Subgroups — C₂×#20288350 #28291975

Order 13568.

14438 conjugacy classes of subgroups.

The subgroup lattice has 580 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	1	yes	maximal, characteristic, complemented
2	#20288350	C₂	1	32	yes	maximal, direct product factor
3	C₂×#16811267	C₂	1	8	yes	maximal, complemented
4	C₂×D₄×C₂²⋊D₅₃	C₂	1	8	yes	maximal, complemented
5	C₂×#16811190	C₂	1	2	yes	maximal, complemented
6	C₂×#16811182	C₂	1	4	yes	maximal, complemented
7	C₂×#16811134	C₂	1	2	yes	maximal, complemented
8	C₂²×#13133186	C₂	1	4	yes	maximal, complemented
9	C₂×#16811098	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#13133262	—	2	16	no	—
12	C₂×#13133262	C₂²	1	16	yes	complemented
13	C₂²×#11719945	C₂²	1	2	yes	complemented
14	C₂²×#11719947	C₂²	1	4	yes	complemented
15	C₂³×C₂²⋊D₅₃	C₂²	1	4	yes	complemented
16	C₂³×C₂²⋊D₅₃	C₂²	1	8	yes	complemented
17	C₂²×D₄×D₅₃	C₂²	1	4	yes	complemented
18	#16811267	C₂²	1	128	yes	complemented
19	D₄×C₂²⋊D₅₃	C₂²	1	128	yes	complemented
20	#16811190	C₂²	1	32	yes	complemented
21	#16811182	C₂²	1	64	yes	complemented
22	C₂×#13133213	C₂²	1	4	yes	complemented
23	C₂×#13133204	C₂²	1	4	yes	complemented
24	#16811134	C₂²	1	32	yes	complemented
25	C₂×#13133200	C₂²	1	4	yes	complemented
26	C₂×#13133198	C₂²	1	8	yes	complemented
27	C₂×#13133196	C₂²	1	4	yes	complemented
28	C₂×#13133191	C₂²	1	4	yes	complemented
29	C₂×#13133189	C₂²	1	16	yes	complemented
30	C₂×#13133188	C₂²	1	8	yes	complemented
31	C₂×#13133186	C₂²	1	4	yes	complemented
32	C₂×#13133186	C₂²	1	8	yes	complemented
33	C₂×#13133186	C₂²	1	16	yes	complemented
34	C₂×#13133186	C₂²	1	64	yes	complemented
35	C₂×#13133185	C₂²	1	8	yes	complemented
36	#16811098	C₂²	1	16	yes	complemented
37	C₂×D₄×Dic₅₃	C₂²	1	8	yes	complemented
38	C₂×C₄×C₂²⋊D₅₃	C₂²	1	8	yes	complemented
39	C₂×#13133176	C₂²	1	2	yes	complemented
40	C₂²×C₂²⋊Dic₅₃	C₂²	1	4	yes	complemented
41	C₂²×C₂²⋊Dic₅₃	C₂²	1	8	yes	complemented
42	C₂⁶×C₅₃	C₂²	1	2	yes	complemented
43	C₂³×D₄×C₅₃	—	2	16	no	—
44	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
45	C₂³×D₄×C₅₃	C₂²	1	2	yes	complemented
46	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
47	C₂³×D₄×C₅₃	C₂²	1	8	yes	complemented
48	C₂³×D₄×C₅₃	C₂²	1	16	yes	complemented
49	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
50	C₂⁴×D₅₃	C₂³	1	4	yes	complemented
51	#13133262	—	2	128	no	—
52	C₂×#11719945	—	2	16	no	—
53	C₂×#11719945	C₂³	1	16	yes	complemented
54	C₂×#11719947	—	2	32	no	—
55	C₂×#11719947	C₂³	1	32	yes	complemented
56	C₂²×Dic₁₀₆	C₂³	1	2	yes	complemented
57	C₂²×D₂₁₂	C₂³	1	2	yes	complemented
58	C₂²×C₂²⋊D₅₃	—	2	32	no	—
59	C₂²×C₂²⋊D₅₃	—	4	32	no	—
60	C₂²×C₂²⋊D₅₃	C₂³	1	8	yes	—
61	C₂²×C₂²⋊D₅₃	C₂³	1	4	yes	—
62	C₂²×C₂²⋊D₅₃	C₂³	1	8	yes	complemented
63	C₂²×C₂²⋊D₅₃	C₂³	1	16	yes	complemented
64	C₂²×C₂²⋊D₅₃	C₂³	1	32	yes	complemented
65	C₂²×C₂²⋊D₅₃	C₂³	1	64	yes	complemented
66	C₂×D₄×D₅₃	—	2	32	no	—
67	C₂×D₄×D₅₃	C₂³	1	32	yes	complemented
68	C₂³×Dic₅₃	C₂³	1	4	yes	complemented
