Subgroups — C₄⋊C₄×D₈×C₅₃ #27230192

Order 13568.

1134 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₅₃	C₂	1	2	yes	maximal
2	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	D₄×C₄⋊C₄×C₅₃	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1942	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1862	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1802	C₂	1	2	yes	maximal, complemented
8	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
11	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
12	C₂²×C₄⋊C₄×C₅₃	—	2	2	no	—
13	C₄×D₈×C₅₃	—	2	8	no	—
14	C₄×D₈×C₅₃	C₄	1	8	yes	complemented
15	C₂×#107×C₅₃	C₂²	1	2	yes	—
16	C₂×#107×C₅₃	C₂²	1	2	yes	—
17	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#102×C₅₃	C₂²	1	4	yes	—
20	C₂×#102×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#389	C₂²	1	4	yes	—
24	C₅₃×#385	—	2	4	no	—
25	C₅₃×#384	C₂²	1	2	yes	—
26	C₅₃×#380	—	2	2	no	—
27	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#340	C₂²	1	4	yes	—
29	C₅₃×#336	C₂²	1	2	yes	—
30	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
31	C₂³×C₄×C₅₃	—	2	2	no	—
32	C₂³×C₄×C₅₃	—	2	4	no	—
33	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
34	C₂×D₈×C₅₃	D₄	1	4	yes	—
35	C₂×D₈×C₅₃	Q₈	1	4	yes	—
36	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	C₄×D₄×C₅₃	—	2	4	no	—
40	C₄×D₄×C₅₃	—	2	8	no	—
41	C₄×D₄×C₅₃	—	4	4	no	—
42	C₄×D₄×C₅₃	C₂×C₄	1	8	yes	complemented
43	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
45	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₅₃	—	4	8	no	—
47	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
50	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
51	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
53	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	#107×C₅₃	—	2	4	no	—
56	#107×C₅₃	—	4	2	no	—
57	#107×C₅₃	C₂×C₄	1	4	yes	complemented
58	C₄⋊C₈×C₅₃	—	2	2	no	—
59	#102×C₅₃	—	2	8	no	—
60	#102×C₅₃	C₂×C₄	1	8	yes	complemented
61	C₄×C₈×C₅₃	—	2	4	no	—
62	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
63	#95×C₅₃	—	2	4	no	—
64	#95×C₅₃	D₄	1	2	yes	—
65	C₂⁴×C₅₃	—	2	2	no	—
66	C₂×D₄×C₅₃	—	2	2	no	—
67	C₂×D₄×C₅₃	—	2	2	no	—
68	C₂×D₄×C₅₃	—	2	2	no	—
69	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
70	C₂×D₄×C₅₃	C₂×D₄	1	4	yes	—
71	C₂×D₄×C₅₃	C₂×Q₈	1	4	yes	—
72	C₂²×C₄×C₅₃	—	2	2	no	—
73	C₂²×C₄×C₅₃	—	2	8	no	—
74	C₂²×C₄×C₅₃	—	2	4	no	—
75	C₂²×C₄×C₅₃	—	4	8	no	—
76	C₂²×C₄×C₅₃	—	4	4	no	—
77	C₂²×C₄×C₅₃	—	4	4	no	—
78	C₂²×C₄×C₅₃	—	4	8	no	—
79	C₂²×C₄×C₅₃	—	8	4	no	—
80	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
85	D₈×C₅₃	C₄⋊C₄	1	16	yes	direct product factor
86	C₂×C₈×C₅₃	—	2	1	no	—
87	C₂×C₈×C₅₃	—	2	1	no	—
88	C₂×C₈×C₅₃	—	2	2	no	—
89	C₂×C₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
95	C₄⋊C₄×C₅₃	—	2	2	no	—
96	C₄⋊C₄×C₅₃	—	2	2	no	—
97	C₄⋊C₄×C₅₃	—	2	4	no	—
98	C₄⋊C₄×C₅₃	—	2	4	no	—
99	C₄⋊C₄×C₅₃	—	4	8	no	—
100	C₄⋊C₄×C₅₃	—	4	4	no	—
101	C₄⋊C₄×C₅₃	—	8	2	no	—
102	C₄⋊C₄×C₅₃	D₈	1	4	yes	direct product factor
103	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
104	C₂²⋊C₄×C₅₃	—	2	8	no	—
105	C₂²⋊C₄×C₅₃	—	4	8	no	—
106	C₄²×C₅₃	—	2	2	no	—
107	C₄²×C₅₃	—	2	2	no	—
108	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
109	C₄²×C₅₃	#41	1	2	yes	—
110	C₂³×C₅₃	—	2	4	no	—
111	C₂³×C₅₃	—	2	4	no	—
112	C₂³×C₅₃	—	2	4	no	—
