719921	C₂²	1	2	yes	complemented
44	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
45	C₂³×D₄×C₅₃	C₂²	1	2	yes	complemented
46	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
47	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
48	C₂×#121×C₅₃	C₂²	1	24	yes	complemented
49	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
50	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
51	C₂⁴×D₅₃	—	2	16	no	—
52	C₂⁴×D₅₃	—	2	16	no	—
53	C₂⁴×D₅₃	—	2	16	no	—
54	C₂⁴×D₅₃	—	2	16	no	—
55	C₂⁴×D₅₃	—	4	8	no	—
56	C₂⁴×D₅₃	—	4	8	no	—
57	C₂⁴×D₅₃	C₂³	1	2	yes	—
58	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
59	C₂⁴×D₅₃	C₂³	1	2	yes	—
60	C₂⁴×D₅₃	C₂³	1	24	yes	complemented
61	C₂⁴×D₅₃	C₂³	1	24	yes	complemented
62	C₂²×D₂₁₂	—	2	16	no	—
63	C₂²×D₂₁₂	—	2	16	no	—
64	C₂²×D₂₁₂	—	2	8	no	—
65	C₂²×D₂₁₂	—	2	192	no	—
66	C₂²×D₂₁₂	D₄	1	16	yes	complemented
67	C₂²×D₂₁₂	C₂³	1	12	yes	—
68	C₂²×D₂₁₂	C₂³	1	64	yes	complemented
69	C₂²×D₂₁₂	C₂³	1	24	yes	complemented
70	C₂²×D₂₁₂	C₂³	1	24	yes	complemented
71	C₂²×D₂₁₂	C₂³	1	12	yes	complemented
72	C₂²×C₂²⋊D₅₃	—	2	16	no	—
73	C₂²×C₂²⋊D₅₃	—	2	16	no	—
74	C₂²×C₂²⋊D₅₃	—	4	8	no	—
75	C₂²×C₂²⋊D₅₃	D₄	1	16	yes	complemented
76	C₂²×C₂²⋊D₅₃	C₂³	1	12	yes	—
77	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	complemented
78	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	complemented
79	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	complemented
80	C₂×D₄×D₅₃	—	2	384	no	—
81	C₂×D₄×D₅₃	—	2	192	no	—
82	C₂×D₄×D₅₃	C₂³	1	64	yes	complemented
83	C₂×D₄×D₅₃	C₂³	1	128	yes	complemented
84	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
85	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
86	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
87	C₂²×C₄×D₅₃	—	2	8	no	—
88	C₂²×C₄×D₅₃	—	2	16	no	—
89	C₂²×C₄×D₅₃	C₂³	1	12	yes	complemented
90	C₂²×C₄×D₅₃	C₂³	1	24	yes	complemented
91	#13133213	C₂³	1	64	yes	complemented
92	#13133192	C₂³	1	64	yes	complemented
93	#13133190	C₂³	1	128	yes	complemented
94	#13133187	C₂³	1	128	yes	complemented
95	#13133185	C₂³	1	128	yes	complemented
96	#13133179	C₂³	1	64	yes	complemented
97	C₄×C₂²⋊D₅₃	C₂³	1	64	yes	complemented
98	C₂×#11719919	C₂³	1	12	yes	complemented
99	C₂×C₂²⋊Dic₅₃	C₂³	1	12	yes	complemented
100	C₂×#11719921	C₂³	1	12	yes	complemented
101	C₂×#11719921	C₂³	1	24	yes	complemented
102	C₂×#11719921	C₂³	1	24	yes	complemented
103	C₂×C₄×Dic₅₃	C₂³	1	12	yes	complemented
104	C₂⁵×C₅₃	C₂³	1	2	yes	—
105	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
106	C₂²×D₄×C₅₃	—	2	16	no	—
107	C₂²×D₄×C₅₃	—	2	8	no	—
108	C₂²×D₄×C₅₃	C₂³	1	24	yes	complemented
109	C₂²×D₄×C₅₃	C₂³	1	12	yes	complemented
110	C₂³×C₄×C₅₃	—	2	8	no	—
111	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
112	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
113	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
114	C₂³×C₄×C₅₃	C₂³	1	12	yes	complemented
