Subgroups — C₂²×#16811421 #28297896

Order 13568.

11294 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719945	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16811421	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#13133248	C₂	1	2	yes	maximal, complemented
4	C₂²×#13133247	C₂	1	2	yes	maximal, complemented
5	C₂²×#13133243	C₂	1	2	yes	maximal, complemented
6	C₂²×#13133242	C₂	1	2	yes	maximal, complemented
7	C₂²×#13133237	C₂	1	4	yes	maximal, complemented
8	C₂³×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#11719945	—	2	1	no	—
11	C₂²×#11719945	—	2	1	no	—
12	C₂²×#11719945	—	2	12	no	—
13	C₂²×#11719945	C₂²	1	2	yes	—
14	C₂²×#11719945	C₂²	1	2	yes	complemented
15	C₂²×#11719945	C₂²	1	24	yes	complemented
16	C₂³×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
17	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
18	C₂³×C₂²⋊D₅₃	—	2	1	no	—
19	C₂³×C₄×D₅₃	—	2	1	no	—
20	#16811421	C₂²	1	256	yes	direct product factor
21	C₂×#13133248	C₂²	1	48	yes	complemented
22	C₂×#13133247	C₂²	1	48	yes	complemented
23	C₂×#13133243	C₂²	1	48	yes	complemented
24	C₂×#13133242	C₂²	1	48	yes	complemented
25	C₂×#13133237	C₂²	1	96	yes	complemented
26	C₂²×#11719925	C₂²	1	4	yes	complemented
27	C₂²×#11719923	C₂²	1	4	yes	complemented
28	C₂²×#11719924	C₂²	1	4	yes	complemented
29	C₂²×#11719922	C₂²	1	4	yes	complemented
30	C₂²×#11719917	C₂²	1	2	yes	complemented
31	C₂²×#11719917	C₂²	1	2	yes	complemented
32	C₂²×#11719917	C₂²	1	24	yes	complemented
33	C₂³×#88322	C₂²	1	2	yes	complemented
34	C₂²×#41×C₅₃	—	2	1	no	—
35	C₂²×#41×C₅₃	—	2	1	no	—
36	C₂²×#41×C₅₃	—	2	12	no	—
37	C₂²×#41×C₅₃	C₂²	1	2	yes	—
38	C₂²×#41×C₅₃	C₂²	1	2	yes	complemented
39	C₂²×#41×C₅₃	C₂²	1	24	yes	complemented
40	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
41	C₂³×D₄×C₅₃	—	2	1	no	—
42	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
43	C₂⁴×C₄×C₅₃	—	2	1	no	—
44	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
45	C₂⁴×D₅₃	—	2	1	no	—
46	C₂×#11719945	—	2	12	no	—
47	C₂×#11719945	—	2	12	no	—
48	C₂×#11719945	—	2	24	no	—
49	C₂×#11719945	—	2	96	no	—
50	C₂×#11719945	—	2	24	no	—
51	C₂×#11719945	C₂³	1	24	yes	—
52	C₂×#11719945	C₂³	1	24	yes	complemented
53	C₂×#11719945	C₂³	1	64	yes	complemented
54	C₂²×Dic₁₀₆	—	2	6	no	—
55	C₂²×Dic₁₀₆	—	2	1	no	—
56	C₂²×Dic₁₀₆	C₂³	1	2	yes	—
57	C₂²×Dic₁₀₆	C₂³	1	12	yes	complemented
58	C₂²×D₂₁₂	—	2	6	no	—
59	C₂²×D₂₁₂	—	2	1	no	—
60	C₂²×D₂₁₂	C₂³	1	2	yes	—
61	C₂²×D₂₁₂	C₂³	1	12	yes	complemented
62	C₂²×C₂²⋊D₅₃	—	2	2	no	—
63	C₂²×C₂²⋊D₅₃	—	2	2	no	—
64	C₂²×C₂²⋊D₅₃	—	2	24	no	—
65	C₂³×Dic₅₃	—	2	1	no	—
66	C₂²×C₄×D₅₃	—	2	2	no	—
67	C₂²×C₄×D₅₃	—	2	24	no	—
68	C₂²×C₄×D₅₃	—	2	2	no	—
