Subgroups — C₈×#135×C₅₃ #27225357

Order 13568.

750 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#41×C₅₃	C₂	1	2	yes	maximal, complemented
2	C₄×#135×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₈×Q₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
4	C₈×SD₁₆×C₅₃	C₂	1	4	yes	maximal, complemented
5	C₈×D₈×C₅₃	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1476	C₂	1	2	yes	maximal, complemented
8	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₅₃	C₄	1	1	yes	characteristic
10	C₄×#41×C₅₃	C₂²	1	2	yes	—
11	C₈×Q₈×C₅₃	C₂²	1	4	yes	—
12	C₄×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
13	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×SD₁₆×C₅₃	C₄	1	2	yes	—
15	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
16	C₄×D₈×C₅₃	C₄	1	1	yes	characteristic
17	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₈×D₄×C₅₃	—	2	4	no	—
19	C₈×D₄×C₅₃	C₂²	1	4	yes	—
20	C₅₃×#431	—	2	2	no	—
21	C₅₃×#431	C₂²	1	2	yes	—
22	C₅₃×#427	C₄	1	1	yes	characteristic
23	C₅₃×#416	C₄	1	2	yes	—
24	C₂×C₄×C₈×C₅₃	—	2	2	no	—
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#335	C₂²	1	2	yes	—
29	C₅₃×#334	C₂²	1	2	yes	—
30	C₅₃×Q₈⋊C₈	C₂²	1	4	yes	—
31	C₅₃×D₄⋊C₈	C₂²	1	4	yes	—
32	C₈²×C₅₃	C₂²	1	2	yes	—
33	C₈²×C₅₃	C₂²	1	2	yes	—
34	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
35	#135×C₅₃	C₈	1	8	yes	direct product factor
36	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	—	2	4	no	—
40	C₂²×C₈×C₅₃	—	2	1	no	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	D₄	1	2	yes	—
44	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
45	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
46	C₄×D₄×C₅₃	—	2	2	no	—
47	C₄×D₄×C₅₃	—	2	2	no	—
48	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
49	C₄×D₄×C₅₃	C₂³	1	2	yes	—
50	#117×C₅₃	—	2	2	no	—
51	#117×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₅₃	—	2	2	no	—
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
55	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
56	#106×C₅₃	C₂×C₄	1	2	yes	—
57	C₄⋊C₈×C₅₃	—	2	4	no	—
58	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
59	#103×C₅₃	C₂×C₄	1	2	yes	—
60	#103×C₅₃	C₂×C₄	1	2	yes	—
61	#102×C₅₃	C₂×C₄	1	2	yes	—
62	#102×C₅₃	C₂×C₄	1	2	yes	—
63	C₂²⋊C₈×C₅₃	—	2	4	no	—
64	C₂²⋊C₈×C₅₃	—	2	2	no	—
65	C₄×C₈×C₅₃	—	2	4	no	—
66	C₄×C₈×C₅₃	—	2	2	no	—
67	C₄×C₈×C₅₃	—	2	1	no	—
68	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
71	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
72	C₄×C₈×C₅₃	C₂³	1	2	yes	—
73	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₅₃	C₂³	1	2	yes	—
76	#41×C₅₃	—	2	4	no	—
77	#41×C₅₃	C₂×C₈	1	8	yes	complemented
78	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂×D₄×C₅₃	—	2	2	no	—
80	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₅₃	—	2	2	no	—
82	C₂²×C₄×C₅₃	—	2	2	no	—
83	C₂²×C₄×C₅₃	—	2	2	no	—
84	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
87	Q₁₆×C₅₃	C₂×C₈	1	4	yes	complemented
88	SD₁₆×C₅₃	C₂×C₈	1	8	yes	complemented
89	D₈×C₅₃	C₂×C₈	1	4	yes	complemented
90	C₂×C₈×C₅₃	—	2	4	no	—
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	—	2	4	no	—
94	C₂×C₈×C₅₃	—	2	4	no	—
95	C₂×C₈×C₅₃	—	4	8	no	—
96	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	complemented
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	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
103	C₄⋊C₄×C₅₃	—	2	2	no	—
104	C₄⋊C₄×C₅₃	—	2	2	no	—
105	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
106	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
107	C₂²⋊C₄×C₅₃	—	2	2	no	—
108	C₂²⋊C₄×C₅₃	—	2	2	no	—
109	C₂²⋊C₄×C₅₃	—	2	2	no	—
110	C₄²×C₅₃	—	2	2	no	—
111	C₄²×C₅₃	—	2	2	no	—
112	C₄²×C₅₃	—	2	2	no	—
113	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₅₃	#41	1	1	yes	characteristic
116	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
117	C₂³×C₅₃	—	2	2	no	—
118	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
119	Q₈×C₅₃	—	2	2	no	—
120	Q₈×C₅₃	C₂²×C₈	1	4	yes	—
121	D₄×C₅₃	—	2	8	no	—
122	D₄×C₅₃	—	2	2	no	—
123	D₄×C₅₃	C₂²×C₈	1	4	yes	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	8	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	4	no	—