115	#121×C₅₃	C₂³	1	64	yes	complemented
116	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	complemented
117	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	complemented
118	C₂⁴×C₅₃	—	2	4	no	—
119	C₂⁴×C₅₃	—	2	4	no	—
120	C₂⁴×C₅₃	—	2	8	no	—
121	C₂⁴×C₅₃	—	2	8	no	—
122	C₂⁴×C₅₃	—	2	8	no	—
123	C₂⁴×C₅₃	C₂×D₄	1	8	yes	complemented
124	C₂⁴×C₅₃	C₂⁴	1	12	yes	—
125	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
126	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
127	C₂³×D₅₃	—	2	4	no	—
128	C₂³×D₅₃	—	2	8	no	—
129	C₂³×D₅₃	—	2	4	no	—
130	C₂³×D₅₃	—	2	8	no	—
131	C₂³×D₅₃	—	2	8	no	—
132	C₂³×D₅₃	—	2	8	no	—
133	C₂³×D₅₃	—	2	96	no	—
134	C₂³×D₅₃	—	2	8	no	—
135	C₂³×D₅₃	—	2	96	no	—
136	C₂³×D₅₃	—	2	96	no	—
137	C₂³×D₅₃	—	2	96	no	—
138	C₂³×D₅₃	—	4	48	no	—
139	C₂³×D₅₃	—	4	48	no	—
140	C₂³×D₅₃	—	4	128	no	—
141	C₂³×D₅₃	—	4	128	no	—
142	C₂³×D₅₃	C₂×D₄	1	16	yes	complemented
143	C₂³×D₅₃	C₂×D₄	1	8	yes	complemented
144	C₂³×D₅₃	C₂⁴	1	12	yes	—
145	C₂³×D₅₃	C₂⁴	1	6	yes	—
146	C₂³×D₅₃	C₂⁴	1	12	yes	—
147	C₂³×D₅₃	C₂⁴	1	32	yes	complemented
148	C₂³×D₅₃	C₂⁴	1	32	yes	complemented
149	C₂×D₄×C₅₃	—	2	96	no	—
150	C₂×D₄×C₅₃	—	2	48	no	—
151	C₂×D₄×C₅₃	C₂⁴	1	32	yes	complemented
152	C₂×D₄×C₅₃	C₂⁴	1	16	yes	complemented
153	C₂²×C₄×C₅₃	—	2	4	no	—
154	C₂²×C₄×C₅₃	—	2	8	no	—
155	C₂²×C₄×C₅₃	—	2	48	no	—
156	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	complemented
157	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
158	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
159	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
160	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	complemented
161	C₂×C₂²⋊D₅₃	—	2	96	no	—
162	C₂×C₂²⋊D₅₃	—	2	96	no	—
163	C₂×C₂²⋊D₅₃	—	4	48	no	—
164	C₂×C₂²⋊D₅₃	C₂×D₄	1	96	yes	complemented
165	C₂×C₂²⋊D₅₃	C₂⁴	1	16	yes	—
166	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	complemented
167	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	complemented
168	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	complemented
169	C₂²×Dic₅₃	—	2	4	no	—
170	C₂²×Dic₅₃	—	2	4	no	—
171	C₂²×Dic₅₃	C₂×D₄	1	8	yes	complemented
172	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
173	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
174	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
175	D₄×D₅₃	—	2	512	no	—
176	D₄×D₅₃	—	2	256	no	—
177	C₂×D₂₁₂	—	2	96	no	—
178	C₂×D₂₁₂	—	2	96	no	—
179	C₂×D₂₁₂	—	2	48	no	—
180	C₂×D₂₁₂	—	2	256	no	—
181	C₂×D₂₁₂	C₂×D₄	1	96	yes	complemented
182	C₂×D₂₁₂	C₂⁴	1	16	yes	—
183	C₂×D₂₁₂	C₂⁴	1	32	yes	complemented
184	C₂×D₂₁₂	C₂⁴	1	32	yes	complemented
185	C₂×D₂₁₂	C₂⁴	1	16	yes	complemented
186	C₂×C₄×D₅₃	—	2	48	no	—
187	C₂×C₄×D₅₃	—	2	96	no	—
188	C₂×C₄×D₅₃	C₂⁴	1	16	yes	complemented
189	C₂×C₄×D₅₃	C₂⁴	1	32	yes	complemented
190	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	complemented
191	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