69	C₂³×Dic₅₃	C₂³	1	8	yes	complemented
70	C₂²×C₄×D₅₃	C₂³	1	4	yes	complemented
71	#13133213	C₂³	1	32	yes	complemented
72	#13133204	C₂³	1	32	yes	complemented
73	#13133200	C₂³	1	32	yes	complemented
74	#13133198	C₂³	1	64	yes	complemented
75	#13133196	C₂³	1	32	yes	complemented
76	#13133191	C₂³	1	32	yes	complemented
77	#13133189	C₂³	1	128	yes	complemented
78	#13133188	C₂³	1	64	yes	complemented
79	#13133186	C₂³	1	32	yes	complemented
80	#13133186	C₂³	1	128	yes	complemented
81	#13133186	C₂³	1	64	yes	complemented
82	#13133185	C₂³	1	64	yes	complemented
83	D₄×Dic₅₃	C₂³	1	64	yes	complemented
84	C₄×C₂²⋊D₅₃	C₂³	1	64	yes	complemented
85	#13133176	C₂³	1	16	yes	complemented
86	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	complemented
87	C₂×#11719919	C₂³	1	8	yes	complemented
88	C₂×C₂²⋊Dic₅₃	C₂³	1	4	yes	—
89	C₂×C₂²⋊Dic₅₃	C₂³	1	8	yes	—
90	C₂×C₂²⋊Dic₅₃	C₂³	1	8	yes	complemented
91	C₂×C₂²⋊Dic₅₃	C₂³	1	16	yes	complemented
92	C₂×C₂²⋊Dic₅₃	C₂³	1	64	yes	complemented
93	C₂×C₂²⋊Dic₅₃	C₂³	1	32	yes	complemented
94	C₂×#11719921	C₂³	1	8	yes	complemented
95	C₂×C₄×Dic₅₃	C₂³	1	4	yes	complemented
96	C₂⁵×C₅₃	—	2	16	no	—
97	C₂⁵×C₅₃	—	4	16	no	—
98	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
99	C₂⁵×C₅₃	C₂³	1	8	yes	—
100	C₂⁵×C₅₃	C₂³	1	2	yes	—
101	C₂⁵×C₅₃	C₂³	1	2	yes	—
102	C₂⁵×C₅₃	C₂³	1	16	yes	complemented
103	C₂²×D₄×C₅₃	—	2	16	no	—
104	C₂²×D₄×C₅₃	—	2	64	no	—
105	C₂²×D₄×C₅₃	—	2	128	no	—
106	C₂²×D₄×C₅₃	—	2	8	no	—
107	C₂²×D₄×C₅₃	D₄	1	16	yes	complemented
108	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
109	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
110	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
111	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
112	C₂²×D₄×C₅₃	C₂³	1	64	yes	complemented
113	C₂²×D₄×C₅₃	C₂³	1	16	yes	complemented
114	C₂²×D₄×C₅₃	C₂³	1	8	yes	complemented
115	C₂³×C₄×C₅₃	—	2	8	no	—
116	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
117	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
118	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
119	C₂³×C₄×C₅₃	C₂³	1	8	yes	complemented
120	C₂⁴×C₅₃	—	2	4	no	—
121	C₂⁴×C₅₃	—	2	8	no	—
122	C₂⁴×C₅₃	—	2	32	no	—
123	C₂⁴×C₅₃	—	2	8	no	—
124	C₂⁴×C₅₃	—	2	64	no	—
125	C₂⁴×C₅₃	—	2	32	no	—
126	C₂⁴×C₅₃	—	4	8	no	—
127	C₂⁴×C₅₃	—	4	16	no	—
128	C₂⁴×C₅₃	—	4	8	no	—
129	C₂⁴×C₅₃	—	4	64	no	—
130	C₂⁴×C₅₃	—	4	128	no	—
131	C₂⁴×C₅₃	C₂×D₄	1	16	yes	complemented
132	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
133	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
134	C₂⁴×C₅₃	C₂⁴	1	32	yes	—
135	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
136	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
137	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
138	C₂⁴×C₅₃	C₂⁴	1	8	yes	—
139	C₂⁴×C₅₃	C₂⁴	1	8	yes	—
140	C₂³×D₅₃	—	2	16	no	—
141	C₂³×D₅₃	—	4	16	no	—
142	C₂³×D₅₃	C₂⁴	1	4	yes	—
143	C₂³×D₅₃	C₂⁴	1	16	yes	complemented
144	C₂×D₄×C₅₃	—	2	32	no	—
145	C₂×D₄×C₅₃	—	2	64	no	—
