Subgroups — C₂×#20274838 #27384264

Order 13568.

786 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133153	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810578	C₂	1	1	yes	maximal, characteristic
3	#20274838	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810393	C₂	1	2	yes	maximal
5	C₂×#16810379	C₂	1	2	yes	maximal
6	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₂²	1	2	yes	—
8	C₂²×#11719917	C₂²	1	1	yes	characteristic
9	C₂×#13133176	C₂²	1	1	yes	characteristic
10	C₂×#13133153	—	2	1	no	—
11	C₂×#13133153	—	2	1	no	—
12	C₂×#13133153	—	2	2	no	—
13	C₂×#13133153	C₂²	1	4	yes	—
14	#16810578	C₄	1	4	yes	complemented
15	#16810578	C₂²	1	4	yes	—
16	C₂×#13133127	C₄	1	2	yes	complemented
17	C₂×C₈×Dic₅₃	C₄	1	2	yes	complemented
18	#16810393	C₂²	1	8	yes	—
19	#16810379	C₂²	1	8	yes	—
20	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#117×C₅₃	C₂²	1	2	yes	—
22	C₅₃×#427	—	2	2	no	—
23	C₅₃×#427	C₂²	1	4	yes	—
24	C₂×#107×C₅₃	—	2	1	no	—
25	C₂×#106×C₅₃	—	2	1	no	—
26	C₂×#11719917	—	2	2	no	—
27	C₂×#11719917	—	2	2	no	—
28	C₂×#11719917	—	2	1	no	—
29	C₂×#11719917	—	2	1	no	—
30	C₂×#11719917	—	2	1	no	—
31	C₂×#11719917	—	2	1	no	—
32	C₂×#11719917	—	2	2	no	—
33	C₂×#11719917	—	2	2	no	—
34	C₂×#11719917	—	2	2	no	—
35	C₂×#11719917	C₂×C₄	1	2	yes	—
36	C₂×#11719917	C₂³	1	4	yes	—
37	C₂×#11719917	C₂³	1	2	yes	—
38	C₂²×#88322	—	2	2	no	—
39	C₂²×#88322	C₂×C₄	1	2	yes	—
40	#13133176	—	2	1	no	—
41	#13133176	—	2	1	no	—
42	#13133176	C₂×C₄	1	2	yes	—
43	#13133176	C₂³	1	2	yes	—
44	C₂×C₄⋊Dic₅₃	—	2	1	no	—
45	C₂×C₂²⋊Dic₅₃	—	2	1	no	—
46	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
47	#13133153	—	2	4	no	—
48	#13133153	—	2	4	no	—
49	#13133127	C₂×C₄	1	8	yes	complemented
50	C₈×Dic₅₃	C₂×C₄	1	8	yes	complemented
51	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
55	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₅₃	D₄	1	2	yes	—
58	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
59	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
61	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
62	#117×C₅₃	—	2	2	no	—
63	#117×C₅₃	—	2	2	no	—
64	#117×C₅₃	—	2	2	no	—
65	#117×C₅₃	C₂³	1	4	yes	—
66	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
68	C₂×C₄²×C₅₃	—	2	2	no	—
69	#107×C₅₃	—	2	4	no	—
70	#106×C₅₃	—	2	4	no	—
71	C₂⁴×C₅₃	#41	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	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
87	C₂²×Dic₅₃	—	2	2	no	—
88	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
91	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
94	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
95	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
96	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
97	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
98	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
103	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
104	C₄⋊C₄×C₅₃	—	2	4	no	—
105	C₄⋊C₄×C₅₃	—	4	2	no	—
106	C₂²⋊C₄×C₅₃	—	2	4	no	—
107	C₂²⋊C₄×C₅₃	—	4	2	no	—
108	C₄²×C₅₃	—	2	4	no	—
109	C₄²×C₅₃	—	4	2	no	—
110	C₂²⋊Dic₅₃	—	2	2	no	—
111	C₂²⋊Dic₅₃	—	2	2	no	—
112	C₄⋊Dic₅₃	—	2	2	no	—
113	C₄⋊Dic₅₃	—	2	2	no	—
114	C₄×Dic₅₃	—	2	2	no	—
115	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
116	#11719917	—	2	2	no	—
117	#11719917	—	2	2	no	—
118	#11719917	—	2	2	no	—
119	#11719917	—	2	2	no	—
120	#11719917	—	2	4	no	—
121	#11719917	—	2	4	no	—
122	#11719917	—	4	2	no	—
123	#11719917	—	4	2	no	—
124	C₂×#88322	—	2	2	no	—
125	C₂×#88322	—	2	2	no	—
126	C₂×#88322	—	2	4	no	—
127	C₂×#88322	—	4	2	no	—
128	C₂×#88322	—	4	2	no	—
129	C₂×#88322	C₂²×C₄	1	4	yes	—
