Subgroups — C₈×#13133179 #27382402

Order 13568.

1422 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810688	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×D₅₃	C₂	1	2	yes	maximal, complemented
4	C₄×#13133179	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄⋊C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×#11719921	C₂	1	2	yes	maximal, complemented
8	#20274485	C₂	1	2	yes	maximal, complemented
9	#20274472	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₈×D₅₃	C₂²	1	2	yes	—
13	C₂²×C₈×D₅₃	C₂²	1	2	yes	—
14	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
15	C₂×#13133179	C₄	1	2	yes	—
16	C₂×C₄²×D₅₃	C₂²	1	2	yes	—
17	C₈×D₂₁₂	C₄	1	8	yes	complemented
18	#16810688	C₄	1	8	yes	complemented
19	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
20	C₂×#13133129	C₂²	1	2	yes	—
21	C₂×#13133129	C₂²	1	2	yes	—
22	C₂×C₄×#88322	C₂²	1	2	yes	—
23	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
24	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
26	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
29	C₄×C₈×D₅₃	—	2	8	no	—
30	C₄⋊C₄×Dic₅₃	C₄	1	2	yes	—
31	C₄×#11719921	C₂²	1	2	yes	—
32	#16810470	C₄	1	4	yes	—
33	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
34	#16810378	C₂²	1	2	yes	—
35	#16810377	C₂²	1	2	yes	—
36	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#336	C₂²	1	2	yes	—
41	C₂²×D₂₁₂	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×D₅₃	C₂×C₄	1	4	yes	—
43	C₂²×C₄×D₅₃	C₂³	1	2	yes	—
44	C₂²×#88322	C₂³	1	1	yes	characteristic
45	C₂²×#88322	C₂³	1	2	yes	—
46	C₂²×#88322	C₂³	1	1	yes	characteristic
47	C₂×C₈×D₅₃	—	2	4	no	—
48	C₂×C₈×D₅₃	—	2	4	no	—
49	C₂×C₈×D₅₃	—	2	4	no	—
50	C₂×C₈×D₅₃	—	2	4	no	—
51	C₂×C₈×D₅₃	C₂×C₄	1	8	yes	complemented
52	C₂×C₈×D₅₃	C₂×C₄	1	8	yes	complemented
53	C₄×D₂₁₂	C₂×C₄	1	4	yes	—
54	C₄×D₂₁₂	C₂×C₄	1	4	yes	—
55	#13133179	C₈	1	16	yes	direct product factor
56	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×#11719921	C₂×C₄	1	2	yes	—
58	C₂×#11719921	C₂×C₄	1	4	yes	—
59	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
60	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
61	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
62	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
63	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
64	C₄²×D₅₃	—	2	4	no	—
65	C₄²×D₅₃	—	2	4	no	—
66	C₄⋊#88322	C₂×C₄	1	4	yes	complemented
67	#13133129	C₂×C₄	1	8	yes	complemented
68	#13133129	C₂×C₄	1	8	yes	complemented
69	C₄×#88322	—	2	4	no	—
70	C₄×#88322	C₂×C₄	1	4	yes	complemented
71	C₈×Dic₅₃	—	2	2	no	—
72	C₈×Dic₅₃	—	2	4	no	—
73	C₈×Dic₅₃	—	2	2	no	—
74	C₈×Dic₅₃	—	2	2	no	—
75	C₈×Dic₅₃	D₄	1	4	yes	complemented
76	#13133124	C₂×C₄	1	4	yes	—
77	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
79	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
81	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
82	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
84	C₄×C₈×C₅₃	—	2	4	no	—
85	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
