Subgroups — C₂²×#16811264 #28291021

Order 13568.

14462 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719949	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16811264	C₂	1	48	yes	maximal, direct product factor
4	C₂²×#13133206	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#13133202	C₂	1	2	yes	maximal, complemented
6	C₂²×#13133192	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#13133192	C₂	1	2	yes	maximal, complemented
8	C₂²×#13133187	C₂	1	2	yes	maximal, complemented
9	C₂²×#13133179	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×C₂²⋊D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×C₂²⋊C₄×D₅₃	C₂	1	2	yes	maximal, complemented
12	C₂²×C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂⁵×D₅₃	C₂²	1	2	yes	—
14	C₂²×#11719949	—	2	16	no	—
15	C₂²×#11719949	C₂²	1	24	yes	complemented
16	C₂³×D₂₁₂	—	2	16	no	—
17	C₂³×D₂₁₂	C₂²	1	2	yes	—
18	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic
19	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic
20	C₂³×D₂₁₂	C₂²	1	24	yes	complemented
21	C₂³×D₂₁₂	C₂²	1	2	yes	complemented
22	C₂³×D₂₁₂	C₂²	1	2	yes	complemented
23	C₂³×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
24	C₂³×C₄×D₅₃	C₂²	1	2	yes	complemented
25	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic, complemented
26	C₂³×C₄×D₅₃	C₂²	1	2	yes	complemented
27	#16811264	C₂²	1	256	yes	direct product factor
28	C₂×#13133206	C₂²	1	24	yes	complemented
29	C₂×#13133202	C₂²	1	48	yes	complemented
30	C₂×#13133192	C₂²	1	24	yes	complemented
31	C₂×#13133192	C₂²	1	48	yes	complemented
32	C₂×#13133187	C₂²	1	48	yes	complemented
33	C₂×#13133179	C₂²	1	24	yes	complemented
34	C₂×C₄×C₂²⋊D₅₃	C₂²	1	24	yes	complemented
35	C₂²×#11719919	C₂²	1	1	yes	characteristic, complemented
36	C₂²×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic
37	C₂²×#11719921	C₂²	1	2	yes	complemented
38	C₂²×#11719921	C₂²	1	2	yes	complemented
39	C₂²×#11719921	C₂²	1	1	yes	characteristic, complemented
40	C₂²×#11719921	C₂²	1	2	yes	complemented
41	C₂²×#11719921	C₂²	1	2	yes	complemented
42	C₂×C₂²⋊C₄×D₅₃	C₂²	1	48	yes	complemented
43	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
44	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
45	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
46	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	24	yes	complemented
47	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
48	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
49	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
50	C₂⁴×D₅₃	—	2	16	no	—
51	C₂⁴×D₅₃	—	2	16	no	—
52	C₂⁴×D₅₃	—	4	8	no	—
53	C₂⁴×D₅₃	C₂³	1	2	yes	—
54	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
55	C₂⁴×D₅₃	C₂³	1	24	yes	—
56	C₂⁴×D₅₃	C₂³	1	2	yes	—
