Subgroups — C₂×#20288491 #28292410

Order 13568.

12982 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133262	C₂	1	2	yes	maximal, complemented
2	#20288491	C₂	1	32	yes	maximal, direct product factor
3	C₂×#16811293	C₂	1	4	yes	maximal, complemented
4	C₂×#16811258	C₂	1	4	yes	maximal, complemented
5	C₂×D₄×C₂²⋊D₅₃	C₂	1	8	yes	maximal, complemented
6	C₂×#16811190	C₂	1	2	yes	maximal, complemented
7	C₂×#16811186	C₂	1	4	yes	maximal, complemented
8	C₂×#16811173	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16811171	C₂	1	2	yes	maximal, complemented
10	C₂×#127×D₅₃	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×#11719945	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#127×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#13133262	—	2	32	no	—
14	C₂×#13133262	C₂²	1	32	yes	complemented
15	C₂²×#11719945	C₂²	1	1	yes	characteristic
16	C₂²×#11719945	C₂²	1	2	yes	complemented
17	C₂²×#11719947	C₂²	1	4	yes	complemented
18	C₂²×#11719947	C₂²	1	4	yes	complemented
19	C₂³×C₂²⋊D₅₃	C₂²	1	8	yes	complemented
20	C₂²×D₄×D₅₃	C₂²	1	4	yes	complemented
21	C₂²×D₄×D₅₃	C₂²	1	4	yes	complemented
22	C₂²×D₄×D₅₃	C₂²	1	8	yes	complemented
23	#16811293	C₂²	1	64	yes	complemented
24	#16811258	C₂²	1	64	yes	complemented
25	D₄×C₂²⋊D₅₃	C₂²	1	128	yes	complemented
26	#16811190	C₂²	1	32	yes	complemented
27	#16811186	C₂²	1	64	yes	complemented
28	#16811173	C₂²	1	16	yes	complemented
29	#16811171	C₂²	1	32	yes	complemented
30	C₂×#13133217	C₂²	1	1	yes	characteristic, complemented
31	C₂×#13133214	C₂²	1	1	yes	characteristic, complemented
32	C₂×#13133213	C₂²	1	4	yes	complemented
33	C₂×#13133213	C₂²	1	8	yes	complemented
34	#127×D₅₃	C₂²	1	32	yes	complemented
35	C₂×#13133205	C₂²	1	2	yes	complemented
36	C₂×#13133204	C₂²	1	4	yes	complemented
37	C₂×#13133191	C₂²	1	4	yes	complemented
38	C₂×#13133189	C₂²	1	16	yes	complemented
39	C₂×#13133188	C₂²	1	8	yes	complemented
40	C₂×#13133188	C₂²	1	8	yes	complemented
41	C₂×#13133186	C₂²	1	8	yes	complemented
42	C₂×#13133185	C₂²	1	16	yes	complemented
43	C₄×#11719945	C₂²	1	16	yes	complemented
44	C₂×C₄×Dic₁₀₆	C₂²	1	2	yes	complemented
45	C₂×C₄×D₂₁₂	C₂²	1	2	yes	complemented
46	C₂×C₄×D₂₁₂	C₂²	1	4	yes	complemented
47	C₂×D₄×Dic₅₃	C₂²	1	8	yes	complemented
48	C₂×C₄×C₂²⋊D₅₃	C₂²	1	4	yes	complemented
49	C₂×#13133176	C₂²	1	1	yes	characteristic, complemented
50	C₂×#13133174	C₂²	1	2	yes	complemented
51	C₂×C₄²×D₅₃	C₂²	1	2	yes	complemented
52	C₂³×D₄×C₅₃	C₂²	1	2	yes	complemented
53	C₂³×D₄×C₅₃	C₂²	1	4	yes	complemented
54	C₂×#127×C₅₃	C₂²	1	2	yes	complemented
55	C₂×#127×C₅₃	C₂²	1	4	yes	complemented
56	C₂×#127×C₅₃	C₂²	1	2	yes	complemented
57	C₂×#127×C₅₃	C₂²	1	16	yes	complemented
58	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
59	C₂⁴×D₅₃	C₂³	1	8	yes	—
60	#13133262	—	2	256	no	—
61	C₂×#11719945	—	2	16	no	—
62	C₂×#11719945	D₄	1	16	yes	complemented
63	C₂×#11719945	C₂³	1	8	yes	—
64	C₂×#11719945	C₂³	1	16	yes	complemented
65	C₂×#11719947	—	2	32	no	—
66	C₂×#11719947	—	2	32	no	—
67	C₂×#11719947	C₂³	1	32	yes	complemented