131	C₂×C₄×C₅₃	—	4	4	no	—
132	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₅₃	#131	1	2	yes	—
137	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
140	C₈×C₅₃	—	2	4	no	—
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	—	2	1	no	—
143	C₈×C₅₃	—	2	2	no	—
144	C₈×C₅₃	—	4	4	no	—
145	C₈×C₅₃	C₂²×C₈	1	2	yes	—
146	C₈×C₅₃	C₂²×C₈	1	2	yes	—
147	C₈×C₅₃	#135	1	4	yes	direct product factor
148	C₈×#135	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	2	1	no	—
151	C₂²×C₅₃	—	2	4	no	—
152	C₂²×C₅₃	—	4	2	no	—
153	C₂²×C₅₃	—	4	2	no	—
154	C₂²×C₅₃	C₈×D₄	1	2	yes	—
155	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
156	C₄×C₅₃	—	2	4	no	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	—	4	2	no	—
160	C₄×C₅₃	—	4	2	no	—
161	C₄×C₅₃	C₈×D₄	1	2	yes	—
162	C₄×C₅₃	#443	1	2	yes	—
163	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
164	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
165	C₄×C₅₃	C₂×#135	1	2	yes	—
166	C₈×#41	C₂×C₅₃	1	2	yes	complemented
167	C₄×#135	C₂×C₅₃	1	1	yes	characteristic
168	C₈×Q₁₆	C₂×C₅₃	1	2	yes	complemented
169	C₈×SD₁₆	C₂×C₅₃	1	4	yes	complemented
170	C₈×D₈	C₂×C₅₃	1	2	yes	complemented
171	#1564	C₂×C₅₃	1	1	yes	characteristic, complemented
172	#1476	C₂×C₅₃	1	2	yes	complemented
173	C₂×C₈²	C₂×C₅₃	1	1	yes	characteristic, complemented
174	C₂×C₅₃	—	2	2	no	—
175	C₂×C₅₃	—	4	4	no	—
176	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
177	C₂×C₅₃	C₄×#135	1	1	yes	characteristic
178	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
179	C₂×#135	C₄×C₅₃	1	1	yes	characteristic
180	C₄×#41	C₂²×C₅₃	1	2	yes	—
181	C₈×Q₈	C₂²×C₅₃	1	4	yes	—
182	C₄×Q₁₆	C₄×C₅₃	1	1	yes	characteristic
183	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
184	C₄×SD₁₆	C₄×C₅₃	1	2	yes	—
185	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	—
186	C₄×D₈	C₄×C₅₃	1	1	yes	characteristic
187	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic
188	C₈×D₄	—	2	4	no	—
189	C₈×D₄	C₂²×C₅₃	1	4	yes	—
190	#431	—	2	2	no	—
191	#431	C₂²×C₅₃	1	2	yes	—
192	#427	C₄×C₅₃	1	1	yes	characteristic
193	#416	C₄×C₅₃	1	2	yes	—
194	C₂×C₄×C₈	—	2	2	no	—
195	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
196	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
197	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
198	#335	C₂²×C₅₃	1	2	yes	—
199	#334	C₂²×C₅₃	1	2	yes	—
200	Q₈⋊C₈	C₂²×C₅₃	1	4	yes	—
201	D₄⋊C₈	C₂²×C₅₃	1	4	yes	—
202	C₈²	C₂²×C₅₃	1	2	yes	—
203	C₈²	C₂²×C₅₃	1	2	yes	—
204	C₅₃	C₈×#135	1	1	yes	characteristic, direct product factor, Sylow
205	C₂×#41	C₂×C₄×C₅₃	1	2	yes	—
206	#135	C₈×C₅₃	1	8	yes	direct product factor
207	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
208	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
209	C₂×D₈	C₂×C₄×C₅₃	1	1	yes	characteristic
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	2	4	no	—
212	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
213	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
214	C₂²×C₈	D₄×C₅₃	1	2	yes	—
215	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
216	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
217	C₄×D₄	—	2	2	no	—
218	C₄×D₄	—	2	2	no	—
219	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
220	C₄×D₄	C₂³×C₅₃	1	2	yes	—
221	#117	—	2	2	no	—
222	#117	C₂×C₄×C₅₃	1	2	yes	—
223	C₂×C₄²	—	2	2	no	—
224	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
225	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
226	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
227	#106	C₂×C₄×C₅₃	1	2	yes	—
228	C₄⋊C₈	—	2	4	no	—
229	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
230	#103	C₂×C₄×C₅₃	1	2	yes	—
231	#103	C₂×C₄×C₅₃	1	2	yes	—
232	#102	C₂×C₄×C₅₃	1	2	yes	—
233	#102	C₂×C₄×C₅₃	1	2	yes	—
234	C₂²⋊C₈	—	2	2	no	—
235	C₂²⋊C₈	—	2	4	no	—
236	C₄×C₈	—	2	4	no	—
237	C₄×C₈	—	2	2	no	—
238	C₄×C₈	—	2	1	no	—
239	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
240	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
241	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
