Subgroups — C₂²×#16811182 #28290809

Order 13568.

15230 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₂²⋊D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16811182	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#13133200	C₂	1	2	yes	maximal, complemented
4	C₂²×#13133198	C₂	1	2	yes	maximal, complemented
5	C₂²×#13133189	C₂	1	4	yes	maximal, complemented
6	C₂²×#13133186	C₂	1	2	yes	maximal, complemented
7	C₂²×#13133185	C₂	1	2	yes	maximal, complemented
8	C₂³×C₂²⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic, complemented
11	C₂³×C₂²⋊D₅₃	—	2	16	no	—
12	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	—
13	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
14	C₂³×C₂²⋊D₅₃	C₂²	1	4	yes	complemented
15	C₂³×C₂²⋊D₅₃	C₂²	1	24	yes	complemented
16	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
17	C₂⁴×Dic₅₃	C₂²	1	2	yes	complemented
18	#16811182	C₂²	1	256	yes	direct product factor
19	C₂×#13133200	C₂²	1	48	yes	complemented
20	C₂×#13133198	C₂²	1	48	yes	complemented
21	C₂×#13133189	C₂²	1	96	yes	complemented
22	C₂×#13133186	C₂²	1	48	yes	complemented
23	C₂×#13133185	C₂²	1	48	yes	complemented
24	C₂²×#11719919	C₂²	1	4	yes	complemented
25	C₂²×C₂²⋊Dic₅₃	C₂²	1	2	yes	complemented
26	C₂²×C₂²⋊Dic₅₃	C₂²	1	2	yes	complemented
27	C₂²×C₂²⋊Dic₅₃	C₂²	1	4	yes	complemented
28	C₂²×C₂²⋊Dic₅₃	C₂²	1	24	yes	complemented
29	C₂²×#11719921	C₂²	1	4	yes	complemented
30	C₂⁶×C₅₃	C₂²	1	1	yes	characteristic
31	C₂⁶×C₅₃	C₂²	1	1	yes	characteristic, complemented
32	C₂³×D₄×C₅₃	—	2	16	no	—
33	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
34	C₂³×D₄×C₅₃	C₂²	1	2	yes	complemented
35	C₂³×D₄×C₅₃	C₂²	1	24	yes	complemented
36	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
37	C₂⁴×D₅₃	—	2	8	no	—
38	C₂⁴×D₅₃	—	4	8	no	—
39	C₂⁴×D₅₃	C₂³	1	2	yes	—
40	C₂⁴×D₅₃	C₂³	1	12	yes	complemented
41	C₂²×C₂²⋊D₅₃	—	2	16	no	—
42	C₂²×C₂²⋊D₅₃	—	2	16	no	—
43	C₂²×C₂²⋊D₅₃	—	2	192	no	—
44	C₂²×C₂²⋊D₅₃	—	4	16	no	—
45	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	—
46	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	complemented
47	C₂²×C₂²⋊D₅₃	C₂³	1	48	yes	complemented
48	C₂²×C₂²⋊D₅₃	C₂³	1	64	yes	complemented
49	C₂³×Dic₅₃	—	2	8	no	—
50	C₂³×Dic₅₃	—	4	8	no	—
51	C₂³×Dic₅₃	C₂³	1	4	yes	—
52	C₂³×Dic₅₃	C₂³	1	2	yes	—
53	C₂³×Dic₅₃	C₂³	1	12	yes	complemented
54	C₂³×Dic₅₃	C₂³	1	24	yes	complemented
55	#13133200	C₂³	1	128	yes	complemented
56	#13133198	C₂³	1	128	yes	complemented
57	#13133189	C₂³	1	256	yes	complemented
58	#13133186	C₂³	1	128	yes	complemented