68	C₂×#11719947	C₂³	1	32	yes	complemented
69	C₂²×Dic₁₀₆	C₂³	1	2	yes	—
70	C₂²×Dic₁₀₆	C₂³	1	1	yes	characteristic
71	C₂²×D₂₁₂	C₂³	1	4	yes	—
72	C₂²×D₂₁₂	C₂³	1	2	yes	—
73	C₂²×D₂₁₂	C₂³	1	1	yes	characteristic
74	C₂²×C₂²⋊D₅₃	—	2	64	no	—
75	C₂²×C₂²⋊D₅₃	C₂³	1	8	yes	—
76	C₂²×C₂²⋊D₅₃	C₂³	1	4	yes	—
77	C₂²×C₂²⋊D₅₃	C₂³	1	8	yes	—
78	C₂²×C₂²⋊D₅₃	C₂³	1	16	yes	—
79	C₂²×C₂²⋊D₅₃	C₂³	1	2	yes	—
80	C₂²×C₂²⋊D₅₃	C₂³	1	64	yes	complemented
81	C₂×D₄×D₅₃	—	2	32	no	—
82	C₂×D₄×D₅₃	—	2	32	no	—
83	C₂×D₄×D₅₃	—	4	32	no	—
84	C₂×D₄×D₅₃	C₂³	1	32	yes	complemented
85	C₂×D₄×D₅₃	C₂³	1	32	yes	complemented
86	C₂×D₄×D₅₃	C₂³	1	64	yes	complemented
87	C₂³×Dic₅₃	C₂³	1	8	yes	—
88	C₂²×C₄×D₅₃	C₂³	1	4	yes	—
89	C₂²×C₄×D₅₃	C₂³	1	4	yes	—
90	C₂²×C₄×D₅₃	C₂³	1	2	yes	—
91	#13133217	C₂³	1	8	yes	complemented
92	#13133214	C₂³	1	8	yes	complemented
93	#13133213	C₂³	1	64	yes	complemented
94	#13133213	C₂³	1	32	yes	complemented
95	#13133205	C₂³	1	16	yes	complemented
96	#13133204	C₂³	1	32	yes	complemented
97	#13133191	C₂³	1	32	yes	complemented
98	#13133189	C₂³	1	128	yes	complemented
99	#13133188	C₂³	1	64	yes	complemented
100	#13133188	C₂³	1	64	yes	complemented
101	#13133186	C₂³	1	64	yes	complemented
102	#13133185	C₂³	1	128	yes	complemented
103	C₄×Dic₁₀₆	C₂³	1	16	yes	complemented
104	C₄×D₂₁₂	C₂³	1	32	yes	complemented
105	C₄×D₂₁₂	C₂³	1	16	yes	complemented
106	D₄×Dic₅₃	C₂³	1	64	yes	complemented
107	C₄×C₂²⋊D₅₃	C₂³	1	32	yes	complemented
108	#13133176	C₂³	1	8	yes	complemented
109	#13133174	C₂³	1	16	yes	complemented
110	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
111	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
112	C₂×#11719919	C₂³	1	4	yes	—
113	C₂×C₂²⋊Dic₅₃	C₂³	1	2	yes	—
114	C₂×C₂²⋊Dic₅₃	C₂³	1	16	yes	—
115	C₂×#11719921	C₂³	1	4	yes	—
116	C₂×#11719921	C₂³	1	8	yes	—
117	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
118	C₂×C₄×Dic₅₃	C₂³	1	4	yes	—
119	C₄²×D₅₃	C₂³	1	16	yes	complemented
120	C₂⁵×C₅₃	C₂³	1	4	yes	—
121	C₂²×D₄×C₅₃	—	2	16	no	—
122	C₂²×D₄×C₅₃	—	4	16	no	—
123	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
124	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
125	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
126	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
127	C₂²×D₄×C₅₃	C₂³	1	16	yes	complemented
128	C₂²×D₄×C₅₃	C₂³	1	32	yes	complemented
129	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
130	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
131	#127×C₅₃	C₂³	1	32	yes	complemented
132	#127×C₅₃	C₂³	1	16	yes	complemented
133	#127×C₅₃	C₂³	1	16	yes	complemented
134	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
135	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
136	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
137	C₂×C₄²×C₅₃	C₂³	1	8	yes	complemented
138	C₂⁴×C₅₃	—	2	16	no	—
139	C₂⁴×C₅₃	—	4	16	no	—
140	C₂⁴×C₅₃	C₂⁴	1	8	yes	—
141	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
142	C₂⁴×C₅₃	C₂⁴	1	16	yes	—
143	C₂³×D₅₃	—	2	32	no	—
144	C₂³×D₅₃	—	4	32	no	—