130	C₂³×C₅₃	—	4	2	no	—
131	C₂³×C₅₃	#117	1	1	yes	characteristic
132	C₂³×C₅₃	#117	1	1	yes	characteristic
133	C₂³×C₅₃	#124	1	1	yes	characteristic
134	C₂³×C₅₃	#124	1	1	yes	characteristic
135	C₂³×C₅₃	#125	1	1	yes	characteristic
136	C₂³×C₅₃	#125	1	1	yes	characteristic
137	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₅₃	—	4	4	no	—
139	C₂×C₄×C₅₃	—	8	4	no	—
140	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
145	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
146	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
147	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
148	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
149	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
150	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
151	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
152	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
153	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
154	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
155	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
156	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
157	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
158	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
159	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
160	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
161	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
168	C₂×Dic₅₃	—	2	4	no	—
169	C₂×Dic₅₃	—	4	4	no	—
170	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
171	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
172	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
173	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
174	#88322	—	2	4	no	—
175	#88322	—	4	4	no	—
176	C₂×#1833	—	53	1	no	maximal, Sylow
177	C₂²×C₅₃	—	2	2	no	—
178	C₂²×C₅₃	—	2	2	no	—
179	C₂²×C₅₃	—	2	2	no	—
180	C₂²×C₅₃	—	4	1	no	—
181	C₂²×C₅₃	—	4	1	no	—
182	C₂²×C₅₃	—	4	1	no	—
183	C₂²×C₅₃	—	4	1	no	—
184	C₂²×C₅₃	#382	1	1	yes	characteristic
185	C₂²×C₅₃	#382	1	1	yes	characteristic
186	C₂²×C₅₃	#382	1	1	yes	characteristic
187	C₂²×C₅₃	#382	1	1	yes	characteristic
188	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
189	C₂²×C₅₃	C₂×#124	1	1	yes	characteristic
190	C₂²×C₅₃	C₂×#125	1	1	yes	characteristic
191	C₄×C₅₃	—	8	4	no	—
192	C₄×C₅₃	#382	1	1	yes	characteristic
193	C₄×C₅₃	#382	1	1	yes	characteristic
194	C₄×C₅₃	#382	1	1	yes	characteristic
195	C₄×C₅₃	#382	1	1	yes	characteristic
196	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
197	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
198	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
199	C₄×C₅₃	C₂×#125	1	1	yes	characteristic
200	Dic₅₃	—	4	4	no	—
201	C₂²×#108	—	53	1	no	—
202	C₂×#427	D₅₃	1	1	yes	characteristic
203	C₂×#405	—	53	1	no	—
204	#1833	—	53	8	no	—
205	C₂×#344	—	53	2	no	—
206	C₂×H(3,ℤ/4)	—	53	2	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	—	2	1	no	—
213	C₂×C₅₃	#1833	1	2	yes	complemented
214	C₂×C₅₃	C₂×#382	1	1	yes	characteristic
215	C₂²×M₁₆	—	53	1	no	—
216	C₂²×M₁₆	—	53	2	no	—
217	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
218	C₂×#117	—	53	1	no	—
219	C₂×#117	C₂×D₅₃	1	2	yes	—
220	C₂×#108	—	53	4	no	—
221	C₂×#108	—	106	1	no	—
222	C₂×#108	—	106	1	no	—
223	C₂×#108	—	106	2	no	—
224	#427	—	2	2	no	—
225	#427	C₂×D₅₃	1	4	yes	—
226	C₂×#107	—	2	1	no	—
227	C₂×#106	—	2	1	no	—
228	#405	—	53	4	no	—
229	#405	—	53	4	no	—
230	C₂×#97	—	53	2	no	—
231	C₂×C₄×C₈	—	53	2	no	—
232	#344	—	53	8	no	—
233	H(3,ℤ/4)	—	53	8	no	—
234	C₅₃	C₂×#1833	1	1	yes	characteristic, complemented, Sylow
235	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
