Subgroups — C₂×#16811262 #20288159

Order 6784.

1900 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11719947	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811262	C₂	1	16	yes	maximal, direct product factor
4	C₂×#13133203	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13133202	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13133198	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13133195	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133190	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133189	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13133185	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#13133185	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×C₂²⋊D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#13133177	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×#11719921	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₂²⋊C₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#121×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂⁴×D₅₃	C₂²	1	1	yes	characteristic
18	C₂⁴×D₅₃	C₂²	1	1	yes	characteristic
19	C₂×#11719947	—	2	8	no	—
20	C₂×#11719947	C₂²	1	8	yes	complemented
21	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
22	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
24	C₂²×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
26	C₂²×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic, complemented
27	C₂²×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic, complemented
28	C₂×D₄×D₅₃	—	2	8	no	—
29	C₂×D₄×D₅₃	C₂²	1	8	yes	complemented
30	C₂³×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
31	C₂³×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
32	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
33	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic, complemented
34	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic, complemented
35	#13133203	C₂²	1	8	yes	complemented
36	#13133202	C₂²	1	8	yes	complemented
37	#13133198	C₂²	1	8	yes	complemented
38	#13133195	C₂²	1	8	yes	complemented
39	#13133190	C₂²	1	8	yes	complemented
40	#13133189	C₂²	1	8	yes	complemented
41	#13133185	C₂²	1	8	yes	complemented
42	#13133185	C₂²	1	8	yes	complemented
43	C₄×C₂²⋊D₅₃	C₂²	1	8	yes	complemented
44	#13133177	C₂²	1	8	yes	complemented
45	C₂×#11719919	C₂²	1	1	yes	characteristic, complemented
46	C₂×#11719919	C₂²	1	1	yes	characteristic, complemented
47	C₂×#11719919	C₂²	1	1	yes	characteristic, complemented
