Subgroups — C₂²×#16811184 #28290810

Order 13568.

15230 conjugacy classes of subgroups.

The subgroup lattice has 576 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	1	yes	maximal, characteristic, complemented
3	C₂×#16811184	C₂	1	48	yes	maximal, direct product factor
4	C₂²×#13133202	C₂	1	2	yes	maximal, complemented
5	C₂²×#13133200	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#13133192	C₂	1	2	yes	maximal, complemented
7	C₂²×#13133190	C₂	1	2	yes	maximal, complemented
8	C₂²×#13133187	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#13133187	C₂	1	2	yes	maximal, complemented
10	C₂²×#13133185	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×C₂²⋊C₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#123×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
14	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic, complemented
15	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic, complemented
16	C₂³×D₂₁₂	—	2	16	no	—
17	C₂³×D₂₁₂	C₂²	1	2	yes	—
18	C₂³×D₂₁₂	C₂²	1	2	yes	complemented
19	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
20	C₂³×D₂₁₂	C₂²	1	2	yes	complemented
21	C₂³×D₂₁₂	C₂²	1	24	yes	complemented
22	C₂³×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
23	C₂³×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic, complemented
24	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
25	C₂²×D₄×D₅₃	—	2	16	no	—
26	C₂²×D₄×D₅₃	C₂²	1	24	yes	complemented
27	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic, complemented
28	C₂³×C₄×D₅₃	C₂²	1	2	yes	complemented
29	#16811184	C₂²	1	256	yes	direct product factor
30	C₂×#13133202	C₂²	1	48	yes	complemented
31	C₂×#13133200	C₂²	1	24	yes	complemented
32	C₂×#13133192	C₂²	1	48	yes	complemented
33	C₂×#13133190	C₂²	1	48	yes	complemented
34	C₂×#13133187	C₂²	1	24	yes	complemented
35	C₂×#13133187	C₂²	1	48	yes	complemented
36	C₂×#13133185	C₂²	1	24	yes	complemented
37	C₂²×#11719919	C₂²	1	2	yes	complemented
38	C₂²×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
39	C₂²×#11719921	C₂²	1	2	yes	complemented
40	C₂²×#11719921	C₂²	1	2	yes	complemented
41	C₂²×#11719921	C₂²	1	2	yes	complemented
42	C₂²×#11719921	C₂²	1	2	yes	complemented
43	C₂×C₂²⋊C₄×D₅₃	C₂²	1	24	yes	complemented
44	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
45	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
46	C₂×#123×C₅₃	C₂²	1	24	yes	complemented
47	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
48	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
49	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
50	C₂⁴×D₅₃	—	2	8	no	—
51	C₂⁴×D₅₃	—	2	8	no	—
52	C₂⁴×D₅₃	—	4	8	no	—
53	C₂⁴×D₅₃	—	4	8	no	—
54	C₂⁴×D₅₃	—	4	8	no	—
55	C₂⁴×D₅₃	D₄	1	16	yes	complemented
56	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
57	C₂⁴×D₅₃	C₂³	1	2	yes	—
58	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