86	#95×C₅₃	C₂×C₄	1	2	yes	—
87	C₂³×D₅₃	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
90	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
92	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
93	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
94	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
95	C₂×D₂₁₂	C₄²	1	4	yes	—
96	C₂×D₂₁₂	C₂×C₈	1	8	yes	—
97	C₂×C₄×D₅₃	—	2	2	no	—
98	C₂×C₄×D₅₃	—	2	2	no	—
99	C₂×C₄×D₅₃	—	2	2	no	—
100	C₂×C₄×D₅₃	—	2	2	no	—
101	C₂×C₄×D₅₃	—	2	8	no	—
102	C₂×C₄×D₅₃	—	2	8	no	—
103	C₂×C₄×D₅₃	C₂×C₈	1	16	yes	complemented
104	C₂×C₄×D₅₃	C₂²×C₄	1	4	yes	—
105	C₂×C₄×D₅₃	C₂²×C₄	1	4	yes	—
106	C₂×C₈×C₅₃	—	2	2	no	—
107	C₂×C₈×C₅₃	—	2	1	no	—
108	C₂×C₈×C₅₃	—	2	1	no	—
109	C₂×C₈×C₅₃	—	2	2	no	—
110	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
112	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
113	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
114	C₂×C₈×C₅₃	#41	1	2	yes	—
115	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	complemented
116	C₄²×C₅₃	—	2	2	no	—
117	C₄²×C₅₃	—	2	2	no	—
118	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
119	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
120	#11719921	C₄²	1	8	yes	—
121	#11719921	C₂×C₈	1	16	yes	complemented
122	C₄⋊Dic₅₃	C₄²	1	4	yes	—
123	C₄×Dic₅₃	—	2	1	no	—
124	C₄×Dic₅₃	—	2	1	no	—
125	C₄×Dic₅₃	—	2	8	no	—
126	C₄×Dic₅₃	—	2	4	no	—
127	C₄×Dic₅₃	C₂×C₈	1	8	yes	complemented
128	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
129	C₄×Dic₅₃	#41	1	2	yes	—
130	C₂×#88322	—	2	2	no	—
131	C₂×#88322	—	2	1	no	—
132	C₂×#88322	—	2	1	no	—
133	C₂×#88322	—	2	2	no	—
134	C₂×#88322	C₂²×C₄	1	2	yes	—
135	C₂×#88322	C₂²×C₄	1	4	yes	—
136	C₂×#88322	C₂²×C₄	1	2	yes	—
137	C₂×#88322	C₂×D₄	1	2	yes	—
138	C₂×#88322	#41	1	2	yes	—
139	C₈×D₅₃	—	2	16	no	—
140	C₈×D₅₃	—	2	16	no	—
141	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
142	C₂²×D₅₃	—	2	2	no	—
143	C₂²×D₅₃	—	2	2	no	—
144	C₂²×D₅₃	—	2	4	no	—
145	C₂²×D₅₃	C₂×C₄²	1	4	yes	—
146	C₂²×D₅₃	C₂²×C₈	1	8	yes	—
147	C₂×C₄×C₅₃	—	2	4	no	—
148	C₂×C₄×C₅₃	—	2	4	no	—
149	C₂×C₄×C₅₃	—	2	2	no	—
150	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
151	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
152	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
153	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
154	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
160	C₂×Dic₅₃	—	2	2	no	—
161	C₂×Dic₅₃	—	2	2	no	—
162	C₂×Dic₅₃	—	2	2	no	—
163	C₂×Dic₅₃	—	2	2	no	—
164	C₂×Dic₅₃	C₂×C₄²	1	4	yes	—
165	C₂×Dic₅₃	C₄×D₄	1	4	yes	—
166	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
167	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
168	C₂×Dic₅₃	#131	1	4	yes	—
169	D₂₁₂	C₄×C₈	1	16	yes	complemented
170	C₄×D₅₃	—	2	4	no	—
171	C₄×D₅₃	—	2	4	no	—
172	C₄×D₅₃	—	2	4	no	—
173	C₄×D₅₃	—	2	4	no	—
174	C₄×D₅₃	—	2	32	no	—
175	C₈×C₅₃	—	2	4	no	—
176	C₈×C₅₃	—	2	2	no	—
177	C₈×C₅₃	C₄×D₄	1	4	yes	complemented
178	#88322	—	2	4	no	—
179	#88322	—	2	2	no	—