57	C₂×#11719949	—	2	192	no	—
58	C₂×#11719949	C₂³	1	64	yes	complemented
59	C₂²×D₂₁₂	—	2	16	no	—
60	C₂²×D₂₁₂	—	2	16	no	—
61	C₂²×D₂₁₂	—	2	8	no	—
62	C₂²×D₂₁₂	—	2	192	no	—
63	C₂²×D₂₁₂	—	4	8	no	—
64	C₂²×D₂₁₂	D₄	1	16	yes	complemented
65	C₂²×D₂₁₂	C₂³	1	24	yes	—
66	C₂²×D₂₁₂	C₂³	1	12	yes	—
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×C₂²⋊D₅₃	—	2	8	no	—
72	C₂²×C₂²⋊D₅₃	C₂³	1	12	yes	—
73	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
74	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
75	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
76	C₂²×C₄×D₅₃	—	2	16	no	—
77	C₂²×C₄×D₅₃	—	2	8	no	—
78	C₂²×C₄×D₅₃	—	4	8	no	—
79	C₂²×C₄×D₅₃	C₂³	1	24	yes	complemented
80	C₂²×C₄×D₅₃	C₂³	1	12	yes	complemented
81	C₂²×C₄×D₅₃	C₂³	1	24	yes	complemented
82	#13133206	C₂³	1	64	yes	complemented
83	#13133202	C₂³	1	128	yes	complemented
84	#13133192	C₂³	1	64	yes	complemented
85	#13133192	C₂³	1	128	yes	complemented
86	#13133187	C₂³	1	128	yes	complemented
87	#13133179	C₂³	1	64	yes	complemented
88	C₄×C₂²⋊D₅₃	C₂³	1	64	yes	complemented
89	C₂×#11719919	C₂³	1	12	yes	complemented
90	C₂×C₂²⋊Dic₅₃	C₂³	1	12	yes	—
91	C₂×#11719921	C₂³	1	24	yes	complemented
92	C₂×#11719921	C₂³	1	24	yes	complemented
93	C₂×#11719921	C₂³	1	12	yes	complemented
94	C₂×#11719921	C₂³	1	24	yes	complemented
95	C₂×#11719921	C₂³	1	24	yes	complemented
96	C₂²⋊C₄×D₅₃	C₂³	1	128	yes	complemented
97	C₂×C₄×Dic₅₃	C₂³	1	12	yes	complemented
98	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
99	C₂²×Q₈×C₅₃	—	2	8	no	—
100	C₂²×Q₈×C₅₃	C₂³	1	12	yes	complemented
101	C₂³×C₄×C₅₃	—	2	8	no	—
102	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
103	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
104	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
105	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
106	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
107	C₂²⋊Q₈×C₅₃	C₂³	1	64	yes	complemented
108	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	complemented
109	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	complemented
110	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	—
111	C₂⁴×C₅₃	—	2	4	no	—
112	C₂⁴×C₅₃	—	2	4	no	—
113	C₂⁴×C₅₃	#41	1	8	yes	complemented
114	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
115	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
116	C₂³×D₅₃	—	2	8	no	—
117	C₂³×D₅₃	—	2	8	no	—
118	C₂³×D₅₃	—	2	4	no	—
119	C₂³×D₅₃	—	2	8	no	—
120	C₂³×D₅₃	—	2	4	no	—
121	C₂³×D₅₃	—	2	8	no	—
122	C₂³×D₅₃	—	2	96	no	—
123	C₂³×D₅₃	—	2	4	no	—
124	C₂³×D₅₃	—	2	8	no	—
125	C₂³×D₅₃	—	2	96	no	—
126	C₂³×D₅₃	—	4	48	no	—
127	C₂³×D₅₃	—	4	128	no	—
128	C₂³×D₅₃	C₂×D₄	1	16	yes	—
129	C₂³×D₅₃	C₂⁴	1	12	yes	—
130	C₂³×D₅₃	C₂⁴	1	6	yes	—
131	C₂³×D₅₃	C₂⁴	1	32	yes	—
132	C₂³×D₅₃	C₂⁴	1	12	yes	—