146	C₂×D₄×C₅₃	—	2	256	no	—
147	C₂×D₄×C₅₃	—	2	16	no	—
148	C₂×D₄×C₅₃	—	2	32	no	—
149	C₂×D₄×C₅₃	C₂×D₄	1	32	yes	complemented
150	C₂×D₄×C₅₃	C₂×D₄	1	64	yes	complemented
151	C₂×D₄×C₅₃	C₂⁴	1	32	yes	—
152	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
153	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
154	C₂²×C₄×C₅₃	—	2	4	no	—
155	C₂²×C₄×C₅₃	—	2	16	no	—
156	C₂²×C₄×C₅₃	—	2	32	no	—
157	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	complemented
158	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
160	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
161	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
163	C₂×C₂²⋊D₅₃	—	2	32	no	—
164	C₂×C₂²⋊D₅₃	—	2	16	no	—
165	C₂×C₂²⋊D₅₃	—	2	128	no	—
166	C₂×C₂²⋊D₅₃	—	2	32	no	—
167	C₂×C₂²⋊D₅₃	—	4	32	no	—
168	C₂×C₂²⋊D₅₃	—	4	128	no	—
169	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
170	C₂×C₂²⋊D₅₃	C₂⁴	1	16	yes	—
171	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	complemented
172	C₂×C₂²⋊D₅₃	C₂⁴	1	64	yes	complemented
173	C₂²×Dic₅₃	—	2	16	no	—
174	C₂²×Dic₅₃	—	4	16	no	—
175	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
176	C₂²×Dic₅₃	C₂⁴	1	4	yes	—
177	C₂²×Dic₅₃	C₂⁴	1	16	yes	complemented
178	C₂²×Dic₅₃	C₂⁴	1	32	yes	complemented
179	#11719947	—	2	128	no	—
180	D₄×D₅₃	—	2	128	no	—
181	#11719945	—	2	64	no	—
182	C₂×D₂₁₂	—	2	8	no	—
183	C₂×D₂₁₂	C₂⁴	1	8	yes	complemented
184	C₂×C₄×D₅₃	—	2	16	no	—
185	C₂×C₄×D₅₃	C₂⁴	1	16	yes	complemented
186	C₂×Dic₁₀₆	—	2	8	no	—
187	C₂×Dic₁₀₆	C₂⁴	1	8	yes	complemented
188	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	—
189	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	—
190	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	complemented
191	C₂²⋊Dic₅₃	C₂⁴	1	64	yes	complemented
192	#11719921	C₂⁴	1	32	yes	complemented
193	C₄⋊Dic₅₃	C₂⁴	1	16	yes	complemented
194	#11719919	C₂⁴	1	32	yes	complemented
195	C₄×Dic₅₃	C₂⁴	1	16	yes	complemented
196	C₂³×C₅₃	—	2	32	no	—
197	C₂³×C₅₃	—	2	8	no	—
198	C₂³×C₅₃	—	2	4	no	—
199	C₂³×C₅₃	—	2	16	no	—
200	C₂³×C₅₃	—	2	4	no	—
201	C₂³×C₅₃	—	2	4	no	—
202	C₂³×C₅₃	—	2	16	no	—
203	C₂³×C₅₃	—	2	64	no	—
204	C₂³×C₅₃	—	2	16	no	—
205	C₂³×C₅₃	—	2	8	no	—
206	C₂³×C₅₃	—	2	16	no	—
207	C₂³×C₅₃	—	2	64	no	—
208	C₂³×C₅₃	—	4	32	no	—
209	C₂³×C₅₃	—	4	16	no	—
210	C₂³×C₅₃	—	4	32	no	—
211	C₂³×C₅₃	—	4	16	no	—
212	C₂³×C₅₃	—	4	128	no	—
213	C₂³×C₅₃	—	4	256	no	—
214	C₂³×C₅₃	—	4	32	no	—
215	C₂³×C₅₃	C₂²×D₄	1	16	yes	—
216	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
217	C₂³×C₅₃	C₂²×D₄	1	32	yes	complemented
218	C₂³×C₅₃	H(5,𝔽₂)	1	8	yes	complemented
219	C₂³×C₅₃	C₂⁵	1	8	yes	—
220	C₂³×C₅₃	C₂⁵	1	8	yes	—
221	C₂³×C₅₃	C₂⁵	1	2	yes	—
222	C₂³×C₅₃	C₂⁵	1	4	yes	—
223	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
224	C₂²×D₅₃	—	2	8	no	—
225	C₂²×D₅₃	—	2	16	no	—
226	C₂²×D₅₃	—	2	32	no	—
227	C₂²×D₅₃	—	4	32	no	—
228	C₂²×D₅₃	—	8	32	no	—
229	C₂²×D₅₃	C₂⁵	1	8	yes	—
230	D₄×C₅₃	—	2	64	no	—
231	D₄×C₅₃	—	2	32	no	—
232	D₄×C₅₃	C₂²×D₄	1	64	yes	complemented