145	C₂³×D₅₃	C₂⁴	1	4	yes	—
146	C₂³×D₅₃	C₂⁴	1	2	yes	—
147	C₂³×D₅₃	C₂⁴	1	32	yes	—
148	C₂×D₄×C₅₃	—	2	16	no	—
149	C₂×D₄×C₅₃	—	2	16	no	—
150	C₂×D₄×C₅₃	—	2	64	no	—
151	C₂×D₄×C₅₃	—	2	32	no	—
152	C₂×D₄×C₅₃	—	4	64	no	—
153	C₂×D₄×C₅₃	—	4	16	no	—
154	C₂×D₄×C₅₃	C₂⁴	1	32	yes	—
155	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
156	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
157	C₂×D₄×C₅₃	C₂⁴	1	32	yes	—
158	C₂²×C₄×C₅₃	—	2	8	no	—
159	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	complemented
160	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
162	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
163	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
164	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
165	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
166	C₂×C₂²⋊D₅₃	—	2	16	no	—
167	C₂×C₂²⋊D₅₃	—	2	32	no	—
168	C₂×C₂²⋊D₅₃	—	2	256	no	—
169	C₂×C₂²⋊D₅₃	—	2	32	no	—
170	C₂×C₂²⋊D₅₃	—	4	32	no	—
171	C₂×C₂²⋊D₅₃	C₂×D₄	1	16	yes	complemented
172	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
173	C₂×C₂²⋊D₅₃	C₂⁴	1	16	yes	—
174	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
175	C₂×C₂²⋊D₅₃	C₂⁴	1	64	yes	—
176	C₂×C₂²⋊D₅₃	C₂⁴	1	8	yes	—
177	C₂²×Dic₅₃	—	2	32	no	—
178	C₂²×Dic₅₃	C₂⁴	1	4	yes	—
179	C₂²×Dic₅₃	C₂⁴	1	4	yes	—
180	C₂²×Dic₅₃	C₂⁴	1	32	yes	—
181	C₂²×Dic₅₃	C₂⁴	1	2	yes	—
182	#11719947	—	2	128	no	—
183	#11719947	—	2	128	no	—
184	D₄×D₅₃	—	2	128	no	—
185	D₄×D₅₃	—	2	128	no	—
186	D₄×D₅₃	—	4	128	no	—
187	#11719945	—	2	64	no	—
188	#11719945	C₂×D₄	1	64	yes	complemented
189	C₂×D₂₁₂	—	2	8	no	—
190	C₂×D₂₁₂	—	2	16	no	—
191	C₂×D₂₁₂	C₂×D₄	1	8	yes	complemented
192	C₂×D₂₁₂	C₂⁴	1	16	yes	—
193	C₂×D₂₁₂	C₂⁴	1	8	yes	—
194	C₂×D₂₁₂	C₂⁴	1	4	yes	—
195	C₂×C₄×D₅₃	—	2	16	no	—
196	C₂×C₄×D₅₃	—	2	16	no	—
197	C₂×C₄×D₅₃	C₂×D₄	1	16	yes	complemented
198	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
199	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
200	C₂×C₄×D₅₃	C₂⁴	1	8	yes	—
201	C₂×Dic₁₀₆	—	2	8	no	—
202	C₂×Dic₁₀₆	C₂×D₄	1	8	yes	complemented
203	C₂×Dic₁₀₆	C₂⁴	1	8	yes	—
204	C₂×Dic₁₀₆	C₂⁴	1	4	yes	—
205	C₄²×C₅₃	C₂⁴	1	4	yes	—
206	C₄²×C₅₃	C₂⁴	1	4	yes	—
207	C₄²×C₅₃	C₂⁴	1	8	yes	—
208	C₂²⋊Dic₅₃	C₂⁴	1	8	yes	—
209	C₂²⋊Dic₅₃	C₂⁴	1	64	yes	—
210	#11719921	C₂⁴	1	16	yes	—
211	#11719921	C₂⁴	1	32	yes	—
212	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
213	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
214	#11719919	C₂⁴	1	16	yes	—
215	C₄×Dic₅₃	C₂⁴	1	8	yes	—
216	C₄×Dic₅₃	C₂⁴	1	16	yes	—
217	C₂³×C₅₃	—	2	4	no	—
218	C₂³×C₅₃	—	2	16	no	—
219	C₂³×C₅₃	—	2	32	no	—
220	C₂³×C₅₃	—	2	32	no	—
221	C₂³×C₅₃	—	4	32	no	—
222	C₂³×C₅₃	—	8	32	no	—
223	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
224	C₂³×C₅₃	C₂⁵	1	16	yes	—
225	C₂³×C₅₃	C₂⁵	1	2	yes	—
226	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
227	C₂²×D₅₃	—	2	8	no	—
228	C₂²×D₅₃	—	2	8	no	—
229	C₂²×D₅₃	—	2	16	no	—
230	C₂²×D₅₃	—	2	64	no	—
231	C₂²×D₅₃	—	4	64	no	—
232	C₂²×D₅₃	—	8	64	no	—
233	C₂²×D₅₃	C₂²×D₄	1	8	yes	—