59	#13133185	C₂³	1	128	yes	complemented
60	C₂×#11719919	C₂³	1	48	yes	complemented
61	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	complemented
62	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	complemented
63	C₂×C₂²⋊Dic₅₃	C₂³	1	48	yes	complemented
64	C₂×C₂²⋊Dic₅₃	C₂³	1	64	yes	complemented
65	C₂×#11719921	C₂³	1	48	yes	complemented
66	C₂⁵×C₅₃	—	2	8	no	—
67	C₂⁵×C₅₃	—	4	8	no	—
68	C₂⁵×C₅₃	—	4	8	no	—
69	C₂⁵×C₅₃	D₄	1	16	yes	complemented
70	C₂⁵×C₅₃	C₂³	1	2	yes	—
71	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
72	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
73	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
74	C₂⁵×C₅₃	C₂³	1	12	yes	—
75	C₂⁵×C₅₃	C₂³	1	12	yes	complemented
76	C₂²×D₄×C₅₃	—	2	16	no	—
77	C₂²×D₄×C₅₃	—	2	16	no	—
78	C₂²×D₄×C₅₃	—	2	192	no	—
79	C₂²×D₄×C₅₃	C₂³	1	24	yes	—
80	C₂²×D₄×C₅₃	C₂³	1	24	yes	complemented
81	C₂²×D₄×C₅₃	C₂³	1	64	yes	complemented
82	C₂³×C₄×C₅₃	—	2	8	no	—
83	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
84	C₂³×C₄×C₅₃	C₂³	1	12	yes	complemented
85	C₂⁴×C₅₃	—	2	8	no	—
86	C₂⁴×C₅₃	—	2	8	no	—
87	C₂⁴×C₅₃	—	2	8	no	—
88	C₂⁴×C₅₃	—	2	8	no	—
89	C₂⁴×C₅₃	—	2	48	no	—
90	C₂⁴×C₅₃	—	2	16	no	—
91	C₂⁴×C₅₃	—	4	64	no	—
92	C₂⁴×C₅₃	—	4	48	no	—
93	C₂⁴×C₅₃	—	4	48	no	—
94	C₂⁴×C₅₃	—	4	64	no	—
95	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
96	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
97	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
98	C₂⁴×C₅₃	C₂×D₄	1	96	yes	complemented
99	C₂⁴×C₅₃	C₂⁴	1	12	yes	—
100	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
101	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
102	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
103	C₂⁴×C₅₃	C₂⁴	1	16	yes	—
104	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
105	C₂⁴×C₅₃	C₂⁴	1	16	yes	complemented
106	C₂³×D₅₃	—	2	8	no	—
107	C₂³×D₅₃	—	2	48	no	—
108	C₂³×D₅₃	—	2	16	no	—
109	C₂³×D₅₃	—	4	48	no	—
110	C₂³×D₅₃	—	4	64	no	—
111	C₂³×D₅₃	C₂⁴	1	12	yes	—
112	C₂³×D₅₃	C₂⁴	1	16	yes	complemented
113	C₂×D₄×C₅₃	—	2	96	no	—
114	C₂×D₄×C₅₃	—	2	96	no	—
115	C₂×D₄×C₅₃	—	2	256	no	—
116	C₂×D₄×C₅₃	C₂⁴	1	32	yes	—
117	C₂×D₄×C₅₃	C₂⁴	1	32	yes	complemented
118	C₂²×C₄×C₅₃	—	2	8	no	—
119	C₂²×C₄×C₅₃	—	2	48	no	—
120	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
121	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	complemented
122	C₂×C₂²⋊D₅₃	—	2	96	no	—
123	C₂×C₂²⋊D₅₃	—	2	96	no	—
124	C₂×C₂²⋊D₅₃	—	2	256	no	—
125	C₂×C₂²⋊D₅₃	—	4	96	no	—
126	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