113	C₂³×C₅₃	—	4	2	no	—
114	C₂³×C₅₃	—	4	2	no	—
115	C₂³×C₅₃	—	4	2	no	—
116	C₂³×C₅₃	—	4	2	no	—
117	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
118	D₄×C₅₃	—	2	4	no	—
119	D₄×C₅₃	—	2	4	no	—
120	D₄×C₅₃	—	2	4	no	—
121	D₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	complemented
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	2	1	no	—
126	C₂×C₄×C₅₃	—	2	4	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	4	2	no	—
131	C₂×C₄×C₅₃	—	4	2	no	—
132	C₂×C₄×C₅₃	—	4	16	no	—
133	C₂×C₄×C₅₃	—	4	4	no	—
134	C₂×C₄×C₅₃	—	4	4	no	—
135	C₂×C₄×C₅₃	—	4	4	no	—
136	C₂×C₄×C₅₃	—	8	16	no	—
137	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
138	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
139	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
140	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
141	C₂×C₄×C₅₃	#135	1	2	yes	—
142	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
148	C₈×C₅₃	—	2	2	no	—
149	C₈×C₅₃	—	4	2	no	—
150	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	complemented
151	C₄⋊C₄×D₈	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂²×C₅₃	—	2	8	no	—
153	C₂²×C₅₃	—	4	4	no	—
154	C₂²×C₅₃	—	4	4	no	—
155	C₂²×C₅₃	—	4	4	no	—
156	C₂²×C₅₃	—	4	4	no	—
157	C₂²×C₅₃	—	4	4	no	—
158	C₂²×C₅₃	—	4	4	no	—
159	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
160	C₂²×C₅₃	D₄²	1	1	yes	characteristic
161	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
162	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
163	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
164	C₂²×C₅₃	#576	1	1	yes	characteristic
165	C₂²×C₅₃	#578	1	1	yes	characteristic
166	C₄×C₅₃	—	2	2	no	—
167	C₄×C₅₃	—	2	4	no	—
168	C₄×C₅₃	—	4	4	no	—
169	C₄×C₅₃	—	4	2	no	—
170	C₄×C₅₃	—	8	4	no	—
171	C₄×C₅₃	C₄×D₈	1	4	yes	complemented
172	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
173	C₄×C₅₃	#520	1	1	yes	characteristic
174	C₄×C₅₃	#547	1	1	yes	characteristic
175	C₄×C₅₃	#554	1	1	yes	characteristic
176	C₂×C₄×D₈	C₂×C₅₃	1	2	yes	—
177	C₂×C₄×D₈	C₂×C₅₃	1	1	yes	characteristic
178	D₄×C₄⋊C₄	C₂×C₅₃	1	2	yes	complemented
179	#1942	C₂×C₅₃	1	2	yes	complemented
180	#1862	C₂×C₅₃	1	4	yes	complemented
181	#1845	C₂×C₅₃	1	1	yes	characteristic, complemented
182	#1802	C₂×C₅₃	1	2	yes	complemented
183	C₈×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
184	C₂×C₅₃	—	4	8	no	—
185	C₂×C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic
186	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
187	C₂×C₅₃	#2951	1	1	yes	characteristic
188	C₂×C₅₃	D₄×D₈	1	1	yes	characteristic
189	C₂×C₅₃	#3283	1	1	yes	characteristic
190	C₂×C₅₃	Q₈×D₈	1	1	yes	characteristic
191	C₂×C₅₃	#3380	1	1	yes	characteristic
192	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
193	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
194	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
195	C₂²×C₄⋊C₄	—	2	2	no	—
196	C₄×D₈	—	2	8	no	—
197	C₄×D₈	C₄×C₅₃	1	8	yes	complemented
198	C₂×#107	C₂²×C₅₃	1	2	yes	—
199	C₂×#107	C₂²×C₅₃	1	2	yes	—
200	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
201	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
202	C₂×#102	C₂²×C₅₃	1	4	yes	—
203	C₂×#102	C₂²×C₅₃	1	2	yes	—
204	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
205	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
