Subgroups — C₂×#16811361 #20288830

Order 6784.

1344 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811361	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13133213	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13133212	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13133210	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13133202	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13133201	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13133200	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133191	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133190	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13133189	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#13133189	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133188	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#13133187	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#13133185	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#13133185	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	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
19	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
20	C₂²×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
22	C₂²×C₂²⋊D₅₃	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, complemented
26	C₂²×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic, complemented
27	C₂³×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
28	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic, complemented
29	#13133213	C₂²	1	8	yes	complemented
30	#13133212	C₂²	1	8	yes	complemented
31	#13133210	C₂²	1	8	yes	complemented
32	#13133202	C₂²	1	8	yes	complemented
33	#13133201	C₂²	1	8	yes	complemented
34	#13133200	C₂²	1	8	yes	complemented
35	#13133191	C₂²	1	8	yes	complemented
36	#13133190	C₂²	1	8	yes	complemented
37	#13133189	C₂²	1	8	yes	complemented
38	#13133189	C₂²	1	8	yes	complemented
39	#13133188	C₂²	1	8	yes	complemented
40	#13133187	C₂²	1	8	yes	complemented
41	#13133185	C₂²	1	8	yes	complemented
42	#13133185	C₂²	1	8	yes	complemented
43	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
44	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic, 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
49	C₂×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic
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
55	C₂×#11719921	C₂²	1	1	yes	characteristic
56	C₂×#11719921	C₂²	1	1	yes	characteristic