180	#88322	C₄×D₄	1	4	yes	complemented
181	C₄×C₈×D₄	—	53	1	no	maximal, Sylow
182	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
183	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
184	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
185	C₂²×C₅₃	C₂³×C₈	1	2	yes	—
186	C₂²×C₅₃	C₂×#131	1	2	yes	—
187	C₂×D₅₃	—	2	4	no	—
188	C₂×D₅₃	—	2	8	no	—
189	C₂×D₅₃	—	2	8	no	—
190	C₂×D₅₃	—	2	4	no	—
191	C₂×D₅₃	C₂×C₄×C₈	1	8	yes	complemented
192	C₄×C₅₃	—	2	2	no	—
193	C₄×C₅₃	—	2	8	no	—
194	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	complemented
195	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
196	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
197	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
198	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
199	Dic₅₃	—	2	4	no	—
200	Dic₅₃	—	2	4	no	—
201	Dic₅₃	C₈×D₄	1	8	yes	complemented
202	C₂×C₈×D₄	—	53	1	no	—
203	C₂×C₈×D₄	—	53	1	no	—
204	C₂²×C₄×C₈	—	53	2	no	—
205	C₄²×D₄	—	53	1	no	—
206	C₈×C₄⋊C₄	—	53	1	no	—
207	C₈×C₄⋊C₄	D₅₃	1	1	yes	characteristic, complemented
208	C₄×C₄⋊C₈	—	53	1	no	—
209	C₈×C₂²⋊C₄	—	53	2	no	—
210	C₈×C₂²⋊C₄	—	53	2	no	—
211	C₄×C₂²⋊C₈	—	53	2	no	—
212	C₄²×C₈	—	53	1	no	—
213	C₂×C₅₃	C₄²×D₄	1	1	yes	characteristic
214	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
215	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
216	C₂×C₅₃	C₂×C₈×D₄	1	2	yes	—
217	C₂×C₅₃	C₈×#41	1	2	yes	—
218	D₅₃	—	2	8	no	—
219	C₂³×C₈	—	53	2	no	—
220	C₂³×C₈	—	53	2	no	—
221	C₂×C₄×D₄	—	53	2	no	—
222	C₂×C₄×D₄	—	53	1	no	—
223	C₂²×C₄²	—	53	2	no	—
224	C₈×D₄	—	53	8	no	—
225	C₈×D₄	—	53	8	no	—
226	C₂×C₄⋊C₈	—	53	1	no	—
227	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
228	C₂×C₂²⋊C₈	—	53	2	no	—
229	C₂×C₂²⋊C₈	—	53	2	no	—
230	C₂×C₄×C₈	—	53	1	no	—
231	C₂×C₄×C₈	—	53	1	no	—
232	C₂×C₄×C₈	—	53	2	no	—
233	C₂×C₄×C₈	—	53	1	no	—
234	C₂×C₄×C₈	—	53	1	no	—
235	C₂×C₄×C₈	—	53	1	no	—
236	C₂×C₄×C₈	—	53	2	no	—
237	C₂×C₄×C₈	—	106	8	no	—
238	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
239	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
240	C₄×C₄⋊C₄	—	53	2	no	—
241	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
242	C₄×C₂²⋊C₄	—	53	2	no	—
243	C₄×C₂²⋊C₄	—	53	4	no	—
244	C₄³	—	53	1	no	—
245	#336	—	53	2	no	—
246	#336	—	53	2	no	—
247	#336	C₂×D₅₃	1	2	yes	—
248	C₅₃	C₄×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
249	C₂²×D₄	—	53	1	no	—
250	C₂³×C₄	—	53	4	no	—
251	C₂³×C₄	—	53	2	no	—
252	C₂²×C₈	—	53	1	no	—
253	C₂²×C₈	—	53	2	no	—
254	C₂²×C₈	—	53	1	no	—
255	C₂²×C₈	—	53	8	no	—
256	C₂²×C₈	—	53	8	no	—
257	C₂²×C₈	—	106	4	no	—
258	C₂²×C₈	—	106	4	no	—
259	C₂²×C₈	—	106	4	no	—
260	C₂²×C₈	—	106	4	no	—
261	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
262	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
263	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
264	C₄×D₄	—	53	16	no	—
265	C₄×D₄	—	53	4	no	—
266	C₄×D₄	—	53	4	no	—
267	C₂×C₄⋊C₄	—	53	1	no	—
268	C₂×C₄⋊C₄	C₄×D₅₃	1	2	yes	—
269	C₂×C₂²⋊C₄	—	53	2	no	—