233	C₂×C₄×C₅₃	—	2	8	no	—
234	C₂×C₄×C₅₃	—	2	4	no	—
235	C₂×C₄×C₅₃	—	2	32	no	—
236	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
237	C₂×C₄×C₅₃	C₂²×D₄	1	16	yes	complemented
238	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
239	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
240	C₂²⋊D₅₃	—	2	32	no	—
241	C₂²⋊D₅₃	—	2	64	no	—
242	C₂²⋊D₅₃	—	2	64	no	—
243	C₂²⋊D₅₃	—	4	64	no	—
244	C₂×Dic₅₃	—	2	8	no	—
245	C₂×Dic₅₃	—	2	32	no	—
246	C₂×Dic₅₃	—	2	16	no	—
247	C₂×Dic₅₃	—	4	32	no	—
248	C₂×Dic₅₃	C₂⁵	1	8	yes	—
249	C₂×Dic₅₃	C₂⁵	1	16	yes	—
250	D₂₁₂	—	2	16	no	—
251	C₄×D₅₃	—	2	32	no	—
252	Dic₁₀₆	—	2	16	no	—
253	C₂×#3484	—	53	1	no	maximal, Sylow
254	C₂²×C₅₃	—	2	32	no	—
255	C₂²×C₅₃	—	2	4	no	—
256	C₂²×C₅₃	—	2	16	no	—
257	C₂²×C₅₃	—	2	4	no	—
258	C₂²×C₅₃	—	2	4	no	—
259	C₂²×C₅₃	—	2	32	no	—
260	C₂²×C₅₃	—	2	8	no	—
261	C₂²×C₅₃	—	2	16	no	—
262	C₂²×C₅₃	—	2	8	no	—
263	C₂²×C₅₃	—	2	16	no	—
264	C₂²×C₅₃	—	4	32	no	—
265	C₂²×C₅₃	—	4	128	no	—
266	C₂²×C₅₃	—	4	32	no	—
267	C₂²×C₅₃	—	4	16	no	—
268	C₂²×C₅₃	C₂³×D₄	1	16	yes	—
269	C₂²×C₅₃	C₂³×D₄	1	4	yes	—
270	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
271	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
272	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	8	yes	complemented
273	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
274	C₂×D₅₃	—	2	8	no	—
275	C₂×D₅₃	—	2	16	no	—
276	C₂×D₅₃	—	4	16	no	—
277	C₂×D₅₃	—	8	32	no	—
278	C₄×C₅₃	—	2	4	no	—
279	C₄×C₅₃	C₂³×D₄	1	8	yes	—
280	Dic₅₃	—	2	8	no	—
281	Dic₅₃	—	2	16	no	—
282	C₂²×H(5,𝔽₂)	—	53	1	no	—
283	C₂⁴×D₄	D₅₃	1	1	yes	characteristic, complemented
284	#3484	—	53	32	no	—
285	C₂×#546	—	53	8	no	—
286	C₂×D₄²	—	53	8	no	—
287	C₂×#535	—	53	2	no	—
288	C₂×#534	—	53	4	no	—
289	C₂×#525	—	53	2	no	—
290	C₂²×#120	—	53	4	no	—
291	C₂×#518	—	53	1	no	—
292	C₂×C₅₃	—	2	16	no	—
293	C₂×C₅₃	—	2	4	no	—
294	C₂×C₅₃	—	2	8	no	—
295	C₂×C₅₃	—	4	8	no	—
296	C₂×C₅₃	#3484	1	4	yes	complemented
297	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
298	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	2	yes	—
299	D₅₃	—	4	8	no	—
300	C₂⁶	C₂×D₅₃	1	2	yes	complemented
301	C₂×H(5,𝔽₂)	—	53	16	no	—
302	C₂×H(5,𝔽₂)	—	106	16	no	—
303	C₂²×#41	—	53	4	no	—
304	C₂²×#41	—	53	2	no	—
305	C₂³×D₄	—	2	16	no	—
306	C₂³×D₄	—	53	4	no	—
307	C₂³×D₄	—	53	8	no	—
308	C₂³×D₄	—	53	4	no	—
309	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic
310	C₂³×D₄	C₂×D₅₃	1	2	yes	complemented
311	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic, complemented
312	C₂³×D₄	C₂×D₅₃	1	8	yes	complemented
313	C₂³×D₄	C₂×D₅₃	1	16	yes	complemented
314	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
315	#546	—	53	128	no	—
316	D₄²	—	53	128	no	—
317	#535	—	53	32	no	—
318	#534	—	53	64	no	—
319	C₂×#127	—	53	4	no	—
320	C₂×#124	—	53	4	no	—
321	#525	—	53	32	no	—
322	C₂×#123	—	53	8	no	—
323	C₂×#123	—	53	4	no	—
324	C₂×C₂²⋊Q₈	—	53	4	no	—
325	C₂×#121	—	53	8	no	—
326	C₂×#121	—	53	16	no	—
327	C₂×#121	—	53	4	no	—
328	C₂×#120	—	53	4	no	—