57	C₂×#11719921	C₂²	1	1	yes	characteristic, complemented
58	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
59	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
60	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
61	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
62	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
63	#121×C₅₃	C₂²	1	8	yes	complemented
64	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
65	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
67	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
68	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
69	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
70	C₂³×D₅₃	—	4	2	no	—
71	C₂³×D₅₃	—	4	2	no	—
72	C₂³×D₅₃	—	4	2	no	—
73	C₂³×D₅₃	C₂³	1	1	yes	characteristic
74	C₂³×D₅₃	C₂³	1	1	yes	characteristic
75	C₂³×D₅₃	C₂³	1	1	yes	characteristic
76	C₂³×D₅₃	C₂³	1	4	yes	—
77	C₂×D₄×C₅₃	—	4	2	no	—
78	C₂×D₄×C₅₃	—	4	2	no	—
79	C₂×D₄×C₅₃	—	4	2	no	—
80	C₂×D₄×C₅₃	C₂³	1	4	yes	—
81	C₂×D₄×C₅₃	C₂³	1	4	yes	—
82	C₂×D₄×C₅₃	C₂³	1	4	yes	complemented
83	C₂²×C₄×C₅₃	—	4	2	no	—
84	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂³	1	4	yes	complemented
89	C₂×C₂²⋊D₅₃	—	4	2	no	—
90	C₂×C₂²⋊D₅₃	—	4	2	no	—
91	C₂×C₂²⋊D₅₃	—	4	2	no	—
92	C₂×C₂²⋊D₅₃	—	4	2	no	—
93	C₂×C₂²⋊D₅₃	—	4	2	no	—
94	C₂×C₂²⋊D₅₃	—	4	2	no	—
95	C₂×C₂²⋊D₅₃	—	4	2	no	—
96	C₂×C₂²⋊D₅₃	C₂³	1	4	yes	—
97	C₂×C₂²⋊D₅₃	C₂³	1	4	yes	—
98	C₂×C₂²⋊D₅₃	C₂³	1	4	yes	—
99	C₂×C₂²⋊D₅₃	C₂³	1	4	yes	—
100	C₂×C₂²⋊D₅₃	C₂³	1	4	yes	—
101	C₂×C₂²⋊D₅₃	C₂³	1	4	yes	complemented
102	C₂×C₂²⋊D₅₃	C₂³	1	4	yes	complemented
103	C₂²×Dic₅₃	—	4	2	no	—
104	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
105	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
106	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
107	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
108	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
109	C₂²×Dic₅₃	C₂³	1	4	yes	complemented
110	C₂×D₂₁₂	—	4	2	no	—
111	C₂×D₂₁₂	—	4	2	no	—
112	C₂×D₂₁₂	C₂³	1	4	yes	—
113	C₂×D₂₁₂	C₂³	1	4	yes	—
114	C₂×C₄×D₅₃	—	4	2	no	—
115	C₂×C₄×D₅₃	C₂³	1	4	yes	complemented
116	C₄⋊C₄×C₅₃	C₂³	1	4	yes	complemented
117	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
118	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	complemented
119	C₂²⋊Dic₅₃	C₂³	1	4	yes	—
120	C₂²⋊Dic₅₃	C₂³	1	4	yes	—
121	C₂²⋊Dic₅₃	C₂³	1	4	yes	complemented
122	#11719921	C₂³	1	4	yes	—
123	#11719921	C₂³	1	4	yes	—
124	#11719921	C₂³	1	4	yes	—