234	C₂²×D₅₃	C₂⁵	1	8	yes	—
235	C₂²×D₅₃	C₂⁵	1	4	yes	—
236	D₄×C₅₃	—	2	32	no	—
237	D₄×C₅₃	—	2	32	no	—
238	D₄×C₅₃	—	2	64	no	—
239	D₄×C₅₃	—	4	32	no	—
240	C₂×C₄×C₅₃	—	2	4	no	—
241	C₂×C₄×C₅₃	—	2	16	no	—
242	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
243	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
244	C₂×C₄×C₅₃	C₂²×D₄	1	16	yes	complemented
245	C₂×C₄×C₅₃	H(5,𝔽₂)	1	8	yes	complemented
246	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
247	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
248	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
249	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
250	C₂²⋊D₅₃	—	2	32	no	—
251	C₂²⋊D₅₃	—	2	64	no	—
252	C₂²⋊D₅₃	—	2	64	no	—
253	C₂²⋊D₅₃	—	4	64	no	—
254	C₂²⋊D₅₃	C₂²×D₄	1	32	yes	complemented
255	C₂×Dic₅₃	—	2	8	no	—
256	C₂×Dic₅₃	—	2	8	no	—
257	C₂×Dic₅₃	—	2	64	no	—
258	C₂×Dic₅₃	C₂²×D₄	1	8	yes	—
259	C₂×Dic₅₃	C₂⁵	1	8	yes	—
260	C₂×Dic₅₃	C₂⁵	1	8	yes	—
261	C₂×Dic₅₃	C₂⁵	1	4	yes	—
262	D₂₁₂	—	2	16	no	—
263	D₂₁₂	—	2	32	no	—
264	D₂₁₂	C₂²×D₄	1	16	yes	complemented
265	C₄×D₅₃	—	2	32	no	—
266	C₄×D₅₃	—	2	32	no	—
267	C₄×D₅₃	C₂²×D₄	1	32	yes	complemented
268	Dic₁₀₆	—	2	16	no	—
269	Dic₁₀₆	C₂²×D₄	1	16	yes	complemented
270	C₂×#3498	—	53	1	no	maximal, Sylow
271	C₂²×C₅₃	—	2	4	no	—
272	C₂²×C₅₃	—	2	8	no	—
273	C₂²×C₅₃	—	2	16	no	—
274	C₂²×C₅₃	—	2	32	no	—
275	C₂²×C₅₃	—	4	32	no	—
276	C₂²×C₅₃	—	8	32	no	—
277	C₂²×C₅₃	C₂³×D₄	1	4	yes	—
278	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
279	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
280	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
281	C₂×D₅₃	—	2	8	no	—
282	C₂×D₅₃	—	2	16	no	—
283	C₂×D₅₃	—	2	8	no	—
284	C₂×D₅₃	—	2	16	no	—
285	C₂×D₅₃	—	4	16	no	—
286	C₂×D₅₃	—	8	64	no	—
287	C₂×D₅₃	C₂³×D₄	1	8	yes	—
288	C₄×C₅₃	—	2	4	no	—
289	C₄×C₅₃	C₂³×D₄	1	4	yes	—
290	C₄×C₅₃	C₂³×D₄	1	4	yes	—
291	C₄×C₅₃	C₂×H(5,𝔽₂)	1	8	yes	complemented
292	Dic₅₃	—	2	8	no	—
293	Dic₅₃	—	2	8	no	—
294	Dic₅₃	C₂³×D₄	1	8	yes	—
295	C₂²×H(5,𝔽₂)	—	53	2	no	—
296	#3498	—	53	32	no	—
297	C₂×#550	—	53	4	no	—
298	C₂×D₄²	—	53	8	no	—
299	C₂×D₄²	—	53	4	no	—
300	C₂×#535	—	53	2	no	—
301	C₂×#535	—	53	4	no	—
302	C₂×#532	—	53	1	no	—
303	C₂×#532	—	53	2	no	—
304	C₂²×#127	—	53	2	no	—
305	C₂²×#127	D₅₃	1	1	yes	characteristic, complemented
306	C₂×C₄×#41	—	53	1	no	—
307	C₂×C₅₃	—	2	4	no	—
308	C₂×C₅₃	—	4	16	no	—
309	C₂×C₅₃	#3498	1	4	yes	complemented
310	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
311	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	2	yes	—
312	D₅₃	—	2	8	no	—
313	D₅₃	—	4	8	no	—
314	C₂×H(5,𝔽₂)	—	53	32	no	—
315	C₂×H(5,𝔽₂)	—	106	32	no	—
316	C₂²×#41	—	53	4	no	—
317	C₂²×#41	—	53	4	no	—
318	C₂²×#41	—	53	1	no	—
319	C₂²×#41	—	53	2	no	—
320	C₂³×D₄	—	53	8	no	—
321	C₂³×D₄	—	53	4	no	—
322	C₂³×D₄	—	53	4	no	—
323	C₂³×D₄	—	53	8	no	—
324	C₂³×D₄	C₂×D₅₃	1	2	yes	complemented
325	C₂³×D₄	C₂×D₅₃	1	4	yes	complemented
326	#550	—	53	64	no	—
327	D₄²	—	53	128	no	—
328	D₄²	—	53	64	no	—
329	#535	—	53	32	no	—
330	#535	—	53	64	no	—
331	#532	—	53	32	no	—