59	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
60	C₂⁴×D₅₃	C₂³	1	12	yes	—
61	C₂⁴×D₅₃	C₂³	1	12	yes	complemented
62	C₂⁴×D₅₃	C₂³	1	12	yes	complemented
63	C₂²×D₂₁₂	—	2	8	no	—
64	C₂²×D₂₁₂	—	2	16	no	—
65	C₂²×D₂₁₂	—	2	16	no	—
66	C₂²×D₂₁₂	—	2	192	no	—
67	C₂²×D₂₁₂	—	4	8	no	—
68	C₂²×D₂₁₂	C₂³	1	24	yes	—
69	C₂²×D₂₁₂	C₂³	1	64	yes	complemented
70	C₂²×D₂₁₂	C₂³	1	24	yes	complemented
71	C₂²×D₂₁₂	C₂³	1	12	yes	complemented
72	C₂²×D₂₁₂	C₂³	1	24	yes	complemented
73	C₂²×C₂²⋊D₅₃	—	2	8	no	—
74	C₂²×C₂²⋊D₅₃	—	2	8	no	—
75	C₂²×C₂²⋊D₅₃	—	4	8	no	—
76	C₂²×C₂²⋊D₅₃	C₂³	1	12	yes	—
77	C₂²×C₂²⋊D₅₃	C₂³	1	12	yes	complemented
78	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	complemented
79	C₂×D₄×D₅₃	—	2	192	no	—
80	C₂×D₄×D₅₃	C₂³	1	64	yes	complemented
81	C₂³×Dic₅₃	C₂³	1	2	yes	—
82	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
83	C₂²×C₄×D₅₃	—	2	8	no	—
84	C₂²×C₄×D₅₃	—	4	8	no	—
85	C₂²×C₄×D₅₃	C₂³	1	12	yes	complemented
86	C₂²×C₄×D₅₃	C₂³	1	24	yes	complemented
87	#13133202	C₂³	1	128	yes	complemented
88	#13133200	C₂³	1	64	yes	complemented
89	#13133192	C₂³	1	128	yes	complemented
90	#13133190	C₂³	1	128	yes	complemented
91	#13133187	C₂³	1	128	yes	complemented
92	#13133187	C₂³	1	64	yes	complemented
93	#13133185	C₂³	1	64	yes	complemented
94	C₂×#11719919	C₂³	1	24	yes	complemented
95	C₂×C₂²⋊Dic₅₃	C₂³	1	12	yes	complemented
96	C₂×#11719921	C₂³	1	24	yes	complemented
97	C₂×#11719921	C₂³	1	24	yes	complemented
98	C₂×#11719921	C₂³	1	24	yes	complemented
99	C₂×#11719921	C₂³	1	24	yes	complemented
100	C₂²⋊C₄×D₅₃	C₂³	1	64	yes	complemented
101	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
102	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
103	C₂²×D₄×C₅₃	—	2	8	no	—
104	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
105	C₂³×C₄×C₅₃	—	2	8	no	—
106	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
107	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
108	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
109	C₂³×C₄×C₅₃	C₂³	1	12	yes	complemented
110	#123×C₅₃	C₂³	1	64	yes	complemented
111	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	complemented
112	C₂×C₂²⋊C₄×C₅₃	C₂³	1	12	yes	complemented
113	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	complemented
114	C₂⁴×C₅₃	—	2	8	no	—
115	C₂⁴×C₅₃	—	2	4	no	—
116	C₂⁴×C₅₃	—	2	8	no	—
117	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
118	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
119	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
120	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
121	C₂³×D₅₃	—	2	8	no	—
122	C₂³×D₅₃	—	2	8	no	—
123	C₂³×D₅₃	—	2	4	no	—
124	C₂³×D₅₃	—	2	8	no	—
125	C₂³×D₅₃	—	2	48	no	—
126	C₂³×D₅₃	—	2	8	no	—
127	C₂³×D₅₃	—	2	8	no	—
128	C₂³×D₅₃	—	2	8	no	—
129	C₂³×D₅₃	—	2	48	no	—
130	C₂³×D₅₃	—	4	64	no	—
131	C₂³×D₅₃	—	4	48	no	—
132	C₂³×D₅₃	—	4	48	no	—
133	C₂³×D₅₃	—	4	48	no	—