192	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	complemented
193	#11719921	C₂⁴	1	16	yes	complemented
194	#11719921	C₂⁴	1	32	yes	complemented
195	#11719921	C₂⁴	1	32	yes	complemented
196	#11719919	C₂⁴	1	16	yes	complemented
197	C₄×Dic₅₃	C₂⁴	1	16	yes	complemented
198	C₂³×C₅₃	—	2	32	no	—
199	C₂³×C₅₃	—	2	12	no	—
200	C₂³×C₅₃	—	2	12	no	—
201	C₂³×C₅₃	—	2	24	no	—
202	C₂³×C₅₃	—	2	24	no	—
203	C₂³×C₅₃	—	2	24	no	—
204	C₂³×C₅₃	—	4	32	no	—
205	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
206	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
207	C₂³×C₅₃	C₂²×D₄	1	24	yes	complemented
208	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
209	C₂³×C₅₃	C₂⁵	1	8	yes	—
210	C₂³×C₅₃	C₂⁵	1	4	yes	—
211	C₂³×C₅₃	C₂⁵	1	3	yes	—
212	C₂²×D₅₃	—	2	32	no	—
213	C₂²×D₅₃	—	2	64	no	—
214	C₂²×D₅₃	—	2	12	no	—
215	C₂²×D₅₃	—	2	24	no	—
216	C₂²×D₅₃	—	2	12	no	—
217	C₂²×D₅₃	—	2	24	no	—
218	C₂²×D₅₃	—	2	24	no	—
219	C₂²×D₅₃	—	2	24	no	—
220	C₂²×D₅₃	—	2	64	no	—
221	C₂²×D₅₃	—	2	24	no	—
222	C₂²×D₅₃	—	2	64	no	—
223	C₂²×D₅₃	—	2	64	no	—
224	C₂²×D₅₃	—	2	64	no	—
225	C₂²×D₅₃	—	4	32	no	—
226	C₂²×D₅₃	—	4	32	no	—
227	C₂²×D₅₃	—	4	384	no	—
228	C₂²×D₅₃	—	4	384	no	—
229	C₂²×D₅₃	—	4	32	no	—
230	C₂²×D₅₃	C₂²×D₄	1	24	yes	complemented
231	C₂²×D₅₃	C₂²×D₄	1	48	yes	complemented
232	C₂²×D₅₃	C₂⁵	1	8	yes	—
233	C₂²×D₅₃	C₂⁵	1	8	yes	—
234	C₂²×D₅₃	C₂⁵	1	4	yes	—
235	D₄×C₅₃	—	2	64	no	—
236	D₄×C₅₃	—	2	32	no	—
237	C₂×C₄×C₅₃	—	2	24	no	—
238	C₂×C₄×C₅₃	—	2	12	no	—
239	C₂×C₄×C₅₃	—	2	32	no	—
240	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	complemented
241	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
242	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
243	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
244	C₂²⋊D₅₃	—	2	64	no	—
245	C₂²⋊D₅₃	—	2	64	no	—
246	C₂²⋊D₅₃	—	4	32	no	—
247	C₂²⋊D₅₃	C₂²×D₄	1	64	yes	complemented
248	C₂×Dic₅₃	—	2	12	no	—
249	C₂×Dic₅₃	—	2	12	no	—
250	C₂×Dic₅₃	C₂²×D₄	1	24	yes	complemented
251	C₂×Dic₅₃	C₂⁵	1	4	yes	—
252	C₂×Dic₅₃	C₂⁵	1	4	yes	—
253	C₂×Dic₅₃	C₂⁵	1	4	yes	—
254	D₂₁₂	—	2	64	no	—
255	D₂₁₂	—	2	64	no	—
256	D₂₁₂	—	2	32	no	—
257	D₂₁₂	C₂²×D₄	1	64	yes	complemented
258	C₄×D₅₃	—	2	32	no	—
259	C₄×D₅₃	—	2	64	no	—
260	C₂²×D₄²	—	53	1	no	maximal, Sylow
261	C₂²×C₅₃	—	2	4	no	—
262	C₂²×C₅₃	—	2	4	no	—
263	C₂²×C₅₃	—	2	48	no	—
264	C₂²×C₅₃	—	2	8	no	—
265	C₂²×C₅₃	—	2	8	no	—
266	C₂²×C₅₃	—	2	8	no	—
267	C₂²×C₅₃	—	4	48	no	—
268	C₂²×C₅₃	D₄²	1	16	yes	complemented
269	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
270	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
271	C₂²×C₅₃	C₂³×D₄	1	8	yes	complemented
272	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
273	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
274	C₂×D₅₃	—	2	4	no	—
275	C₂×D₅₃	—	2	8	no	—
276	C₂×D₅₃	—	2	8	no	—
277	C₂×D₅₃	—	2	4	no	—
278	C₂×D₅₃	—	2	48	no	—
279	C₂×D₅₃	—	2	96	no	—