48	C₂×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
49	C₂×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
50	C₂×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
51	C₂×#11719921	C₂²	1	1	yes	characteristic
52	C₂×#11719921	C₂²	1	1	yes	characteristic
53	C₂×#11719921	C₂²	1	1	yes	characteristic
54	C₂×#11719921	C₂²	1	1	yes	characteristic, complemented
55	C₂×#11719921	C₂²	1	1	yes	characteristic, complemented
56	C₂×#11719921	C₂²	1	1	yes	characteristic, complemented
57	C₂×#11719921	C₂²	1	1	yes	characteristic, complemented
58	C₂×#11719921	C₂²	1	8	yes	complemented
59	C₂²⋊C₄×D₅₃	C₂²	1	8	yes	complemented
60	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
61	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
62	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
63	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
64	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
65	#121×C₅₃	C₂²	1	8	yes	complemented
66	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
67	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
68	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
69	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
70	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
71	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
72	C₂³×D₅₃	—	2	4	no	—
73	C₂³×D₅₃	—	2	4	no	—
74	C₂³×D₅₃	—	2	4	no	—
75	C₂³×D₅₃	—	2	4	no	—
76	C₂³×D₅₃	—	4	2	no	—
77	C₂³×D₅₃	—	4	2	no	—
78	C₂³×D₅₃	C₂³	1	1	yes	characteristic
79	C₂³×D₅₃	C₂³	1	4	yes	—
80	C₂³×D₅₃	C₂³	1	4	yes	—
81	C₂³×D₅₃	C₂³	1	1	yes	characteristic
82	C₂³×D₅₃	C₂³	1	1	yes	characteristic
83	C₂×D₄×C₅₃	—	2	4	no	—
84	C₂×D₄×C₅₃	—	2	4	no	—
85	C₂×D₄×C₅₃	—	4	2	no	—
86	C₂×D₄×C₅₃	C₂³	1	4	yes	—
87	C₂×D₄×C₅₃	C₂³	1	4	yes	complemented
88	C₂×D₄×C₅₃	C₂³	1	4	yes	complemented
89	C₂²×C₄×C₅₃	—	4	2	no	—
90	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂³	1	4	yes	complemented
95	C₂×C₂²⋊D₅₃	—	2	4	no	—
96	C₂×C₂²⋊D₅₃	—	2	4	no	—
97	C₂×C₂²⋊D₅₃	—	2	4	no	—
98	C₂×C₂²⋊D₅₃	—	4	2	no	—
99	C₂×C₂²⋊D₅₃	D₄	1	8	yes	complemented
100	C₂×C₂²⋊D₅₃	C₂³	1	4	yes	—
101	C₂×C₂²⋊D₅₃	C₂³	1	4	yes	—
102	C₂×C₂²⋊D₅₃	C₂³	1	4	yes	—
103	C₂×C₂²⋊D₅₃	C₂³	1	4	yes	complemented
104	C₂×C₂²⋊D₅₃	C₂³	1	4	yes	complemented
105	C₂²×Dic₅₃	—	2	4	no	—
106	C₂²×Dic₅₃	—	2	4	no	—
107	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
108	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
109	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