127	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	complemented
128	C₂×C₂²⋊D₅₃	C₂⁴	1	64	yes	complemented
129	C₂²×Dic₅₃	—	2	8	no	—
130	C₂²×Dic₅₃	—	2	48	no	—
131	C₂²×Dic₅₃	—	2	16	no	—
132	C₂²×Dic₅₃	—	4	48	no	—
133	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
134	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
135	C₂²×Dic₅₃	C₂⁴	1	16	yes	complemented
136	C₂²×Dic₅₃	C₂⁴	1	32	yes	complemented
137	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	complemented
138	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	complemented
139	C₂²⋊Dic₅₃	C₂⁴	1	64	yes	complemented
140	#11719921	C₂⁴	1	64	yes	complemented
141	#11719919	C₂⁴	1	64	yes	complemented
142	C₂³×C₅₃	—	2	32	no	—
143	C₂³×C₅₃	—	2	32	no	—
144	C₂³×C₅₃	—	2	32	no	—
145	C₂³×C₅₃	—	2	24	no	—
146	C₂³×C₅₃	—	2	24	no	—
147	C₂³×C₅₃	—	2	24	no	—
148	C₂³×C₅₃	—	2	24	no	—
149	C₂³×C₅₃	—	2	32	no	—
150	C₂³×C₅₃	—	2	48	no	—
151	C₂³×C₅₃	—	4	192	no	—
152	C₂³×C₅₃	—	4	32	no	—
153	C₂³×C₅₃	—	4	32	no	—
154	C₂³×C₅₃	—	4	192	no	—
155	C₂³×C₅₃	—	4	32	no	—
156	C₂³×C₅₃	C₂²×D₄	1	24	yes	—
157	C₂³×C₅₃	C₂²×D₄	1	24	yes	—
158	C₂³×C₅₃	C₂²×D₄	1	24	yes	—
159	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
160	C₂³×C₅₃	C₂²×D₄	1	64	yes	complemented
161	C₂³×C₅₃	H(5,𝔽₂)	1	8	yes	—
162	C₂³×C₅₃	C₂⁵	1	8	yes	—
163	C₂³×C₅₃	C₂⁵	1	4	yes	—
164	C₂³×C₅₃	C₂⁵	1	4	yes	—
165	C₂³×C₅₃	C₂⁵	1	4	yes	—
166	C₂³×C₅₃	C₂⁵	1	3	yes	—
167	C₂²×D₅₃	—	2	24	no	—
168	C₂²×D₅₃	—	2	32	no	—
169	C₂²×D₅₃	—	2	48	no	—
170	C₂²×D₅₃	—	4	32	no	—
171	C₂²×D₅₃	—	4	192	no	—
172	C₂²×D₅₃	—	4	32	no	—
173	C₂²×D₅₃	C₂⁵	1	8	yes	—
174	D₄×C₅₃	—	2	64	no	—
175	D₄×C₅₃	—	2	64	no	—
176	C₂×C₄×C₅₃	—	2	24	no	—
177	C₂×C₄×C₅₃	—	2	32	no	—
178	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
179	C₂²⋊D₅₃	—	2	64	no	—
180	C₂²⋊D₅₃	—	2	64	no	—
181	C₂²⋊D₅₃	—	4	64	no	—
182	C₂×Dic₅₃	—	2	24	no	—
183	C₂×Dic₅₃	—	2	32	no	—
184	C₂×Dic₅₃	—	2	48	no	—
185	C₂×Dic₅₃	—	4	32	no	—
186	C₂×Dic₅₃	C₂⁵	1	8	yes	—
187	C₂×Dic₅₃	C₂⁵	1	16	yes	—
188	C₂²×#534	—	53	1	no	maximal, Sylow
189	C₂²×C₅₃	—	2	8	no	—
190	C₂²×C₅₃	—	2	8	no	—
191	C₂²×C₅₃	—	2	8	no	—
192	C₂²×C₅₃	—	2	48	no	—
193	C₂²×C₅₃	—	2	48	no	—
194	C₂²×C₅₃	—	2	48	no	—
195	C₂²×C₅₃	—	2	8	no	—
196	C₂²×C₅₃	—	2	16	no	—
197	C₂²×C₅₃	—	4	64	no	—
198	C₂²×C₅₃	—	4	64	no	—
199	C₂²×C₅₃	—	4	48	no	—
200	C₂²×C₅₃	#534	1	16	yes	complemented
201	C₂²×C₅₃	C₂³×D₄	1	8	yes	—
202	C₂²×C₅₃	C₂³×D₄	1	8	yes	—
203	C₂²×C₅₃	C₂³×D₄	1	8	yes	—
204	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
205	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	12	yes	—