329	C₂×#120	—	53	8	no	—
330	C₂×#120	—	53	16	no	—
331	C₂×#120	—	53	64	no	—
332	C₂×#120	—	53	8	no	—
333	#518	—	53	16	no	—
334	C₂×C₄×D₄	—	53	8	no	—
335	C₂×C₄×D₄	—	53	8	no	—
336	C₂×#117	—	53	2	no	—
337	C₂²×C₂²⋊C₄	—	53	4	no	—
338	C₂²×C₂²⋊C₄	—	53	8	no	—
339	C₅₃	C₂×#3484	1	1	yes	characteristic, complemented, Sylow
340	C₂⁵	—	2	16	no	—
341	C₂⁵	—	4	16	no	—
342	C₂⁵	—	53	4	no	—
343	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
344	C₂⁵	C₂²×D₅₃	1	8	yes	—
345	C₂⁵	C₂²×D₅₃	1	2	yes	—
346	C₂⁵	C₂²×D₅₃	1	2	yes	—
347	C₂⁵	C₂²×D₅₃	1	16	yes	complemented
348	H(5,𝔽₂)	—	106	128	no	—
349	C₂×#41	—	53	32	no	—
350	C₂×#41	—	53	16	no	—
351	C₂×#41	—	106	32	no	—
352	C₂×#41	—	106	16	no	—
353	C₂²×Q₈	—	53	2	no	—
354	C₂²×D₄	—	2	16	no	—
355	C₂²×D₄	—	2	64	no	—
356	C₂²×D₄	—	2	128	no	—
357	C₂²×D₄	—	2	8	no	—
358	C₂²×D₄	—	53	8	no	—
359	C₂²×D₄	—	53	4	no	—
360	C₂²×D₄	—	53	8	no	—
361	C₂²×D₄	—	53	32	no	—
362	C₂²×D₄	—	53	16	no	—
363	C₂²×D₄	—	53	64	no	—
364	C₂²×D₄	—	53	32	no	—
365	C₂²×D₄	—	53	2	no	—
366	C₂²×D₄	—	106	32	no	—
367	C₂²×D₄	—	106	32	no	—
368	C₂²×D₄	—	212	32	no	—
369	C₂²×D₄	C₂²⋊D₅₃	1	16	yes	complemented
370	C₂²×D₄	C₂²×D₅₃	1	8	yes	—
371	C₂²×D₄	C₂²×D₅₃	1	4	yes	—
372	C₂²×D₄	C₂²×D₅₃	1	4	yes	—
373	C₂²×D₄	C₂²×D₅₃	1	8	yes	—
374	C₂²×D₄	C₂²×D₅₃	1	16	yes	complemented
375	C₂²×D₄	C₂²×D₅₃	1	8	yes	complemented
376	C₂²×D₄	C₂²×D₅₃	1	64	yes	complemented
377	C₂³×C₄	—	2	8	no	—
378	C₂³×C₄	—	53	4	no	—
379	C₂³×C₄	—	53	8	no	—
380	C₂³×C₄	—	53	4	no	—
381	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
382	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
383	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
384	C₂³×C₄	C₂²×D₅₃	1	8	yes	complemented
385	#127	—	53	32	no	—
386	#124	—	53	32	no	—
387	#123	—	53	64	no	—
388	#123	—	53	32	no	—
389	C₂²⋊Q₈	—	53	32	no	—
390	#121	—	53	128	no	—
391	#121	—	53	64	no	—
392	#121	—	53	32	no	—
393	#120	—	53	32	no	—
394	#120	—	53	64	no	—
395	#120	—	53	128	no	—
396	#120	—	53	64	no	—
397	C₄×D₄	—	53	64	no	—
398	C₄×D₄	—	53	64	no	—
399	#117	—	53	16	no	—
400	C₂×C₄⋊C₄	—	53	4	no	—
401	C₂×C₄⋊C₄	—	53	8	no	—
402	C₂×C₂²⋊C₄	—	53	4	no	—
403	C₂×C₂²⋊C₄	—	53	8	no	—
404	C₂×C₂²⋊C₄	—	53	8	no	—
405	C₂×C₂²⋊C₄	—	53	32	no	—
406	C₂×C₂²⋊C₄	—	53	16	no	—
407	C₂×C₂²⋊C₄	—	53	64	no	—
408	C₂×C₂²⋊C₄	—	53	8	no	—
409	C₂×C₄²	—	53	4	no	—
410	C₂⁴	—	2	4	no	—
411	C₂⁴	—	2	8	no	—
412	C₂⁴	—	2	32	no	—
413	C₂⁴	—	2	8	no	—
414	C₂⁴	—	2	64	no	—
415	C₂⁴	—	2	32	no	—
416	C₂⁴	—	4	8	no	—
417	C₂⁴	—	4	16	no	—
418	C₂⁴	—	4	8	no	—
419	C₂⁴	—	4	64	no	—
420	C₂⁴	—	4	128	no	—
421	C₂⁴	—	53	4	no	—
422	C₂⁴	—	53	16	no	—
423	C₂⁴	—	106	16	no	—
424	C₂⁴	—	212	16	no	—
425	C₂⁴	C₂×C₂²⋊D₅₃	1	16	yes	complemented
426	C₂⁴	C₂³×D₅₃	1	4	yes	—
427	C₂⁴	C₂³×D₅₃	1	4	yes	—
428	C₂⁴	C₂³×D₅₃	1	32	yes	—
429	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
430	C₂⁴	C₂³×D₅₃	1	4	yes	—
431	C₂⁴	C₂³×D₅₃	1	2	yes	—
432	C₂⁴	C₂³×D₅₃	1	8	yes	—
433	C₂⁴	C₂³×D₅₃	1	8	yes	—
434	#41	—	106	128	no	—
435	#41	—	106	64	no	—
436	C₂×Q₈	—	53	8	no	—