270	C₂×C₂²⋊C₄	—	53	4	no	—
271	C₂×C₄²	—	53	2	no	—
272	C₂×C₄²	—	53	1	no	—
273	C₂×C₄²	—	53	2	no	—
274	C₂×C₄²	—	53	4	no	—
275	C₂×C₄²	—	53	1	no	—
276	C₂×C₄²	—	106	4	no	—
277	C₂×C₄²	—	106	4	no	—
278	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
279	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
280	C₄⋊C₈	—	53	4	no	—
281	C₄⋊C₈	C₄×D₅₃	1	4	yes	complemented
282	C₂²⋊C₈	—	53	8	no	—
283	C₂²⋊C₈	—	53	8	no	—
284	C₄×C₈	—	2	4	no	—
285	C₄×C₈	—	53	4	no	—
286	C₄×C₈	—	53	4	no	—
287	C₄×C₈	—	106	4	no	—
288	C₄×C₈	—	106	2	no	—
289	C₄×C₈	—	106	4	no	—
290	C₄×C₈	—	106	2	no	—
291	C₄×C₈	—	106	2	no	—
292	C₄×C₈	C₄×D₅₃	1	4	yes	complemented
293	#95	—	53	4	no	—
294	#95	C₄×D₅₃	1	2	yes	—
295	C₂⁴	—	53	2	no	—
296	C₂×D₄	—	53	4	no	—
297	C₂×D₄	—	53	8	no	—
298	C₂²×C₄	—	53	2	no	—
299	C₂²×C₄	—	53	2	no	—
300	C₂²×C₄	—	53	2	no	—
301	C₂²×C₄	—	53	16	no	—
302	C₂²×C₄	—	53	4	no	—
303	C₂²×C₄	—	53	4	no	—
304	C₂²×C₄	—	106	8	no	—
305	C₂²×C₄	—	106	8	no	—
306	C₂²×C₄	—	106	2	no	—
307	C₂²×C₄	—	106	2	no	—
308	C₂²×C₄	—	106	2	no	—
309	C₂²×C₄	—	106	2	no	—
310	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
311	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
312	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
313	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	53	2	no	—
319	C₂×C₈	—	53	2	no	—
320	C₂×C₈	—	53	2	no	—
321	C₂×C₈	—	53	2	no	—
322	C₂×C₈	—	53	4	no	—
323	C₂×C₈	—	106	16	no	—
324	C₂×C₈	—	106	2	no	—
325	C₂×C₈	—	106	1	no	—
326	C₂×C₈	—	106	1	no	—
327	C₂×C₈	—	106	2	no	—
328	C₂×C₈	—	106	16	no	—
329	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
330	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
331	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
332	C₂×C₈	D₄×D₅₃	1	2	yes	—
333	C₂×C₈	#11719949	1	2	yes	—
334	C₄⋊C₄	—	53	4	no	—
335	C₄⋊C₄	C₈×D₅₃	1	8	yes	complemented
336	C₂²⋊C₄	—	53	8	no	—
337	C₂²⋊C₄	—	53	16	no	—
338	C₄²	—	2	2	no	—
339	C₄²	—	2	2	no	—
340	C₄²	—	53	8	no	—
341	C₄²	—	53	2	no	—
342	C₄²	—	53	2	no	—
343	C₄²	—	106	8	no	—
344	C₄²	—	106	4	no	—
345	C₄²	—	106	1	no	—
346	C₄²	—	106	1	no	—
347	C₄²	C₂×C₄×D₅₃	1	2	yes	—
348	C₄²	C₂×C₄×D₅₃	1	2	yes	—
349	C₂³	—	53	4	no	—
350	C₂³	—	53	8	no	—
351	C₂³	—	106	2	no	—
352	C₂³	—	106	2	no	—
353	C₂³	—	106	4	no	—
354	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
355	D₄	—	53	16	no	—
356	C₂×C₄	—	2	4	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	4	no	—
359	C₂×C₄	—	53	4	no	—
360	C₂×C₄	—	53	4	no	—
361	C₂×C₄	—	53	4	no	—
362	C₂×C₄	—	53	4	no	—
363	C₂×C₄	—	53	4	no	—
364	C₂×C₄	—	106	32	no	—
365	C₂×C₄	—	106	2	no	—
366	C₂×C₄	—	106	2	no	—
367	C₂×C₄	—	106	2	no	—
368	C₂×C₄	—	106	2	no	—
369	C₂×C₄	—	106	4	no	—
370	C₂×C₄	—	106	4	no	—
371	C₂×C₄	—	106	4	no	—
372	C₂×C₄	—	106	4	no	—
373	C₂×C₄	C₄²×D₅₃	1	2	yes	—
374	C₂×C₄	C₄²×D₅₃	1	2	yes	—
375	C₂×C₄	C₂×C₈×D₅₃	1	8	yes	—
376	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
377	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
378	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