206	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
207	C₂×D₅₃	—	2	8	no	—
208	C₂×D₅₃	—	2	16	no	—
209	C₂×D₅₃	—	4	64	no	—
210	C₂×D₅₃	—	4	48	no	—
211	C₄×C₅₃	—	2	8	no	—
212	Dic₅₃	—	2	8	no	—
213	Dic₅₃	—	2	16	no	—
214	C₂⁴×D₄	—	53	1	no	—
215	C₂⁴×D₄	D₅₃	1	1	yes	characteristic, complemented
216	C₂×#534	—	53	48	no	—
217	C₂²×#123	—	53	2	no	—
218	C₂²×#123	—	53	2	no	—
219	C₂²×#121	—	53	4	no	—
220	C₂²×#120	—	53	2	no	—
221	C₂²×#120	—	53	2	no	—
222	C₂³×C₂²⋊C₄	—	53	1	no	—
223	C₂×C₅₃	—	2	8	no	—
224	C₂×C₅₃	—	2	8	no	—
225	C₂×C₅₃	—	2	8	no	—
226	C₂×C₅₃	—	4	8	no	—
227	C₂×C₅₃	C₂×#534	1	12	yes	complemented
228	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
229	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	2	yes	—
230	D₅₃	—	4	8	no	—
231	C₂⁶	—	53	1	no	—
232	C₂⁶	C₂×D₅₃	1	1	yes	characteristic
233	C₂⁶	C₂×D₅₃	1	1	yes	characteristic, complemented
234	C₂³×D₄	—	2	16	no	—
235	C₂³×D₄	—	53	2	no	—
236	C₂³×D₄	—	53	2	no	—
237	C₂³×D₄	—	53	24	no	—
238	C₂³×D₄	—	53	4	no	—
239	C₂³×D₄	—	106	16	no	—
240	C₂³×D₄	C₂×D₅₃	1	2	yes	—
241	C₂³×D₄	C₂×D₅₃	1	2	yes	complemented
242	C₂³×D₄	C₂×D₅₃	1	24	yes	complemented
243	C₂⁴×C₄	—	53	1	no	—
244	C₂⁴×C₄	—	53	2	no	—
245	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
246	#534	—	53	256	no	—
247	C₂×#123	—	53	48	no	—
248	C₂×#123	—	53	48	no	—
249	C₂×#121	—	53	96	no	—
250	C₂×#120	—	53	48	no	—
251	C₂×#120	—	53	48	no	—
252	C₂²×C₄⋊C₄	—	53	4	no	—
253	C₂²×C₂²⋊C₄	—	53	2	no	—
254	C₂²×C₂²⋊C₄	—	53	2	no	—
255	C₂²×C₂²⋊C₄	—	53	24	no	—
256	C₂²×C₂²⋊C₄	—	53	4	no	—
257	C₂²×C₂²⋊C₄	—	53	4	no	—
258	C₅₃	C₂²×#534	1	1	yes	characteristic, complemented, Sylow
259	C₂⁵	—	2	8	no	—
260	C₂⁵	—	4	8	no	—
261	C₂⁵	—	4	8	no	—
262	C₂⁵	—	53	2	no	—
263	C₂⁵	—	53	12	no	—
264	C₂⁵	—	106	8	no	—
265	C₂⁵	—	212	8	no	—
266	C₂⁵	C₂²⋊D₅₃	1	16	yes	complemented
267	C₂⁵	C₂²×D₅₃	1	2	yes	—
268	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
269	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
270	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
271	C₂⁵	C₂²×D₅₃	1	12	yes	—
272	C₂⁵	C₂²×D₅₃	1	12	yes	complemented
273	C₂²×D₄	—	2	16	no	—
274	C₂²×D₄	—	2	16	no	—
275	C₂²×D₄	—	2	192	no	—
276	C₂²×D₄	—	53	24	no	—
277	C₂²×D₄	—	53	24	no	—
278	C₂²×D₄	—	53	64	no	—
279	C₂²×D₄	—	53	48	no	—
280	C₂²×D₄	—	106	16	no	—
281	C₂²×D₄	—	106	16	no	—
282	C₂²×D₄	—	106	192	no	—
283	C₂²×D₄	—	212	16	no	—
284	C₂²×D₄	C₂²×D₅₃	1	24	yes	—
285	C₂²×D₄	C₂²×D₅₃	1	24	yes	complemented
286	C₂²×D₄	C₂²×D₅₃	1	64	yes	complemented
287	C₂³×C₄	—	2	8	no	—
288	C₂³×C₄	—	53	12	no	—
