Subgroups — C₂×#20272816 #27299000

Order 13568.

586 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133157	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810661	C₂	1	1	yes	maximal, characteristic
3	#20272816	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133176	—	2	1	no	—
6	C₂×#13133157	—	2	1	no	—
7	C₂×#13133157	—	2	1	no	—
8	C₂×#13133157	—	2	1	no	—
9	C₂×#13133157	—	2	1	no	—
10	C₂×#13133157	C₄	1	2	yes	complemented
11	C₂×#13133157	C₂²	1	2	yes	—
12	C₂²×#11719908	C₄	1	2	yes	complemented
13	#16810661	—	2	1	no	—
14	#16810661	—	2	1	no	—
15	#16810661	C₄	1	2	yes	complemented
16	#16810661	C₂²	1	2	yes	—
17	C₂×#13133151	—	2	1	no	—
18	C₂×#13133149	C₄	1	2	yes	complemented
19	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#117×C₅₃	—	2	1	no	—
21	C₅₃×#427	—	2	1	no	—
22	C₅₃×#427	—	2	1	no	—
23	C₅₃×#427	C₄	1	2	yes	complemented
24	C₅₃×#427	C₂²	1	2	yes	—
25	C₂×#107×C₅₃	—	2	1	no	—
26	C₂×#106×C₅₃	C₄	1	2	yes	complemented
27	#13133176	—	2	2	no	—
28	#13133176	—	2	2	no	—
29	#13133176	—	4	1	no	—
30	#13133176	—	4	1	no	—
31	C₂×C₄⋊Dic₅₃	—	2	2	no	—
32	C₂×C₂²⋊Dic₅₃	—	4	1	no	—
33	C₂×C₄×Dic₅₃	—	4	1	no	—
34	#13133157	—	2	2	no	—
35	#13133157	—	2	2	no	—
36	#13133157	—	2	2	no	—
37	#13133157	—	2	2	no	—
38	C₂×#11719908	—	2	2	no	—
39	C₂×#11719908	—	2	2	no	—
40	C₂×#11719908	C₂×C₄	1	4	yes	complemented
41	#13133151	—	2	2	no	—
42	#13133151	—	2	2	no	—
43	#13133149	—	2	2	no	—
44	#13133149	C₂×C₄	1	4	yes	complemented
45	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
58	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
59	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
60	#117×C₅₃	—	2	2	no	—
61	#117×C₅₃	—	2	2	no	—
62	#117×C₅₃	—	4	1	no	—
63	#117×C₅₃	—	4	1	no	—
64	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
66	C₂×C₄²×C₅₃	—	4	1	no	—
67	#107×C₅₃	—	2	2	no	—
68	#107×C₅₃	—	2	2	no	—
69	#106×C₅₃	—	2	2	no	—
70	#106×C₅₃	C₂×C₄	1	4	yes	complemented
71	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	—	4	2	no	—
73	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×Dic₅₃	—	4	2	no	—
88	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
116	C₄⋊C₄×C₅₃	—	2	4	no	—
117	C₄⋊C₄×C₅₃	—	4	2	no	—
118	C₂²⋊C₄×C₅₃	—	4	2	no	—
119	C₂²⋊C₄×C₅₃	—	4	2	no	—
120	C₄²×C₅₃	—	4	2	no	—
121	C₄²×C₅₃	—	4	2	no	—
122	C₂²⋊Dic₅₃	—	4	2	no	—
123	C₂²⋊Dic₅₃	—	4	2	no	—
124	C₄⋊Dic₅₃	—	2	4	no	—
125	C₄⋊Dic₅₃	—	4	2	no	—
126	C₄×Dic₅₃	—	4	2	no	—
127	C₄×Dic₅₃	—	4	2	no	—
128	#11719908	—	2	4	no	—
129	C₂³×C₅₃	—	4	2	no	—
130	C₂³×C₅₃	#102	1	1	yes	characteristic
131	C₂³×C₅₃	#102	1	1	yes	characteristic
132	C₂³×C₅₃	#102	1	1	yes	characteristic
133	C₂³×C₅₃	#102	1	1	yes	characteristic
134	C₂³×C₅₃	#107	1	1	yes	characteristic
135	C₂³×C₅₃	#107	1	1	yes	characteristic
136	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₅₃	—	4	4	no	—
138	C₂×C₄×C₅₃	—	8	4	no	—
139	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
140	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
141	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
142	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
143	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
144	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
145	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
146	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
147	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
148	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
149	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
150	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
151	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
152	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