280	C₂×D₅₃	—	2	8	no	—
281	C₂×D₅₃	—	2	8	no	—
282	C₂×D₅₃	—	2	8	no	—
283	C₂×D₅₃	—	4	128	no	—
284	C₂×D₅₃	—	4	128	no	—
285	C₂×D₅₃	—	4	48	no	—
286	C₂×D₅₃	C₂³×D₄	1	8	yes	complemented
287	C₂×D₅₃	C₂³×D₄	1	16	yes	complemented
288	C₄×C₅₃	—	2	8	no	—
289	C₄×C₅₃	—	2	4	no	—
290	C₄×C₅₃	C₂³×D₄	1	8	yes	complemented
291	Dic₅₃	—	2	4	no	—
292	Dic₅₃	—	2	4	no	—
293	Dic₅₃	C₂³×D₄	1	8	yes	complemented
294	C₂⁴×D₄	—	53	2	no	—
295	C₂⁴×D₄	—	53	1	no	—
296	C₂⁴×D₄	—	53	1	no	—
297	C₂×D₄²	—	53	48	no	—
298	C₂²×#127	—	53	1	no	—
299	C₂²×#121	—	53	1	no	—
300	C₂²×#121	—	53	2	no	—
301	C₂²×#121	D₅₃	1	1	yes	characteristic, complemented
302	C₂²×#120	—	53	2	no	—
303	C₂²×#120	—	53	2	no	—
304	C₂²×C₄×D₄	—	53	1	no	—
305	C₂²×C₄×D₄	—	53	1	no	—
306	C₂×C₅₃	—	2	8	no	—
307	C₂×C₅₃	—	4	8	no	—
308	C₂×C₅₃	C₂×D₄²	1	12	yes	complemented
309	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
310	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
311	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
312	D₅₃	—	2	8	no	—
313	D₅₃	—	2	16	no	—
314	D₅₃	—	4	8	no	—
315	C₂⁶	—	53	2	no	—
316	C₂⁶	—	53	2	no	—
317	C₂³×D₄	—	53	1	no	—
318	C₂³×D₄	—	53	2	no	—
319	C₂³×D₄	—	53	2	no	—
320	C₂³×D₄	—	53	2	no	—
321	C₂³×D₄	—	53	48	no	—
322	C₂³×D₄	—	53	24	no	—
323	C₂³×D₄	—	53	1	no	—
324	C₂³×D₄	—	53	2	no	—
325	C₂³×D₄	—	53	2	no	—
326	C₂³×D₄	—	53	1	no	—
327	C₂³×D₄	—	53	24	no	—
328	C₂³×D₄	—	106	16	no	—
329	C₂³×D₄	—	106	32	no	—
330	C₂³×D₄	—	106	16	no	—
331	C₂³×D₄	C₂×D₅₃	1	2	yes	complemented
332	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic, complemented
333	C₂⁴×C₄	—	53	2	no	—
334	C₂⁴×C₄	—	53	1	no	—
335	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
336	D₄²	—	53	256	no	—
337	C₂×#127	—	53	24	no	—
338	C₂×#121	—	53	24	no	—
339	C₂×#121	—	53	48	no	—
340	C₂×#121	C₂×D₅₃	1	24	yes	complemented
341	C₂×#120	—	53	48	no	—
342	C₂×#120	—	53	48	no	—
343	C₂×C₄×D₄	—	53	24	no	—
344	C₂×C₄×D₄	—	53	24	no	—
345	C₂²×C₄⋊C₄	—	53	1	no	—
346	C₂²×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
347	C₂²×C₂²⋊C₄	—	53	1	no	—
348	C₂²×C₂²⋊C₄	—	53	2	no	—
349	C₂²×C₂²⋊C₄	—	53	1	no	—
350	C₂²×C₂²⋊C₄	—	53	2	no	—
351	C₂²×C₂²⋊C₄	C₂×D₅₃	1	2	yes	complemented
352	C₂²×C₄²	—	53	1	no	—
353	C₅₃	C₂²×D₄²	1	1	yes	characteristic, complemented, Sylow
354	C₂⁵	—	53	1	no	—
355	C₂⁵	—	53	2	no	—
356	C₂⁵	—	53	2	no	—
357	C₂⁵	—	53	24	no	—
358	C₂⁵	—	53	24	no	—
359	C₂⁵	—	106	16	no	—
360	C₂⁵	—	106	16	no	—
361	C₂⁵	—	106	16	no	—
362	C₂⁵	—	106	16	no	—
363	C₂⁵	—	212	8	no	—
364	C₂⁵	—	212	8	no	—
365	C₂⁵	C₂²×D₅₃	1	2	yes	—
366	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
367	C₂²×D₄	—	2	16	no	—
368	C₂²×D₄	—	2	8	no	—
369	C₂²×D₄	—	53	16	no	—
370	C₂²×D₄	—	53	12	no	—
371	C₂²×D₄	—	53	24	no	—
372	C₂²×D₄	—	53	24	no	—
373	C₂²×D₄	—	53	24	no	—
374	C₂²×D₄	—	53	128	no	—
375	C₂²×D₄	—	53	64	no	—
376	C₂²×D₄	—	53	16	no	—
377	C₂²×D₄	—	53	12	no	—
378	C₂²×D₄	—	53	24	no	—
379	C₂²×D₄	—	53	24	no	—