69	#13133248	C₂³	1	128	yes	complemented
70	#13133247	C₂³	1	128	yes	complemented
71	#13133243	C₂³	1	128	yes	complemented
72	#13133242	C₂³	1	128	yes	complemented
73	#13133237	C₂³	1	256	yes	complemented
74	C₂×#11719925	C₂³	1	48	yes	complemented
75	C₂×#11719923	C₂³	1	48	yes	complemented
76	C₂×#11719924	C₂³	1	48	yes	complemented
77	C₂×#11719922	C₂³	1	48	yes	complemented
78	C₂×#11719917	C₂³	1	24	yes	complemented
79	C₂×#11719917	C₂³	1	24	yes	complemented
80	C₂×#11719917	C₂³	1	64	yes	complemented
81	C₂²×#88322	C₂³	1	4	yes	—
82	C₂²×#88322	C₂³	1	24	yes	complemented
83	C₂⁵×C₅₃	—	2	1	no	—
84	C₂⁵×C₅₃	D₄	1	1	yes	characteristic
85	C₂×#41×C₅₃	—	2	12	no	—
86	C₂×#41×C₅₃	—	2	24	no	—
87	C₂×#41×C₅₃	—	2	24	no	—
88	C₂×#41×C₅₃	—	2	12	no	—
89	C₂×#41×C₅₃	—	2	96	no	—
90	C₂×#41×C₅₃	C₂³	1	24	yes	—
91	C₂×#41×C₅₃	C₂³	1	24	yes	complemented
92	C₂×#41×C₅₃	C₂³	1	64	yes	complemented
93	C₂²×Q₈×C₅₃	—	2	1	no	—
94	C₂²×Q₈×C₅₃	—	2	6	no	—
95	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
96	C₂²×Q₈×C₅₃	C₂³	1	12	yes	complemented
97	C₂²×D₄×C₅₃	—	2	2	no	—
98	C₂²×D₄×C₅₃	—	2	2	no	—
99	C₂²×D₄×C₅₃	—	2	24	no	—
100	C₂²×D₄×C₅₃	—	2	1	no	—
101	C₂²×D₄×C₅₃	—	2	6	no	—
102	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
103	C₂²×D₄×C₅₃	C₂³	1	12	yes	complemented
104	C₂³×C₄×C₅₃	—	2	1	no	—
105	C₂³×C₄×C₅₃	—	2	2	no	—
106	C₂³×C₄×C₅₃	—	2	2	no	—
107	C₂³×C₄×C₅₃	—	2	24	no	—
108	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
109	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
110	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
111	C₂³×C₄×C₅₃	D₄	1	12	yes	—
112	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
113	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
114	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
115	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
116	C₂⁴×C₅₃	—	2	2	no	—
117	C₂⁴×C₅₃	—	2	12	no	—
118	C₂⁴×C₅₃	—	4	4	no	—
119	C₂⁴×C₅₃	—	4	8	no	—
120	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
122	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
123	C₂⁴×C₅₃	C₂×D₄	1	12	yes	—
124	C₂³×D₅₃	—	2	2	no	—
125	C₂³×D₅₃	—	2	12	no	—
126	C₂³×D₅₃	—	4	8	no	—
127	#41×C₅₃	—	2	48	no	—
128	#41×C₅₃	—	2	64	no	—
129	#41×C₅₃	—	2	64	no	—
130	#41×C₅₃	—	2	48	no	—
131	#41×C₅₃	C₂⁴	1	32	yes	—
132	#41×C₅₃	C₂⁴	1	32	yes	complemented
133	C₂×Q₈×C₅₃	—	2	6	no	—
134	C₂×Q₈×C₅₃	—	2	12	no	—
135	C₂×Q₈×C₅₃	—	2	24	no	—
136	C₂×Q₈×C₅₃	C₂⁴	1	12	yes	—
137	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	complemented
138	C₂×D₄×C₅₃	—	2	24	no	—
139	C₂×D₄×C₅₃	—	2	24	no	—
140	C₂×D₄×C₅₃	—	2	64	no	—
141	C₂×D₄×C₅₃	—	2	6	no	—
142	C₂×D₄×C₅₃	—	2	12	no	—
143	C₂×D₄×C₅₃	—	2	24	no	—