332	#532	—	53	16	no	—
333	C₂×C₄⋊Q₈	—	53	1	no	—
334	C₂×#127	—	53	4	no	—
335	C₂×#127	—	53	8	no	—
336	C₂×#127	—	53	32	no	—
337	C₂×#127	—	53	1	no	—
338	C₂×#127	C₂×D₅₃	1	2	yes	complemented
339	C₂×#127	C₂×D₅₃	1	2	yes	complemented
340	C₂×#127	C₂×D₅₃	1	4	yes	complemented
341	C₂×#127	C₂×D₅₃	1	16	yes	complemented
342	C₂×#124	—	53	4	no	—
343	C₂×#124	—	53	2	no	—
344	C₂×#121	—	53	8	no	—
345	C₂×#121	—	53	8	no	—
346	C₂×#121	—	53	16	no	—
347	C₂×#121	—	53	4	no	—
348	C₂×#120	—	53	8	no	—
349	C₂×#120	—	53	16	no	—
350	C₄×#41	—	53	16	no	—
351	C₂×C₄×Q₈	—	53	2	no	—
352	C₂×C₄×D₄	—	53	8	no	—
353	C₂×C₄×D₄	—	53	4	no	—
354	C₂×C₄×D₄	—	53	2	no	—
355	C₂×C₄×D₄	—	53	4	no	—
356	C₂×#117	—	53	1	no	—
357	C₂×#117	—	53	2	no	—
358	C₂²×C₄²	—	53	2	no	—
359	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic, complemented
360	C₅₃	C₂×#3498	1	1	yes	characteristic, complemented, Sylow
361	C₂⁵	—	53	8	no	—
362	C₂⁵	C₂²×D₅₃	1	4	yes	—
363	H(5,𝔽₂)	—	106	256	no	—
364	C₂×#41	—	53	32	no	—
365	C₂×#41	—	53	32	no	—
366	C₂×#41	—	53	16	no	—
367	C₂×#41	—	53	8	no	—
368	C₂×#41	—	53	16	no	—
369	C₂×#41	—	106	32	no	—
370	C₂×#41	—	106	32	no	—
371	C₂×#41	—	106	16	no	—
372	C₂²×Q₈	—	53	1	no	—
373	C₂²×Q₈	—	53	2	no	—
374	C₂²×D₄	—	2	16	no	—
375	C₂²×D₄	—	4	16	no	—
376	C₂²×D₄	—	53	2	no	—
377	C₂²×D₄	—	53	8	no	—
378	C₂²×D₄	—	53	4	no	—
379	C₂²×D₄	—	53	8	no	—
380	C₂²×D₄	—	53	64	no	—
381	C₂²×D₄	—	53	16	no	—
382	C₂²×D₄	—	53	32	no	—
383	C₂²×D₄	—	53	32	no	—
384	C₂²×D₄	—	53	64	no	—
385	C₂²×D₄	—	53	1	no	—
386	C₂²×D₄	—	53	4	no	—
387	C₂²×D₄	—	53	2	no	—
388	C₂²×D₄	—	106	32	no	—
389	C₂²×D₄	—	106	32	no	—
390	C₂²×D₄	—	106	64	no	—
391	C₂²×D₄	—	212	32	no	—
392	C₂²×D₄	C₂²×D₅₃	1	8	yes	—
393	C₂²×D₄	C₂²×D₅₃	1	4	yes	—
394	C₂²×D₄	C₂²×D₅₃	1	4	yes	—
395	C₂²×D₄	C₂²×D₅₃	1	8	yes	—
396	C₂²×D₄	C₂²×D₅₃	1	16	yes	complemented
397	C₂²×D₄	C₂²×D₅₃	1	32	yes	complemented
398	C₂³×C₄	—	53	2	no	—
399	C₂³×C₄	—	53	8	no	—
400	C₂³×C₄	—	53	4	no	—
401	C₂³×C₄	—	53	4	no	—
402	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
403	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
404	C₄⋊Q₈	—	53	8	no	—
405	#127	—	53	32	no	—
406	#127	—	53	64	no	—
407	#127	—	53	8	no	—
408	#127	C₂²×D₅₃	1	16	yes	complemented
409	#127	C₂²×D₅₃	1	16	yes	complemented
410	#127	C₂²×D₅₃	1	32	yes	complemented
411	#124	—	53	32	no	—
412	#124	—	53	16	no	—
413	#121	—	53	64	no	—
414	#121	—	53	64	no	—
415	#121	—	53	128	no	—
416	#121	—	53	32	no	—
417	#120	—	53	64	no	—
418	#120	—	53	128	no	—
419	C₄×Q₈	—	53	16	no	—
420	C₄×D₄	—	53	64	no	—
421	C₄×D₄	—	53	32	no	—
422	C₄×D₄	—	53	16	no	—
423	C₄×D₄	—	53	32	no	—
424	#117	—	53	8	no	—
425	#117	—	53	16	no	—
426	C₂×C₄⋊C₄	—	53	4	no	—
427	C₂×C₄⋊C₄	—	53	2	no	—
428	C₂×C₄⋊C₄	—	53	4	no	—
429	C₂×C₂²⋊C₄	—	53	2	no	—
430	C₂×C₂²⋊C₄	—	53	16	no	—
431	C₂×C₂²⋊C₄	—	53	4	no	—
432	C₂×C₂²⋊C₄	—	53	8	no	—
433	C₂×C₄²	—	53	2	no	—
434	C₂×C₄²	—	53	4	no	—
435	C₂×C₄²	—	53	16	no	—
436	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
437	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