380	C₂²×D₄	—	53	12	no	—
381	C₂²×D₄	—	53	64	no	—
382	C₂²×D₄	—	106	192	no	—
383	C₂²×D₄	—	106	16	no	—
384	C₂²×D₄	—	106	16	no	—
385	C₂²×D₄	—	106	16	no	—
386	C₂²×D₄	—	106	384	no	—
387	C₂²×D₄	—	106	192	no	—
388	C₂²×D₄	—	106	16	no	—
389	C₂²×D₄	—	106	8	no	—
390	C₂²×D₄	—	212	8	no	—
391	C₂²×D₄	C₂²×D₅₃	1	24	yes	complemented
392	C₂²×D₄	C₂²×D₅₃	1	12	yes	complemented
393	C₂³×C₄	—	2	8	no	—
394	C₂³×C₄	—	53	1	no	—
395	C₂³×C₄	—	53	1	no	—
396	C₂³×C₄	—	53	1	no	—
397	C₂³×C₄	—	53	24	no	—
398	C₂³×C₄	—	53	12	no	—
399	C₂³×C₄	—	106	16	no	—
400	C₂³×C₄	—	106	8	no	—
401	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
402	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
403	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
404	C₂³×C₄	C₂²×D₅₃	1	12	yes	complemented
405	#127	—	53	64	no	—
406	#121	—	53	64	no	—
407	#121	—	53	128	no	—
408	#121	C₂²×D₅₃	1	64	yes	complemented
409	#120	—	53	128	no	—
410	#120	—	53	128	no	—
411	C₄×D₄	—	53	64	no	—
412	C₄×D₄	—	53	64	no	—
413	C₂×C₄⋊C₄	—	53	12	no	—
414	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	complemented
415	C₂×C₂²⋊C₄	—	53	12	no	—
416	C₂×C₂²⋊C₄	—	53	24	no	—
417	C₂×C₂²⋊C₄	—	53	12	no	—
418	C₂×C₂²⋊C₄	—	53	24	no	—
419	C₂×C₂²⋊C₄	C₂²×D₅₃	1	24	yes	complemented
420	C₂×C₄²	—	53	12	no	—
421	C₂⁴	—	2	4	no	—
422	C₂⁴	—	2	4	no	—
423	C₂⁴	—	2	8	no	—
424	C₂⁴	—	2	8	no	—
425	C₂⁴	—	2	8	no	—
426	C₂⁴	—	53	8	no	—
427	C₂⁴	—	53	16	no	—
428	C₂⁴	—	53	6	no	—
429	C₂⁴	—	53	12	no	—
430	C₂⁴	—	53	12	no	—
431	C₂⁴	—	53	32	no	—
432	C₂⁴	—	53	32	no	—
433	C₂⁴	—	106	96	no	—
434	C₂⁴	—	106	96	no	—
435	C₂⁴	—	106	96	no	—
436	C₂⁴	—	106	8	no	—
437	C₂⁴	—	106	4	no	—
438	C₂⁴	—	106	8	no	—
439	C₂⁴	—	106	4	no	—
440	C₂⁴	—	106	8	no	—
441	C₂⁴	—	106	8	no	—
442	C₂⁴	—	106	96	no	—
443	C₂⁴	—	106	8	no	—
444	C₂⁴	—	212	48	no	—
445	C₂⁴	—	212	48	no	—
446	C₂⁴	—	212	128	no	—
447	C₂⁴	—	212	128	no	—
448	C₂⁴	D₄×D₅₃	1	8	yes	complemented
449	C₂⁴	C₂³×D₅₃	1	12	yes	—
450	C₂⁴	C₂³×D₅₃	1	6	yes	—
451	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
452	C₂×D₄	—	2	96	no	—
453	C₂×D₄	—	2	48	no	—
454	C₂×D₄	—	53	96	no	—
455	C₂×D₄	—	53	16	no	—
456	C₂×D₄	—	53	32	no	—
457	C₂×D₄	—	53	32	no	—
458	C₂×D₄	—	53	32	no	—
459	C₂×D₄	—	53	96	no	—
460	C₂×D₄	—	53	16	no	—
461	C₂×D₄	—	53	32	no	—
462	C₂×D₄	—	53	32	no	—
463	C₂×D₄	—	53	16	no	—
464	C₂×D₄	—	106	256	no	—
465	C₂×D₄	—	106	96	no	—
466	C₂×D₄	—	106	96	no	—
467	C₂×D₄	—	106	96	no	—
468	C₂×D₄	—	106	512	no	—
469	C₂×D₄	—	106	256	no	—
470	C₂×D₄	—	106	96	no	—
471	C₂×D₄	—	106	48	no	—
472	C₂×D₄	—	212	48	no	—
473	C₂×D₄	C₂³×D₅₃	1	32	yes	complemented
474	C₂×D₄	C₂³×D₅₃	1	16	yes	complemented
475	C₂²×C₄	—	2	4	no	—
476	C₂²×C₄	—	2	8	no	—
477	C₂²×C₄	—	2	48	no	—
478	C₂²×C₄	—	53	8	no	—
479	C₂²×C₄	—	53	6	no	—
480	C₂²×C₄	—	53	6	no	—
481	C₂²×C₄	—	53	6	no	—
482	C₂²×C₄	—	53	32	no	—
483	C₂²×C₄	—	53	16	no	—
484	C₂²×C₄	—	106	96	no	—
485	C₂²×C₄	—	106	4	no	—
486	C₂²×C₄	—	106	4	no	—