125	#11719921	C₂³	1	4	yes	—
126	#11719921	C₂³	1	4	yes	—
127	#11719921	C₂³	1	4	yes	—
128	#11719921	C₂³	1	4	yes	complemented
129	C₄⋊Dic₅₃	C₂³	1	4	yes	complemented
130	C₄⋊Dic₅₃	C₂³	1	4	yes	complemented
131	#11719919	C₂³	1	4	yes	complemented
132	#11719919	C₂³	1	4	yes	complemented
133	#11719919	C₂³	1	4	yes	complemented
134	C₄×Dic₅₃	C₂³	1	4	yes	complemented
135	C₂³×C₅₃	—	2	2	no	—
136	C₂³×C₅₃	—	2	2	no	—
137	C₂³×C₅₃	—	2	2	no	—
138	C₂³×C₅₃	—	2	2	no	—
139	C₂³×C₅₃	—	2	2	no	—
140	C₂³×C₅₃	—	2	2	no	—
141	C₂³×C₅₃	—	2	2	no	—
142	C₂³×C₅₃	—	2	2	no	—
143	C₂³×C₅₃	—	2	2	no	—
144	C₂³×C₅₃	C₂⁴	1	2	yes	—
145	C₂³×C₅₃	C₂⁴	1	2	yes	—
146	C₂³×C₅₃	C₂⁴	1	2	yes	—
147	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
148	C₂²×D₅₃	—	2	2	no	—
149	C₂²×D₅₃	—	2	2	no	—
150	C₂²×D₅₃	—	2	2	no	—
151	C₂²×D₅₃	—	2	2	no	—
152	C₂²×D₅₃	—	2	2	no	—
153	C₂²×D₅₃	—	2	2	no	—
154	C₂²×D₅₃	—	2	2	no	—
155	C₂²×D₅₃	—	2	2	no	—
156	C₂²×D₅₃	—	2	2	no	—
157	C₂²×D₅₃	—	4	4	no	—
158	C₂²×D₅₃	—	4	4	no	—
159	C₂²×D₅₃	—	4	4	no	—
160	C₂²×D₅₃	—	4	16	no	—
161	C₂²×D₅₃	C₂⁴	1	2	yes	—
162	C₂²×D₅₃	C₂⁴	1	2	yes	—
163	C₂²×D₅₃	C₂⁴	1	2	yes	—
164	D₄×C₅₃	—	4	4	no	—
165	D₄×C₅₃	—	4	4	no	—
166	D₄×C₅₃	—	4	4	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	2	2	no	—
169	C₂×C₄×C₅₃	—	2	2	no	—
170	C₂×C₄×C₅₃	—	2	2	no	—
171	C₂×C₄×C₅₃	—	4	4	no	—
172	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
173	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
174	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
175	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
176	C₂²⋊D₅₃	—	4	4	no	—
177	C₂²⋊D₅₃	—	4	4	no	—
178	C₂²⋊D₅₃	—	4	4	no	—
179	C₂²⋊D₅₃	—	4	4	no	—
180	C₂²⋊D₅₃	—	4	4	no	—
181	C₂²⋊D₅₃	—	4	4	no	—
182	C₂²⋊D₅₃	—	4	4	no	—
183	C₂×Dic₅₃	—	2	2	no	—
184	C₂×Dic₅₃	—	2	2	no	—
185	C₂×Dic₅₃	—	2	2	no	—
186	C₂×Dic₅₃	—	2	2	no	—
187	C₂×Dic₅₃	—	2	2	no	—
188	C₂×Dic₅₃	—	4	4	no	—
189	C₂×Dic₅₃	C₂⁴	1	2	yes	—
190	C₂×Dic₅₃	C₂⁴	1	2	yes	—
191	C₂×Dic₅₃	C₂⁴	1	2	yes	—
192	C₂×Dic₅₃	C₂⁴	1	2	yes	—
193	C₂×Dic₅₃	C₂⁴	1	2	yes	—
194	D₂₁₂	—	4	4	no	—
195	D₂₁₂	—	4	4	no	—
196	C₄×D₅₃	—	4	4	no	—
197	C₂²×C₅₃	—	2	2	no	—
198	C₂²×C₅₃	—	2	2	no	—
199	C₂²×C₅₃	—	2	2	no	—
200	C₂²×C₅₃	—	2	2	no	—
201	C₂²×C₅₃	—	2	2	no	—
202	C₂²×C₅₃	—	2	2	no	—
203	C₂²×C₅₃	—	2	2	no	—
204	C₂²×C₅₃	—	2	2	no	—
205	C₂²×C₅₃	—	2	2	no	—
206	C₂²×C₅₃	—	4	4	no	—
207	C₂²×C₅₃	—	4	4	no	—
208	C₂²×C₅₃	—	4	4	no	—
209	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
210	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