133	C₂×Q₈×C₅₃	—	2	48	no	—
134	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	complemented
135	C₂²×C₄×C₅₃	—	2	4	no	—
136	C₂²×C₄×C₅₃	—	2	8	no	—
137	C₂²×C₄×C₅₃	—	2	8	no	—
138	C₂²×C₄×C₅₃	—	2	48	no	—
139	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
140	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
141	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
142	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
143	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
144	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
145	C₂×C₂²⋊D₅₃	—	2	48	no	—
146	C₂×C₂²⋊D₅₃	C₂⁴	1	16	yes	—
147	C₂²×Dic₅₃	—	2	4	no	—
148	C₂²×Dic₅₃	—	2	4	no	—
149	C₂²×Dic₅₃	—	2	4	no	—
150	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
151	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
152	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
153	#11719949	—	2	256	no	—
154	C₂×D₂₁₂	—	2	96	no	—
155	C₂×D₂₁₂	—	2	96	no	—
156	C₂×D₂₁₂	—	2	48	no	—
157	C₂×D₂₁₂	—	2	256	no	—
158	C₂×D₂₁₂	—	4	48	no	—
159	C₂×D₂₁₂	C₂×D₄	1	96	yes	complemented
160	C₂×D₂₁₂	C₂⁴	1	32	yes	—
161	C₂×D₂₁₂	C₂⁴	1	16	yes	—
162	C₂×D₂₁₂	C₂⁴	1	16	yes	—
163	C₂×D₂₁₂	C₂⁴	1	32	yes	complemented
164	C₂×D₂₁₂	C₂⁴	1	32	yes	complemented
165	C₂×C₄×D₅₃	—	2	96	no	—
166	C₂×C₄×D₅₃	—	2	48	no	—
167	C₂×C₄×D₅₃	—	4	48	no	—
168	C₂×C₄×D₅₃	C₂⁴	1	32	yes	complemented
169	C₂×C₄×D₅₃	C₂⁴	1	16	yes	complemented
170	C₂×C₄×D₅₃	C₂⁴	1	32	yes	complemented
171	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	complemented
172	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
173	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
174	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	—
175	#11719921	C₂⁴	1	32	yes	complemented
176	#11719921	C₂⁴	1	32	yes	complemented
177	#11719921	C₂⁴	1	16	yes	complemented
178	#11719921	C₂⁴	1	32	yes	complemented
179	#11719921	C₂⁴	1	32	yes	complemented
180	#11719919	C₂⁴	1	16	yes	complemented
181	C₄×Dic₅₃	C₂⁴	1	16	yes	complemented
182	C₂³×C₅₃	—	2	32	no	—
183	C₂³×C₅₃	—	2	12	no	—
184	C₂³×C₅₃	—	2	12	no	—
185	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
186	C₂³×C₅₃	C₂×#41	1	4	yes	—
187	C₂³×C₅₃	C₂×#41	1	24	yes	complemented
188	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
189	C₂³×C₅₃	C₂⁵	1	4	yes	—
190	C₂³×C₅₃	C₂⁵	1	3	yes	—
191	C₂²×D₅₃	—	2	64	no	—
192	C₂²×D₅₃	—	2	24	no	—
193	C₂²×D₅₃	—	2	24	no	—
194	C₂²×D₅₃	—	2	12	no	—
195	C₂²×D₅₃	—	2	24	no	—
196	C₂²×D₅₃	—	2	12	no	—
197	C₂²×D₅₃	—	2	24	no	—
198	C₂²×D₅₃	—	2	64	no	—
199	C₂²×D₅₃	—	2	12	no	—
200	C₂²×D₅₃	—	2	24	no	—
201	C₂²×D₅₃	—	2	64	no	—
202	C₂²×D₅₃	—	4	32	no	—
203	C₂²×D₅₃	—	4	384	no	—
204	C₂²×D₅₃	—	4	16	no	—
205	C₂²×D₅₃	—	4	32	no	—
206	C₂²×D₅₃	C₂²×D₄	1	48	yes	—
207	C₂²×D₅₃	C₂⁵	1	4	yes	—
208	C₂²×D₅₃	C₂⁵	1	8	yes	—