487	C₂²×C₄	—	106	48	no	—
488	C₂²×C₄	D₄×D₅₃	1	8	yes	complemented
489	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
490	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
491	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
492	C₂²×C₄	C₂³×D₅₃	1	16	yes	complemented
493	C₄⋊C₄	—	53	16	no	—
494	C₄⋊C₄	C₂³×D₅₃	1	16	yes	complemented
495	C₂²⋊C₄	—	53	16	no	—
496	C₂²⋊C₄	—	53	32	no	—
497	C₂²⋊C₄	—	53	16	no	—
498	C₂²⋊C₄	—	53	32	no	—
499	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	complemented
500	C₄²	—	53	16	no	—
501	C₂³	—	2	12	no	—
502	C₂³	—	2	12	no	—
503	C₂³	—	2	32	no	—
504	C₂³	—	2	24	no	—
505	C₂³	—	2	24	no	—
506	C₂³	—	2	24	no	—
507	C₂³	—	4	32	no	—
508	C₂³	—	53	24	no	—
509	C₂³	—	53	48	no	—
510	C₂³	—	53	8	no	—
511	C₂³	—	53	4	no	—
512	C₂³	—	53	8	no	—
513	C₂³	—	106	64	no	—
514	C₂³	—	106	64	no	—
515	C₂³	—	106	64	no	—
516	C₂³	—	106	24	no	—
517	C₂³	—	106	12	no	—
518	C₂³	—	106	24	no	—
519	C₂³	—	106	12	no	—
520	C₂³	—	106	64	no	—
521	C₂³	—	106	32	no	—
522	C₂³	—	106	24	no	—
523	C₂³	—	106	24	no	—
524	C₂³	—	106	64	no	—
525	C₂³	—	106	24	no	—
526	C₂³	—	212	32	no	—
527	C₂³	—	212	32	no	—
528	C₂³	—	212	384	no	—
529	C₂³	—	212	384	no	—
530	C₂³	—	212	32	no	—
531	C₂³	C₂×D₄×D₅₃	1	4	yes	—
532	C₂³	C₂×D₄×D₅₃	1	4	yes	—
533	C₂³	C₂×D₄×D₅₃	1	24	yes	complemented
534	C₂³	#13133263	1	4	yes	—
535	C₂³	C₂⁴×D₅₃	1	8	yes	—
536	C₂³	C₂⁴×D₅₃	1	4	yes	—
537	C₂³	C₂⁴×D₅₃	1	3	yes	—
538	D₄	—	2	64	no	—
539	D₄	—	2	32	no	—
540	D₄	—	53	64	no	—
541	D₄	—	53	64	no	—
542	D₄	—	106	64	no	—
543	D₄	—	106	64	no	—
544	D₄	—	106	64	no	—
545	D₄	—	106	64	no	—
546	D₄	—	106	32	no	—
547	D₄	—	212	32	no	—
548	C₂×C₄	—	2	12	no	—
549	C₂×C₄	—	2	24	no	—
550	C₂×C₄	—	2	32	no	—
551	C₂×C₄	—	53	24	no	—
552	C₂×C₄	—	53	4	no	—
553	C₂×C₄	—	53	4	no	—
554	C₂×C₄	—	53	4	no	—
555	C₂×C₄	—	106	64	no	—
556	C₂×C₄	—	106	12	no	—
557	C₂×C₄	—	106	12	no	—
558	C₂×C₄	—	106	32	no	—
559	C₂×C₄	C₂×D₄×D₅₃	1	24	yes	complemented
560	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
561	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
562	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
563	C₂²	—	2	48	no	—
564	C₂²	—	2	4	no	—
565	C₂²	—	2	4	no	—
566	C₂²	—	2	8	no	—
567	C₂²	—	2	8	no	—
568	C₂²	—	2	8	no	—
569	C₂²	—	4	48	no	—
570	C₂²	—	53	8	no	—
571	C₂²	—	53	16	no	—
572	C₂²	—	106	96	no	—
573	C₂²	—	106	48	no	—
574	C₂²	—	106	8	no	—
575	C₂²	—	106	4	no	—
576	C₂²	—	106	8	no	—
577	C₂²	—	106	4	no	—
578	C₂²	—	106	8	no	—
579	C₂²	—	106	8	no	—
580	C₂²	—	106	8	no	—
581	C₂²	—	212	128	no	—
582	C₂²	—	212	128	no	—
583	C₂²	—	212	48	no	—
584	C₂²	#16811257	1	16	yes	direct product factor
585	C₂²	C₂²×D₄×D₅₃	1	6	yes	—
586	C₂²	C₂²×D₄×D₅₃	1	6	yes	—
587	C₂²	C₂²×D₄×D₅₃	1	8	yes	complemented
588	C₂²	C₂×#13133263	1	6	yes	—
589	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
590	C₄	—	2	4	no	—
591	C₄	—	2	8	no	—
592	C₄	—	53	8	no	—
593	C₄	—	106	4	no	—
594	C₄	—	106	4	no	—