236	C₂×M₁₆	—	53	2	no	—
237	C₂×M₁₆	—	53	2	no	—
238	C₂×M₁₆	—	53	4	no	—
239	C₂×M₁₆	—	106	1	no	—
240	C₂×M₁₆	—	106	1	no	—
241	C₂×M₁₆	—	106	1	no	—
242	C₂×M₁₆	—	106	1	no	—
243	C₂×M₁₆	—	106	2	no	—
244	C₂×M₁₆	—	106	2	no	—
245	C₂×M₁₆	—	106	2	no	—
246	C₂×M₁₆	—	106	2	no	—
247	C₂×M₁₆	—	106	2	no	—
248	C₂²×C₈	—	53	2	no	—
249	C₂²×C₈	—	106	2	no	—
250	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
251	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
252	C₂²×C₈	Dic₁₀₆	1	2	yes	—
253	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
254	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
255	C₂²×C₈	C₄×D₅₃	1	2	yes	—
256	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
257	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
258	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
259	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
260	#117	—	2	2	no	—
261	#117	—	2	2	no	—
262	#117	—	2	2	no	—
263	#117	—	53	2	no	—
264	#117	—	53	2	no	—
265	#117	—	106	1	no	—
266	#117	—	106	1	no	—
267	#117	C₂²×D₅₃	1	4	yes	—
268	C₂×C₄⋊C₄	—	2	2	no	—
269	C₂×C₄⋊C₄	—	106	1	no	—
270	C₂×C₂²⋊C₄	—	2	2	no	—
271	C₂×C₂²⋊C₄	—	106	1	no	—
272	C₂×C₄²	—	2	2	no	—
273	C₂×C₄²	—	53	2	no	—
274	#108	—	106	4	no	—
275	#108	—	106	4	no	—
276	#107	—	2	4	no	—
277	#106	—	2	4	no	—
278	#97	—	53	8	no	—
279	C₄×C₈	—	53	8	no	—
280	C₂⁴	#11719947	1	1	yes	characteristic
281	C₂²×C₄	—	4	2	no	—
282	C₂²×C₄	—	106	2	no	—
283	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
284	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
285	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
286	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
287	C₂²×C₄	#11719947	1	1	yes	characteristic
288	C₂²×C₄	#11719947	1	1	yes	characteristic
289	C₂²×C₄	#11719947	1	1	yes	characteristic
290	C₂²×C₄	#11719949	1	1	yes	characteristic
291	C₂²×C₄	#11719949	1	1	yes	characteristic
292	C₂²×C₄	#11719949	1	1	yes	characteristic
293	C₂²×C₄	#11719949	1	1	yes	characteristic
294	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
295	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
296	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
297	M₁₆	—	106	2	no	—
298	M₁₆	—	106	2	no	—
299	M₁₆	—	106	2	no	—
300	M₁₆	—	106	2	no	—
301	M₁₆	—	106	4	no	—
302	M₁₆	—	106	4	no	—
303	M₁₆	—	212	2	no	—
304	M₁₆	—	212	2	no	—
305	C₂×C₈	—	53	4	no	—
306	C₂×C₈	—	106	2	no	—
307	C₂×C₈	—	106	2	no	—
308	C₂×C₈	—	106	4	no	—
309	C₂×C₈	—	212	2	no	—
310	C₂×C₈	—	212	2	no	—
311	C₂×C₈	#11719919	1	1	yes	characteristic
312	C₂×C₈	#11719919	1	1	yes	characteristic
313	C₂×C₈	#11719919	1	2	yes	—
314	C₂×C₈	#11719919	1	1	yes	characteristic
315	C₂×C₈	#11719919	1	1	yes	characteristic
316	C₂×C₈	#11719919	1	2	yes	—
317	C₂×C₈	#11719919	1	2	yes	—
318	C₂×C₈	#11719919	1	2	yes	—
319	C₂×C₈	#11719919	1	2	yes	—
320	C₂×C₈	#11719919	1	2	yes	—
321	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
322	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
323	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
324	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
325	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
326	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
327	C₄⋊C₄	—	2	4	no	—
328	C₄⋊C₄	—	4	2	no	—
329	C₄⋊C₄	—	106	2	no	—
330	C₄⋊C₄	—	106	2	no	—
331	C₂²⋊C₄	—	2	4	no	—
332	C₂²⋊C₄	—	4	2	no	—
333	C₂²⋊C₄	—	106	2	no	—
334	C₂²⋊C₄	—	106	2	no	—
335	C₄²	—	2	4	no	—
336	C₄²	—	4	2	no	—
337	C₄²	—	53	4	no	—
338	C₄²	—	106	2	no	—
339	C₂³	—	4	2	no	—
340	C₂³	#13133175	1	1	yes	characteristic
341	C₂³	#13133175	1	1	yes	characteristic
342	C₂³	#13133204	1	1	yes	characteristic