209	C₂²×D₅₃	C₂⁵	1	8	yes	—
210	Q₈×C₅₃	—	2	32	no	—
211	C₂×C₄×C₅₃	—	2	24	no	—
212	C₂×C₄×C₅₃	—	2	12	no	—
213	C₂×C₄×C₅₃	—	2	24	no	—
214	C₂×C₄×C₅₃	—	2	32	no	—
215	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
216	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
217	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
218	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
219	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
220	C₂²⋊D₅₃	—	2	32	no	—
221	C₂×Dic₅₃	—	2	12	no	—
222	C₂×Dic₅₃	—	2	12	no	—
223	C₂×Dic₅₃	—	2	12	no	—
224	C₂×Dic₅₃	C₂⁵	1	4	yes	—
225	C₂×Dic₅₃	C₂⁵	1	4	yes	—
226	C₂×Dic₅₃	C₂⁵	1	4	yes	—
227	D₂₁₂	—	2	64	no	—
228	D₂₁₂	—	2	64	no	—
229	D₂₁₂	—	2	32	no	—
230	D₂₁₂	—	4	32	no	—
231	D₂₁₂	C₂²×D₄	1	64	yes	complemented
232	C₄×D₅₃	—	2	64	no	—
233	C₄×D₅₃	—	2	32	no	—
234	C₄×D₅₃	—	4	32	no	—
235	C₂²×#546	—	53	1	no	maximal, Sylow
236	C₂²×C₅₃	—	2	4	no	—
237	C₂²×C₅₃	—	2	4	no	—
238	C₂²×C₅₃	—	2	48	no	—
239	C₂²×C₅₃	#546	1	16	yes	complemented
240	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
241	C₂²×C₅₃	C₂²×#41	1	6	yes	—
242	C₂²×C₅₃	C₂²×#41	1	8	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	4	no	—
249	C₂×D₅₃	—	2	8	no	—
250	C₂×D₅₃	—	2	4	no	—
251	C₂×D₅₃	—	2	8	no	—
252	C₂×D₅₃	—	2	4	no	—
253	C₂×D₅₃	—	2	8	no	—
254	C₂×D₅₃	—	4	128	no	—
255	C₂×D₅₃	—	4	24	no	—
256	C₂×D₅₃	—	4	48	no	—
257	C₂×D₅₃	C₂³×D₄	1	16	yes	—
258	C₄×C₅₃	—	2	8	no	—
259	C₄×C₅₃	—	2	4	no	—
260	C₄×C₅₃	—	2	8	no	—
261	C₄×C₅₃	C₂³×D₄	1	8	yes	—
262	Dic₅₃	—	2	4	no	—
263	Dic₅₃	—	2	4	no	—
264	Dic₅₃	—	2	4	no	—
265	C₂³×#41	—	53	1	no	—
266	C₂⁴×D₄	—	53	1	no	—
267	C₂×#546	—	53	48	no	—
268	C₂²×#124	—	53	1	no	—
269	C₂²×#123	—	53	2	no	—
270	C₂²×C₂²⋊Q₈	D₅₃	1	1	yes	characteristic, complemented
271	C₂²×#121	—	53	1	no	—
272	C₂²×#121	—	53	2	no	—
273	C₂²×#120	—	53	2	no	—
274	C₂²×C₄×D₄	—	53	1	no	—
275	C₂²×C₄×D₄	—	53	1	no	—
276	C₂³×C₂²⋊C₄	—	53	2	no	—
277	C₂×C₅₃	—	2	8	no	—
278	C₂×C₅₃	C₂×#546	1	12	yes	complemented
279	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
280	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
281	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
282	D₅₃	—	2	16	no	—
283	D₅₃	—	4	4	no	—
284	D₅₃	—	4	8	no	—
285	C₂⁶	—	53	2	no	—
286	C₂²×#41	—	53	24	no	—
287	C₂²×#41	—	106	16	no	—
288	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
289	C₂³×D₄	—	53	1	no	—
290	C₂³×D₄	—	53	1	no	—
291	C₂³×D₄	—	53	2	no	—
292	C₂³×D₄	—	53	1	no	—
293	C₂³×D₄	—	53	24	no	—
294	C₂³×D₄	—	53	2	no	—
295	C₂³×D₄	—	53	2	no	—
296	C₂³×D₄	—	106	16	no	—
297	C₂⁴×C₄	—	53	2	no	—
298	C₂⁴×C₄	—	53	1	no	—
299	C₂⁴×C₄	—	53	2	no	—
300	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
301	#546	—	53	256	no	—
302	C₂×#124	—	53	24	no	—
303	C₂×#123	—	53	48	no	—
304	C₂×C₂²⋊Q₈	C₂×D₅₃	1	24	yes	complemented