110	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
111	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
112	C₂²×Dic₅₃	C₂³	1	4	yes	complemented
113	C₂²×Dic₅₃	C₂³	1	4	yes	complemented
114	#11719947	—	2	32	no	—
115	D₄×D₅₃	—	2	32	no	—
116	C₂×D₂₁₂	—	2	4	no	—
117	C₂×D₂₁₂	C₂³	1	4	yes	—
118	C₂×C₄×D₅₃	—	2	4	no	—
119	C₂×C₄×D₅₃	—	2	4	no	—
120	C₂×C₄×D₅₃	—	4	2	no	—
121	C₂×C₄×D₅₃	C₂³	1	4	yes	—
122	C₂×C₄×D₅₃	C₂³	1	4	yes	complemented
123	C₂×C₄×D₅₃	C₂³	1	4	yes	complemented
124	C₂×Dic₁₀₆	—	2	4	no	—
125	C₂×Dic₁₀₆	C₂³	1	4	yes	complemented
126	C₄⋊C₄×C₅₃	C₂³	1	4	yes	complemented
127	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	complemented
128	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	complemented
129	C₂²⋊Dic₅₃	C₂³	1	4	yes	complemented
130	C₂²⋊Dic₅₃	C₂³	1	4	yes	complemented
131	C₂²⋊Dic₅₃	C₂³	1	4	yes	complemented
132	#11719921	C₂³	1	4	yes	—
133	#11719921	C₂³	1	4	yes	—
134	#11719921	C₂³	1	4	yes	—
135	#11719921	C₂³	1	4	yes	complemented
136	#11719921	C₂³	1	4	yes	complemented
137	#11719921	C₂³	1	4	yes	complemented
138	#11719921	C₂³	1	4	yes	complemented
139	#11719919	C₂³	1	4	yes	complemented
140	#11719919	C₂³	1	4	yes	complemented
141	#11719919	C₂³	1	4	yes	complemented
142	C₄×Dic₅₃	C₂³	1	4	yes	complemented
143	C₂³×C₅₃	—	2	2	no	—
144	C₂³×C₅₃	—	2	2	no	—
145	C₂³×C₅₃	—	2	2	no	—
146	C₂³×C₅₃	—	2	2	no	—
147	C₂³×C₅₃	—	2	2	no	—
148	C₂³×C₅₃	—	2	2	no	—
149	C₂³×C₅₃	—	2	2	no	—
150	C₂³×C₅₃	—	2	2	no	—
151	C₂³×C₅₃	C₂×D₄	1	4	yes	—
152	C₂³×C₅₃	C₂⁴	1	2	yes	—
153	C₂³×C₅₃	C₂⁴	1	2	yes	—
154	C₂³×C₅₃	C₂⁴	1	2	yes	—
155	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
156	C₂²×D₅₃	—	2	2	no	—
157	C₂²×D₅₃	—	2	2	no	—
158	C₂²×D₅₃	—	2	2	no	—
159	C₂²×D₅₃	—	2	2	no	—
160	C₂²×D₅₃	—	2	2	no	—
161	C₂²×D₅₃	—	2	8	no	—
162	C₂²×D₅₃	—	2	2	no	—
163	C₂²×D₅₃	—	2	2	no	—
164	C₂²×D₅₃	—	2	8	no	—
165	C₂²×D₅₃	—	2	8	no	—
166	C₂²×D₅₃	—	2	8	no	—
167	C₂²×D₅₃	—	4	4	no	—
168	C₂²×D₅₃	—	4	4	no	—
169	C₂²×D₅₃	—	4	16	no	—
170	C₂²×D₅₃	—	4	16	no	—
171	C₂²×D₅₃	C₂×D₄	1	4	yes	—
172	C₂²×D₅₃	#41	1	4	yes	complemented
173	C₂²×D₅₃	C₂⁴	1	2	yes	—
174	C₂²×D₅₃	C₂⁴	1	2	yes	—
175	C₂²×D₅₃	C₂⁴	1	2	yes	—
176	D₄×C₅₃	—	2	8	no	—
177	D₄×C₅₃	—	2	8	no	—
178	D₄×C₅₃	—	4	4	no	—
179	C₂×C₄×C₅₃	—	2	2	no	—
180	C₂×C₄×C₅₃	—	2	2	no	—
181	C₂×C₄×C₅₃	—	2	2	no	—
182	C₂×C₄×C₅₃	—	2	2	no	—
183	C₂×C₄×C₅₃	—	4	4	no	—
184	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
185	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