134	C₂³×D₅₃	—	4	64	no	—
135	C₂³×D₅₃	—	4	64	no	—
136	C₂³×D₅₃	C₂×D₄	1	8	yes	—
137	C₂³×D₅₃	C₂×D₄	1	8	yes	—
138	C₂³×D₅₃	C₂×D₄	1	96	yes	complemented
139	C₂³×D₅₃	C₂⁴	1	6	yes	—
140	C₂³×D₅₃	C₂⁴	1	12	yes	—
141	C₂³×D₅₃	C₂⁴	1	6	yes	—
142	C₂³×D₅₃	C₂⁴	1	6	yes	—
143	C₂³×D₅₃	C₂⁴	1	16	yes	—
144	C₂³×D₅₃	C₂⁴	1	16	yes	complemented
145	C₂³×D₅₃	C₂⁴	1	16	yes	complemented
146	C₂×D₄×C₅₃	—	2	48	no	—
147	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
148	C₂²×C₄×C₅₃	—	2	8	no	—
149	C₂²×C₄×C₅₃	—	2	48	no	—
150	C₂²×C₄×C₅₃	—	2	4	no	—
151	C₂²×C₄×C₅₃	—	2	8	no	—
152	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
153	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
154	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
155	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	complemented
156	C₂×C₂²⋊D₅₃	—	2	48	no	—
157	C₂×C₂²⋊D₅₃	—	2	48	no	—
158	C₂×C₂²⋊D₅₃	—	4	48	no	—
159	C₂×C₂²⋊D₅₃	C₂⁴	1	16	yes	—
160	C₂×C₂²⋊D₅₃	C₂⁴	1	16	yes	complemented
161	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	complemented
162	C₂²×Dic₅₃	—	2	4	no	—
163	C₂²×Dic₅₃	—	2	8	no	—
164	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
165	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
166	D₄×D₅₃	—	2	256	no	—
167	C₂×D₂₁₂	—	2	48	no	—
168	C₂×D₂₁₂	—	2	96	no	—
169	C₂×D₂₁₂	—	2	96	no	—
170	C₂×D₂₁₂	—	2	256	no	—
171	C₂×D₂₁₂	—	4	48	no	—
172	C₂×D₂₁₂	C₂⁴	1	32	yes	—
173	C₂×D₂₁₂	C₂⁴	1	32	yes	complemented
174	C₂×D₂₁₂	C₂⁴	1	16	yes	complemented
175	C₂×D₂₁₂	C₂⁴	1	32	yes	complemented
176	C₂×C₄×D₅₃	—	2	48	no	—
177	C₂×C₄×D₅₃	—	4	48	no	—
178	C₂×C₄×D₅₃	C₂⁴	1	16	yes	complemented
179	C₂×C₄×D₅₃	C₂⁴	1	32	yes	complemented
180	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
181	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	complemented
182	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
183	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	complemented
184	#11719921	C₂⁴	1	32	yes	complemented
185	#11719921	C₂⁴	1	32	yes	complemented
186	#11719921	C₂⁴	1	32	yes	complemented
187	#11719921	C₂⁴	1	32	yes	complemented
188	#11719919	C₂⁴	1	32	yes	complemented
189	C₂³×C₅₃	—	2	32	no	—
190	C₂³×C₅₃	—	2	24	no	—
191	C₂³×C₅₃	—	2	12	no	—
192	C₂³×C₅₃	—	2	24	no	—
193	C₂³×C₅₃	—	4	16	no	—
194	C₂³×C₅₃	C₂²×D₄	1	24	yes	—
195	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
196	C₂³×C₅₃	H(5,𝔽₂)	1	8	yes	—
197	C₂³×C₅₃	C₂⁵	1	4	yes	—
198	C₂³×C₅₃	C₂⁵	1	4	yes	—
199	C₂³×C₅₃	C₂⁵	1	3	yes	—
200	C₂²×D₅₃	—	2	32	no	—
201	C₂²×D₅₃	—	2	32	no	—
202	C₂²×D₅₃	—	2	24	no	—
203	C₂²×D₅₃	—	2	24	no	—
204	C₂²×D₅₃	—	2	24	no	—
205	C₂²×D₅₃	—	2	32	no	—
206	C₂²×D₅₃	—	2	12	no	—
207	C₂²×D₅₃	—	2	24	no	—
208	C₂²×D₅₃	—	2	24	no	—
209	C₂²×D₅₃	—	2	24	no	—
210	C₂²×D₅₃	—	2	32	no	—
211	C₂²×D₅₃	—	4	192	no	—
212	C₂²×D₅₃	—	4	32	no	—
213	C₂²×D₅₃	—	4	32	no	—
214	C₂²×D₅₃	—	4	32	no	—
215	C₂²×D₅₃	—	4	192	no	—
216	C₂²×D₅₃	—	4	192	no	—
217	C₂²×D₅₃	—	4	16	no	—