595	C₄	C₂²×D₄×D₅₃	1	8	yes	complemented
596	C₂	—	2	8	no	—
597	C₂	—	4	8	no	—
598	C₂	—	106	16	no	—
599	C₂	—	106	8	no	—
600	C₂	—	212	8	no	—
601	C₂	C₂×#16811257	1	12	yes	direct product factor
602	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
603	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
604	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×D₄²
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. C₂⁴×D₂₁₂	C₂	semidirect
2. C₂³×D₄×D₅₃	C₂	semidirect
3. C₂³×D₄×D₅₃	C₂	semidirect
4. C₂×#16811257	C₂	direct (N × H)
5. C₂²×#13133213	C₂	semidirect
6. C₂²×#13133192	C₂	semidirect
7. C₂²×#13133190	C₂	semidirect
8. C₂²×#13133187	C₂	semidirect
9. C₂²×#13133185	C₂	semidirect
10. C₂²×#13133179	C₂	semidirect
11. C₂²×C₄×C₂²⋊D₅₃	C₂	semidirect
12. C₂²×#121×C₅₃	C₂	semidirect
13. C₂⁵×D₅₃	C₂²	semidirect
14. C₂⁵×D₅₃	C₂²	semidirect
17. C₂³×D₂₁₂	C₂²	semidirect
18. C₂³×D₂₁₂	C₂²	semidirect
19. C₂³×D₂₁₂	C₂²	semidirect
20. C₂³×D₂₁₂	C₂²	semidirect
22. C₂³×C₂²⋊D₅₃	C₂²	semidirect
23. C₂³×C₂²⋊D₅₃	C₂²	semidirect
24. C₂³×C₂²⋊D₅₃	C₂²	semidirect
27. C₂²×D₄×D₅₃	C₂²	semidirect
28. C₂²×D₄×D₅₃	C₂²	semidirect
29. C₂³×C₄×D₅₃	C₂²	semidirect
30. C₂³×C₄×D₅₃	C₂²	semidirect
31. #16811257	C₂²	direct (N × H)
32. C₂×#13133213	C₂²	semidirect
33. C₂×#13133192	C₂²	semidirect
34. C₂×#13133190	C₂²	semidirect
35. C₂×#13133187	C₂²	semidirect
36. C₂×#13133185	C₂²	semidirect
37. C₂×#13133179	C₂²	semidirect
38. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
39. C₂²×#11719919	C₂²	semidirect
40. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
41. C₂²×#11719921	C₂²	semidirect
42. C₂²×#11719921	C₂²	semidirect
43. C₂²×#11719921	C₂²	semidirect
44. C₂²×C₄×Dic₅₃	C₂²	semidirect
45. C₂³×D₄×C₅₃	C₂²	semidirect
46. C₂³×D₄×C₅₃	C₂²	semidirect
47. C₂⁴×C₄×C₅₃	C₂²	semidirect
48. C₂×#121×C₅₃	C₂²	semidirect
49. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
50. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
60. C₂⁴×D₅₃	C₂³	semidirect
61. C₂⁴×D₅₃	C₂³	semidirect
66. C₂²×D₂₁₂	D₄	semidirect
68. C₂²×D₂₁₂	C₂³	semidirect
69. C₂²×D₂₁₂	C₂³	semidirect
70. C₂²×D₂₁₂	C₂³	semidirect
71. C₂²×D₂₁₂	C₂³	semidirect
75. C₂²×C₂²⋊D₅₃	D₄	semidirect
77. C₂²×C₂²⋊D₅₃	C₂³	semidirect
78. C₂²×C₂²⋊D₅₃	C₂³	semidirect
79. C₂²×C₂²⋊D₅₃	C₂³	semidirect
82. C₂×D₄×D₅₃	C₂³	semidirect
83. C₂×D₄×D₅₃	C₂³	semidirect
89. C₂²×C₄×D₅₃	C₂³	semidirect
90. C₂²×C₄×D₅₃	C₂³	semidirect
91. #13133213	C₂³	semidirect
92. #13133192	C₂³	semidirect
93. #13133190	C₂³	semidirect
94. #13133187	C₂³	semidirect
95. #13133185	C₂³	semidirect
96. #13133179	C₂³	semidirect
97. C₄×C₂²⋊D₅₃	C₂³	semidirect
98. C₂×#11719919	C₂³	semidirect
99. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
100. C₂×#11719921	C₂³	semidirect
101. C₂×#11719921	C₂³	semidirect
102. C₂×#11719921	C₂³	semidirect
103. C₂×C₄×Dic₅₃	C₂³	semidirect
108. C₂²×D₄×C₅₃	C₂³	semidirect