211	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
212	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
213	C₂×D₅₃	—	2	2	no	—
214	C₂×D₅₃	—	2	2	no	—
215	C₂×D₅₃	—	2	2	no	—
216	C₂×D₅₃	—	2	2	no	—
217	C₂×D₅₃	—	2	2	no	—
218	C₂×D₅₃	—	2	2	no	—
219	C₂×D₅₃	—	2	2	no	—
220	C₂×D₅₃	—	2	2	no	—
221	C₂×D₅₃	—	2	2	no	—
222	C₂×D₅₃	—	4	16	no	—
223	C₂×D₅₃	—	4	4	no	—
224	C₂×D₅₃	—	4	4	no	—
225	C₂×D₅₃	—	4	4	no	—
226	C₄×C₅₃	—	2	2	no	—
227	C₄×C₅₃	—	2	2	no	—
228	C₄×C₅₃	—	2	2	no	—
229	C₄×C₅₃	—	2	2	no	—
230	Dic₅₃	—	2	2	no	—
231	Dic₅₃	—	2	2	no	—
232	Dic₅₃	—	2	2	no	—
233	Dic₅₃	—	2	2	no	—
234	Dic₅₃	—	2	2	no	—
235	C₂×#560	—	53	1	no	maximal, Sylow
236	C₂×C₅₃	—	4	2	no	—
237	C₂×C₅₃	—	4	2	no	—
238	C₂×C₅₃	—	4	2	no	—
239	C₂×C₅₃	#560	1	4	yes	complemented
240	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
241	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
242	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
243	D₅₃	—	4	2	no	—
244	D₅₃	—	4	2	no	—
245	D₅₃	—	4	2	no	—
246	#560	—	53	16	no	—
247	C₂×#127	—	53	1	no	—
248	C₂×#126	—	53	1	no	—
249	C₂×#126	—	53	1	no	—
250	C₂×#123	—	53	1	no	—
251	C₂×#123	—	53	1	no	—
252	C₂×#123	—	53	1	no	—
253	C₂×#121	—	53	1	no	—
254	C₂×#121	—	53	1	no	—
255	C₂×#121	—	53	1	no	—
256	C₂×#121	—	53	1	no	—
257	C₂×#121	—	53	1	no	—
258	C₂×#121	D₅₃	1	1	yes	characteristic, complemented
259	C₂×#120	—	53	1	no	—
260	C₂×#120	—	53	1	no	—
261	C₂×#120	—	53	1	no	—
262	C₅₃	C₂×#560	1	1	yes	characteristic, complemented, Sylow
263	C₂⁵	—	53	1	no	—
264	C₂²×D₄	—	53	1	no	—
265	C₂²×D₄	—	53	1	no	—
266	C₂²×D₄	—	53	1	no	—
267	C₂²×D₄	—	53	1	no	—
268	C₂²×D₄	—	53	1	no	—
269	C₂²×D₄	—	53	1	no	—
270	C₂²×D₄	—	53	1	no	—
271	C₂²×D₄	—	53	1	no	—
272	C₂²×D₄	—	53	1	no	—
273	C₂²×D₄	C₂×D₅₃	1	1	yes	characteristic
274	C₂²×D₄	C₂×D₅₃	1	1	yes	characteristic
275	C₂²×D₄	C₂×D₅₃	1	1	yes	characteristic, complemented
276	C₂³×C₄	—	53	1	no	—
277	C₂³×C₄	—	53	1	no	—
278	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
279	#127	—	53	8	no	—
280	#126	—	53	8	no	—
281	#126	—	53	8	no	—
282	#123	—	53	8	no	—
283	#123	—	53	8	no	—
284	#123	—	53	8	no	—
285	#121	—	53	8	no	—
286	#121	—	53	8	no	—
287	#121	—	53	8	no	—
288	#121	—	53	8	no	—
289	#121	—	53	8	no	—
290	#121	C₂×D₅₃	1	8	yes	complemented
291	#120	—	53	8	no	—
292	#120	—	53	8	no	—
293	#120	—	53	8	no	—
294	C₂×C₄⋊C₄	—	53	1	no	—
295	C₂×C₄⋊C₄	—	53	1	no	—
296	C₂×C₄⋊C₄	—	53	1	no	—
297	C₂×C₄⋊C₄	—	53	1	no	—
298	C₂×C₄⋊C₄	—	53	1	no	—
299	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