438	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
439	C₂×C₄²	C₂²×D₅₃	1	8	yes	complemented
440	C₂⁴	—	2	16	no	—
441	C₂⁴	—	4	16	no	—
442	C₂⁴	—	53	2	no	—
443	C₂⁴	—	53	4	no	—
444	C₂⁴	—	53	32	no	—
445	C₂⁴	—	106	32	no	—
446	C₂⁴	—	212	32	no	—
447	C₂⁴	C₂³×D₅₃	1	8	yes	—
448	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
449	C₂⁴	C₂³×D₅₃	1	16	yes	—
450	#41	—	53	64	no	—
451	#41	—	106	128	no	—
452	#41	—	106	128	no	—
453	#41	—	106	64	no	—
454	C₂×Q₈	—	53	8	no	—
455	C₂×Q₈	—	53	4	no	—
456	C₂×Q₈	—	53	8	no	—
457	C₂×Q₈	—	106	8	no	—
458	C₂×D₄	—	2	16	no	—
459	C₂×D₄	—	2	16	no	—
460	C₂×D₄	—	2	64	no	—
461	C₂×D₄	—	2	32	no	—
462	C₂×D₄	—	4	64	no	—
463	C₂×D₄	—	4	16	no	—
464	C₂×D₄	—	53	16	no	—
465	C₂×D₄	—	53	8	no	—
466	C₂×D₄	—	53	32	no	—
467	C₂×D₄	—	53	16	no	—
468	C₂×D₄	—	53	32	no	—
469	C₂×D₄	—	53	64	no	—
470	C₂×D₄	—	53	8	no	—
471	C₂×D₄	—	53	4	no	—
472	C₂×D₄	—	53	16	no	—
473	C₂×D₄	—	53	8	no	—
474	C₂×D₄	—	106	128	no	—
475	C₂×D₄	—	106	32	no	—
476	C₂×D₄	—	106	128	no	—
477	C₂×D₄	—	106	256	no	—
478	C₂×D₄	—	106	32	no	—
479	C₂×D₄	—	106	16	no	—
480	C₂×D₄	—	106	8	no	—
481	C₂×D₄	—	106	16	no	—
482	C₂×D₄	—	212	128	no	—
483	C₂×D₄	—	212	32	no	—
484	C₂×D₄	C₂³×D₅₃	1	32	yes	—
485	C₂×D₄	C₂³×D₅₃	1	16	yes	—
486	C₂×D₄	C₂³×D₅₃	1	16	yes	—
487	C₂×D₄	C₂³×D₅₃	1	32	yes	—
488	C₂²×C₄	—	2	8	no	—
489	C₂²×C₄	—	53	2	no	—
490	C₂²×C₄	—	53	16	no	—
491	C₂²×C₄	—	53	8	no	—
492	C₂²×C₄	—	53	32	no	—
493	C₂²×C₄	—	53	4	no	—
494	C₂²×C₄	—	53	4	no	—
495	C₂²×C₄	—	53	16	no	—
496	C₂²×C₄	—	53	16	no	—
497	C₂²×C₄	—	106	32	no	—
498	C₂²×C₄	—	106	16	no	—
499	C₂²×C₄	—	106	16	no	—
500	C₂²×C₄	D₄×D₅₃	1	8	yes	complemented
501	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
502	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
503	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
504	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
505	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
506	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
507	C₄⋊C₄	—	53	16	no	—
508	C₄⋊C₄	—	53	8	no	—
509	C₄⋊C₄	—	53	16	no	—
510	C₂²⋊C₄	—	53	8	no	—
511	C₂²⋊C₄	—	53	64	no	—
512	C₂²⋊C₄	—	53	16	no	—
513	C₂²⋊C₄	—	53	32	no	—
514	C₄²	—	53	8	no	—
515	C₄²	—	53	16	no	—
516	C₄²	C₂³×D₅₃	1	4	yes	—
517	C₄²	C₂³×D₅₃	1	4	yes	—
518	C₄²	C₂³×D₅₃	1	8	yes	—
519	C₂³	—	2	4	no	—
520	C₂³	—	2	16	no	—
521	C₂³	—	2	32	no	—
522	C₂³	—	2	32	no	—
523	C₂³	—	4	32	no	—
524	C₂³	—	8	32	no	—
525	C₂³	—	53	8	no	—
526	C₂³	—	53	4	no	—
527	C₂³	—	53	8	no	—
528	C₂³	—	106	64	no	—
529	C₂³	—	106	8	no	—
530	C₂³	—	106	8	no	—
531	C₂³	—	106	16	no	—
532	C₂³	—	424	64	no	—
533	C₂³	—	212	64	no	—
534	C₂³	C₂×D₄×D₅₃	1	4	yes	—
535	C₂³	C₂⁴×D₅₃	1	16	yes	—
536	C₂³	C₂⁴×D₅₃	1	2	yes	—
537	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
538	Q₈	—	53	16	no	—
539	Q₈	—	106	16	no	—
540	D₄	—	2	32	no	—
541	D₄	—	2	32	no	—
542	D₄	—	2	64	no	—
543	D₄	—	4	32	no	—
544	D₄	—	53	32	no	—
545	D₄	—	53	16	no	—
546	D₄	—	106	64	no	—
547	D₄	—	106	64	no	—
548	D₄	—	106	32	no	—
549	D₄	—	106	16	no	—
550	D₄	—	106	32	no	—