437	C₂×Q₈	—	106	8	no	—
438	C₂×D₄	—	2	32	no	—
439	C₂×D₄	—	2	64	no	—
440	C₂×D₄	—	2	256	no	—
441	C₂×D₄	—	2	16	no	—
442	C₂×D₄	—	2	32	no	—
443	C₂×D₄	—	53	32	no	—
444	C₂×D₄	—	53	16	no	—
445	C₂×D₄	—	53	32	no	—
446	C₂×D₄	—	53	64	no	—
447	C₂×D₄	—	53	8	no	—
448	C₂×D₄	—	106	16	no	—
449	C₂×D₄	—	106	32	no	—
450	C₂×D₄	—	106	128	no	—
451	C₂×D₄	—	106	32	no	—
452	C₂×D₄	—	106	128	no	—
453	C₂×D₄	—	106	8	no	—
454	C₂×D₄	—	212	32	no	—
455	C₂×D₄	—	212	128	no	—
456	C₂×D₄	C₂×C₂²⋊D₅₃	1	32	yes	complemented
457	C₂×D₄	C₂×C₂²⋊D₅₃	1	64	yes	complemented
458	C₂×D₄	C₂³×D₅₃	1	32	yes	—
459	C₂×D₄	C₂³×D₅₃	1	16	yes	—
460	C₂×D₄	C₂³×D₅₃	1	16	yes	—
461	C₂²×C₄	—	2	4	no	—
462	C₂²×C₄	—	2	16	no	—
463	C₂²×C₄	—	2	32	no	—
464	C₂²×C₄	—	53	16	no	—
465	C₂²×C₄	—	53	8	no	—
466	C₂²×C₄	—	53	4	no	—
467	C₂²×C₄	—	53	32	no	—
468	C₂²×C₄	—	53	16	no	—
469	C₂²×C₄	—	106	16	no	—
470	C₂²×C₄	—	106	16	no	—
471	C₂²×C₄	—	212	16	no	—
472	C₂²×C₄	C₂×C₂²⋊D₅₃	1	8	yes	complemented
473	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
474	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
475	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
476	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
477	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
478	C₄⋊C₄	—	53	16	no	—
479	C₄⋊C₄	—	53	32	no	—
480	C₂²⋊C₄	—	53	16	no	—
481	C₂²⋊C₄	—	53	32	no	—
482	C₂²⋊C₄	—	53	32	no	—
483	C₂²⋊C₄	—	53	64	no	—
484	C₂²⋊C₄	—	53	32	no	—
485	C₄²	—	53	16	no	—
486	C₂³	—	2	32	no	—
487	C₂³	—	2	8	no	—
488	C₂³	—	2	4	no	—
489	C₂³	—	2	16	no	—
490	C₂³	—	2	4	no	—
491	C₂³	—	2	4	no	—
492	C₂³	—	2	16	no	—
493	C₂³	—	2	64	no	—
494	C₂³	—	2	16	no	—
495	C₂³	—	2	8	no	—
496	C₂³	—	2	16	no	—
497	C₂³	—	2	64	no	—
498	C₂³	—	4	32	no	—
499	C₂³	—	4	16	no	—
500	C₂³	—	4	32	no	—
501	C₂³	—	4	16	no	—
502	C₂³	—	4	128	no	—
503	C₂³	—	4	256	no	—
504	C₂³	—	4	32	no	—
505	C₂³	—	53	8	no	—
506	C₂³	—	106	32	no	—
507	C₂³	—	106	8	no	—
508	C₂³	—	106	16	no	—
509	C₂³	—	424	32	no	—
510	C₂³	—	212	32	no	—
511	C₂³	C₂²×C₂²⋊D₅₃	1	4	yes	—
512	C₂³	C₂²×C₂²⋊D₅₃	1	16	yes	—
513	C₂³	C₂²×C₂²⋊D₅₃	1	32	yes	complemented
514	C₂³	#13133262	1	8	yes	complemented
515	C₂³	C₂⁴×D₅₃	1	8	yes	—
516	C₂³	C₂⁴×D₅₃	1	2	yes	—
517	C₂³	C₂⁴×D₅₃	1	8	yes	—
518	C₂³	C₂⁴×D₅₃	1	4	yes	—
519	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
520	Q₈	—	106	16	no	—
521	D₄	—	2	64	no	—
522	D₄	—	2	32	no	—
523	D₄	—	106	32	no	—
524	D₄	—	106	64	no	—
525	D₄	—	106	64	no	—
526	D₄	—	106	16	no	—
527	D₄	—	212	64	no	—
528	D₄	C₂²×C₂²⋊D₅₃	1	64	yes	complemented
529	C₂×C₄	—	2	8	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	32	no	—
532	C₂×C₄	—	53	16	no	—
533	C₂×C₄	—	53	8	no	—
534	C₂×C₄	—	106	8	no	—
535	C₂×C₄	—	106	32	no	—
536	C₂×C₄	—	106	16	no	—
537	C₂×C₄	—	106	32	no	—
538	C₂×C₄	—	212	32	no	—
539	C₂×C₄	C₂²×C₂²⋊D₅₃	1	8	yes	—
540	C₂×C₄	C₂²×C₂²⋊D₅₃	1	16	yes	complemented
541	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
542	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
543	C₂²	—	2	32	no	—
544	C₂²	—	2	8	no	—