144	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
145	C₂×D₄×C₅₃	C₂⁴	1	16	yes	complemented
146	C₂²×C₄×C₅₃	—	2	2	no	—
147	C₂²×C₄×C₅₃	—	2	12	no	—
148	C₂²×C₄×C₅₃	—	2	24	no	—
149	C₂²×C₄×C₅₃	—	2	24	no	—
150	C₂²×C₄×C₅₃	—	2	64	no	—
151	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
153	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
155	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
156	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
157	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
158	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
159	C₂²×C₄×C₅₃	C₂×D₄	1	48	yes	—
160	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
162	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
163	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
164	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
165	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
166	C₂×C₂²⋊D₅₃	—	2	24	no	—
167	C₂×C₂²⋊D₅₃	—	2	24	no	—
168	C₂×C₂²⋊D₅₃	—	2	64	no	—
169	C₂²×Dic₅₃	—	2	2	no	—
170	C₂²×Dic₅₃	—	2	12	no	—
171	#11719945	—	2	48	no	—
172	#11719945	—	2	48	no	—
173	#11719945	—	2	64	no	—
174	#11719945	—	2	64	no	—
175	#11719945	C₂⁴	1	32	yes	—
176	#11719945	C₂⁴	1	32	yes	complemented
177	C₂×D₂₁₂	—	2	6	no	—
178	C₂×D₂₁₂	—	2	24	no	—
179	C₂×D₂₁₂	—	2	12	no	—
180	C₂×D₂₁₂	C₂⁴	1	12	yes	—
181	C₂×D₂₁₂	C₂⁴	1	16	yes	complemented
182	C₂×C₄×D₅₃	—	2	24	no	—
183	C₂×C₄×D₅₃	—	2	64	no	—
184	C₂×C₄×D₅₃	—	2	24	no	—
185	C₂×Dic₁₀₆	—	2	6	no	—
186	C₂×Dic₁₀₆	—	2	24	no	—
187	C₂×Dic₁₀₆	—	2	12	no	—
188	C₂×Dic₁₀₆	C₂⁴	1	12	yes	—
189	C₂×Dic₁₀₆	C₂⁴	1	16	yes	complemented
190	#11719925	C₂⁴	1	64	yes	complemented
191	#11719924	C₂⁴	1	64	yes	complemented
192	#11719923	C₂⁴	1	64	yes	complemented
193	#11719922	C₂⁴	1	64	yes	complemented
194	#11719917	C₂⁴	1	32	yes	complemented
195	#11719917	C₂⁴	1	32	yes	complemented
196	C₂×#88322	C₂⁴	1	24	yes	—
197	C₂×#88322	C₂⁴	1	32	yes	complemented
198	C₂³×C₅₃	—	2	4	no	—
199	C₂³×C₅₃	—	2	4	no	—
200	C₂³×C₅₃	—	2	4	no	—
201	C₂³×C₅₃	—	2	12	no	—
202	C₂³×C₅₃	—	2	16	no	—
203	C₂³×C₅₃	—	4	24	no	—
204	C₂³×C₅₃	—	4	48	no	—
205	C₂³×C₅₃	—	4	8	no	—
206	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
207	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
208	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
209	C₂³×C₅₃	C₂²×D₄	1	16	yes	—
210	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
211	C₂²×D₅₃	—	2	12	no	—
212	C₂²×D₅₃	—	2	16	no	—
213	C₂²×D₅₃	—	4	48	no	—
214	C₂²×D₅₃	—	4	8	no	—
215	Q₈×C₅₃	—	2	12	no	—
216	Q₈×C₅₃	—	2	16	no	—
217	Q₈×C₅₃	C₂⁵	1	8	yes	—
218	D₄×C₅₃	—	2	32	no	—
219	D₄×C₅₃	—	2	32	no	—
220	D₄×C₅₃	—	2	12	no	—
221	D₄×C₅₃	—	2	16	no	—
222	D₄×C₅₃	C₂⁵	1	8	yes	—