305	C₂×#121	—	53	24	no	—
306	C₂×#121	—	53	48	no	—
307	C₂×#120	—	53	48	no	—
308	C₂×C₄×D₄	—	53	24	no	—
309	C₂×C₄×D₄	—	53	24	no	—
310	C₂²×C₄⋊C₄	—	53	1	no	—
311	C₂²×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
312	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	complemented
313	C₂²×C₂²⋊C₄	—	53	1	no	—
314	C₂²×C₂²⋊C₄	—	53	2	no	—
315	C₂²×C₂²⋊C₄	—	53	2	no	—
316	C₂²×C₂²⋊C₄	—	53	2	no	—
317	C₂²×C₂²⋊C₄	—	53	1	no	—
318	C₂²×C₂²⋊C₄	—	53	2	no	—
319	C₂²×C₂²⋊C₄	—	53	48	no	—
320	C₂²×C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
321	C₂²×C₄²	—	53	1	no	—
322	C₅₃	C₂²×#546	1	1	yes	characteristic, complemented, Sylow
323	C₂⁵	—	53	2	no	—
324	C₂⁵	—	53	2	no	—
325	C₂⁵	—	53	1	no	—
326	C₂⁵	—	53	24	no	—
327	C₂⁵	—	106	16	no	—
328	C₂⁵	—	106	16	no	—
329	C₂⁵	—	212	8	no	—
330	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
331	C₂×#41	—	53	64	no	—
332	C₂×#41	—	106	192	no	—
333	C₂²×Q₈	—	2	8	no	—
334	C₂²×Q₈	C₂²×D₅₃	1	12	yes	complemented
335	C₂²×D₄	—	53	12	no	—
336	C₂²×D₄	—	53	16	no	—
337	C₂²×D₄	—	53	12	no	—
338	C₂²×D₄	—	53	24	no	—
339	C₂²×D₄	—	53	24	no	—
340	C₂²×D₄	—	53	12	no	—
341	C₂²×D₄	—	53	64	no	—
342	C₂²×D₄	—	53	24	no	—
343	C₂²×D₄	—	106	192	no	—
344	C₂²×D₄	—	106	16	no	—
345	C₂²×D₄	—	106	16	no	—
346	C₂²×D₄	—	106	8	no	—
347	C₂²×D₄	—	106	8	no	—
348	C₂²×D₄	—	212	8	no	—
349	C₂³×C₄	—	2	8	no	—
350	C₂³×C₄	—	53	1	no	—
351	C₂³×C₄	—	53	1	no	—
352	C₂³×C₄	—	53	1	no	—
353	C₂³×C₄	—	53	24	no	—
354	C₂³×C₄	—	53	12	no	—
355	C₂³×C₄	—	53	24	no	—
356	C₂³×C₄	—	106	16	no	—
357	C₂³×C₄	—	106	8	no	—
358	C₂³×C₄	—	212	8	no	—
359	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
360	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
361	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
362	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
363	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
364	#124	—	53	64	no	—
365	#123	—	53	128	no	—
366	C₂²⋊Q₈	C₂²×D₅₃	1	64	yes	complemented
367	#121	—	53	64	no	—
368	#121	—	53	128	no	—
369	#120	—	53	128	no	—
370	C₄×D₄	—	53	64	no	—
371	C₄×D₄	—	53	64	no	—
372	C₂×C₄⋊C₄	—	53	12	no	—
373	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	complemented
374	C₂×C₄⋊C₄	C₂²×D₅₃	1	24	yes	complemented
375	C₂×C₂²⋊C₄	—	53	12	no	—
376	C₂×C₂²⋊C₄	—	53	24	no	—
377	C₂×C₂²⋊C₄	—	53	24	no	—
378	C₂×C₂²⋊C₄	—	53	24	no	—
379	C₂×C₂²⋊C₄	—	53	12	no	—
380	C₂×C₂²⋊C₄	—	53	24	no	—
381	C₂×C₂²⋊C₄	—	53	128	no	—
382	C₂×C₂²⋊C₄	C₂²×D₅₃	1	24	yes	—
383	C₂×C₄²	—	53	12	no	—
384	C₂⁴	—	2	4	no	—
385	C₂⁴	—	2	4	no	—
386	C₂⁴	—	53	16	no	—
387	C₂⁴	—	53	12	no	—
388	C₂⁴	—	53	12	no	—
389	C₂⁴	—	53	6	no	—
390	C₂⁴	—	53	32	no	—
391	C₂⁴	—	106	96	no	—
392	C₂⁴	—	106	8	no	—
393	C₂⁴	—	106	8	no	—
394	C₂⁴	—	106	4	no	—
395	C₂⁴	—	106	8	no	—
396	C₂⁴	—	106	4	no	—