186	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
187	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
188	C₂²⋊D₅₃	—	2	8	no	—
189	C₂²⋊D₅₃	—	2	8	no	—
190	C₂²⋊D₅₃	—	2	8	no	—
191	C₂²⋊D₅₃	—	4	4	no	—
192	C₂²⋊D₅₃	C₂×D₄	1	16	yes	complemented
193	C₂×Dic₅₃	—	2	2	no	—
194	C₂×Dic₅₃	—	2	2	no	—
195	C₂×Dic₅₃	—	2	8	no	—
196	C₂×Dic₅₃	—	2	2	no	—
197	C₂×Dic₅₃	—	2	2	no	—
198	C₂×Dic₅₃	—	2	8	no	—
199	C₂×Dic₅₃	C₂×D₄	1	4	yes	—
200	C₂×Dic₅₃	C₂⁴	1	2	yes	—
201	C₂×Dic₅₃	C₂⁴	1	2	yes	—
202	C₂×Dic₅₃	C₂⁴	1	2	yes	—
203	C₂×Dic₅₃	C₂⁴	1	2	yes	—
204	C₂×Dic₅₃	C₂⁴	1	2	yes	—
205	D₂₁₂	—	2	8	no	—
206	C₄×D₅₃	—	2	8	no	—
207	C₄×D₅₃	—	2	8	no	—
208	C₄×D₅₃	—	4	4	no	—
209	Dic₁₀₆	—	2	8	no	—
210	C₂²×C₅₃	—	2	8	no	—
211	C₂²×C₅₃	—	2	2	no	—
212	C₂²×C₅₃	—	2	2	no	—
213	C₂²×C₅₃	—	2	2	no	—
214	C₂²×C₅₃	—	2	2	no	—
215	C₂²×C₅₃	—	2	2	no	—
216	C₂²×C₅₃	—	2	2	no	—
217	C₂²×C₅₃	—	2	2	no	—
218	C₂²×C₅₃	—	2	2	no	—
219	C₂²×C₅₃	—	4	4	no	—
220	C₂²×C₅₃	—	4	4	no	—
221	C₂²×C₅₃	C₂²×D₄	1	4	yes	—
222	C₂²×C₅₃	C₂²×D₄	1	2	yes	—
223	C₂²×C₅₃	C₂×#41	1	2	yes	—
224	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
225	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
226	C₂×D₅₃	—	2	8	no	—
227	C₂×D₅₃	—	2	8	no	—
228	C₂×D₅₃	—	2	2	no	—
229	C₂×D₅₃	—	2	2	no	—
230	C₂×D₅₃	—	2	2	no	—
231	C₂×D₅₃	—	2	2	no	—
232	C₂×D₅₃	—	2	2	no	—
233	C₂×D₅₃	—	2	2	no	—
234	C₂×D₅₃	—	2	2	no	—
235	C₂×D₅₃	—	4	16	no	—
236	C₂×D₅₃	—	4	16	no	—
237	C₂×D₅₃	—	4	4	no	—
238	C₂×D₅₃	C₂²×D₄	1	4	yes	—
239	C₂×D₅₃	C₂×#41	1	4	yes	complemented
240	C₄×C₅₃	—	2	2	no	—
241	C₄×C₅₃	—	2	2	no	—
242	C₄×C₅₃	—	2	2	no	—
243	C₄×C₅₃	—	2	2	no	—
244	Dic₅₃	—	2	2	no	—
245	Dic₅₃	—	2	2	no	—
246	Dic₅₃	—	2	2	no	—
247	Dic₅₃	—	2	2	no	—
248	Dic₅₃	C₂²×D₄	1	4	yes	—
249	C₂×#546	—	53	1	no	maximal, Sylow
250	C₂×C₅₃	—	2	4	no	—
251	C₂×C₅₃	—	4	2	no	—
252	C₂×C₅₃	—	4	2	no	—
253	C₂×C₅₃	#546	1	4	yes	complemented
254	C₂×C₅₃	C₂³×D₄	1	1	yes	characteristic
255	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
256	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
257	D₅₃	—	2	4	no	—
258	D₅₃	—	2	4	no	—
259	D₅₃	—	4	2	no	—
260	C₂²×#41	—	53	1	no	—
261	C₂³×D₄	—	53	1	no	—
262	#546	—	53	16	no	—
263	C₂×#124	—	53	1	no	—
264	C₂×#123	—	53	1	no	—
265	C₂×#123	—	53	1	no	—
266	C₂×C₂²⋊Q₈	—	53	1	no	—
267	C₂×#121	—	53	1	no	—
268	C₂×#121	—	53	1	no	—
269	C₂×#121	D₅₃	1	1	yes	characteristic, complemented
270	C₂×#120	—	53	1	no	—
271	C₂×#120	—	53	1	no	—
272	C₂×C₄×D₄	—	53	1	no	—
273	C₂×C₄×D₄	—	53	1	no	—
274	C₂²×C₂²⋊C₄	—	53	1	no	—