153	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
154	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
155	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
156	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
157	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
158	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
159	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
160	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
167	C₂×Dic₅₃	—	4	4	no	—
168	C₂×Dic₅₃	—	8	4	no	—
169	C₈×C₅₃	#95	1	1	yes	characteristic
170	C₈×C₅₃	#95	1	1	yes	characteristic
171	C₈×C₅₃	#95	1	1	yes	characteristic
172	C₈×C₅₃	#95	1	1	yes	characteristic
173	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
174	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
175	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂×#1384	—	53	1	no	maximal, Sylow
178	C₂²×C₅₃	—	2	2	no	—
179	C₂²×C₅₃	—	2	2	no	—
180	C₂²×C₅₃	—	2	2	no	—
181	C₂²×C₅₃	—	4	1	no	—
182	C₂²×C₅₃	—	4	1	no	—
183	C₂²×C₅₃	—	4	1	no	—
184	C₂²×C₅₃	—	4	1	no	—
185	C₂²×C₅₃	#340	1	1	yes	characteristic
186	C₂²×C₅₃	#340	1	1	yes	characteristic
187	C₂²×C₅₃	#340	1	1	yes	characteristic
188	C₂²×C₅₃	#340	1	1	yes	characteristic
189	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
190	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
191	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
192	C₄×C₅₃	—	8	4	no	—
193	C₄×C₅₃	#340	1	1	yes	characteristic
194	C₄×C₅₃	#340	1	1	yes	characteristic
195	C₄×C₅₃	#340	1	1	yes	characteristic
196	C₄×C₅₃	#340	1	1	yes	characteristic
197	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
198	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
199	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
200	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
201	Dic₅₃	—	8	4	no	—
202	C₂²×M₃₂	—	53	1	no	—
203	C₂×#427	—	53	1	no	—
204	C₂×#427	D₅₃	1	1	yes	characteristic
205	#1384	—	53	4	no	—
206	C₂×C₅₃	—	2	1	no	—
207	C₂×C₅₃	—	2	1	no	—
208	C₂×C₅₃	—	2	1	no	—
209	C₂×C₅₃	—	2	1	no	—
210	C₂×C₅₃	—	2	1	no	—
211	C₂×C₅₃	—	2	1	no	—
212	C₂×C₅₃	#1384	1	2	yes	complemented
213	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
214	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
215	C₂×#117	—	2	1	no	—
216	C₂×#117	—	106	1	no	—
217	C₂×M₃₂	—	53	2	no	—
218	C₂×M₃₂	—	53	2	no	—
219	C₂×M₃₂	—	106	1	no	—
220	C₂×M₃₂	—	106	1	no	—
221	C₂×M₃₂	—	106	1	no	—
222	C₂×M₃₂	—	106	1	no	—
223	C₂²×C₁₆	—	53	2	no	—
224	#427	—	2	1	no	—
225	#427	—	2	1	no	—
226	#427	—	53	2	no	—
227	#427	—	53	2	no	—
228	#427	—	106	1	no	—
229	#427	—	106	1	no	—
230	#427	Dic₅₃	1	2	yes	complemented
231	#427	C₂×D₅₃	1	2	yes	—
232	C₂×#107	—	2	1	no	—
233	C₂×#107	—	106	1	no	—
234	C₂×#106	—	53	2	no	—
235	C₂×#106	Dic₅₃	1	2	yes	complemented
236	C₅₃	C₂×#1384	1	1	yes	characteristic, complemented, Sylow
237	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
238	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
239	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
240	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
241	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
242	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
243	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
244	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
245	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
246	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
247	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
248	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
249	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
250	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