551	D₄	—	212	64	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	16	no	—
554	C₂×C₄	—	53	8	no	—
555	C₂×C₄	—	53	4	no	—
556	C₂×C₄	—	53	32	no	—
557	C₂×C₄	—	53	8	no	—
558	C₂×C₄	—	53	8	no	—
559	C₂×C₄	—	106	64	no	—
560	C₂×C₄	—	106	8	no	—
561	C₂×C₄	—	106	8	no	—
562	C₂×C₄	—	106	32	no	—
563	C₂×C₄	—	106	32	no	—
564	C₂×C₄	C₂×D₄×D₅₃	1	4	yes	—
565	C₂×C₄	C₂×D₄×D₅₃	1	4	yes	—
566	C₂×C₄	C₂×D₄×D₅₃	1	16	yes	complemented
567	C₂×C₄	#13133262	1	8	yes	complemented
568	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
569	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
570	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
571	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
572	C₂²	—	2	8	no	—
573	C₂²	—	2	4	no	—
574	C₂²	—	2	16	no	—
575	C₂²	—	2	32	no	—
576	C₂²	—	4	32	no	—
577	C₂²	—	8	32	no	—
578	C₂²	—	53	8	no	—
579	C₂²	—	106	16	no	—
580	C₂²	—	106	8	no	—
581	C₂²	—	106	8	no	—
582	C₂²	—	106	16	no	—
583	C₂²	—	424	64	no	—
584	C₂²	—	212	16	no	—
585	C₂²	C₂²×D₄×D₅₃	1	4	yes	—
586	C₂²	C₂²×D₄×D₅₃	1	2	yes	—
587	C₂²	C₂×#13133262	1	4	yes	—
588	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
589	C₄	—	2	4	no	—
590	C₄	—	53	8	no	—
591	C₄	—	106	8	no	—
592	C₄	—	106	8	no	—
593	C₄	C₂²×D₄×D₅₃	1	4	yes	—
594	C₄	C₂²×D₄×D₅₃	1	4	yes	—
595	C₄	C₂×#13133262	1	8	yes	complemented
596	C₂	—	2	4	no	—
597	C₂	—	4	16	no	—
598	C₂	—	106	8	no	—
599	C₂	—	212	8	no	—
600	C₂	#20288491	1	4	yes	direct product factor
601	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
602	C₂	C₂²×#13133262	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×#127×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂²×#127	D₅₃
hypocenter	C₅₃	C₂×#3498
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×#127

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133262	C₂	semidirect
2. #20288491	C₂	direct (N × H)
3. C₂×#16811293	C₂	semidirect
4. C₂×#16811258	C₂	semidirect
5. C₂×D₄×C₂²⋊D₅₃	C₂	semidirect
6. C₂×#16811190	C₂	semidirect
7. C₂×#16811186	C₂	semidirect
8. C₂×#16811173	C₂	semidirect
9. C₂×#16811171	C₂	semidirect
10. C₂×#127×D₅₃	C₂	semidirect
11. C₂×C₄×#11719945	C₂	semidirect
12. C₂²×#127×C₅₃	C₂	semidirect
14. C₂×#13133262	C₂²	semidirect
16. C₂²×#11719945	C₂²	semidirect
17. C₂²×#11719947	C₂²	semidirect
18. C₂²×#11719947	C₂²	semidirect
19. C₂³×C₂²⋊D₅₃	C₂²	semidirect
20. C₂²×D₄×D₅₃	C₂²	semidirect
21. C₂²×D₄×D₅₃	C₂²	semidirect
22. C₂²×D₄×D₅₃	C₂²	semidirect
23. #16811293	C₂²	semidirect
24. #16811258	C₂²	semidirect
25. D₄×C₂²⋊D₅₃	C₂²	semidirect
26. #16811190	C₂²	semidirect
27. #16811186	C₂²	semidirect
28. #16811173	C₂²	semidirect
29. #16811171	C₂²	semidirect
30. C₂×#13133217	C₂²	semidirect
31. C₂×#13133214	C₂²	semidirect
32. C₂×#13133213	C₂²	semidirect
33. C₂×#13133213	C₂²	semidirect
34. #127×D₅₃	C₂²	semidirect
35. C₂×#13133205	C₂²	semidirect
36. C₂×#13133204	C₂²	semidirect
37. C₂×#13133191	C₂²	semidirect
38. C₂×#13133189	C₂²	semidirect
39. C₂×#13133188	C₂²	semidirect
40. C₂×#13133188	C₂²	semidirect
41. C₂×#13133186	C₂²	semidirect
42. C₂×#13133185	C₂²	semidirect