397	C₂⁴	—	106	8	no	—
398	C₂⁴	—	106	96	no	—
399	C₂⁴	—	106	4	no	—
400	C₂⁴	—	106	8	no	—
401	C₂⁴	—	212	48	no	—
402	C₂⁴	—	212	128	no	—
403	C₂⁴	#11719949	1	8	yes	complemented
404	C₂⁴	C₂³×D₅₃	1	6	yes	—
405	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
406	#41	—	106	256	no	—
407	C₂×Q₈	—	2	48	no	—
408	C₂×Q₈	C₂³×D₅₃	1	16	yes	complemented
409	C₂×D₄	—	53	16	no	—
410	C₂×D₄	—	53	96	no	—
411	C₂×D₄	—	53	16	no	—
412	C₂×D₄	—	53	32	no	—
413	C₂×D₄	—	53	16	no	—
414	C₂×D₄	—	53	32	no	—
415	C₂×D₄	—	53	32	no	—
416	C₂×D₄	—	106	256	no	—
417	C₂×D₄	—	106	96	no	—
418	C₂×D₄	—	106	96	no	—
419	C₂×D₄	—	106	48	no	—
420	C₂×D₄	—	106	48	no	—
421	C₂×D₄	—	212	48	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	2	8	no	—
424	C₂²×C₄	—	2	8	no	—
425	C₂²×C₄	—	2	48	no	—
426	C₂²×C₄	—	53	6	no	—
427	C₂²×C₄	—	53	6	no	—
428	C₂²×C₄	—	53	6	no	—
429	C₂²×C₄	—	53	32	no	—
430	C₂²×C₄	—	53	16	no	—
431	C₂²×C₄	—	53	32	no	—
432	C₂²×C₄	—	106	96	no	—
433	C₂²×C₄	—	106	4	no	—
434	C₂²×C₄	—	106	4	no	—
435	C₂²×C₄	—	106	4	no	—
436	C₂²×C₄	—	106	48	no	—
437	C₂²×C₄	—	212	48	no	—
438	C₂²×C₄	D₄×D₅₃	1	8	yes	—
439	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
440	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
441	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
442	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
443	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
444	C₄⋊C₄	—	53	16	no	—
445	C₄⋊C₄	C₂³×D₅₃	1	16	yes	complemented
446	C₄⋊C₄	C₂³×D₅₃	1	32	yes	complemented
447	C₂²⋊C₄	—	53	16	no	—
448	C₂²⋊C₄	—	53	32	no	—
449	C₂²⋊C₄	—	53	32	no	—
450	C₂²⋊C₄	—	53	32	no	—
451	C₂²⋊C₄	—	53	16	no	—
452	C₂²⋊C₄	—	53	32	no	—
453	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	—
454	C₄²	—	53	16	no	—
455	C₂³	—	2	32	no	—
456	C₂³	—	2	12	no	—
457	C₂³	—	2	12	no	—
458	C₂³	—	53	48	no	—
459	C₂³	—	53	8	no	—
460	C₂³	—	53	8	no	—
461	C₂³	—	53	4	no	—
462	C₂³	—	106	64	no	—
463	C₂³	—	106	64	no	—
464	C₂³	—	106	24	no	—
465	C₂³	—	106	24	no	—
466	C₂³	—	106	12	no	—
467	C₂³	—	106	24	no	—
468	C₂³	—	106	12	no	—
469	C₂³	—	106	24	no	—
470	C₂³	—	106	64	no	—
471	C₂³	—	106	12	no	—
472	C₂³	—	106	24	no	—
473	C₂³	—	212	32	no	—
474	C₂³	—	212	384	no	—
475	C₂³	—	212	16	no	—
476	C₂³	—	212	32	no	—
477	C₂³	C₂×D₄×D₅₃	1	4	yes	—
478	C₂³	C₂×#11719949	1	4	yes	—
479	C₂³	C₂×#11719949	1	24	yes	complemented
480	C₂³	#13133263	1	4	yes	—
481	C₂³	C₂⁴×D₅₃	1	4	yes	—
482	C₂³	C₂⁴×D₅₃	1	3	yes	—
483	Q₈	—	2	32	no	—
484	D₄	—	53	64	no	—
485	D₄	—	106	64	no	—
486	D₄	—	106	64	no	—
487	D₄	—	106	32	no	—
488	D₄	—	106	32	no	—
489	D₄	—	212	32	no	—
490	C₂×C₄	—	2	12	no	—
491	C₂×C₄	—	2	24	no	—
492	C₂×C₄	—	2	24	no	—
493	C₂×C₄	—	2	32	no	—
494	C₂×C₄	—	53	4	no	—
495	C₂×C₄	—	53	4	no	—
496	C₂×C₄	—	53	4	no	—
497	C₂×C₄	—	106	64	no	—
498	C₂×C₄	—	106	12	no	—