223	C₂×C₄×C₅₃	—	2	12	no	—
224	C₂×C₄×C₅₃	—	2	16	no	—
225	C₂×C₄×C₅₃	—	2	32	no	—
226	C₂×C₄×C₅₃	—	2	32	no	—
227	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
228	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
229	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
230	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
231	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
232	C₂×C₄×C₅₃	C₂²×D₄	1	16	yes	—
233	C₂×C₄×C₅₃	C₂²×D₄	1	16	yes	—
234	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
235	C₂×C₄×C₅₃	C₂²×D₄	1	16	yes	—
236	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
237	C₂×C₄×C₅₃	C₂⁵	1	3	yes	—
238	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
239	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
240	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
241	C₂²⋊D₅₃	—	2	32	no	—
242	C₂²⋊D₅₃	—	2	32	no	—
243	C₂×Dic₅₃	—	2	12	no	—
244	C₂×Dic₅₃	—	2	16	no	—
245	D₂₁₂	—	2	12	no	—
246	D₂₁₂	—	2	16	no	—
247	D₂₁₂	C₂⁵	1	8	yes	—
248	C₄×D₅₃	—	2	32	no	—
249	C₄×D₅₃	—	2	32	no	—
250	Dic₁₀₆	—	2	12	no	—
251	Dic₁₀₆	—	2	16	no	—
252	Dic₁₀₆	C₂⁵	1	8	yes	—
253	#88322	C₂⁵	1	16	yes	—
254	C₂²×#575	—	53	1	no	maximal, Sylow
255	C₂²×C₅₃	—	2	12	no	—
256	C₂²×C₅₃	—	2	12	no	—
257	C₂²×C₅₃	—	2	12	no	—
258	C₂²×C₅₃	—	2	8	no	—
259	C₂²×C₅₃	—	4	16	no	—
260	C₂²×C₅₃	—	4	32	no	—
261	C₂²×C₅₃	—	4	24	no	—
262	C₂²×C₅₃	#575	1	4	yes	complemented
263	C₂²×C₅₃	C₂³×D₄	1	4	yes	—
264	C₂²×C₅₃	C₂³×D₄	1	4	yes	—
265	C₂²×C₅₃	C₂³×D₄	1	4	yes	—
266	C₂²×C₅₃	C₂³×D₄	1	3	yes	—
267	C₂×D₅₃	—	2	8	no	—
268	C₂×D₅₃	—	4	32	no	—
269	C₂×D₅₃	—	4	24	no	—
270	C₄×C₅₃	—	2	8	no	—
271	C₄×C₅₃	C₂³×D₄	1	4	yes	—
272	C₄×C₅₃	C₂³×D₄	1	3	yes	—
273	C₄×C₅₃	C₂³×D₄	1	4	yes	—
274	C₄×C₅₃	C₂³×D₄	1	4	yes	—
275	C₄×C₅₃	C₂⁶	1	1	yes	characteristic
276	Dic₅₃	—	2	8	no	—
277	C₂³×#41	—	53	1	no	—
278	C₂³×#41	D₅₃	1	1	yes	characteristic, complemented
279	C₂×#575	—	53	48	no	—
280	C₂²×#137	—	53	2	no	—
281	C₂²×#137	—	53	2	no	—
282	C₂²×AGL(1,ℤ/8)	—	53	2	no	—
283	C₂²×AGL(1,ℤ/8)	—	53	2	no	—
284	C₂²×#135	—	53	4	no	—
285	C₂³×M₁₆	—	53	1	no	—
286	C₂×C₅₃	—	2	4	no	—
287	C₂×C₅₃	—	2	4	no	—
288	C₂×C₅₃	—	2	4	no	—
289	C₂×C₅₃	—	4	8	no	—
290	C₂×C₅₃	C₂×#575	1	6	yes	complemented
291	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
292	D₅₃	—	4	8	no	—
293	C₂²×#41	—	2	1	no	—
294	C₂²×#41	—	2	12	no	—
295	C₂²×#41	—	2	1	no	—
296	C₂²×#41	—	53	2	no	—
297	C₂²×#41	—	53	2	no	—
298	C₂²×#41	—	53	24	no	—
299	C₂²×#41	—	106	1	no	—
300	C₂²×#41	—	106	12	no	—
301	C₂²×#41	—	106	1	no	—
302	C₂²×#41	C₂×D₅₃	1	2	yes	—
303	C₂²×#41	C₂×D₅₃	1	2	yes	complemented
304	C₂²×#41	C₂×D₅₃	1	24	yes	complemented
305	C₂³×Q₈	—	53	1	no	—
306	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