251	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
252	#117	—	2	2	no	—
253	#117	—	2	2	no	—
254	#117	—	4	1	no	—
255	#117	—	4	1	no	—
256	#117	—	106	2	no	—
257	#117	—	106	2	no	—
258	#117	—	212	1	no	—
259	#117	—	212	1	no	—
260	C₂×C₄⋊C₄	—	2	2	no	—
261	C₂×C₄⋊C₄	—	106	2	no	—
262	C₂×C₂²⋊C₄	—	4	1	no	—
263	C₂×C₂²⋊C₄	—	212	1	no	—
264	C₂×C₄²	—	4	1	no	—
265	C₂×C₄²	—	212	1	no	—
266	M₃₂	—	106	2	no	—
267	M₃₂	—	106	2	no	—
268	M₃₂	—	106	2	no	—
269	M₃₂	—	106	2	no	—
270	C₂×C₁₆	—	53	4	no	—
271	C₂×C₁₆	—	106	2	no	—
272	C₂×C₁₆	—	106	2	no	—
273	#107	—	2	2	no	—
274	#107	—	2	2	no	—
275	#107	—	106	2	no	—
276	#107	—	106	2	no	—
277	#106	—	2	2	no	—
278	#106	—	53	4	no	—
279	#106	—	106	2	no	—
280	#106	C₂×Dic₅₃	1	4	yes	complemented
281	C₂⁴	#11719922	1	1	yes	characteristic
282	C₂²×C₄	—	4	2	no	—
283	C₂²×C₄	—	212	2	no	—
284	C₂²×C₄	#11719919	1	1	yes	characteristic
285	C₂²×C₄	#11719919	1	1	yes	characteristic
286	C₂²×C₄	#11719921	1	1	yes	characteristic
287	C₂²×C₄	#11719921	1	1	yes	characteristic
288	C₂²×C₄	#11719922	1	1	yes	characteristic
289	C₂²×C₄	#11719923	1	1	yes	characteristic
290	C₂²×C₄	#11719923	1	1	yes	characteristic
291	C₂²×C₄	#11719924	1	1	yes	characteristic
292	C₂²×C₄	#11719924	1	1	yes	characteristic
293	C₂²×C₄	#11719925	1	1	yes	characteristic
294	C₂²×C₄	#11719925	1	1	yes	characteristic
295	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
296	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
297	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
298	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
299	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
300	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
301	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
302	C₂×C₈	#11719919	1	1	yes	characteristic
303	C₂×C₈	#11719919	1	1	yes	characteristic
304	C₂×C₈	#11719919	1	1	yes	characteristic
305	C₂×C₈	#11719919	1	1	yes	characteristic
306	C₂×C₈	#11719919	1	1	yes	characteristic
307	C₂×C₈	#11719919	1	1	yes	characteristic
308	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
309	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
310	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
311	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
312	C₂×C₈	#11719921	1	1	yes	characteristic
313	C₂×C₈	#11719921	1	1	yes	characteristic
314	C₂×C₈	#11719921	1	1	yes	characteristic
315	C₂×C₈	#11719921	1	1	yes	characteristic
316	C₂×C₈	#11719921	1	1	yes	characteristic
317	C₂×C₈	#11719921	1	1	yes	characteristic
318	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
319	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
320	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
321	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
322	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
323	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
324	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
325	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
326	C₄⋊C₄	—	2	4	no	—
327	C₄⋊C₄	—	4	2	no	—
328	C₄⋊C₄	—	106	4	no	—
329	C₄⋊C₄	—	212	2	no	—
330	C₂²⋊C₄	—	4	2	no	—
331	C₂²⋊C₄	—	4	2	no	—
332	C₂²⋊C₄	—	212	2	no	—
333	C₂²⋊C₄	—	212	2	no	—
334	C₄²	—	4	2	no	—
335	C₄²	—	4	2	no	—
336	C₄²	—	212	2	no	—
337	C₄²	—	212	2	no	—
338	C₁₆	—	106	4	no	—
339	C₂³	—	4	2	no	—
340	C₂³	#13133137	1	1	yes	characteristic
341	C₂³	#13133137	1	1	yes	characteristic
342	C₂³	#13133138	1	1	yes	characteristic
343	C₂³	#13133138	1	1	yes	characteristic
344	C₂³	#13133150	1	1	yes	characteristic
345	C₂³	#13133150	1	1	yes	characteristic
346	C₂³	C₂×#11719922	1	1	yes	characteristic
347	C₂×C₄	—	4	4	no	—
348	C₂×C₄	—	8	4	no	—
349	C₂×C₄	—	424	4	no	—
350	C₂×C₄	—	212	4	no	—
351	C₂×C₄	#13133124	1	1	yes	characteristic