242	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₅₃	1	2	yes	—
244	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
245	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
246	C₄×C₈	C₂³×C₅₃	1	2	yes	—
247	#41	—	2	4	no	—
248	#41	C₂×C₈×C₅₃	1	8	yes	complemented
249	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
257	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
258	Q₁₆	C₂×C₈×C₅₃	1	4	yes	complemented
259	SD₁₆	C₂×C₈×C₅₃	1	8	yes	complemented
260	D₈	C₂×C₈×C₅₃	1	4	yes	complemented
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	4	8	no	—
267	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	complemented
268	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
269	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
270	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
271	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
272	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
273	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
277	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
278	C₂²⋊C₄	—	2	2	no	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
285	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
286	C₄²	#41×C₅₃	1	1	yes	characteristic
287	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
288	C₂³	—	2	2	no	—
289	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
290	Q₈	—	2	2	no	—
291	Q₈	C₂²×C₈×C₅₃	1	4	yes	—
292	D₄	—	2	8	no	—
293	D₄	—	2	2	no	—
294	D₄	C₂²×C₈×C₅₃	1	4	yes	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	8	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
304	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
305	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
307	C₂×C₄	#131×C₅₃	1	2	yes	—
308	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
309	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
310	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
311	C₈	—	2	4	no	—
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	2	no	—
315	C₈	—	4	4	no	—
316	C₈	C₂²×C₈×C₅₃	1	2	yes	—
317	C₈	C₂²×C₈×C₅₃	1	2	yes	—
318	C₈	#135×C₅₃	1	4	yes	direct product factor
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	4	no	—
322	C₂²	—	4	2	no	—
323	C₂²	—	4	2	no	—
324	C₂²	C₈×D₄×C₅₃	1	2	yes	—
325	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	4	no	—
329	C₄	—	4	2	no	—
330	C₄	—	4	2	no	—
331	C₄	C₈×D₄×C₅₃	1	2	yes	—
332	C₄	C₅₃×#443	1	2	yes	—
333	C₄	C₂³×C₈×C₅₃	1	1	yes	characteristic
334	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
335	C₄	C₂×#135×C₅₃	1	2	yes	—
336	C₂	—	2	2	no	—
337	C₂	—	4	4	no	—
338	C₂	C₂×C₈×D₄×C₅₃	1	1	yes	characteristic
339	C₂	C₄×#135×C₅₃	1	1	yes	characteristic
340	C₂	C₂×C₅₃×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈×C₅₃	D₄
commutator	C₄	C₂³×C₈×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#135×C₅₃	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈×#41×C₅₃	C₂	semidirect
3. C₈×Q₁₆×C₅₃	C₂	semidirect
4. C₈×SD₁₆×C₅₃	C₂	semidirect
5. C₈×D₈×C₅₃	C₂	semidirect
6. C₅₃×#1564	C₂	semidirect
7. C₅₃×#1476	C₂	semidirect
8. C₂×C₈²×C₅₃	C₂	semidirect
35. #135×C₅₃	C₈	direct (N × H)
77. #41×C₅₃	C₂×C₈	semidirect
87. Q₁₆×C₅₃	C₂×C₈	semidirect
88. SD₁₆×C₅₃	C₂×C₈	semidirect
89. D₈×C₅₃	C₂×C₈	semidirect
96. C₂×C₈×C₅₃	C₂×C₈	semidirect
147. C₈×C₅₃	#135	direct (N × H)
148. C₈×#135	C₅₃	direct (N × H)
166. C₈×#41	C₂×C₅₃	semidirect
168. C₈×Q₁₆	C₂×C₅₃	semidirect
169. C₈×SD₁₆	C₂×C₅₃	semidirect
170. C₈×D₈	C₂×C₅₃	semidirect
171. #1564	C₂×C₅₃	semidirect
172. #1476	C₂×C₅₃	semidirect
173. C₂×C₈²	C₂×C₅₃	semidirect
204. C₅₃	C₈×#135	direct (N × H)
206. #135	C₈×C₅₃	direct (N × H)
248. #41	C₂×C₈×C₅₃	semidirect
258. Q₁₆	C₂×C₈×C₅₃	semidirect
259. SD₁₆	C₂×C₈×C₅₃	semidirect
260. D₈	C₂×C₈×C₅₃	semidirect
267. C₂×C₈	C₂×C₈×C₅₃	semidirect
318. C₈	#135×C₅₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈×#41×C₅₃	2
C₄×#135×C₅₃	2
C₈×Q₁₆×C₅₃	2
C₈×SD₁₆×C₅₃	2
C₈×D₈×C₅₃	2
C₅₃×#1564	2
C₅₃×#1476	2
C₂×C₈²×C₅₃	2
C₈×#135	53

Subgroups — C8×#135×C53 #27225357

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#135×C₅₃ #27225357