307	C₂³×D₄	—	2	1	no	—
308	C₂³×D₄	—	53	1	no	—
309	C₂³×D₄	—	106	1	no	—
310	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic, complemented
311	C₂⁴×C₄	—	2	1	no	—
312	C₂⁴×C₄	—	106	1	no	—
313	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
314	#575	—	53	256	no	—
315	C₂×#137	—	53	48	no	—
316	C₂×#137	—	53	48	no	—
317	C₂×AGL(1,ℤ/8)	—	53	48	no	—
318	C₂×AGL(1,ℤ/8)	—	53	48	no	—
319	C₂×#135	—	53	96	no	—
320	C₂²×Q₁₆	—	53	4	no	—
321	C₂²×SD₁₆	—	53	4	no	—
322	C₂²×SD₁₆	—	53	4	no	—
323	C₂²×D₈	—	53	4	no	—
324	C₂²×M₁₆	—	53	2	no	—
325	C₂²×M₁₆	—	53	2	no	—
326	C₂²×M₁₆	—	53	24	no	—
327	C₂³×C₈	—	53	2	no	—
328	C₅₃	C₂²×#575	1	1	yes	characteristic, complemented, Sylow
329	C₂⁵	—	2	1	no	—
330	C₂⁵	—	106	1	no	—
331	C₂⁵	C₂²⋊D₅₃	1	1	yes	characteristic
332	C₂×#41	—	2	12	no	—
333	C₂×#41	—	2	24	no	—
334	C₂×#41	—	2	24	no	—
335	C₂×#41	—	2	12	no	—
336	C₂×#41	—	2	96	no	—
337	C₂×#41	—	53	24	no	—
338	C₂×#41	—	53	24	no	—
339	C₂×#41	—	53	64	no	—
340	C₂×#41	—	106	24	no	—
341	C₂×#41	—	106	12	no	—
342	C₂×#41	—	106	12	no	—
343	C₂×#41	—	106	24	no	—
344	C₂×#41	—	106	96	no	—
345	C₂×#41	C₂²×D₅₃	1	24	yes	—
346	C₂×#41	C₂²×D₅₃	1	24	yes	complemented
347	C₂×#41	C₂²×D₅₃	1	64	yes	complemented
348	C₂²×Q₈	—	2	1	no	—
349	C₂²×Q₈	—	2	6	no	—
350	C₂²×Q₈	—	53	2	no	—
351	C₂²×Q₈	—	53	12	no	—
352	C₂²×Q₈	—	106	1	no	—
353	C₂²×Q₈	—	106	6	no	—
354	C₂²×Q₈	C₂²×D₅₃	1	2	yes	—
355	C₂²×Q₈	C₂²×D₅₃	1	12	yes	complemented
356	C₂²×D₄	—	2	2	no	—
357	C₂²×D₄	—	2	2	no	—
358	C₂²×D₄	—	2	24	no	—
359	C₂²×D₄	—	2	1	no	—
360	C₂²×D₄	—	2	6	no	—
361	C₂²×D₄	—	53	2	no	—
362	C₂²×D₄	—	53	12	no	—
363	C₂²×D₄	—	106	2	no	—
364	C₂²×D₄	—	106	2	no	—
365	C₂²×D₄	—	106	24	no	—
366	C₂²×D₄	—	106	1	no	—
367	C₂²×D₄	—	106	6	no	—
368	C₂²×D₄	C₂²×D₅₃	1	2	yes	—
369	C₂²×D₄	C₂²×D₅₃	1	12	yes	complemented
370	C₂³×C₄	—	2	1	no	—
371	C₂³×C₄	—	2	2	no	—
372	C₂³×C₄	—	2	24	no	—
373	C₂³×C₄	—	2	2	no	—
374	C₂³×C₄	—	106	1	no	—
375	C₂³×C₄	—	106	2	no	—
376	C₂³×C₄	—	106	24	no	—
377	C₂³×C₄	—	106	2	no	—
378	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
379	C₂³×C₄	C₂²⋊D₅₃	1	12	yes	—
380	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
381	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
382	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
383	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
384	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
385	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
386	#137	—	53	128	no	—
387	#137	—	53	128	no	—
388	AGL(1,ℤ/8)	—	53	128	no	—
389	AGL(1,ℤ/8)	—	53	128	no	—
390	#135	—	53	256	no	—
391	C₂×Q₁₆	—	53	48	no	—
392	C₂×SD₁₆	—	53	48	no	—
393	C₂×SD₁₆	—	53	48	no	—
394	C₂×D₈	—	53	48	no	—
395	C₂×M₁₆	—	53	24	no	—