218	C₂²×D₅₃	—	4	32	no	—
219	C₂²×D₅₃	C₂²×D₄	1	24	yes	—
220	C₂²×D₅₃	C₂²×D₄	1	24	yes	—
221	C₂²×D₅₃	C₂²×D₄	1	64	yes	complemented
222	C₂²×D₅₃	C₂⁵	1	8	yes	—
223	C₂²×D₅₃	C₂⁵	1	4	yes	—
224	C₂²×D₅₃	C₂⁵	1	4	yes	—
225	C₂²×D₅₃	C₂⁵	1	4	yes	—
226	D₄×C₅₃	—	2	32	no	—
227	C₂×C₄×C₅₃	—	2	12	no	—
228	C₂×C₄×C₅₃	—	2	24	no	—
229	C₂×C₄×C₅₃	—	2	32	no	—
230	C₂×C₄×C₅₃	—	2	24	no	—
231	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
232	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
233	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
234	C₂²⋊D₅₃	—	2	32	no	—
235	C₂²⋊D₅₃	—	2	32	no	—
236	C₂²⋊D₅₃	—	4	32	no	—
237	C₂×Dic₅₃	—	2	12	no	—
238	C₂×Dic₅₃	—	2	24	no	—
239	C₂×Dic₅₃	C₂⁵	1	4	yes	—
240	C₂×Dic₅₃	C₂⁵	1	8	yes	—
241	D₂₁₂	—	2	64	no	—
242	D₂₁₂	—	2	64	no	—
243	D₂₁₂	—	2	32	no	—
244	D₂₁₂	—	4	32	no	—
245	C₄×D₅₃	—	2	32	no	—
246	C₄×D₅₃	—	4	32	no	—
247	C₂²×#534	—	53	1	no	maximal, Sylow
248	C₂²×C₅₃	—	2	8	no	—
249	C₂²×C₅₃	—	2	48	no	—
250	C₂²×C₅₃	—	2	4	no	—
251	C₂²×C₅₃	—	2	8	no	—
252	C₂²×C₅₃	—	4	24	no	—
253	C₂²×C₅₃	#534	1	16	yes	complemented
254	C₂²×C₅₃	C₂³×D₄	1	8	yes	—
255	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
256	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	12	yes	—
257	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
258	C₂×D₅₃	—	2	8	no	—
259	C₂×D₅₃	—	2	8	no	—
260	C₂×D₅₃	—	2	48	no	—
261	C₂×D₅₃	—	2	48	no	—
262	C₂×D₅₃	—	2	8	no	—
263	C₂×D₅₃	—	2	4	no	—
264	C₂×D₅₃	—	2	8	no	—
265	C₂×D₅₃	—	2	8	no	—
266	C₂×D₅₃	—	2	8	no	—
267	C₂×D₅₃	—	4	64	no	—
268	C₂×D₅₃	—	4	64	no	—
269	C₂×D₅₃	—	4	64	no	—
270	C₂×D₅₃	—	4	48	no	—
271	C₂×D₅₃	—	4	24	no	—
272	C₂×D₅₃	C₂³×D₄	1	8	yes	—
273	C₂×D₅₃	C₂³×D₄	1	8	yes	—
274	C₄×C₅₃	—	2	4	no	—
275	C₄×C₅₃	—	2	8	no	—
276	C₄×C₅₃	—	2	8	no	—
277	Dic₅₃	—	2	4	no	—
278	Dic₅₃	—	2	8	no	—
279	C₂⁴×D₄	—	53	1	no	—
280	C₂⁴×D₄	—	53	1	no	—
281	C₂×#534	—	53	48	no	—
282	C₂²×#123	—	53	1	no	—
283	C₂²×#123	—	53	2	no	—
284	C₂²×#123	D₅₃	1	1	yes	characteristic, complemented
285	C₂²×#121	—	53	2	no	—
286	C₂²×#121	—	53	2	no	—
287	C₂²×#120	—	53	1	no	—
288	C₂²×#120	—	53	2	no	—
289	C₂²×#120	—	53	1	no	—
290	C₂³×C₂²⋊C₄	—	53	1	no	—
291	C₂×C₅₃	—	2	8	no	—
292	C₂×C₅₃	—	4	4	no	—
293	C₂×C₅₃	C₂×#534	1	12	yes	complemented
294	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
295	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	2	yes	—
296	D₅₃	—	2	8	no	—
297	D₅₃	—	2	8	no	—
298	D₅₃	—	4	8	no	—
299	D₅₃	—	4	4	no	—
300	C₂⁶	—	53	1	no	—
301	C₂⁶	—	53	1	no	—
302	C₂⁶	—	53	1	no	—
303	C₂³×D₄	—	53	1	no	—
304	C₂³×D₄	—	53	1	no	—
305	C₂³×D₄	—	53	2	no	—
306	C₂³×D₄	—	53	24	no	—
307	C₂³×D₄	—	53	2	no	—
308	C₂³×D₄	—	53	1	no	—
309	C₂³×D₄	—	53	2	no	—
310	C₂³×D₄	—	53	2	no	—
311	C₂³×D₄	—	53	24	no	—
312	C₂³×D₄	—	106	16	no	—
313	C₂³×D₄	—	106	16	no	—
314	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic
315	C₂⁴×C₄	—	53	1	no	—
316	C₂⁴×C₄	—	53	2	no	—
317	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