300	C₂×C₂²⋊C₄	—	53	1	no	—
301	C₂×C₂²⋊C₄	—	53	1	no	—
302	C₂×C₂²⋊C₄	—	53	1	no	—
303	C₂×C₂²⋊C₄	—	53	1	no	—
304	C₂×C₂²⋊C₄	—	53	1	no	—
305	C₂×C₂²⋊C₄	—	53	1	no	—
306	C₂×C₂²⋊C₄	—	53	1	no	—
307	C₂×C₂²⋊C₄	—	53	1	no	—
308	C₂×C₂²⋊C₄	—	53	1	no	—
309	C₂×C₂²⋊C₄	—	53	1	no	—
310	C₂×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
311	C₂×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
312	C₂×C₄²	—	53	1	no	—
313	C₂⁴	—	53	1	no	—
314	C₂⁴	—	53	1	no	—
315	C₂⁴	—	53	1	no	—
316	C₂⁴	—	53	4	no	—
317	C₂⁴	—	212	2	no	—
318	C₂⁴	—	212	2	no	—
319	C₂⁴	—	212	2	no	—
320	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
321	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
322	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
323	C₂×D₄	—	4	2	no	—
324	C₂×D₄	—	4	2	no	—
325	C₂×D₄	—	4	2	no	—
326	C₂×D₄	—	53	4	no	—
327	C₂×D₄	—	53	4	no	—
328	C₂×D₄	—	53	4	no	—
329	C₂×D₄	—	53	4	no	—
330	C₂×D₄	—	53	4	no	—
331	C₂×D₄	—	53	4	no	—
332	C₂×D₄	—	53	4	no	—
333	C₂×D₄	—	53	4	no	—
334	C₂×D₄	—	53	4	no	—
335	C₂×D₄	—	212	2	no	—
336	C₂×D₄	—	212	2	no	—
337	C₂×D₄	—	212	2	no	—
338	C₂×D₄	—	212	2	no	—
339	C₂×D₄	—	212	2	no	—
340	C₂×D₄	—	212	2	no	—
341	C₂×D₄	—	212	2	no	—
342	C₂×D₄	—	212	2	no	—
343	C₂×D₄	—	212	2	no	—
344	C₂×D₄	C₂²×D₅₃	1	4	yes	—
345	C₂×D₄	C₂²×D₅₃	1	4	yes	—
346	C₂×D₄	C₂²×D₅₃	1	4	yes	complemented
347	C₂²×C₄	—	4	2	no	—
348	C₂²×C₄	—	53	1	no	—
349	C₂²×C₄	—	53	1	no	—
350	C₂²×C₄	—	53	1	no	—
351	C₂²×C₄	—	53	1	no	—
352	C₂²×C₄	—	53	1	no	—
353	C₂²×C₄	—	53	4	no	—
354	C₂²×C₄	—	53	4	no	—
355	C₂²×C₄	—	212	2	no	—
356	C₂²×C₄	—	212	2	no	—
357	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
358	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
359	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
360	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
361	C₂²×C₄	C₂²×D₅₃	1	4	yes	complemented
362	C₄⋊C₄	—	53	4	no	—
363	C₄⋊C₄	—	53	4	no	—
364	C₄⋊C₄	—	53	4	no	—
365	C₄⋊C₄	—	53	4	no	—
366	C₄⋊C₄	—	53	4	no	—
367	C₄⋊C₄	C₂²×D₅₃	1	4	yes	complemented
368	C₂²⋊C₄	—	53	4	no	—
369	C₂²⋊C₄	—	53	4	no	—
370	C₂²⋊C₄	—	53	4	no	—
371	C₂²⋊C₄	—	53	4	no	—
372	C₂²⋊C₄	—	53	4	no	—
373	C₂²⋊C₄	—	53	4	no	—
374	C₂²⋊C₄	—	53	4	no	—
375	C₂²⋊C₄	—	53	4	no	—
376	C₂²⋊C₄	—	53	4	no	—
377	C₂²⋊C₄	—	53	4	no	—
378	C₂²⋊C₄	C₂²×D₅₃	1	4	yes	—
379	C₂²⋊C₄	C₂²×D₅₃	1	4	yes	complemented
380	C₄²	—	53	4	no	—
381	C₂³	—	2	2	no	—
382	C₂³	—	2	2	no	—
383	C₂³	—	2	2	no	—
384	C₂³	—	2	2	no	—
385	C₂³	—	2	2	no	—
386	C₂³	—	2	2	no	—
387	C₂³	—	2	2	no	—