275	C₂²×C₂²⋊C₄	—	53	1	no	—
276	C₅₃	C₂×#546	1	1	yes	characteristic, complemented, Sylow
277	C₂⁵	—	53	1	no	—
278	C₂⁵	—	53	1	no	—
279	C₂×#41	—	53	8	no	—
280	C₂×#41	—	106	8	no	—
281	C₂²×Q₈	—	53	1	no	—
282	C₂²×D₄	—	53	1	no	—
283	C₂²×D₄	—	53	1	no	—
284	C₂²×D₄	—	53	1	no	—
285	C₂²×D₄	—	53	1	no	—
286	C₂²×D₄	—	53	1	no	—
287	C₂²×D₄	—	53	8	no	—
288	C₂²×D₄	—	53	1	no	—
289	C₂²×D₄	—	106	8	no	—
290	C₂²×D₄	C₂×D₅₃	1	1	yes	characteristic
291	C₂²×D₄	C₂×D₅₃	1	1	yes	characteristic, complemented
292	C₂²×D₄	C₂×D₅₃	1	1	yes	characteristic, complemented
293	C₂³×C₄	—	53	1	no	—
294	C₂³×C₄	—	53	1	no	—
295	C₂³×C₄	—	53	1	no	—
296	C₂³×C₄	—	53	1	no	—
297	C₂³×C₄	—	53	1	no	—
298	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
299	#124	—	53	8	no	—
300	#123	—	53	8	no	—
301	#123	—	53	8	no	—
302	C₂²⋊Q₈	—	53	8	no	—
303	#121	—	53	8	no	—
304	#121	—	53	8	no	—
305	#121	C₂×D₅₃	1	8	yes	complemented
306	#120	—	53	8	no	—
307	#120	—	53	8	no	—
308	C₄×D₄	—	53	8	no	—
309	C₄×D₄	—	53	8	no	—
310	C₂×C₄⋊C₄	—	53	1	no	—
311	C₂×C₄⋊C₄	—	53	1	no	—
312	C₂×C₄⋊C₄	—	53	1	no	—
313	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
314	C₂×C₂²⋊C₄	—	53	1	no	—
315	C₂×C₂²⋊C₄	—	53	1	no	—
316	C₂×C₂²⋊C₄	—	53	1	no	—
317	C₂×C₂²⋊C₄	—	53	1	no	—
318	C₂×C₂²⋊C₄	—	53	1	no	—
319	C₂×C₂²⋊C₄	—	53	1	no	—
320	C₂×C₂²⋊C₄	—	53	1	no	—
321	C₂×C₂²⋊C₄	—	53	1	no	—
322	C₂×C₂²⋊C₄	—	53	1	no	—
323	C₂×C₂²⋊C₄	—	53	1	no	—
324	C₂×C₂²⋊C₄	—	53	8	no	—
325	C₂×C₂²⋊C₄	—	53	8	no	—
326	C₂×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
327	C₂×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
328	C₂×C₄²	—	53	1	no	—
329	C₂⁴	—	53	1	no	—
330	C₂⁴	—	53	1	no	—
331	C₂⁴	—	53	1	no	—
332	C₂⁴	—	53	4	no	—
333	C₂⁴	—	53	4	no	—
334	C₂⁴	—	106	4	no	—
335	C₂⁴	—	106	4	no	—
336	C₂⁴	—	106	4	no	—
337	C₂⁴	—	106	4	no	—
338	C₂⁴	—	212	2	no	—
339	C₂⁴	—	212	2	no	—
340	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
341	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
342	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
343	#41	—	106	32	no	—
344	C₂×Q₈	—	53	4	no	—
345	C₂×Q₈	—	106	4	no	—
346	C₂×D₄	—	2	4	no	—
347	C₂×D₄	—	2	4	no	—
348	C₂×D₄	—	4	2	no	—
349	C₂×D₄	—	53	8	no	—
350	C₂×D₄	—	53	4	no	—
351	C₂×D₄	—	53	4	no	—
352	C₂×D₄	—	53	4	no	—
353	C₂×D₄	—	53	4	no	—
354	C₂×D₄	—	53	4	no	—
355	C₂×D₄	—	53	4	no	—
356	C₂×D₄	—	106	4	no	—
357	C₂×D₄	—	106	4	no	—
358	C₂×D₄	—	106	32	no	—
359	C₂×D₄	—	106	4	no	—
360	C₂×D₄	—	106	4	no	—
361	C₂×D₄	—	212	2	no	—
362	C₂×D₄	C₂²×D₅₃	1	4	yes	—