206	#389	C₂²×C₅₃	1	4	yes	—
207	#385	—	2	4	no	—
208	#384	C₂²×C₅₃	1	2	yes	—
209	#380	—	2	2	no	—
210	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
211	#340	C₂²×C₅₃	1	4	yes	—
212	#336	C₂²×C₅₃	1	2	yes	—
213	C₅₃	C₄⋊C₄×D₈	1	1	yes	characteristic, direct product factor, Sylow
214	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
215	C₂³×C₄	—	2	2	no	—
216	C₂³×C₄	—	2	4	no	—
217	C₂×D₈	C₂×C₄×C₅₃	1	4	yes	—
218	C₂×D₈	D₄×C₅₃	1	4	yes	—
219	C₂×D₈	Q₈×C₅₃	1	4	yes	—
220	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
221	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
222	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
223	C₄×D₄	—	2	8	no	—
224	C₄×D₄	—	2	4	no	—
225	C₄×D₄	—	4	4	no	—
226	C₄×D₄	C₂×C₄×C₅₃	1	8	yes	complemented
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	2	4	no	—
229	C₂×C₄⋊C₄	—	2	2	no	—
230	C₂×C₄⋊C₄	—	4	8	no	—
231	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
232	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
233	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
234	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
235	C₂×C₂²⋊C₄	—	2	2	no	—
236	C₂×C₂²⋊C₄	—	2	4	no	—
237	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
238	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
239	#107	—	2	4	no	—
240	#107	—	4	2	no	—
241	#107	C₂×C₄×C₅₃	1	4	yes	complemented
242	C₄⋊C₈	—	2	2	no	—
243	#102	—	2	8	no	—
244	#102	C₂×C₄×C₅₃	1	8	yes	complemented
245	C₄×C₈	—	2	4	no	—
246	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	complemented
247	#95	—	2	4	no	—
248	#95	D₄×C₅₃	1	2	yes	—
249	C₂⁴	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
254	C₂×D₄	C₂×D₄×C₅₃	1	4	yes	—
255	C₂×D₄	C₂×Q₈×C₅₃	1	4	yes	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	8	no	—
258	C₂²×C₄	—	2	4	no	—
259	C₂²×C₄	—	4	8	no	—
260	C₂²×C₄	—	4	4	no	—
261	C₂²×C₄	—	4	4	no	—
262	C₂²×C₄	—	4	8	no	—
263	C₂²×C₄	—	8	4	no	—
264	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
265	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
266	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
267	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
268	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
269	D₈	C₄⋊C₄×C₅₃	1	16	yes	direct product factor
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
276	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
277	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
278	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	4	no	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	4	8	no	—
284	C₄⋊C₄	—	4	4	no	—
285	C₄⋊C₄	—	8	2	no	—
286	C₄⋊C₄	D₈×C₅₃	1	4	yes	direct product factor
287	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
288	C₂²⋊C₄	—	2	8	no	—
289	C₂²⋊C₄	—	4	8	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	2	no	—
292	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
293	C₄²	#41×C₅₃	1	2	yes	—
294	C₂³	—	2	4	no	—
295	C₂³	—	2	4	no	—
296	C₂³	—	2	4	no	—
297	C₂³	—	4	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
302	D₄	—	2	4	no	—
303	D₄	—	2	4	no	—
304	D₄	—	2	4	no	—
305	D₄	C₂×C₄⋊C₄×C₅₃	1	8	yes	complemented
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	16	no	—
317	C₂×C₄	—	4	4	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	4	4	no	—