388	C₂³	—	2	2	no	—
389	C₂³	—	2	2	no	—
390	C₂³	—	53	2	no	—
391	C₂³	—	53	2	no	—
392	C₂³	—	53	2	no	—
393	C₂³	—	106	2	no	—
394	C₂³	—	106	2	no	—
395	C₂³	—	106	2	no	—
396	C₂³	—	106	2	no	—
397	C₂³	—	106	2	no	—
398	C₂³	—	106	2	no	—
399	C₂³	—	106	2	no	—
400	C₂³	—	106	2	no	—
401	C₂³	—	106	2	no	—
402	C₂³	—	212	4	no	—
403	C₂³	—	212	4	no	—
404	C₂³	—	212	4	no	—
405	C₂³	—	212	16	no	—
406	C₂³	C₂³×D₅₃	1	2	yes	—
407	C₂³	C₂³×D₅₃	1	2	yes	—
408	C₂³	C₂³×D₅₃	1	2	yes	—
409	C₂³	C₂³×D₅₃	1	1	yes	characteristic
410	D₄	—	4	4	no	—
411	D₄	—	4	4	no	—
412	D₄	—	4	4	no	—
413	D₄	—	212	4	no	—
414	D₄	—	212	4	no	—
415	D₄	—	212	4	no	—
416	D₄	—	212	4	no	—
417	D₄	—	212	4	no	—
418	D₄	—	212	4	no	—
419	D₄	—	212	4	no	—
420	D₄	—	212	4	no	—
421	D₄	—	212	4	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	4	4	no	—
427	C₂×C₄	—	53	2	no	—
428	C₂×C₄	—	53	2	no	—
429	C₂×C₄	—	53	2	no	—
430	C₂×C₄	—	53	2	no	—
431	C₂×C₄	—	53	2	no	—
432	C₂×C₄	—	106	2	no	—
433	C₂×C₄	—	106	2	no	—
434	C₂×C₄	—	106	2	no	—
435	C₂×C₄	—	106	2	no	—
436	C₂×C₄	—	106	2	no	—
437	C₂×C₄	—	212	4	no	—
438	C₂×C₄	—	212	4	no	—
439	C₂×C₄	C₂³×D₅₃	1	2	yes	—
440	C₂×C₄	C₂³×D₅₃	1	2	yes	—
441	C₂×C₄	C₂³×D₅₃	1	2	yes	—
442	C₂×C₄	C₂³×D₅₃	1	2	yes	—
443	C₂²	—	2	2	no	—
444	C₂²	—	2	2	no	—
445	C₂²	—	2	2	no	—
446	C₂²	—	2	2	no	—
447	C₂²	—	2	2	no	—
448	C₂²	—	2	2	no	—
449	C₂²	—	2	2	no	—
450	C₂²	—	2	2	no	—
451	C₂²	—	2	2	no	—
452	C₂²	—	4	4	no	—
453	C₂²	—	4	4	no	—
454	C₂²	—	4	4	no	—
455	C₂²	—	106	2	no	—
456	C₂²	—	106	2	no	—
457	C₂²	—	106	2	no	—
458	C₂²	—	106	2	no	—
459	C₂²	—	106	2	no	—
460	C₂²	—	106	2	no	—
461	C₂²	—	106	2	no	—
462	C₂²	—	106	2	no	—
463	C₂²	—	106	2	no	—
464	C₂²	—	212	16	no	—
465	C₂²	—	212	4	no	—
466	C₂²	—	212	4	no	—
467	C₂²	—	212	4	no	—
468	C₂²	#13133262	1	2	yes	—
469	C₂²	#13133262	1	2	yes	—
470	C₂²	#13133263	1	2	yes	—
471	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
472	C₄	—	2	2	no	—
473	C₄	—	2	2	no	—
474	C₄	—	2	2	no	—
475	C₄	—	2	2	no	—
476	C₄	—	106	2	no	—
477	C₄	—	106	2	no	—
478	C₄	—	106	2	no	—
479	C₄	—	106	2	no	—
480	C₄	—	106	2	no	—
481	C₂	—	4	2	no	—
482	C₂	—	4	2	no	—
483	C₂	—	4	2	no	—
484	C₂	—	212	2	no	—
485	C₂	—	212	2	no	—
486	C₂	—	212	2	no	—
487	C₂	#16811361	1	4	yes	direct product factor
488	C₂	C₂×#13133262	1	1	yes	characteristic
489	C₂	C₂×#13133262	1	1	yes	characteristic
490	C₂	C₂×#13133263	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×#560
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. #16811361	C₂	direct (N × H)