363	C₂×D₄	C₂²×D₅₃	1	4	yes	complemented
364	C₂×D₄	C₂²×D₅₃	1	4	yes	complemented
365	C₂²×C₄	—	4	2	no	—
366	C₂²×C₄	—	53	1	no	—
367	C₂²×C₄	—	53	4	no	—
368	C₂²×C₄	—	53	4	no	—
369	C₂²×C₄	—	53	1	no	—
370	C₂²×C₄	—	53	1	no	—
371	C₂²×C₄	—	53	1	no	—
372	C₂²×C₄	—	53	1	no	—
373	C₂²×C₄	—	53	4	no	—
374	C₂²×C₄	—	53	4	no	—
375	C₂²×C₄	—	53	4	no	—
376	C₂²×C₄	—	106	4	no	—
377	C₂²×C₄	—	106	4	no	—
378	C₂²×C₄	—	106	4	no	—
379	C₂²×C₄	—	106	4	no	—
380	C₂²×C₄	—	212	2	no	—
381	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
382	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
383	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
384	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
385	C₂²×C₄	C₂²×D₅₃	1	4	yes	complemented
386	C₄⋊C₄	—	53	4	no	—
387	C₄⋊C₄	—	53	4	no	—
388	C₄⋊C₄	—	53	4	no	—
389	C₄⋊C₄	C₂²×D₅₃	1	4	yes	complemented
390	C₂²⋊C₄	—	53	4	no	—
391	C₂²⋊C₄	—	53	4	no	—
392	C₂²⋊C₄	—	53	4	no	—
393	C₂²⋊C₄	—	53	4	no	—
394	C₂²⋊C₄	—	53	4	no	—
395	C₂²⋊C₄	—	53	4	no	—
396	C₂²⋊C₄	—	53	4	no	—
397	C₂²⋊C₄	—	53	4	no	—
398	C₂²⋊C₄	—	53	4	no	—
399	C₂²⋊C₄	—	53	4	no	—
400	C₂²⋊C₄	C₂²×D₅₃	1	4	yes	complemented
401	C₂²⋊C₄	C₂²×D₅₃	1	4	yes	complemented
402	C₄²	—	53	4	no	—
403	C₂³	—	2	2	no	—
404	C₂³	—	2	2	no	—
405	C₂³	—	2	2	no	—
406	C₂³	—	2	2	no	—
407	C₂³	—	2	2	no	—
408	C₂³	—	2	2	no	—
409	C₂³	—	2	2	no	—
410	C₂³	—	2	2	no	—
411	C₂³	—	53	4	no	—
412	C₂³	—	53	4	no	—
413	C₂³	—	53	2	no	—
414	C₂³	—	53	2	no	—
415	C₂³	—	53	2	no	—
416	C₂³	—	106	8	no	—
417	C₂³	—	106	8	no	—
418	C₂³	—	106	8	no	—
419	C₂³	—	106	2	no	—
420	C₂³	—	106	2	no	—
421	C₂³	—	106	2	no	—
422	C₂³	—	106	2	no	—
423	C₂³	—	106	8	no	—
424	C₂³	—	106	2	no	—
425	C₂³	—	106	2	no	—
426	C₂³	—	106	2	no	—
427	C₂³	—	212	4	no	—
428	C₂³	—	212	4	no	—
429	C₂³	—	212	16	no	—
430	C₂³	—	212	16	no	—
431	C₂³	D₄×D₅₃	1	4	yes	—
432	C₂³	C₂³×D₅₃	1	2	yes	—
433	C₂³	C₂³×D₅₃	1	2	yes	—
434	C₂³	C₂³×D₅₃	1	2	yes	—
435	C₂³	C₂³×D₅₃	1	1	yes	characteristic
436	Q₈	—	106	8	no	—
437	D₄	—	2	8	no	—
438	D₄	—	2	8	no	—
439	D₄	—	4	4	no	—
440	D₄	—	53	16	no	—
441	D₄	—	106	8	no	—
442	D₄	—	106	8	no	—
443	D₄	—	106	8	no	—
444	D₄	—	106	8	no	—
445	D₄	—	212	4	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	4	4	no	—
451	C₂×C₄	—	53	4	no	—
452	C₂×C₄	—	53	2	no	—
453	C₂×C₄	—	53	2	no	—
454	C₂×C₄	—	53	2	no	—
455	C₂×C₄	—	53	2	no	—
456	C₂×C₄	—	53	2	no	—
457	C₂×C₄	—	106	8	no	—
458	C₂×C₄	—	106	8	no	—
459	C₂×C₄	—	106	8	no	—
460	C₂×C₄	—	106	2	no	—
461	C₂×C₄	—	106	2	no	—
462	C₂×C₄	—	106	8	no	—
463	C₂×C₄	—	106	2	no	—