318	#534	—	53	256	no	—
319	C₂×#123	—	53	24	no	—
320	C₂×#123	—	53	48	no	—
321	C₂×#123	C₂×D₅₃	1	24	yes	complemented
322	C₂×#121	—	53	48	no	—
323	C₂×#121	—	53	48	no	—
324	C₂×#120	—	53	24	no	—
325	C₂×#120	—	53	48	no	—
326	C₂×#120	—	53	24	no	—
327	C₂²×C₄⋊C₄	—	53	2	no	—
328	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	complemented
329	C₂²×C₂²⋊C₄	—	53	1	no	—
330	C₂²×C₂²⋊C₄	—	53	2	no	—
331	C₂²×C₂²⋊C₄	—	53	2	no	—
332	C₂²×C₂²⋊C₄	—	53	2	no	—
333	C₂²×C₂²⋊C₄	—	53	2	no	—
334	C₂²×C₂²⋊C₄	—	53	24	no	—
335	C₂²×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
336	C₂²×C₂²⋊C₄	C₂×D₅₃	1	2	yes	complemented
337	C₅₃	C₂²×#534	1	1	yes	characteristic, complemented, Sylow
338	C₂⁵	—	53	1	no	—
339	C₂⁵	—	53	1	no	—
340	C₂⁵	—	53	16	no	—
341	C₂⁵	—	53	12	no	—
342	C₂⁵	—	53	1	no	—
343	C₂⁵	—	53	2	no	—
344	C₂⁵	—	53	12	no	—
345	C₂⁵	—	53	12	no	—
346	C₂⁵	—	106	8	no	—
347	C₂⁵	—	106	8	no	—
348	C₂⁵	—	212	8	no	—
349	C₂⁵	—	212	8	no	—
350	C₂⁵	—	212	8	no	—
351	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
352	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
353	C₂²×D₄	—	2	8	no	—
354	C₂²×D₄	—	53	12	no	—
355	C₂²×D₄	—	53	12	no	—
356	C₂²×D₄	—	53	24	no	—
357	C₂²×D₄	—	53	64	no	—
358	C₂²×D₄	—	53	24	no	—
359	C₂²×D₄	—	53	12	no	—
360	C₂²×D₄	—	53	24	no	—
361	C₂²×D₄	—	53	24	no	—
362	C₂²×D₄	—	53	64	no	—
363	C₂²×D₄	—	106	16	no	—
364	C₂²×D₄	—	106	16	no	—
365	C₂²×D₄	—	106	192	no	—
366	C₂²×D₄	—	106	8	no	—
367	C₂²×D₄	—	106	192	no	—
368	C₂²×D₄	—	106	8	no	—
369	C₂²×D₄	—	106	8	no	—
370	C₂²×D₄	—	212	8	no	—
371	C₂²×D₄	—	212	8	no	—
372	C₂²×D₄	C₂²×D₅₃	1	12	yes	—
373	C₂³×C₄	—	2	8	no	—
374	C₂³×C₄	—	53	2	no	—
375	C₂³×C₄	—	53	1	no	—
376	C₂³×C₄	—	53	12	no	—
377	C₂³×C₄	—	53	24	no	—
378	C₂³×C₄	—	106	8	no	—
379	C₂³×C₄	—	212	8	no	—
380	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
381	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
382	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
383	C₂³×C₄	C₂²×D₅₃	1	12	yes	complemented
384	#123	—	53	64	no	—
385	#123	—	53	128	no	—
386	#123	C₂²×D₅₃	1	64	yes	complemented
387	#121	—	53	128	no	—
388	#121	—	53	128	no	—
389	#120	—	53	64	no	—
390	#120	—	53	128	no	—
391	#120	—	53	64	no	—
392	C₂×C₄⋊C₄	—	53	24	no	—
393	C₂×C₄⋊C₄	C₂²×D₅₃	1	24	yes	complemented
394	C₂×C₂²⋊C₄	—	53	12	no	—
395	C₂×C₂²⋊C₄	—	53	24	no	—
396	C₂×C₂²⋊C₄	—	53	24	no	—
397	C₂×C₂²⋊C₄	—	53	24	no	—
398	C₂×C₂²⋊C₄	—	53	24	no	—
399	C₂×C₂²⋊C₄	—	53	64	no	—
400	C₂×C₂²⋊C₄	C₂²×D₅₃	1	12	yes	complemented
401	C₂×C₂²⋊C₄	C₂²×D₅₃	1	24	yes	complemented
402	C₂⁴	—	2	8	no	—
403	C₂⁴	—	2	4	no	—
404	C₂⁴	—	2	8	no	—
405	C₂⁴	—	53	8	no	—
406	C₂⁴	—	53	8	no	—
407	C₂⁴	—	53	6	no	—
408	C₂⁴	—	53	6	no	—
409	C₂⁴	—	53	96	no	—
410	C₂⁴	—	53	16	no	—
411	C₂⁴	—	53	6	no	—
412	C₂⁴	—	53	12	no	—
413	C₂⁴	—	53	16	no	—
414	C₂⁴	—	53	16	no	—
415	C₂⁴	—	106	48	no	—
416	C₂⁴	—	106	8	no	—
417	C₂⁴	—	106	8	no	—
418	C₂⁴	—	106	48	no	—