2. C₂×#13133213	C₂	semidirect
3. C₂×#13133212	C₂	semidirect
4. C₂×#13133210	C₂	semidirect
5. C₂×#13133202	C₂	semidirect
6. C₂×#13133201	C₂	semidirect
7. C₂×#13133200	C₂	semidirect
8. C₂×#13133191	C₂	semidirect
9. C₂×#13133190	C₂	semidirect
10. C₂×#13133189	C₂	semidirect
11. C₂×#13133189	C₂	semidirect
12. C₂×#13133188	C₂	semidirect
13. C₂×#13133187	C₂	semidirect
14. C₂×#13133185	C₂	semidirect
15. C₂×#13133185	C₂	semidirect
16. C₂×#121×C₅₃	C₂	semidirect
25. C₂²×C₂²⋊D₅₃	C₂²	semidirect
26. C₂²×C₂²⋊D₅₃	C₂²	semidirect
27. C₂³×Dic₅₃	C₂²	semidirect
28. C₂²×C₄×D₅₃	C₂²	semidirect
29. #13133213	C₂²	semidirect
30. #13133212	C₂²	semidirect
31. #13133210	C₂²	semidirect
32. #13133202	C₂²	semidirect
33. #13133201	C₂²	semidirect
34. #13133200	C₂²	semidirect
35. #13133191	C₂²	semidirect
36. #13133190	C₂²	semidirect
37. #13133189	C₂²	semidirect
38. #13133189	C₂²	semidirect
39. #13133188	C₂²	semidirect
40. #13133187	C₂²	semidirect
41. #13133185	C₂²	semidirect
42. #13133185	C₂²	semidirect
43. C₂×C₄⋊Dic₅₃	C₂²	semidirect
44. C₂×C₄⋊Dic₅₃	C₂²	semidirect
45. C₂×#11719919	C₂²	semidirect
46. C₂×#11719919	C₂²	semidirect
47. C₂×#11719919	C₂²	semidirect
50. C₂×C₂²⋊Dic₅₃	C₂²	semidirect
57. C₂×#11719921	C₂²	semidirect
58. C₂×C₄×Dic₅₃	C₂²	semidirect
61. C₂²×D₄×C₅₃	C₂²	semidirect
62. C₂³×C₄×C₅₃	C₂²	semidirect
63. #121×C₅₃	C₂²	semidirect
64. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
66. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
82. C₂×D₄×C₅₃	C₂³	semidirect
88. C₂²×C₄×C₅₃	C₂³	semidirect
101. C₂×C₂²⋊D₅₃	C₂³	semidirect
102. C₂×C₂²⋊D₅₃	C₂³	semidirect
109. C₂²×Dic₅₃	C₂³	semidirect
115. C₂×C₄×D₅₃	C₂³	semidirect
116. C₄⋊C₄×C₅₃	C₂³	semidirect
118. C₂²⋊C₄×C₅₃	C₂³	semidirect
121. C₂²⋊Dic₅₃	C₂³	semidirect
128. #11719921	C₂³	semidirect
129. C₄⋊Dic₅₃	C₂³	semidirect
130. C₄⋊Dic₅₃	C₂³	semidirect
131. #11719919	C₂³	semidirect
132. #11719919	C₂³	semidirect
133. #11719919	C₂³	semidirect
134. C₄×Dic₅₃	C₂³	semidirect
239. C₂×C₅₃	#560	semidirect
258. C₂×#121	D₅₃	semidirect
262. C₅₃	C₂×#560	semidirect
275. C₂²×D₄	C₂×D₅₃	semidirect
278. C₂³×C₄	C₂×D₅₃	semidirect
290. #121	C₂×D₅₃	semidirect
299. C₂×C₄⋊C₄	C₂×D₅₃	semidirect
311. C₂×C₂²⋊C₄	C₂×D₅₃	semidirect
346. C₂×D₄	C₂²×D₅₃	semidirect
361. C₂²×C₄	C₂²×D₅₃	semidirect
367. C₄⋊C₄	C₂²×D₅₃	semidirect
379. C₂²⋊C₄	C₂²×D₅₃	semidirect
487. C₂	#16811361	direct (N × H)

Subgroups — C2×#16811361 #20288830

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811361 #20288830