343	C₂³	#13133204	1	1	yes	characteristic
344	C₂³	#13133208	1	1	yes	characteristic
345	C₂³	#13133208	1	1	yes	characteristic
346	C₂³	C₂×#11719947	1	1	yes	characteristic
347	C₂×C₄	—	4	4	no	—
348	C₂×C₄	—	8	4	no	—
349	C₂×C₄	—	106	4	no	—
350	C₂×C₄	—	212	4	no	—
351	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
352	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
353	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
354	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
355	C₂×C₄	#13133175	1	1	yes	characteristic
356	C₂×C₄	#13133175	1	1	yes	characteristic
357	C₂×C₄	#13133175	1	1	yes	characteristic
358	C₂×C₄	#13133175	1	1	yes	characteristic
359	C₂×C₄	#13133175	1	1	yes	characteristic
360	C₂×C₄	#13133175	1	1	yes	characteristic
361	C₂×C₄	#13133204	1	1	yes	characteristic
362	C₂×C₄	#13133204	1	1	yes	characteristic
363	C₂×C₄	#13133206	1	1	yes	characteristic
364	C₂×C₄	#13133206	1	1	yes	characteristic
365	C₂×C₄	#13133206	1	1	yes	characteristic
366	C₂×C₄	#13133206	1	1	yes	characteristic
367	C₂×C₄	#13133208	1	1	yes	characteristic
368	C₂×C₄	#13133208	1	1	yes	characteristic
369	C₂×C₄	#13133208	1	1	yes	characteristic
370	C₂×C₄	#13133208	1	1	yes	characteristic
371	C₂×C₄	#13133208	1	1	yes	characteristic
372	C₂×C₄	#13133208	1	1	yes	characteristic
373	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
374	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
375	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
376	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
377	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
378	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
379	C₈	—	106	4	no	—
380	C₈	—	212	4	no	—
381	C₈	C₂×#11719919	1	2	yes	—
382	C₈	C₂×#11719919	1	2	yes	—
383	C₈	C₂×#11719919	1	2	yes	—
384	C₈	C₂×#11719919	1	2	yes	—
385	C₂²	—	2	2	no	—
386	C₂²	—	2	2	no	—
387	C₂²	—	2	2	no	—
388	C₂²	—	4	1	no	—
389	C₂²	—	4	1	no	—
390	C₂²	—	4	1	no	—
391	C₂²	—	4	1	no	—
392	C₂²	#16810488	1	1	yes	characteristic
393	C₂²	#16810488	1	1	yes	characteristic
394	C₂²	#16810488	1	1	yes	characteristic
395	C₂²	#16810488	1	1	yes	characteristic
396	C₂²	C₂×#13133175	1	1	yes	characteristic
397	C₂²	C₂×#13133204	1	1	yes	characteristic
398	C₂²	C₂×#13133208	1	1	yes	characteristic
399	C₄	—	8	4	no	—
400	C₄	—	212	4	no	—
401	C₄	#16810488	1	1	yes	characteristic
402	C₄	#16810488	1	1	yes	characteristic
403	C₄	#16810488	1	1	yes	characteristic
404	C₄	#16810488	1	1	yes	characteristic
405	C₄	C₂²×#11719919	1	1	yes	characteristic
406	C₄	C₂×#13133175	1	1	yes	characteristic
407	C₄	C₂×#13133206	1	1	yes	characteristic
408	C₄	C₂×#13133208	1	1	yes	characteristic
409	C₂	—	2	1	no	—
410	C₂	—	2	1	no	—
411	C₂	—	2	1	no	—
412	C₂	—	2	1	no	—
413	C₂	—	2	1	no	—
414	C₂	—	2	1	no	—
415	C₂	#20274838	1	2	yes	direct product factor
416	C₂	C₂×#16810488	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133175
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#427	C₂
socle	C₂²×C₅₃	#382
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#427	D₅₃
hypocenter	C₅₃	C₂×#1833
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
3. #20274838	C₂	direct (N × H)
14. #16810578	C₄	semidirect
16. C₂×#13133127	C₄	semidirect
17. C₂×C₈×Dic₅₃	C₄	semidirect
49. #13133127	C₂×C₄	semidirect
50. C₈×Dic₅₃	C₂×C₄	semidirect
213. C₂×C₅₃	#1833	semidirect
234. C₅₃	C₂×#1833	semidirect
415. C₂	#20274838	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×#13133153	2
C₂×#16810578	2
#20274838	2
C₂×#16810393	2
C₂×#16810379	2
C₂×C₅₃×#427	2
C₂×#1833	53

Subgroups — C2×#20274838 #27384264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274838 #27384264