545	C₂²	—	2	16	no	—
546	C₂²	—	2	32	no	—
547	C₂²	—	2	4	no	—
548	C₂²	—	2	16	no	—
549	C₂²	—	2	4	no	—
550	C₂²	—	2	4	no	—
551	C₂²	—	2	8	no	—
552	C₂²	—	2	16	no	—
553	C₂²	—	4	32	no	—
554	C₂²	—	4	128	no	—
555	C₂²	—	4	32	no	—
556	C₂²	—	4	16	no	—
557	C₂²	—	106	8	no	—
558	C₂²	—	106	16	no	—
559	C₂²	—	424	32	no	—
560	C₂²	—	212	16	no	—
561	C₂²	C₂³×C₂²⋊D₅₃	1	4	yes	—
562	C₂²	C₂³×C₂²⋊D₅₃	1	16	yes	—
563	C₂²	C₂³×C₂²⋊D₅₃	1	2	yes	—
564	C₂²	C₂×#13133262	1	4	yes	—
565	C₂²	C₂×#13133262	1	8	yes	complemented
566	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
567	C₄	—	2	4	no	—
568	C₄	—	106	8	no	—
569	C₄	—	106	16	no	—
570	C₄	C₂³×C₂²⋊D₅₃	1	8	yes	—
571	C₂	—	2	4	no	—
572	C₂	—	2	8	no	—
573	C₂	—	2	16	no	—
574	C₂	—	4	8	no	—
575	C₂	—	212	8	no	—
576	C₂	#20288350	1	4	yes	direct product factor
577	C₂	C₂⁴×C₂²⋊D₅₃	1	1	yes	characteristic
578	C₂	C₂²×#13133262	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂⁴×D₄×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂⁴×D₄	D₅₃
hypocenter	C₅₃	C₂×#3484
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂⁴×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133262	C₂	semidirect
2. #20288350	C₂	direct (N × H)
3. C₂×#16811267	C₂	semidirect
4. C₂×D₄×C₂²⋊D₅₃	C₂	semidirect
5. C₂×#16811190	C₂	semidirect
6. C₂×#16811182	C₂	semidirect
7. C₂×#16811134	C₂	semidirect
8. C₂²×#13133186	C₂	semidirect
9. C₂×#16811098	C₂	semidirect
10. C₂⁴×D₄×C₅₃	C₂	semidirect
12. C₂×#13133262	C₂²	semidirect
13. C₂²×#11719945	C₂²	semidirect
14. C₂²×#11719947	C₂²	semidirect
15. C₂³×C₂²⋊D₅₃	C₂²	semidirect
16. C₂³×C₂²⋊D₅₃	C₂²	semidirect
17. C₂²×D₄×D₅₃	C₂²	semidirect
18. #16811267	C₂²	semidirect
19. D₄×C₂²⋊D₅₃	C₂²	semidirect
20. #16811190	C₂²	semidirect
21. #16811182	C₂²	semidirect
22. C₂×#13133213	C₂²	semidirect
23. C₂×#13133204	C₂²	semidirect
24. #16811134	C₂²	semidirect
25. C₂×#13133200	C₂²	semidirect
26. C₂×#13133198	C₂²	semidirect
27. C₂×#13133196	C₂²	semidirect
28. C₂×#13133191	C₂²	semidirect
29. C₂×#13133189	C₂²	semidirect
30. C₂×#13133188	C₂²	semidirect
31. C₂×#13133186	C₂²	semidirect
32. C₂×#13133186	C₂²	semidirect
33. C₂×#13133186	C₂²	semidirect
34. C₂×#13133186	C₂²	semidirect
35. C₂×#13133185	C₂²	semidirect
36. #16811098	C₂²	semidirect
37. C₂×D₄×Dic₅₃	C₂²	semidirect
38. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
39. C₂×#13133176	C₂²	semidirect
40. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
41. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
42. C₂⁶×C₅₃	C₂²	semidirect
45. C₂³×D₄×C₅₃	C₂²	semidirect
46. C₂³×D₄×C₅₃	C₂²	semidirect
47. C₂³×D₄×C₅₃	C₂²	semidirect
48. C₂³×D₄×C₅₃	C₂²	semidirect
49. C₂⁴×C₄×C₅₃	C₂²	semidirect
50. C₂⁴×D₅₃	C₂³	semidirect
53. C₂×#11719945	C₂³	semidirect
55. C₂×#11719947	C₂³	semidirect
56. C₂²×Dic₁₀₆	C₂³	semidirect
57. C₂²×D₂₁₂	C₂³	semidirect
62. C₂²×C₂²⋊D₅₃	C₂³	semidirect
63. C₂²×C₂²⋊D₅₃	C₂³	semidirect
64. C₂²×C₂²⋊D₅₃	C₂³	semidirect
65. C₂²×C₂²⋊D₅₃	C₂³	semidirect
67. C₂×D₄×D₅₃	C₂³	semidirect