289	C₂³×C₄	—	53	4	no	—
290	C₂³×C₄	—	53	2	no	—
291	C₂³×C₄	—	53	24	no	—
292	C₂³×C₄	—	106	8	no	—
293	C₂³×C₄	—	212	8	no	—
294	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
295	C₂³×C₄	C₂²×D₅₃	1	12	yes	complemented
296	#123	—	53	128	no	—
297	#123	—	53	128	no	—
298	#121	—	53	256	no	—
299	#120	—	53	128	no	—
300	#120	—	53	128	no	—
301	C₂×C₄⋊C₄	—	53	48	no	—
302	C₂×C₂²⋊C₄	—	53	24	no	—
303	C₂×C₂²⋊C₄	—	53	24	no	—
304	C₂×C₂²⋊C₄	—	53	64	no	—
305	C₂×C₂²⋊C₄	—	53	48	no	—
306	C₂×C₂²⋊C₄	—	53	48	no	—
307	C₂⁴	—	2	8	no	—
308	C₂⁴	—	2	8	no	—
309	C₂⁴	—	2	8	no	—
310	C₂⁴	—	2	8	no	—
311	C₂⁴	—	2	48	no	—
312	C₂⁴	—	2	16	no	—
313	C₂⁴	—	4	64	no	—
314	C₂⁴	—	4	48	no	—
315	C₂⁴	—	4	48	no	—
316	C₂⁴	—	4	64	no	—
317	C₂⁴	—	53	12	no	—
318	C₂⁴	—	53	16	no	—
319	C₂⁴	—	106	48	no	—
320	C₂⁴	—	106	8	no	—
321	C₂⁴	—	106	16	no	—
322	C₂⁴	—	212	48	no	—
323	C₂⁴	—	212	64	no	—
324	C₂⁴	C₂×C₂²⋊D₅₃	1	8	yes	—
325	C₂⁴	C₂×C₂²⋊D₅₃	1	8	yes	—
326	C₂⁴	C₂×C₂²⋊D₅₃	1	8	yes	—
327	C₂⁴	C₂×C₂²⋊D₅₃	1	96	yes	complemented
328	C₂⁴	C₂³×D₅₃	1	12	yes	—
329	C₂⁴	C₂³×D₅₃	1	6	yes	—
330	C₂⁴	C₂³×D₅₃	1	6	yes	—
331	C₂⁴	C₂³×D₅₃	1	6	yes	—
332	C₂⁴	C₂³×D₅₃	1	16	yes	—
333	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
334	C₂⁴	C₂³×D₅₃	1	16	yes	complemented
335	C₂×D₄	—	2	96	no	—
336	C₂×D₄	—	2	96	no	—
337	C₂×D₄	—	2	256	no	—
338	C₂×D₄	—	53	32	no	—
339	C₂×D₄	—	53	32	no	—
340	C₂×D₄	—	53	64	no	—
341	C₂×D₄	—	106	96	no	—
342	C₂×D₄	—	106	96	no	—
343	C₂×D₄	—	106	256	no	—
344	C₂×D₄	—	212	96	no	—
345	C₂×D₄	C₂³×D₅₃	1	32	yes	—
346	C₂×D₄	C₂³×D₅₃	1	32	yes	complemented
347	C₂²×C₄	—	2	8	no	—
348	C₂²×C₄	—	2	48	no	—
349	C₂²×C₄	—	53	24	no	—
350	C₂²×C₄	—	53	12	no	—
351	C₂²×C₄	—	53	16	no	—
352	C₂²×C₄	—	53	32	no	—
353	C₂²×C₄	—	106	48	no	—
354	C₂²×C₄	—	106	8	no	—
355	C₂²×C₄	—	106	16	no	—
356	C₂²×C₄	—	212	48	no	—
357	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
358	C₂²×C₄	C₂³×D₅₃	1	16	yes	complemented
359	C₄⋊C₄	—	53	64	no	—
360	C₂²⋊C₄	—	53	32	no	—
361	C₂²⋊C₄	—	53	32	no	—
362	C₂²⋊C₄	—	53	64	no	—
363	C₂²⋊C₄	—	53	64	no	—
364	C₂³	—	2	32	no	—
365	C₂³	—	2	32	no	—
366	C₂³	—	2	32	no	—
367	C₂³	—	2	24	no	—
368	C₂³	—	2	24	no	—
369	C₂³	—	2	24	no	—
370	C₂³	—	2	24	no	—
371	C₂³	—	2	32	no	—
372	C₂³	—	2	48	no	—
373	C₂³	—	4	192	no	—
374	C₂³	—	4	32	no	—
375	C₂³	—	4	32	no	—
376	C₂³	—	4	192	no	—
377	C₂³	—	4	32	no	—
378	C₂³	—	53	8	no	—
379	C₂³	—	106	32	no	—
380	C₂³	—	106	24	no	—
381	C₂³	—	106	48	no	—
382	C₂³	—	212	32	no	—
383	C₂³	—	212	192	no	—
384	C₂³	—	212	32	no	—
385	C₂³	C₂²×C₂²⋊D₅₃	1	24	yes	—
386	C₂³	C₂²×C₂²⋊D₅₃	1	24	yes	—