320	C₂×C₄	—	8	16	no	—
321	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
322	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
323	C₂×C₄	C₂×D₈×C₅₃	1	4	yes	—
324	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
325	C₂×C₄	#135×C₅₃	1	2	yes	—
326	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
327	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
328	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
329	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
330	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
331	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
332	C₈	—	2	2	no	—
333	C₈	—	4	2	no	—
334	C₈	C₂×C₄⋊C₄×C₅₃	1	4	yes	complemented
335	C₂²	—	2	8	no	—
336	C₂²	—	4	4	no	—
337	C₂²	—	4	4	no	—
338	C₂²	—	4	4	no	—
339	C₂²	—	4	4	no	—
340	C₂²	—	4	4	no	—
341	C₂²	—	4	4	no	—
342	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
343	C₂²	D₄²×C₅₃	1	1	yes	characteristic
344	C₂²	D₄×Q₈×C₅₃	1	1	yes	characteristic
345	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
346	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
347	C₂²	C₅₃×#576	1	1	yes	characteristic
348	C₂²	C₅₃×#578	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	2	4	no	—
351	C₄	—	4	4	no	—
352	C₄	—	4	2	no	—
353	C₄	—	8	4	no	—
354	C₄	C₄×D₈×C₅₃	1	4	yes	complemented
355	C₄	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
356	C₄	C₅₃×#520	1	1	yes	characteristic
357	C₄	C₅₃×#547	1	1	yes	characteristic
358	C₄	C₅₃×#554	1	1	yes	characteristic
359	C₂	—	4	8	no	—
360	C₂	D₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
361	C₂	C₂×C₄×D₈×C₅₃	1	1	yes	characteristic
362	C₂	C₅₃×#2951	1	1	yes	characteristic
363	C₂	D₄×D₈×C₅₃	1	1	yes	characteristic
364	C₂	C₅₃×#3283	1	1	yes	characteristic
365	C₂	Q₈×D₈×C₅₃	1	1	yes	characteristic
366	C₂	C₅₃×#3380	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂²×D₈×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄×C₄⋊C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. D₄×C₄⋊C₄×C₅₃	C₂	semidirect
4. C₅₃×#1942	C₂	semidirect
5. C₅₃×#1862	C₂	semidirect
6. C₅₃×#1845	C₂	semidirect
7. C₅₃×#1802	C₂	semidirect
8. C₈×C₄⋊C₄×C₅₃	C₂	semidirect
14. C₄×D₈×C₅₃	C₄	semidirect
42. C₄×D₄×C₅₃	C₂×C₄	semidirect
57. #107×C₅₃	C₂×C₄	semidirect
60. #102×C₅₃	C₂×C₄	semidirect
62. C₄×C₈×C₅₃	C₂×C₄	semidirect
85. D₈×C₅₃	C₄⋊C₄	direct (N × H)
102. C₄⋊C₄×C₅₃	D₈	direct (N × H)
121. D₄×C₅₃	C₂×C₄⋊C₄	semidirect
150. C₈×C₅₃	C₂×C₄⋊C₄	semidirect
151. C₄⋊C₄×D₈	C₅₃	direct (N × H)
171. C₄×C₅₃	C₄×D₈	semidirect
178. D₄×C₄⋊C₄	C₂×C₅₃	semidirect
179. #1942	C₂×C₅₃	semidirect
180. #1862	C₂×C₅₃	semidirect
181. #1845	C₂×C₅₃	semidirect
182. #1802	C₂×C₅₃	semidirect
183. C₈×C₄⋊C₄	C₂×C₅₃	semidirect
197. C₄×D₈	C₄×C₅₃	semidirect
213. C₅₃	C₄⋊C₄×D₈	direct (N × H)
226. C₄×D₄	C₂×C₄×C₅₃	semidirect
241. #107	C₂×C₄×C₅₃	semidirect
244. #102	C₂×C₄×C₅₃	semidirect
246. C₄×C₈	C₂×C₄×C₅₃	semidirect
269. D₈	C₄⋊C₄×C₅₃	direct (N × H)
286. C₄⋊C₄	D₈×C₅₃	direct (N × H)
305. D₄	C₂×C₄⋊C₄×C₅₃	semidirect
334. C₈	C₂×C₄⋊C₄×C₅₃	semidirect
354. C₄	C₄×D₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₅₃	2
D₄×C₄⋊C₄×C₅₃	2
C₅₃×#1942	2
C₅₃×#1862	2
C₅₃×#1845	2
C₅₃×#1802	2
C₈×C₄⋊C₄×C₅₃	2
C₄⋊C₄×D₈	53

Subgroups — C4⋊C4×D8×C53 #27230192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊C₄×D₈×C₅₃ #27230192