68. C₂³×Dic₅₃	C₂³	semidirect
69. C₂³×Dic₅₃	C₂³	semidirect
70. C₂²×C₄×D₅₃	C₂³	semidirect
71. #13133213	C₂³	semidirect
72. #13133204	C₂³	semidirect
73. #13133200	C₂³	semidirect
74. #13133198	C₂³	semidirect
75. #13133196	C₂³	semidirect
76. #13133191	C₂³	semidirect
77. #13133189	C₂³	semidirect
78. #13133188	C₂³	semidirect
79. #13133186	C₂³	semidirect
80. #13133186	C₂³	semidirect
81. #13133186	C₂³	semidirect
82. #13133185	C₂³	semidirect
83. D₄×Dic₅₃	C₂³	semidirect
84. C₄×C₂²⋊D₅₃	C₂³	semidirect
85. #13133176	C₂³	semidirect
86. C₂×C₄⋊Dic₅₃	C₂³	semidirect
87. C₂×#11719919	C₂³	semidirect
90. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
91. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
92. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
93. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
94. C₂×#11719921	C₂³	semidirect
95. C₂×C₄×Dic₅₃	C₂³	semidirect
102. C₂⁵×C₅₃	C₂³	semidirect
107. C₂²×D₄×C₅₃	D₄	semidirect
112. C₂²×D₄×C₅₃	C₂³	semidirect
113. C₂²×D₄×C₅₃	C₂³	semidirect
114. C₂²×D₄×C₅₃	C₂³	semidirect
119. C₂³×C₄×C₅₃	C₂³	semidirect
131. C₂⁴×C₅₃	C₂×D₄	semidirect
143. C₂³×D₅₃	C₂⁴	semidirect
149. C₂×D₄×C₅₃	C₂×D₄	semidirect
150. C₂×D₄×C₅₃	C₂×D₄	semidirect
157. C₂²×C₄×C₅₃	C₂×D₄	semidirect
171. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
172. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
177. C₂²×Dic₅₃	C₂⁴	semidirect
178. C₂²×Dic₅₃	C₂⁴	semidirect
183. C₂×D₂₁₂	C₂⁴	semidirect
185. C₂×C₄×D₅₃	C₂⁴	semidirect
187. C₂×Dic₁₀₆	C₂⁴	semidirect
190. C₂²⋊Dic₅₃	C₂⁴	semidirect
191. C₂²⋊Dic₅₃	C₂⁴	semidirect
192. #11719921	C₂⁴	semidirect
193. C₄⋊Dic₅₃	C₂⁴	semidirect
194. #11719919	C₂⁴	semidirect
195. C₄×Dic₅₃	C₂⁴	semidirect
217. C₂³×C₅₃	C₂²×D₄	semidirect
218. C₂³×C₅₃	H(5,𝔽₂)	semidirect
232. D₄×C₅₃	C₂²×D₄	semidirect
237. C₂×C₄×C₅₃	C₂²×D₄	semidirect
272. C₂²×C₅₃	C₂×H(5,𝔽₂)	semidirect
283. C₂⁴×D₄	D₅₃	semidirect
296. C₂×C₅₃	#3484	semidirect
300. C₂⁶	C₂×D₅₃	semidirect
310. C₂³×D₄	C₂×D₅₃	semidirect
311. C₂³×D₄	C₂×D₅₃	semidirect
312. C₂³×D₄	C₂×D₅₃	semidirect
313. C₂³×D₄	C₂×D₅₃	semidirect
314. C₂⁴×C₄	C₂×D₅₃	semidirect
339. C₅₃	C₂×#3484	semidirect
347. C₂⁵	C₂²×D₅₃	semidirect
369. C₂²×D₄	C₂²⋊D₅₃	semidirect
374. C₂²×D₄	C₂²×D₅₃	semidirect
375. C₂²×D₄	C₂²×D₅₃	semidirect
376. C₂²×D₄	C₂²×D₅₃	semidirect
384. C₂³×C₄	C₂²×D₅₃	semidirect
425. C₂⁴	C₂×C₂²⋊D₅₃	semidirect
456. C₂×D₄	C₂×C₂²⋊D₅₃	semidirect
457. C₂×D₄	C₂×C₂²⋊D₅₃	semidirect
472. C₂²×C₄	C₂×C₂²⋊D₅₃	semidirect
513. C₂³	C₂²×C₂²⋊D₅₃	semidirect
514. C₂³	#13133262	semidirect
528. D₄	C₂²×C₂²⋊D₅₃	semidirect
540. C₂×C₄	C₂²×C₂²⋊D₅₃	semidirect
565. C₂²	C₂×#13133262	semidirect
576. C₂	#20288350	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
#20288350	2
C₂×#16811267	2
C₂×D₄×C₂²⋊D₅₃	2
C₂×#16811190	2
C₂×#16811182	2
C₂×#16811134	2
C₂²×#13133186	2
C₂×#16811098	2
C₂⁴×D₄×C₅₃	2
C₂×#3484	53

Subgroups — C2×#20288350 #28291975

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20288350 #28291975