43. C₄×#11719945	C₂²	semidirect
44. C₂×C₄×Dic₁₀₆	C₂²	semidirect
45. C₂×C₄×D₂₁₂	C₂²	semidirect
46. C₂×C₄×D₂₁₂	C₂²	semidirect
47. C₂×D₄×Dic₅₃	C₂²	semidirect
48. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
49. C₂×#13133176	C₂²	semidirect
50. C₂×#13133174	C₂²	semidirect
51. C₂×C₄²×D₅₃	C₂²	semidirect
52. C₂³×D₄×C₅₃	C₂²	semidirect
53. C₂³×D₄×C₅₃	C₂²	semidirect
54. C₂×#127×C₅₃	C₂²	semidirect
55. C₂×#127×C₅₃	C₂²	semidirect
56. C₂×#127×C₅₃	C₂²	semidirect
57. C₂×#127×C₅₃	C₂²	semidirect
58. C₂²×C₄²×C₅₃	C₂²	semidirect
62. C₂×#11719945	D₄	semidirect
64. C₂×#11719945	C₂³	semidirect
67. C₂×#11719947	C₂³	semidirect
68. C₂×#11719947	C₂³	semidirect
80. C₂²×C₂²⋊D₅₃	C₂³	semidirect
84. C₂×D₄×D₅₃	C₂³	semidirect
85. C₂×D₄×D₅₃	C₂³	semidirect
86. C₂×D₄×D₅₃	C₂³	semidirect
91. #13133217	C₂³	semidirect
92. #13133214	C₂³	semidirect
93. #13133213	C₂³	semidirect
94. #13133213	C₂³	semidirect
95. #13133205	C₂³	semidirect
96. #13133204	C₂³	semidirect
97. #13133191	C₂³	semidirect
98. #13133189	C₂³	semidirect
99. #13133188	C₂³	semidirect
100. #13133188	C₂³	semidirect
101. #13133186	C₂³	semidirect
102. #13133185	C₂³	semidirect
103. C₄×Dic₁₀₆	C₂³	semidirect
104. C₄×D₂₁₂	C₂³	semidirect
105. C₄×D₂₁₂	C₂³	semidirect
106. D₄×Dic₅₃	C₂³	semidirect
107. C₄×C₂²⋊D₅₃	C₂³	semidirect
108. #13133176	C₂³	semidirect
109. #13133174	C₂³	semidirect
119. C₄²×D₅₃	C₂³	semidirect
127. C₂²×D₄×C₅₃	C₂³	semidirect
128. C₂²×D₄×C₅₃	C₂³	semidirect
131. #127×C₅₃	C₂³	semidirect
132. #127×C₅₃	C₂³	semidirect
133. #127×C₅₃	C₂³	semidirect
137. C₂×C₄²×C₅₃	C₂³	semidirect
159. C₂²×C₄×C₅₃	C₂×D₄	semidirect
171. C₂×C₂²⋊D₅₃	C₂×D₄	semidirect
188. #11719945	C₂×D₄	semidirect
191. C₂×D₂₁₂	C₂×D₄	semidirect
197. C₂×C₄×D₅₃	C₂×D₄	semidirect
202. C₂×Dic₁₀₆	C₂×D₄	semidirect
244. C₂×C₄×C₅₃	C₂²×D₄	semidirect
245. C₂×C₄×C₅₃	H(5,𝔽₂)	semidirect
254. C₂²⋊D₅₃	C₂²×D₄	semidirect
264. D₂₁₂	C₂²×D₄	semidirect
267. C₄×D₅₃	C₂²×D₄	semidirect
269. Dic₁₀₆	C₂²×D₄	semidirect
291. C₄×C₅₃	C₂×H(5,𝔽₂)	semidirect
305. C₂²×#127	D₅₃	semidirect
309. C₂×C₅₃	#3498	semidirect
324. C₂³×D₄	C₂×D₅₃	semidirect
325. C₂³×D₄	C₂×D₅₃	semidirect
338. C₂×#127	C₂×D₅₃	semidirect
339. C₂×#127	C₂×D₅₃	semidirect
340. C₂×#127	C₂×D₅₃	semidirect
341. C₂×#127	C₂×D₅₃	semidirect
359. C₂²×C₄²	C₂×D₅₃	semidirect
360. C₅₃	C₂×#3498	semidirect
396. C₂²×D₄	C₂²×D₅₃	semidirect
397. C₂²×D₄	C₂²×D₅₃	semidirect
408. #127	C₂²×D₅₃	semidirect
409. #127	C₂²×D₅₃	semidirect
410. #127	C₂²×D₅₃	semidirect
439. C₂×C₄²	C₂²×D₅₃	semidirect
500. C₂²×C₄	D₄×D₅₃	semidirect
566. C₂×C₄	C₂×D₄×D₅₃	semidirect
567. C₂×C₄	#13133262	semidirect
595. C₄	C₂×#13133262	semidirect
600. C₂	#20288491	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133262	2
#20288491	2
C₂×#16811293	2
C₂×#16811258	2
C₂×D₄×C₂²⋊D₅₃	2
C₂×#16811190	2
C₂×#16811186	2
C₂×#16811173	2
C₂×#16811171	2
C₂×#127×D₅₃	2
C₂×C₄×#11719945	2
C₂²×#127×C₅₃	2
C₂×#3498	53

Subgroups — C2×#20288491 #28292410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20288491 #28292410