464	C₂×C₄	—	106	2	no	—
465	C₂×C₄	—	212	4	no	—
466	C₂×C₄	C₂³×D₅₃	1	2	yes	—
467	C₂×C₄	C₂³×D₅₃	1	2	yes	—
468	C₂×C₄	C₂³×D₅₃	1	2	yes	—
469	C₂×C₄	C₂³×D₅₃	1	2	yes	—
470	C₂²	—	2	2	no	—
471	C₂²	—	2	2	no	—
472	C₂²	—	2	8	no	—
473	C₂²	—	2	2	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	2	2	no	—
477	C₂²	—	2	2	no	—
478	C₂²	—	2	2	no	—
479	C₂²	—	4	4	no	—
480	C₂²	—	4	4	no	—
481	C₂²	—	53	4	no	—
482	C₂²	—	53	4	no	—
483	C₂²	—	106	2	no	—
484	C₂²	—	106	2	no	—
485	C₂²	—	106	2	no	—
486	C₂²	—	106	2	no	—
487	C₂²	—	106	8	no	—
488	C₂²	—	106	8	no	—
489	C₂²	—	106	2	no	—
490	C₂²	—	106	2	no	—
491	C₂²	—	106	2	no	—
492	C₂²	—	212	16	no	—
493	C₂²	—	212	16	no	—
494	C₂²	—	212	4	no	—
495	C₂²	C₂×D₄×D₅₃	1	4	yes	—
496	C₂²	C₂×D₄×D₅₃	1	2	yes	—
497	C₂²	#41×D₅₃	1	2	yes	—
498	C₂²	#13133262	1	2	yes	—
499	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
500	C₄	—	2	2	no	—
501	C₄	—	2	2	no	—
502	C₄	—	2	2	no	—
503	C₄	—	2	2	no	—
504	C₄	—	53	4	no	—
505	C₄	—	106	2	no	—
506	C₄	—	106	2	no	—
507	C₄	—	106	2	no	—
508	C₄	—	106	2	no	—
509	C₂	—	2	4	no	—
510	C₂	—	4	2	no	—
511	C₂	—	4	2	no	—
512	C₂	—	106	4	no	—
513	C₂	—	106	4	no	—
514	C₂	—	212	2	no	—
515	C₂	#16811262	1	4	yes	direct product factor
516	C₂	C₂²×D₄×D₅₃	1	1	yes	characteristic
517	C₂	C₂×#41×D₅₃	1	1	yes	characteristic
518	C₂	C₂×#13133262	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂⁵
Frattini	C₂²	C₂⁴×D₅₃
Fitting	C₂×#121×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂×#121	D₅₃
hypocenter	C₅₃	C₂×#546
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#121

Complements

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

N	H ≅ G/N	type
1. C₂²×#11719947	C₂	semidirect
2. C₂²×D₄×D₅₃	C₂	semidirect
3. #16811262	C₂	direct (N × H)
4. C₂×#13133203	C₂	semidirect
5. C₂×#13133202	C₂	semidirect
6. C₂×#13133198	C₂	semidirect
7. C₂×#13133195	C₂	semidirect
8. C₂×#13133190	C₂	semidirect
9. C₂×#13133189	C₂	semidirect
10. C₂×#13133185	C₂	semidirect
11. C₂×#13133185	C₂	semidirect
12. C₂×C₄×C₂²⋊D₅₃	C₂	semidirect
13. C₂×#13133177	C₂	semidirect
14. C₂²×#11719921	C₂	semidirect
15. C₂×C₂²⋊C₄×D₅₃	C₂	semidirect
16. C₂×#121×C₅₃	C₂	semidirect
20. C₂×#11719947	C₂²	semidirect
21. C₂²×Dic₁₀₆	C₂²	semidirect
26. C₂²×C₂²⋊D₅₃	C₂²	semidirect
27. C₂²×C₂²⋊D₅₃	C₂²	semidirect
29. C₂×D₄×D₅₃	C₂²	semidirect
30. C₂³×Dic₅₃	C₂²	semidirect
31. C₂³×Dic₅₃	C₂²	semidirect
33. C₂²×C₄×D₅₃	C₂²	semidirect
34. C₂²×C₄×D₅₃	C₂²	semidirect
35. #13133203	C₂²	semidirect
36. #13133202	C₂²	semidirect