499	C₂×C₄	—	106	12	no	—
500	C₂×C₄	—	106	12	no	—
501	C₂×C₄	—	106	32	no	—
502	C₂×C₄	—	212	32	no	—
503	C₂×C₄	C₂×D₄×D₅₃	1	24	yes	—
504	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
505	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
506	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
507	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
508	C₂²	—	2	48	no	—
509	C₂²	—	2	4	no	—
510	C₂²	—	2	4	no	—
511	C₂²	—	53	16	no	—
512	C₂²	—	106	96	no	—
513	C₂²	—	106	8	no	—
514	C₂²	—	106	8	no	—
515	C₂²	—	106	4	no	—
516	C₂²	—	106	8	no	—
517	C₂²	—	106	4	no	—
518	C₂²	—	106	8	no	—
519	C₂²	—	106	4	no	—
520	C₂²	—	106	8	no	—
521	C₂²	—	212	128	no	—
522	C₂²	—	212	48	no	—
523	C₂²	—	212	24	no	—
524	C₂²	#16811264	1	16	yes	direct product factor
525	C₂²	C₂²×D₄×D₅₃	1	6	yes	—
526	C₂²	C₂²×#11719949	1	6	yes	—
527	C₂²	C₂²×#11719949	1	8	yes	complemented
528	C₂²	C₂×#13133263	1	6	yes	—
529	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
530	C₄	—	2	4	no	—
531	C₄	—	2	8	no	—
532	C₄	—	2	8	no	—
533	C₄	—	106	4	no	—
534	C₄	—	106	4	no	—
535	C₄	—	106	4	no	—
536	C₄	C₂²×D₄×D₅₃	1	8	yes	—
537	C₂	—	2	8	no	—
538	C₂	—	106	16	no	—
539	C₂	—	212	8	no	—
540	C₂	—	212	4	no	—
541	C₂	C₂×#16811264	1	12	yes	direct product factor
542	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
543	C₂	C₂³×#11719949	1	1	yes	characteristic
544	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×C₂²⋊Q₈×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂²×C₂²⋊Q₈	D₅₃
hypocenter	C₅₃	C₂²×#546
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂³×#11719949	C₂	semidirect
2. C₂⁴×D₂₁₂	C₂	semidirect
3. C₂×#16811264	C₂	direct (N × H)
4. C₂²×#13133206	C₂	semidirect
5. C₂²×#13133202	C₂	semidirect
6. C₂²×#13133192	C₂	semidirect
7. C₂²×#13133192	C₂	semidirect
8. C₂²×#13133187	C₂	semidirect
9. C₂²×#13133179	C₂	semidirect
10. C₂²×C₄×C₂²⋊D₅₃	C₂	semidirect
11. C₂²×C₂²⋊C₄×D₅₃	C₂	semidirect
12. C₂²×C₂²⋊Q₈×C₅₃	C₂	semidirect
15. C₂²×#11719949	C₂²	semidirect
20. C₂³×D₂₁₂	C₂²	semidirect
21. C₂³×D₂₁₂	C₂²	semidirect
22. C₂³×D₂₁₂	C₂²	semidirect
24. C₂³×C₄×D₅₃	C₂²	semidirect
25. C₂³×C₄×D₅₃	C₂²	semidirect
26. C₂³×C₄×D₅₃	C₂²	semidirect
27. #16811264	C₂²	direct (N × H)
28. C₂×#13133206	C₂²	semidirect
29. C₂×#13133202	C₂²	semidirect
30. C₂×#13133192	C₂²	semidirect
31. C₂×#13133192	C₂²	semidirect
32. C₂×#13133187	C₂²	semidirect
33. C₂×#13133179	C₂²	semidirect
34. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
35. C₂²×#11719919	C₂²	semidirect
37. C₂²×#11719921	C₂²	semidirect
38. C₂²×#11719921	C₂²	semidirect
39. C₂²×#11719921	C₂²	semidirect
40. C₂²×#11719921	C₂²	semidirect
41. C₂²×#11719921	C₂²	semidirect
42. C₂×C₂²⋊C₄×D₅₃	C₂²	semidirect
43. C₂²×C₄×Dic₅₃	C₂²	semidirect
44. C₂³×Q₈×C₅₃	C₂²	semidirect
46. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
47. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
48. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
58. C₂×#11719949	C₂³	semidirect
64. C₂²×D₂₁₂	D₄	semidirect
68. C₂²×D₂₁₂	C₂³	semidirect
69. C₂²×D₂₁₂	C₂³	semidirect
70. C₂²×D₂₁₂	C₂³	semidirect
79. C₂²×C₄×D₅₃	C₂³	semidirect
80. C₂²×C₄×D₅₃	C₂³	semidirect
81. C₂²×C₄×D₅₃	C₂³	semidirect
82. #13133206	C₂³	semidirect
83. #13133202	C₂³	semidirect
84. #13133192	C₂³	semidirect
85. #13133192	C₂³	semidirect
86. #13133187	C₂³	semidirect
87. #13133179	C₂³	semidirect
88. C₄×C₂²⋊D₅₃	C₂³	semidirect
89. C₂×#11719919	C₂³	semidirect
91. C₂×#11719921	C₂³	semidirect
92. C₂×#11719921	C₂³	semidirect
93. C₂×#11719921	C₂³	semidirect
94. C₂×#11719921	C₂³	semidirect
95. C₂×#11719921	C₂³	semidirect
96. C₂²⋊C₄×D₅₃	C₂³	semidirect
97. C₂×C₄×Dic₅₃	C₂³	semidirect
100. C₂²×Q₈×C₅₃	C₂³	semidirect
107. C₂²⋊Q₈×C₅₃	C₂³	semidirect
108. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
109. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
113. C₂⁴×C₅₃	#41	semidirect
134. C₂×Q₈×C₅₃	C₂⁴	semidirect
159. C₂×D₂₁₂	C₂×D₄	semidirect
163. C₂×D₂₁₂	C₂⁴	semidirect
164. C₂×D₂₁₂	C₂⁴	semidirect
168. C₂×C₄×D₅₃	C₂⁴	semidirect
169. C₂×C₄×D₅₃	C₂⁴	semidirect
170. C₂×C₄×D₅₃	C₂⁴	semidirect
171. C₄⋊C₄×C₅₃	C₂⁴	semidirect
172. C₄⋊C₄×C₅₃	C₂⁴	semidirect
175. #11719921	C₂⁴	semidirect
176. #11719921	C₂⁴	semidirect
177. #11719921	C₂⁴	semidirect
178. #11719921	C₂⁴	semidirect
179. #11719921	C₂⁴	semidirect
180. #11719919	C₂⁴	semidirect
181. C₄×Dic₅₃	C₂⁴	semidirect
187. C₂³×C₅₃	C₂×#41	semidirect
231. D₂₁₂	C₂²×D₄	semidirect
239. C₂²×C₅₃	#546	semidirect
242. C₂²×C₅₃	C₂²×#41	semidirect
270. C₂²×C₂²⋊Q₈	D₅₃	semidirect
278. C₂×C₅₃	C₂×#546	semidirect
288. C₂³×Q₈	C₂×D₅₃	semidirect
304. C₂×C₂²⋊Q₈	C₂×D₅₃	semidirect
311. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
312. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
322. C₅₃	C₂²×#546	semidirect
334. C₂²×Q₈	C₂²×D₅₃	semidirect
366. C₂²⋊Q₈	C₂²×D₅₃	semidirect
373. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
374. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
403. C₂⁴	#11719949	semidirect
408. C₂×Q₈	C₂³×D₅₃	semidirect
445. C₄⋊C₄	C₂³×D₅₃	semidirect
446. C₄⋊C₄	C₂³×D₅₃	semidirect
479. C₂³	C₂×#11719949	semidirect
524. C₂²	#16811264	direct (N × H)
527. C₂²	C₂²×#11719949	semidirect
541. C₂	C₂×#16811264	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×#11719949	2
C₂⁴×D₂₁₂	2
C₂×#16811264	2
C₂²×#13133206	2
C₂²×#13133202	2
C₂²×#13133192	2
C₂²×#13133187	2
C₂²×#13133179	2
C₂²×C₄×C₂²⋊D₅₃	2
C₂²×C₂²⋊C₄×D₅₃	2
C₂²×C₂²⋊Q₈×C₅₃	2
C₂²×#546	53

Subgroups — C22×#16811264 #28291021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811264 #28291021