396	C₂×M₁₆	—	53	24	no	—
397	C₂×M₁₆	—	53	64	no	—
398	C₂²×C₈	—	53	4	no	—
399	C₂²×C₈	—	53	24	no	—
400	C₂⁴	—	2	2	no	—
401	C₂⁴	—	2	12	no	—
402	C₂⁴	—	4	4	no	—
403	C₂⁴	—	4	8	no	—
404	C₂⁴	—	106	12	no	—
405	C₂⁴	—	106	2	no	—
406	C₂⁴	—	212	8	no	—
407	C₂⁴	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
408	C₂⁴	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
409	C₂⁴	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
410	C₂⁴	C₂×C₂²⋊D₅₃	1	12	yes	—
411	#41	—	2	48	no	—
412	#41	—	2	64	no	—
413	#41	—	2	64	no	—
414	#41	—	2	48	no	—
415	#41	—	53	32	no	—
416	#41	—	53	32	no	—
417	#41	—	106	64	no	—
418	#41	—	106	48	no	—
419	#41	—	106	48	no	—
420	#41	—	106	64	no	—
421	#41	C₂³×D₅₃	1	32	yes	—
422	#41	C₂³×D₅₃	1	32	yes	complemented
423	C₂×Q₈	—	2	6	no	—
424	C₂×Q₈	—	2	12	no	—
425	C₂×Q₈	—	2	24	no	—
426	C₂×Q₈	—	53	12	no	—
427	C₂×Q₈	—	53	16	no	—
428	C₂×Q₈	—	106	12	no	—
429	C₂×Q₈	—	106	6	no	—
430	C₂×Q₈	—	106	24	no	—
431	C₂×Q₈	C₂³×D₅₃	1	12	yes	—
432	C₂×Q₈	C₂³×D₅₃	1	16	yes	complemented
433	C₂×D₄	—	2	24	no	—
434	C₂×D₄	—	2	24	no	—
435	C₂×D₄	—	2	64	no	—
436	C₂×D₄	—	2	6	no	—
437	C₂×D₄	—	2	12	no	—
438	C₂×D₄	—	2	24	no	—
439	C₂×D₄	—	53	12	no	—
440	C₂×D₄	—	53	16	no	—
441	C₂×D₄	—	106	24	no	—
442	C₂×D₄	—	106	24	no	—
443	C₂×D₄	—	106	64	no	—
444	C₂×D₄	—	106	12	no	—
445	C₂×D₄	—	106	6	no	—
446	C₂×D₄	—	106	24	no	—
447	C₂×D₄	C₂³×D₅₃	1	12	yes	—
448	C₂×D₄	C₂³×D₅₃	1	16	yes	complemented
449	C₂²×C₄	—	2	2	no	—
450	C₂²×C₄	—	2	12	no	—
451	C₂²×C₄	—	2	24	no	—
452	C₂²×C₄	—	2	24	no	—
453	C₂²×C₄	—	2	64	no	—
454	C₂²×C₄	—	106	12	no	—
455	C₂²×C₄	—	106	2	no	—
456	C₂²×C₄	—	106	24	no	—
457	C₂²×C₄	—	106	24	no	—
458	C₂²×C₄	—	106	64	no	—
459	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
460	C₂²×C₄	C₂×C₂²⋊D₅₃	1	12	yes	—
461	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
462	C₂²×C₄	C₂×C₂²⋊D₅₃	1	12	yes	—
463	C₂²×C₄	C₂×C₂²⋊D₅₃	1	6	yes	—
464	C₂²×C₄	C₂×C₂²⋊D₅₃	1	6	yes	—
465	C₂²×C₄	C₂×C₂²⋊D₅₃	1	6	yes	—
466	C₂²×C₄	C₂×C₂²⋊D₅₃	1	12	yes	—
467	C₂²×C₄	C₂×C₂²⋊D₅₃	1	48	yes	—
468	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
469	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
470	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
471	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
472	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
473	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
474	Q₁₆	—	53	64	no	—
475	SD₁₆	—	53	64	no	—
476	SD₁₆	—	53	64	no	—
477	D₈	—	53	64	no	—
478	M₁₆	—	53	32	no	—
479	M₁₆	—	53	32	no	—
480	C₂×C₈	—	53	24	no	—
481	C₂×C₈	—	53	32	no	—
482	C₂³	—	2	4	no	—
483	C₂³	—	2	4	no	—
484	C₂³	—	2	4	no	—
485	C₂³	—	2	12	no	—
486	C₂³	—	2	16	no	—
487	C₂³	—	4	24	no	—