419	C₂⁴	—	106	4	no	—
420	C₂⁴	—	106	8	no	—
421	C₂⁴	—	106	8	no	—
422	C₂⁴	—	106	8	no	—
423	C₂⁴	—	106	8	no	—
424	C₂⁴	—	212	64	no	—
425	C₂⁴	—	212	48	no	—
426	C₂⁴	—	212	48	no	—
427	C₂⁴	—	212	48	no	—
428	C₂⁴	—	212	64	no	—
429	C₂⁴	—	212	64	no	—
430	C₂⁴	D₄×D₅₃	1	8	yes	—
431	C₂⁴	C₂³×D₅₃	1	6	yes	—
432	C₂⁴	C₂³×D₅₃	1	6	yes	—
433	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
434	C₂×D₄	—	2	48	no	—
435	C₂×D₄	—	53	16	no	—
436	C₂×D₄	—	53	16	no	—
437	C₂×D₄	—	53	32	no	—
438	C₂×D₄	—	53	32	no	—
439	C₂×D₄	—	53	16	no	—
440	C₂×D₄	—	53	32	no	—
441	C₂×D₄	—	53	32	no	—
442	C₂×D₄	—	106	96	no	—
443	C₂×D₄	—	106	96	no	—
444	C₂×D₄	—	106	256	no	—
445	C₂×D₄	—	106	48	no	—
446	C₂×D₄	—	106	256	no	—
447	C₂×D₄	—	106	48	no	—
448	C₂×D₄	—	106	48	no	—
449	C₂×D₄	—	212	48	no	—
450	C₂×D₄	—	212	48	no	—
451	C₂×D₄	C₂³×D₅₃	1	16	yes	—
452	C₂²×C₄	—	2	8	no	—
453	C₂²×C₄	—	2	48	no	—
454	C₂²×C₄	—	2	4	no	—
455	C₂²×C₄	—	2	8	no	—
456	C₂²×C₄	—	53	12	no	—
457	C₂²×C₄	—	53	6	no	—
458	C₂²×C₄	—	53	16	no	—
459	C₂²×C₄	—	53	32	no	—
460	C₂²×C₄	—	106	48	no	—
461	C₂²×C₄	—	106	4	no	—
462	C₂²×C₄	—	106	8	no	—
463	C₂²×C₄	—	212	48	no	—
464	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
465	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
466	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
467	C₂²×C₄	C₂³×D₅₃	1	16	yes	complemented
468	C₄⋊C₄	—	53	32	no	—
469	C₄⋊C₄	C₂³×D₅₃	1	32	yes	complemented
470	C₂²⋊C₄	—	53	16	no	—
471	C₂²⋊C₄	—	53	32	no	—
472	C₂²⋊C₄	—	53	32	no	—
473	C₂²⋊C₄	—	53	32	no	—
474	C₂²⋊C₄	—	53	32	no	—
475	C₂²⋊C₄	C₂³×D₅₃	1	16	yes	complemented
476	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	complemented
477	C₂³	—	2	32	no	—
478	C₂³	—	2	24	no	—
479	C₂³	—	2	12	no	—
480	C₂³	—	2	24	no	—
481	C₂³	—	4	16	no	—
482	C₂³	—	53	24	no	—
483	C₂³	—	53	24	no	—
484	C₂³	—	53	4	no	—
485	C₂³	—	53	4	no	—
486	C₂³	—	53	64	no	—
487	C₂³	—	53	8	no	—
488	C₂³	—	53	4	no	—
489	C₂³	—	106	32	no	—
490	C₂³	—	106	32	no	—
491	C₂³	—	106	32	no	—
492	C₂³	—	106	24	no	—
493	C₂³	—	106	24	no	—
494	C₂³	—	106	32	no	—
495	C₂³	—	106	24	no	—
496	C₂³	—	106	12	no	—
497	C₂³	—	106	24	no	—
498	C₂³	—	106	24	no	—
499	C₂³	—	106	24	no	—
500	C₂³	—	212	192	no	—
501	C₂³	—	212	32	no	—
502	C₂³	—	212	32	no	—
503	C₂³	—	212	32	no	—
504	C₂³	—	212	192	no	—
505	C₂³	—	212	192	no	—
506	C₂³	—	212	32	no	—
507	C₂³	—	212	16	no	—
508	C₂³	C₂×D₄×D₅₃	1	24	yes	—
509	C₂³	C₂×D₄×D₅₃	1	4	yes	—
510	C₂³	#13133263	1	8	yes	—
511	C₂³	C₂⁴×D₅₃	1	4	yes	—
512	C₂³	C₂⁴×D₅₃	1	4	yes	—
513	C₂³	C₂⁴×D₅₃	1	3	yes	—
514	D₄	—	2	32	no	—
515	D₄	—	106	64	no	—
516	D₄	—	106	64	no	—
517	D₄	—	106	32	no	—
518	D₄	—	106	32	no	—
519	D₄	—	106	32	no	—
520	D₄	—	212	32	no	—
521	D₄	—	212	32	no	—
522	C₂×C₄	—	2	24	no	—
523	C₂×C₄	—	2	32	no	—
524	C₂×C₄	—	2	12	no	—
525	C₂×C₄	—	2	24	no	—
526	C₂×C₄	—	53	8	no	—
527	C₂×C₄	—	53	4	no	—
528	C₂×C₄	—	106	32	no	—
529	C₂×C₄	—	106	12	no	—
530	C₂×C₄	—	106	24	no	—