379	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
380	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
381	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
382	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
383	C₈	—	2	4	no	—
384	C₈	—	2	2	no	—
385	C₈	—	53	4	no	—
386	C₈	—	106	4	no	—
387	C₈	—	106	2	no	—
388	C₈	#13133179	1	4	yes	direct product factor
389	C₂²	—	53	8	no	—
390	C₂²	—	106	4	no	—
391	C₂²	—	106	8	no	—
392	C₂²	—	106	4	no	—
393	C₂²	—	106	8	no	—
394	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
395	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
396	C₂²	C₄×#11719949	1	1	yes	characteristic
397	C₂²	C₂²×C₈×D₅₃	1	2	yes	—
398	C₂²	#131×D₅₃	1	2	yes	—
399	C₄	—	2	2	no	—
400	C₄	—	2	8	no	—
401	C₄	—	53	8	no	—
402	C₄	—	106	4	no	—
403	C₄	—	106	4	no	—
404	C₄	C₄×C₈×D₅₃	1	4	yes	complemented
405	C₄	C₄×D₄×D₅₃	1	1	yes	characteristic
406	C₄	C₂×#13133179	1	1	yes	characteristic
407	C₄	#16811091	1	1	yes	characteristic
408	C₄	C₄×#11719949	1	1	yes	characteristic
409	C₂	—	106	8	no	—
410	C₂	C₄×#13133179	1	1	yes	characteristic
411	C₂	C₂×C₄×C₈×D₅₃	1	1	yes	characteristic
412	C₂	#20283137	1	1	yes	characteristic
413	C₂	C₈×D₄×D₅₃	1	2	yes	—
414	C₂	C₈×#11719949	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₈×C₄⋊C₄×C₅₃	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂²×D₂₁₂	C₂×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₈×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₄×C₈×D₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×C₈×D₅₃	C₂	semidirect
5. C₄⋊C₄×#88322	C₂	semidirect
6. C₄⋊C₈×Dic₅₃	C₂	semidirect
7. C₈×#11719921	C₂	semidirect
8. #20274485	C₂	semidirect
9. #20274472	C₂	semidirect
10. C₄×C₈×Dic₅₃	C₂	semidirect
11. C₈×C₄⋊C₄×C₅₃	C₂	semidirect
17. C₈×D₂₁₂	C₄	semidirect
18. #16810688	C₄	semidirect
51. C₂×C₈×D₅₃	C₂×C₄	semidirect
52. C₂×C₈×D₅₃	C₂×C₄	semidirect
55. #13133179	C₈	direct (N × H)
66. C₄⋊#88322	C₂×C₄	semidirect
67. #13133129	C₂×C₄	semidirect
68. #13133129	C₂×C₄	semidirect
70. C₄×#88322	C₂×C₄	semidirect
75. C₈×Dic₅₃	D₄	semidirect
83. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
85. C₄×C₈×C₅₃	C₂×C₄	semidirect
103. C₂×C₄×D₅₃	C₂×C₈	semidirect
115. C₄⋊C₄×C₅₃	C₂×C₈	semidirect
121. #11719921	C₂×C₈	semidirect
127. C₄×Dic₅₃	C₂×C₈	semidirect
169. D₂₁₂	C₄×C₈	semidirect
177. C₈×C₅₃	C₄×D₄	semidirect
180. #88322	C₄×D₄	semidirect
191. C₂×D₅₃	C₂×C₄×C₈	semidirect
194. C₄×C₅₃	C₂×C₄×C₈	semidirect
201. Dic₅₃	C₈×D₄	semidirect
207. C₈×C₄⋊C₄	D₅₃	semidirect
248. C₅₃	C₄×C₈×D₄	semidirect
281. C₄⋊C₈	C₄×D₅₃	semidirect
292. C₄×C₈	C₄×D₅₃	semidirect
335. C₄⋊C₄	C₈×D₅₃	semidirect
388. C₈	#13133179	direct (N × H)
404. C₄	C₄×C₈×D₅₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₈×D₂₁₂	2
C₂×#16810688	2
C₂×C₄×C₈×D₅₃	2
C₄×#13133179	2
C₄⋊C₄×#88322	2
C₄⋊C₈×Dic₅₃	2
C₈×#11719921	2
#20274485	2
#20274472	2
C₄×C₈×Dic₅₃	2
C₈×C₄⋊C₄×C₅₃	2
C₄×C₈×D₄	53

Subgroups — C8×#13133179 #27382402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#13133179 #27382402