37. #13133198	C₂²	semidirect
38. #13133195	C₂²	semidirect
39. #13133190	C₂²	semidirect
40. #13133189	C₂²	semidirect
41. #13133185	C₂²	semidirect
42. #13133185	C₂²	semidirect
43. C₄×C₂²⋊D₅₃	C₂²	semidirect
44. #13133177	C₂²	semidirect
45. C₂×#11719919	C₂²	semidirect
46. C₂×#11719919	C₂²	semidirect
47. C₂×#11719919	C₂²	semidirect
48. C₂×C₂²⋊Dic₅₃	C₂²	semidirect
49. C₂×C₂²⋊Dic₅₃	C₂²	semidirect
50. C₂×C₂²⋊Dic₅₃	C₂²	semidirect
54. C₂×#11719921	C₂²	semidirect
55. C₂×#11719921	C₂²	semidirect
56. C₂×#11719921	C₂²	semidirect
57. C₂×#11719921	C₂²	semidirect
58. C₂×#11719921	C₂²	semidirect
59. C₂²⋊C₄×D₅₃	C₂²	semidirect
60. C₂×C₄×Dic₅₃	C₂²	semidirect
62. C₂²×D₄×C₅₃	C₂²	semidirect
63. C₂²×D₄×C₅₃	C₂²	semidirect
64. C₂³×C₄×C₅₃	C₂²	semidirect
65. #121×C₅₃	C₂²	semidirect
66. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
67. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
68. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
87. C₂×D₄×C₅₃	C₂³	semidirect
88. C₂×D₄×C₅₃	C₂³	semidirect
94. C₂²×C₄×C₅₃	C₂³	semidirect
99. C₂×C₂²⋊D₅₃	D₄	semidirect
103. C₂×C₂²⋊D₅₃	C₂³	semidirect
104. C₂×C₂²⋊D₅₃	C₂³	semidirect
112. C₂²×Dic₅₃	C₂³	semidirect
113. C₂²×Dic₅₃	C₂³	semidirect
122. C₂×C₄×D₅₃	C₂³	semidirect
123. C₂×C₄×D₅₃	C₂³	semidirect
125. C₂×Dic₁₀₆	C₂³	semidirect
126. C₄⋊C₄×C₅₃	C₂³	semidirect
127. C₂²⋊C₄×C₅₃	C₂³	semidirect
128. C₂²⋊C₄×C₅₃	C₂³	semidirect
129. C₂²⋊Dic₅₃	C₂³	semidirect
130. C₂²⋊Dic₅₃	C₂³	semidirect
131. C₂²⋊Dic₅₃	C₂³	semidirect
135. #11719921	C₂³	semidirect
136. #11719921	C₂³	semidirect
137. #11719921	C₂³	semidirect
138. #11719921	C₂³	semidirect
139. #11719919	C₂³	semidirect
140. #11719919	C₂³	semidirect
141. #11719919	C₂³	semidirect
142. C₄×Dic₅₃	C₂³	semidirect
172. C₂²×D₅₃	#41	semidirect
192. C₂²⋊D₅₃	C₂×D₄	semidirect
239. C₂×D₅₃	C₂×#41	semidirect
253. C₂×C₅₃	#546	semidirect
269. C₂×#121	D₅₃	semidirect
276. C₅₃	C₂×#546	semidirect
291. C₂²×D₄	C₂×D₅₃	semidirect
292. C₂²×D₄	C₂×D₅₃	semidirect
298. C₂³×C₄	C₂×D₅₃	semidirect
305. #121	C₂×D₅₃	semidirect
313. C₂×C₄⋊C₄	C₂×D₅₃	semidirect
326. C₂×C₂²⋊C₄	C₂×D₅₃	semidirect
327. C₂×C₂²⋊C₄	C₂×D₅₃	semidirect
363. C₂×D₄	C₂²×D₅₃	semidirect
364. C₂×D₄	C₂²×D₅₃	semidirect
385. C₂²×C₄	C₂²×D₅₃	semidirect
389. C₄⋊C₄	C₂²×D₅₃	semidirect
400. C₂²⋊C₄	C₂²×D₅₃	semidirect
401. C₂²⋊C₄	C₂²×D₅₃	semidirect
515. C₂	#16811262	direct (N × H)

Subgroups — C2×#16811262 #20288159

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811262 #20288159