387	C₂³	C₂²×C₂²⋊D₅₃	1	24	yes	—
388	C₂³	C₂²×C₂²⋊D₅₃	1	4	yes	—
389	C₂³	C₂²×C₂²⋊D₅₃	1	64	yes	complemented
390	C₂³	#13133262	1	8	yes	—
391	C₂³	C₂⁴×D₅₃	1	8	yes	—
392	C₂³	C₂⁴×D₅₃	1	4	yes	—
393	C₂³	C₂⁴×D₅₃	1	4	yes	—
394	C₂³	C₂⁴×D₅₃	1	4	yes	—
395	C₂³	C₂⁴×D₅₃	1	3	yes	—
396	D₄	—	2	64	no	—
397	D₄	—	2	64	no	—
398	D₄	—	106	64	no	—
399	D₄	—	106	64	no	—
400	D₄	—	212	64	no	—
401	C₂×C₄	—	2	24	no	—
402	C₂×C₄	—	2	32	no	—
403	C₂×C₄	—	53	16	no	—
404	C₂×C₄	—	53	8	no	—
405	C₂×C₄	—	106	32	no	—
406	C₂×C₄	—	106	24	no	—
407	C₂×C₄	—	106	48	no	—
408	C₂×C₄	—	212	32	no	—
409	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
410	C₂²	—	2	48	no	—
411	C₂²	—	2	48	no	—
412	C₂²	—	2	48	no	—
413	C₂²	—	2	8	no	—
414	C₂²	—	2	8	no	—
415	C₂²	—	2	8	no	—
416	C₂²	—	2	8	no	—
417	C₂²	—	2	16	no	—
418	C₂²	—	4	64	no	—
419	C₂²	—	4	64	no	—
420	C₂²	—	4	48	no	—
421	C₂²	—	106	8	no	—
422	C₂²	—	106	16	no	—
423	C₂²	—	212	64	no	—
424	C₂²	—	212	48	no	—
425	C₂²	#16811182	1	16	yes	direct product factor
426	C₂²	C₂³×C₂²⋊D₅₃	1	8	yes	—
427	C₂²	C₂³×C₂²⋊D₅₃	1	8	yes	—
428	C₂²	C₂³×C₂²⋊D₅₃	1	8	yes	—
429	C₂²	C₂³×C₂²⋊D₅₃	1	6	yes	—
430	C₂²	C₂×#13133262	1	12	yes	—
431	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
432	C₄	—	2	8	no	—
433	C₄	—	106	8	no	—
434	C₄	—	106	16	no	—
435	C₂	—	2	8	no	—
436	C₂	—	2	8	no	—
437	C₂	—	2	8	no	—
438	C₂	—	4	8	no	—
439	C₂	—	212	8	no	—
440	C₂	C₂×#16811182	1	12	yes	direct product factor
441	C₂	C₂⁴×C₂²⋊D₅₃	1	1	yes	characteristic
442	C₂	C₂²×#13133262	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂⁴×D₄×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂⁴×D₄	D₅₃
hypocenter	C₅₃	C₂²×#534
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×D₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₂²⋊D₅₃	C₂	semidirect
2. C₂×#16811182	C₂	direct (N × H)
3. C₂²×#13133200	C₂	semidirect
4. C₂²×#13133198	C₂	semidirect
5. C₂²×#13133189	C₂	semidirect
6. C₂²×#13133186	C₂	semidirect
7. C₂²×#13133185	C₂	semidirect
8. C₂³×C₂²⋊Dic₅₃	C₂	semidirect
9. C₂⁴×D₄×C₅₃	C₂	semidirect
10. C₂⁵×D₅₃	C₂²	semidirect
13. C₂³×C₂²⋊D₅₃	C₂²	semidirect
14. C₂³×C₂²⋊D₅₃	C₂²	semidirect
15. C₂³×C₂²⋊D₅₃	C₂²	semidirect
16. C₂⁴×Dic₅₃	C₂²	semidirect
17. C₂⁴×Dic₅₃	C₂²	semidirect
18. #16811182	C₂²	direct (N × H)
19. C₂×#13133200	C₂²	semidirect
20. C₂×#13133198	C₂²	semidirect
21. C₂×#13133189	C₂²	semidirect
22. C₂×#13133186	C₂²	semidirect
23. C₂×#13133185	C₂²	semidirect
24. C₂²×#11719919	C₂²	semidirect
25. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
26. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
27. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
28. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
29. C₂²×#11719921	C₂²	semidirect
31. C₂⁶×C₅₃	C₂²	semidirect
34. C₂³×D₄×C₅₃	C₂²	semidirect
35. C₂³×D₄×C₅₃	C₂²	semidirect
36. C₂⁴×C₄×C₅₃	C₂²	semidirect
40. C₂⁴×D₅₃	C₂³	semidirect
46. C₂²×C₂²⋊D₅₃	C₂³	semidirect
47. C₂²×C₂²⋊D₅₃	C₂³	semidirect
48. C₂²×C₂²⋊D₅₃	C₂³	semidirect
53. C₂³×Dic₅₃	C₂³	semidirect
54. C₂³×Dic₅₃	C₂³	semidirect
55. #13133200	C₂³	semidirect
56. #13133198	C₂³	semidirect
57. #13133189	C₂³	semidirect
58. #13133186	C₂³	semidirect
59. #13133185	C₂³	semidirect
60. C₂×#11719919	C₂³	semidirect
61. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
62. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
63. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
64. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
65. C₂×#11719921	C₂³	semidirect
69. C₂⁵×C₅₃	D₄	semidirect
75. C₂⁵×C₅₃	C₂³	semidirect
80. C₂²×D₄×C₅₃	C₂³	semidirect
81. C₂²×D₄×C₅₃	C₂³	semidirect
84. C₂³×C₄×C₅₃	C₂³	semidirect
98. C₂⁴×C₅₃	C₂×D₄	semidirect
105. C₂⁴×C₅₃	C₂⁴	semidirect
112. C₂³×D₅₃	C₂⁴	semidirect
117. C₂×D₄×C₅₃	C₂⁴	semidirect
121. C₂²×C₄×C₅₃	C₂⁴	semidirect
127. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
128. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
135. C₂²×Dic₅₃	C₂⁴	semidirect
136. C₂²×Dic₅₃	C₂⁴	semidirect
137. C₂²⋊Dic₅₃	C₂⁴	semidirect
138. C₂²⋊Dic₅₃	C₂⁴	semidirect
139. C₂²⋊Dic₅₃	C₂⁴	semidirect
140. #11719921	C₂⁴	semidirect
141. #11719919	C₂⁴	semidirect
160. C₂³×C₅₃	C₂²×D₄	semidirect
200. C₂²×C₅₃	#534	semidirect
215. C₂⁴×D₄	D₅₃	semidirect
227. C₂×C₅₃	C₂×#534	semidirect
233. C₂⁶	C₂×D₅₃	semidirect
241. C₂³×D₄	C₂×D₅₃	semidirect
242. C₂³×D₄	C₂×D₅₃	semidirect
245. C₂⁴×C₄	C₂×D₅₃	semidirect
258. C₅₃	C₂²×#534	semidirect
266. C₂⁵	C₂²⋊D₅₃	semidirect
272. C₂⁵	C₂²×D₅₃	semidirect
285. C₂²×D₄	C₂²×D₅₃	semidirect
286. C₂²×D₄	C₂²×D₅₃	semidirect
295. C₂³×C₄	C₂²×D₅₃	semidirect
327. C₂⁴	C₂×C₂²⋊D₅₃	semidirect
334. C₂⁴	C₂³×D₅₃	semidirect
346. C₂×D₄	C₂³×D₅₃	semidirect
358. C₂²×C₄	C₂³×D₅₃	semidirect
389. C₂³	C₂²×C₂²⋊D₅₃	semidirect
425. C₂²	#16811182	direct (N × H)
440. C₂	C₂×#16811182	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×C₂²⋊D₅₃	2
C₂×#16811182	2
C₂²×#13133200	2
C₂²×#13133198	2
C₂²×#13133189	2
C₂²×#13133186	2
C₂²×#13133185	2
C₂³×C₂²⋊Dic₅₃	2
C₂⁴×D₄×C₅₃	2
C₂²×#534	53

Subgroups — C22×#16811182 #28290809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811182 #28290809