488	C₂³	—	4	48	no	—
489	C₂³	—	4	8	no	—
490	C₂³	—	106	16	no	—
491	C₂³	—	106	12	no	—
492	C₂³	—	212	48	no	—
493	C₂³	—	212	8	no	—
494	C₂³	C₂²×C₂²⋊D₅₃	1	6	yes	—
495	C₂³	C₂²×C₂²⋊D₅₃	1	6	yes	—
496	C₂³	C₂²×C₂²⋊D₅₃	1	6	yes	—
497	C₂³	C₂²×C₂²⋊D₅₃	1	16	yes	—
498	C₂³	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
499	Q₈	—	2	12	no	—
500	Q₈	—	2	16	no	—
501	Q₈	—	53	8	no	—
502	Q₈	—	106	16	no	—
503	Q₈	—	106	12	no	—
504	Q₈	C₂⁴×D₅₃	1	8	yes	—
505	D₄	—	2	32	no	—
506	D₄	—	2	32	no	—
507	D₄	—	2	12	no	—
508	D₄	—	2	16	no	—
509	D₄	—	53	8	no	—
510	D₄	—	106	32	no	—
511	D₄	—	106	32	no	—
512	D₄	—	106	16	no	—
513	D₄	—	106	12	no	—
514	D₄	C₂⁴×D₅₃	1	8	yes	—
515	C₂×C₄	—	2	12	no	—
516	C₂×C₄	—	2	16	no	—
517	C₂×C₄	—	2	32	no	—
518	C₂×C₄	—	2	32	no	—
519	C₂×C₄	—	106	16	no	—
520	C₂×C₄	—	106	12	no	—
521	C₂×C₄	—	106	32	no	—
522	C₂×C₄	—	106	32	no	—
523	C₂×C₄	C₂²×C₂²⋊D₅₃	1	6	yes	—
524	C₂×C₄	C₂²×C₂²⋊D₅₃	1	6	yes	—
525	C₂×C₄	C₂²×C₂²⋊D₅₃	1	16	yes	—
526	C₂×C₄	C₂²×C₂²⋊D₅₃	1	6	yes	—
527	C₂×C₄	C₂²×C₂²⋊D₅₃	1	16	yes	—
528	C₂×C₄	C₂²×C₂²⋊D₅₃	1	3	yes	—
529	C₂×C₄	C₂²×C₂²⋊D₅₃	1	12	yes	—
530	C₂×C₄	C₂²×C₂²⋊D₅₃	1	12	yes	—
531	C₂×C₄	C₂²×C₂²⋊D₅₃	1	12	yes	—
532	C₂×C₄	C₂²×C₂²⋊D₅₃	1	16	yes	—
533	C₂×C₄	C₂⁴×D₅₃	1	3	yes	—
534	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
535	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
536	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
537	C₈	—	53	16	no	—
538	C₂²	—	2	12	no	—
539	C₂²	—	2	12	no	—
540	C₂²	—	2	12	no	—
541	C₂²	—	2	8	no	—
542	C₂²	—	4	16	no	—
543	C₂²	—	4	32	no	—
544	C₂²	—	4	24	no	—
545	C₂²	—	106	8	no	—
546	C₂²	—	212	32	no	—
547	C₂²	—	212	24	no	—
548	C₂²	#16811421	1	4	yes	direct product factor
549	C₂²	C₂³×C₂²⋊D₅₃	1	4	yes	—
550	C₂²	C₂³×C₂²⋊D₅₃	1	4	yes	—
551	C₂²	C₂³×C₂²⋊D₅₃	1	4	yes	—
552	C₂²	C₂³×C₂²⋊D₅₃	1	3	yes	—
553	C₄	—	2	8	no	—
554	C₄	—	106	8	no	—
555	C₄	C₂³×C₂²⋊D₅₃	1	4	yes	—
556	C₄	C₂³×C₂²⋊D₅₃	1	4	yes	—
557	C₄	C₂³×C₂²⋊D₅₃	1	4	yes	—
558	C₄	C₂³×C₂²⋊D₅₃	1	3	yes	—
559	C₄	C₂⁵×D₅₃	1	1	yes	characteristic
560	C₂	—	2	4	no	—
561	C₂	—	2	4	no	—
562	C₂	—	2	4	no	—
563	C₂	—	4	8	no	—
564	C₂	—	212	8	no	—
565	C₂	C₂×#16811421	1	6	yes	direct product factor
566	C₂	C₂⁴×C₂²⋊D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator, cosocle	C₄×C₅₃	C₂⁶
Frattini	C₄	C₂⁵×D₅₃
Fitting	C₂³×#41×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×D₄
hypercenter	C₂³×#41	D₅₃
hypocenter	C₅₃	C₂²×#575
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂³×#11719945	C₂	semidirect
2. C₂×#16811421	C₂	direct (N × H)
3. C₂²×#13133248	C₂	semidirect
4. C₂²×#13133247	C₂	semidirect
5. C₂²×#13133243	C₂	semidirect
6. C₂²×#13133242	C₂	semidirect
7. C₂²×#13133237	C₂	semidirect