531	C₂×C₄	—	212	32	no	—
532	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
533	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
534	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
535	C₂²	—	2	48	no	—
536	C₂²	—	2	8	no	—
537	C₂²	—	2	4	no	—
538	C₂²	—	2	8	no	—
539	C₂²	—	4	24	no	—
540	C₂²	—	53	8	no	—
541	C₂²	—	53	8	no	—
542	C₂²	—	106	48	no	—
543	C₂²	—	106	48	no	—
544	C₂²	—	106	8	no	—
545	C₂²	—	106	8	no	—
546	C₂²	—	106	8	no	—
547	C₂²	—	106	4	no	—
548	C₂²	—	106	8	no	—
549	C₂²	—	106	8	no	—
550	C₂²	—	106	8	no	—
551	C₂²	—	212	64	no	—
552	C₂²	—	212	64	no	—
553	C₂²	—	212	64	no	—
554	C₂²	—	212	48	no	—
555	C₂²	—	212	24	no	—
556	C₂²	#16811184	1	16	yes	direct product factor
557	C₂²	C₂²×D₄×D₅₃	1	8	yes	—
558	C₂²	C₂²×D₄×D₅₃	1	6	yes	—
559	C₂²	C₂×#13133263	1	12	yes	—
560	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
561	C₄	—	2	8	no	—
562	C₄	—	2	4	no	—
563	C₄	—	2	8	no	—
564	C₄	—	106	4	no	—
565	C₄	—	106	8	no	—
566	C₂	—	2	8	no	—
567	C₂	—	4	4	no	—
568	C₂	—	106	8	no	—
569	C₂	—	106	8	no	—
570	C₂	—	212	8	no	—
571	C₂	—	212	4	no	—
572	C₂	C₂×#16811184	1	12	yes	direct product factor
573	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
574	C₂	C₂²×#13133263	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×#123×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂²×#123	D₅₃
hypocenter	C₅₃	C₂²×#534
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#123

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. C₂×#16811184	C₂	direct (N × H)
4. C₂²×#13133202	C₂	semidirect
5. C₂²×#13133200	C₂	semidirect
6. C₂²×#13133192	C₂	semidirect
7. C₂²×#13133190	C₂	semidirect
8. C₂²×#13133187	C₂	semidirect
9. C₂²×#13133187	C₂	semidirect
10. C₂²×#13133185	C₂	semidirect
11. C₂²×C₂²⋊C₄×D₅₃	C₂	semidirect
12. C₂²×#123×C₅₃	C₂	semidirect
14. C₂⁵×D₅₃	C₂²	semidirect
15. C₂⁵×D₅₃	C₂²	semidirect
18. C₂³×D₂₁₂	C₂²	semidirect
19. C₂³×D₂₁₂	C₂²	semidirect
20. C₂³×D₂₁₂	C₂²	semidirect
21. C₂³×D₂₁₂	C₂²	semidirect
23. C₂³×C₂²⋊D₅₃	C₂²	semidirect
24. C₂³×C₂²⋊D₅₃	C₂²	semidirect
26. C₂²×D₄×D₅₃	C₂²	semidirect
27. C₂³×C₄×D₅₃	C₂²	semidirect
28. C₂³×C₄×D₅₃	C₂²	semidirect
29. #16811184	C₂²	direct (N × H)
30. C₂×#13133202	C₂²	semidirect
31. C₂×#13133200	C₂²	semidirect
32. C₂×#13133192	C₂²	semidirect
33. C₂×#13133190	C₂²	semidirect
34. C₂×#13133187	C₂²	semidirect
35. C₂×#13133187	C₂²	semidirect
36. C₂×#13133185	C₂²	semidirect
37. C₂²×#11719919	C₂²	semidirect
38. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
39. C₂²×#11719921	C₂²	semidirect
40. C₂²×#11719921	C₂²	semidirect
41. C₂²×#11719921	C₂²	semidirect
42. C₂²×#11719921	C₂²	semidirect
43. C₂×C₂²⋊C₄×D₅₃	C₂²	semidirect
45. C₂⁴×C₄×C₅₃	C₂²	semidirect
46. C₂×#123×C₅₃	C₂²	semidirect
47. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
48. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
49. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