352	C₂×C₄	#13133124	1	1	yes	characteristic
353	C₂×C₄	#13133124	1	1	yes	characteristic
354	C₂×C₄	#13133124	1	1	yes	characteristic
355	C₂×C₄	#13133137	1	1	yes	characteristic
356	C₂×C₄	#13133137	1	1	yes	characteristic
357	C₂×C₄	#13133138	1	1	yes	characteristic
358	C₂×C₄	#13133138	1	1	yes	characteristic
359	C₂×C₄	#13133140	1	1	yes	characteristic
360	C₂×C₄	#13133140	1	1	yes	characteristic
361	C₂×C₄	#13133140	1	1	yes	characteristic
362	C₂×C₄	#13133140	1	1	yes	characteristic
363	C₂×C₄	#13133141	1	1	yes	characteristic
364	C₂×C₄	#13133141	1	1	yes	characteristic
365	C₂×C₄	#13133141	1	1	yes	characteristic
366	C₂×C₄	#13133141	1	1	yes	characteristic
367	C₂×C₄	#13133148	1	1	yes	characteristic
368	C₂×C₄	#13133148	1	1	yes	characteristic
369	C₂×C₄	#13133148	1	1	yes	characteristic
370	C₂×C₄	#13133148	1	1	yes	characteristic
371	C₂×C₄	#13133150	1	1	yes	characteristic
372	C₂×C₄	#13133150	1	1	yes	characteristic
373	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
374	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
375	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
376	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
377	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
378	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
379	C₈	#13133124	1	1	yes	characteristic
380	C₈	#13133124	1	1	yes	characteristic
381	C₈	#13133124	1	1	yes	characteristic
382	C₈	#13133124	1	1	yes	characteristic
383	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
384	C₈	C₂×#11719921	1	1	yes	characteristic
385	C₈	C₂×#11719919	1	1	yes	characteristic
386	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
387	C₂²	—	2	2	no	—
388	C₂²	—	2	2	no	—
389	C₂²	—	2	2	no	—
390	C₂²	—	4	1	no	—
391	C₂²	—	4	1	no	—
392	C₂²	—	4	1	no	—
393	C₂²	—	4	1	no	—
394	C₂²	#16810385	1	1	yes	characteristic
395	C₂²	#16810385	1	1	yes	characteristic
396	C₂²	#16810385	1	1	yes	characteristic
397	C₂²	#16810385	1	1	yes	characteristic
398	C₂²	C₂×#13133137	1	1	yes	characteristic
399	C₂²	C₂×#13133138	1	1	yes	characteristic
400	C₂²	C₂×#13133150	1	1	yes	characteristic
401	C₄	—	8	4	no	—
402	C₄	—	424	4	no	—
403	C₄	#16810385	1	1	yes	characteristic
404	C₄	#16810385	1	1	yes	characteristic
405	C₄	#16810385	1	1	yes	characteristic
406	C₄	#16810385	1	1	yes	characteristic
407	C₄	C₂×#13133124	1	1	yes	characteristic
408	C₄	C₂×#13133140	1	1	yes	characteristic
409	C₄	C₂×#13133141	1	1	yes	characteristic
410	C₄	C₂×#13133148	1	1	yes	characteristic
411	C₂	—	2	1	no	—
412	C₂	—	2	1	no	—
413	C₂	—	2	1	no	—
414	C₂	—	2	1	no	—
415	C₂	—	2	1	no	—
416	C₂	—	2	1	no	—
417	C₂	#20272816	1	2	yes	direct product factor
418	C₂	C₂×#16810385	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133148
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#427	C₂
socle	C₂²×C₅₃	#340
prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#427	D₅₃
hypocenter	C₅₃	C₂×#1384
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
3. #20272816	C₂	direct (N × H)
10. C₂×#13133157	C₄	semidirect
12. C₂²×#11719908	C₄	semidirect
15. #16810661	C₄	semidirect
18. C₂×#13133149	C₄	semidirect
23. C₅₃×#427	C₄	semidirect
26. C₂×#106×C₅₃	C₄	semidirect
40. C₂×#11719908	C₂×C₄	semidirect
44. #13133149	C₂×C₄	semidirect
70. #106×C₅₃	C₂×C₄	semidirect
212. C₂×C₅₃	#1384	semidirect
230. #427	Dic₅₃	semidirect
235. C₂×#106	Dic₅₃	semidirect
236. C₅₃	C₂×#1384	semidirect
280. #106	C₂×Dic₅₃	semidirect
417. C₂	#20272816	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×#13133157	2
C₂×#16810661	2
#20272816	2
C₂×C₅₃×#427	2
C₂×#1384	53

Subgroups — C2×#20272816 #27299000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272816 #27299000