8. C₂³×#11719917	C₂	semidirect
9. C₂³×#41×C₅₃	C₂	semidirect
14. C₂²×#11719945	C₂²	semidirect
15. C₂²×#11719945	C₂²	semidirect
16. C₂³×Dic₁₀₆	C₂²	semidirect
17. C₂³×D₂₁₂	C₂²	semidirect
20. #16811421	C₂²	direct (N × H)
21. C₂×#13133248	C₂²	semidirect
22. C₂×#13133247	C₂²	semidirect
23. C₂×#13133243	C₂²	semidirect
24. C₂×#13133242	C₂²	semidirect
25. C₂×#13133237	C₂²	semidirect
26. C₂²×#11719925	C₂²	semidirect
27. C₂²×#11719923	C₂²	semidirect
28. C₂²×#11719924	C₂²	semidirect
29. C₂²×#11719922	C₂²	semidirect
30. C₂²×#11719917	C₂²	semidirect
31. C₂²×#11719917	C₂²	semidirect
32. C₂²×#11719917	C₂²	semidirect
33. C₂³×#88322	C₂²	semidirect
38. C₂²×#41×C₅₃	C₂²	semidirect
39. C₂²×#41×C₅₃	C₂²	semidirect
40. C₂³×Q₈×C₅₃	C₂²	semidirect
42. C₂³×D₄×C₅₃	C₂²	semidirect
52. C₂×#11719945	C₂³	semidirect
53. C₂×#11719945	C₂³	semidirect
57. C₂²×Dic₁₀₆	C₂³	semidirect
61. C₂²×D₂₁₂	C₂³	semidirect
69. #13133248	C₂³	semidirect
70. #13133247	C₂³	semidirect
71. #13133243	C₂³	semidirect
72. #13133242	C₂³	semidirect
73. #13133237	C₂³	semidirect
74. C₂×#11719925	C₂³	semidirect
75. C₂×#11719923	C₂³	semidirect
76. C₂×#11719924	C₂³	semidirect
77. C₂×#11719922	C₂³	semidirect
78. C₂×#11719917	C₂³	semidirect
79. C₂×#11719917	C₂³	semidirect
80. C₂×#11719917	C₂³	semidirect
82. C₂²×#88322	C₂³	semidirect
91. C₂×#41×C₅₃	C₂³	semidirect
92. C₂×#41×C₅₃	C₂³	semidirect
96. C₂²×Q₈×C₅₃	C₂³	semidirect
103. C₂²×D₄×C₅₃	C₂³	semidirect
132. #41×C₅₃	C₂⁴	semidirect
137. C₂×Q₈×C₅₃	C₂⁴	semidirect
145. C₂×D₄×C₅₃	C₂⁴	semidirect
176. #11719945	C₂⁴	semidirect
181. C₂×D₂₁₂	C₂⁴	semidirect
189. C₂×Dic₁₀₆	C₂⁴	semidirect
190. #11719925	C₂⁴	semidirect
191. #11719924	C₂⁴	semidirect
192. #11719923	C₂⁴	semidirect
193. #11719922	C₂⁴	semidirect
194. #11719917	C₂⁴	semidirect
195. #11719917	C₂⁴	semidirect
197. C₂×#88322	C₂⁴	semidirect
262. C₂²×C₅₃	#575	semidirect
278. C₂³×#41	D₅₃	semidirect
290. C₂×C₅₃	C₂×#575	semidirect
303. C₂²×#41	C₂×D₅₃	semidirect
304. C₂²×#41	C₂×D₅₃	semidirect
306. C₂³×Q₈	C₂×D₅₃	semidirect
310. C₂³×D₄	C₂×D₅₃	semidirect
328. C₅₃	C₂²×#575	semidirect
346. C₂×#41	C₂²×D₅₃	semidirect
347. C₂×#41	C₂²×D₅₃	semidirect
355. C₂²×Q₈	C₂²×D₅₃	semidirect
369. C₂²×D₄	C₂²×D₅₃	semidirect
422. #41	C₂³×D₅₃	semidirect
432. C₂×Q₈	C₂³×D₅₃	semidirect
448. C₂×D₄	C₂³×D₅₃	semidirect
548. C₂²	#16811421	direct (N × H)
565. C₂	C₂×#16811421	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×#11719945	2
C₂×#16811421	2
C₂²×#13133248	2
C₂²×#13133247	2
C₂²×#13133243	2
C₂²×#13133242	2
C₂²×#13133237	2
C₂³×#11719917	2
C₂³×#41×C₅₃	2
C₂²×#575	53

Subgroups — C22×#16811421 #28297896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811421 #28297896