55. C₂⁴×D₅₃	D₄	semidirect
61. C₂⁴×D₅₃	C₂³	semidirect
62. C₂⁴×D₅₃	C₂³	semidirect
69. C₂²×D₂₁₂	C₂³	semidirect
70. C₂²×D₂₁₂	C₂³	semidirect
71. C₂²×D₂₁₂	C₂³	semidirect
72. C₂²×D₂₁₂	C₂³	semidirect
77. C₂²×C₂²⋊D₅₃	C₂³	semidirect
78. C₂²×C₂²⋊D₅₃	C₂³	semidirect
80. C₂×D₄×D₅₃	C₂³	semidirect
85. C₂²×C₄×D₅₃	C₂³	semidirect
86. C₂²×C₄×D₅₃	C₂³	semidirect
87. #13133202	C₂³	semidirect
88. #13133200	C₂³	semidirect
89. #13133192	C₂³	semidirect
90. #13133190	C₂³	semidirect
91. #13133187	C₂³	semidirect
92. #13133187	C₂³	semidirect
93. #13133185	C₂³	semidirect
94. C₂×#11719919	C₂³	semidirect
95. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
96. C₂×#11719921	C₂³	semidirect
97. C₂×#11719921	C₂³	semidirect
98. C₂×#11719921	C₂³	semidirect
99. C₂×#11719921	C₂³	semidirect
100. C₂²⋊C₄×D₅₃	C₂³	semidirect
109. C₂³×C₄×C₅₃	C₂³	semidirect
110. #123×C₅₃	C₂³	semidirect
111. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
112. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
113. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
138. C₂³×D₅₃	C₂×D₄	semidirect
144. C₂³×D₅₃	C₂⁴	semidirect
145. C₂³×D₅₃	C₂⁴	semidirect
155. C₂²×C₄×C₅₃	C₂⁴	semidirect
160. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
161. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
173. C₂×D₂₁₂	C₂⁴	semidirect
174. C₂×D₂₁₂	C₂⁴	semidirect
175. C₂×D₂₁₂	C₂⁴	semidirect
178. C₂×C₄×D₅₃	C₂⁴	semidirect
179. C₂×C₄×D₅₃	C₂⁴	semidirect
180. C₄⋊C₄×C₅₃	C₂⁴	semidirect
181. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
182. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
183. C₂²⋊Dic₅₃	C₂⁴	semidirect
184. #11719921	C₂⁴	semidirect
185. #11719921	C₂⁴	semidirect
186. #11719921	C₂⁴	semidirect
187. #11719921	C₂⁴	semidirect
188. #11719919	C₂⁴	semidirect
221. C₂²×D₅₃	C₂²×D₄	semidirect
253. C₂²×C₅₃	#534	semidirect
284. C₂²×#123	D₅₃	semidirect
293. C₂×C₅₃	C₂×#534	semidirect
317. C₂⁴×C₄	C₂×D₅₃	semidirect
321. C₂×#123	C₂×D₅₃	semidirect
328. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
335. C₂²×C₂²⋊C₄	C₂×D₅₃	semidirect
336. C₂²×C₂²⋊C₄	C₂×D₅₃	semidirect
337. C₅₃	C₂²×#534	semidirect
383. C₂³×C₄	C₂²×D₅₃	semidirect
386. #123	C₂²×D₅₃	semidirect
393. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
400. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
401. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
467. C₂²×C₄	C₂³×D₅₃	semidirect
469. C₄⋊C₄	C₂³×D₅₃	semidirect
475. C₂²⋊C₄	C₂³×D₅₃	semidirect
476. C₂²⋊C₄	C₂³×D₅₃	semidirect
556. C₂²	#16811184	direct (N × H)
572. C₂	C₂×#16811184	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×#16811184	2
C₂²×#13133202	2
C₂²×#13133200	2
C₂²×#13133192	2
C₂²×#13133190	2
C₂²×#13133187	2
C₂²×#13133185	2
C₂²×C₂²⋊C₄×D₅₃	2
C₂²×#123×C₅₃	2
C₂²×#534	53

Subgroups — C22×#16811184 #28290810

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811184 #28290810