109. C₂²×D₄×C₅₃	C₂³	semidirect
114. C₂³×C₄×C₅₃	C₂³	semidirect
115. #121×C₅₃	C₂³	semidirect
116. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
117. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
123. C₂⁴×C₅₃	C₂×D₄	semidirect
142. C₂³×D₅₃	C₂×D₄	semidirect
143. C₂³×D₅₃	C₂×D₄	semidirect
147. C₂³×D₅₃	C₂⁴	semidirect
148. C₂³×D₅₃	C₂⁴	semidirect
151. C₂×D₄×C₅₃	C₂⁴	semidirect
152. C₂×D₄×C₅₃	C₂⁴	semidirect
156. C₂²×C₄×C₅₃	C₂×D₄	semidirect
160. C₂²×C₄×C₅₃	C₂⁴	semidirect
164. C₂×C₂²⋊D₅₃	C₂×D₄	semidirect
166. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
167. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
168. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
171. C₂²×Dic₅₃	C₂×D₄	semidirect
181. C₂×D₂₁₂	C₂×D₄	semidirect
183. C₂×D₂₁₂	C₂⁴	semidirect
184. C₂×D₂₁₂	C₂⁴	semidirect
185. C₂×D₂₁₂	C₂⁴	semidirect
188. C₂×C₄×D₅₃	C₂⁴	semidirect
189. C₂×C₄×D₅₃	C₂⁴	semidirect
190. C₄⋊C₄×C₅₃	C₂⁴	semidirect
191. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
192. C₂²⋊Dic₅₃	C₂⁴	semidirect
193. #11719921	C₂⁴	semidirect
194. #11719921	C₂⁴	semidirect
195. #11719921	C₂⁴	semidirect
196. #11719919	C₂⁴	semidirect
197. C₄×Dic₅₃	C₂⁴	semidirect
207. C₂³×C₅₃	C₂²×D₄	semidirect
230. C₂²×D₅₃	C₂²×D₄	semidirect
231. C₂²×D₅₃	C₂²×D₄	semidirect
240. C₂×C₄×C₅₃	C₂²×D₄	semidirect
247. C₂²⋊D₅₃	C₂²×D₄	semidirect
250. C₂×Dic₅₃	C₂²×D₄	semidirect
257. D₂₁₂	C₂²×D₄	semidirect
268. C₂²×C₅₃	D₄²	semidirect
271. C₂²×C₅₃	C₂³×D₄	semidirect
286. C₂×D₅₃	C₂³×D₄	semidirect
287. C₂×D₅₃	C₂³×D₄	semidirect
290. C₄×C₅₃	C₂³×D₄	semidirect
293. Dic₅₃	C₂³×D₄	semidirect
301. C₂²×#121	D₅₃	semidirect
308. C₂×C₅₃	C₂×D₄²	semidirect
331. C₂³×D₄	C₂×D₅₃	semidirect
332. C₂³×D₄	C₂×D₅₃	semidirect
335. C₂⁴×C₄	C₂×D₅₃	semidirect
340. C₂×#121	C₂×D₅₃	semidirect
346. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
351. C₂²×C₂²⋊C₄	C₂×D₅₃	semidirect
353. C₅₃	C₂²×D₄²	semidirect
391. C₂²×D₄	C₂²×D₅₃	semidirect
392. C₂²×D₄	C₂²×D₅₃	semidirect
404. C₂³×C₄	C₂²×D₅₃	semidirect
408. #121	C₂²×D₅₃	semidirect
414. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
419. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
448. C₂⁴	D₄×D₅₃	semidirect
473. C₂×D₄	C₂³×D₅₃	semidirect
474. C₂×D₄	C₂³×D₅₃	semidirect
488. C₂²×C₄	D₄×D₅₃	semidirect
492. C₂²×C₄	C₂³×D₅₃	semidirect
494. C₄⋊C₄	C₂³×D₅₃	semidirect
499. C₂²⋊C₄	C₂³×D₅₃	semidirect
533. C₂³	C₂×D₄×D₅₃	semidirect
559. C₂×C₄	C₂×D₄×D₅₃	semidirect
584. C₂²	#16811257	direct (N × H)
587. C₂²	C₂²×D₄×D₅₃	semidirect
595. C₄	C₂²×D₄×D₅₃	semidirect
601. C₂	C₂×#16811257	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×D₂₁₂	2
C₂³×D₄×D₅₃	2
C₂×#16811257	2
C₂²×#13133213	2
C₂²×#13133192	2
C₂²×#13133190	2
C₂²×#13133187	2
C₂²×#13133185	2
C₂²×#13133179	2
C₂²×C₄×C₂²⋊D₅₃	2
C₂²×#121×C₅₃	2
C₂²×D₄²	53

Subgroups — C22×#16811257 #28291001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811257 #28291001