Subgroups — C₂²×#16810686 #28247921

Order 13568.

3694 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	C₂⁴×#88322	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄×C₂²⋊D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810686	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#13133146	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#13133129	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₂²⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₂²⋊D₅₃	C₄	1	1	yes	characteristic
10	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
11	C₂³×#88322	—	2	8	no	—
12	C₂³×#88322	C₂²	1	1	yes	characteristic
13	C₂³×#88322	C₂²	1	1	yes	characteristic
14	C₂³×#88322	C₂²	1	12	yes	complemented
15	C₂²×C₈×D₅₃	—	2	8	no	—
16	C₂²×C₈×D₅₃	C₂²	1	12	yes	complemented
17	C₂×C₄×C₂²⋊D₅₃	C₄	1	12	yes	—
18	C₂×C₄×C₂²⋊D₅₃	C₂²	1	12	yes	—
19	C₂²×#11719919	C₄	1	1	yes	characteristic
20	C₂²×C₂²⋊Dic₅₃	C₄	1	1	yes	characteristic
21	C₂²×#11719921	C₄	1	1	yes	characteristic
22	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
23	#16810686	C₂²	1	64	yes	direct product factor
24	C₂×#13133146	C₂²	1	12	yes	complemented
25	C₂×#13133129	C₂²	1	12	yes	complemented
26	C₂×C₈×Dic₅₃	C₂²	1	12	yes	complemented
27	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
28	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₈×C₅₃	C₂²	1	12	yes	complemented
31	C₂⁴×D₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₂²⋊D₅₃	C₈	1	16	yes	complemented
33	C₂²×C₂²⋊D₅₃	C₂×C₄	1	12	yes	—
34	C₂³×Dic₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂³×Dic₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂³×Dic₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×D₅₃	—	2	4	no	—
38	C₂²×C₄×D₅₃	—	2	4	no	—
39	C₂²×C₄×D₅₃	C₂×C₄	1	6	yes	—
40	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
41	C₂²×#88322	—	2	48	no	—
42	C₂²×#88322	—	2	4	no	—
43	C₂²×#88322	D₄	1	8	yes	complemented
44	C₂²×#88322	C₂³	1	6	yes	—
45	C₂²×#88322	C₂³	1	6	yes	—
46	C₂²×#88322	C₂³	1	16	yes	complemented
47	C₂×C₈×D₅₃	—	2	48	no	—
48	C₂×C₈×D₅₃	C₂³	1	16	yes	complemented
49	C₄×C₂²⋊D₅₃	C₂×C₄	1	48	yes	—
50	C₄×C₂²⋊D₅₃	C₂³	1	16	yes	—
51	C₂×#11719919	C₂×C₄	1	12	yes	—
52	C₂×C₂²⋊Dic₅₃	C₂×C₄	1	12	yes	—
53	C₂×#11719921	C₂×C₄	1	12	yes	—
54	C₂×C₄×Dic₅₃	C₂×C₄	1	6	yes	—
55	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
56	#13133146	C₂³	1	16	yes	complemented
57	#13133129	C₂³	1	16	yes	complemented
58	C₈×Dic₅₃	C₂³	1	16	yes	complemented
59	C₂⁵×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂³×C₄×C₅₃	—	2	4	no	—
61	C₂³×C₄×C₅₃	—	2	4	no	—
62	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂³×C₄×C₅₃	C₂×C₄	1	6	yes	—
64	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
65	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₅₃	—	2	4	no	—
67	C₂²×C₈×C₅₃	—	2	4	no	—
68	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
69	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
70	C₂²⋊C₈×C₅₃	C₂³	1	16	yes	complemented
71	C₂⁴×C₅₃	—	2	4	no	—
72	C₂⁴×C₅₃	—	2	4	no	—
73	C₂⁴×C₅₃	C₂×C₈	1	8	yes	complemented
74	C₂⁴×C₅₃	C₂²×C₄	1	6	yes	—
75	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂³×D₅₃	—	2	4	no	—
77	C₂³×D₅₃	—	2	4	no	—
78	C₂³×D₅₃	C₂×C₈	1	8	yes	complemented
79	C₂³×D₅₃	C₂²×C₄	1	6	yes	—
80	C₂²×C₄×C₅₃	—	2	4	no	—
81	C₂²×C₄×C₅₃	—	2	24	no	—
82	C₂²×C₄×C₅₃	—	2	12	no	—
83	C₂²×C₄×C₅₃	—	2	12	no	—
84	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
85	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
86	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
87	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
88	C₂²×C₄×C₅₃	#41	1	4	yes	—
89	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
90	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
91	C₂×C₂²⋊D₅₃	C₂×C₈	1	96	yes	complemented
92	C₂×C₂²⋊D₅₃	C₂²×C₄	1	16	yes	—
93	C₂²×Dic₅₃	—	2	4	no	—
94	C₂²×Dic₅₃	—	2	4	no	—
95	C₂²×Dic₅₃	C₂×C₈	1	8	yes	complemented
96	C₂²×Dic₅₃	C₂²×C₄	1	6	yes	—
97	C₂²×Dic₅₃	C₂²×C₄	1	6	yes	—
98	C₂²×Dic₅₃	C₂²×C₄	1	6	yes	—
99	C₂×C₄×D₅₃	—	2	24	no	—
100	C₂×C₄×D₅₃	—	2	12	no	—
101	C₂×C₄×D₅₃	—	2	12	no	—
102	C₂×C₄×D₅₃	C₂²×C₄	1	12	yes	—
103	C₂×C₄×D₅₃	C₂⁴	1	4	yes	—
104	C₂×C₈×C₅₃	—	2	12	no	—
105	C₂×C₈×C₅₃	—	2	12	no	—
106	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
107	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
108	C₂²⋊Dic₅₃	C₂²×C₄	1	16	yes	—
109	#11719921	C₂²×C₄	1	16	yes	—
110	#11719919	C₂²×C₄	1	16	yes	—
111	C₄×Dic₅₃	C₂²×C₄	1	12	yes	—
112	C₄×Dic₅₃	C₂⁴	1	4	yes	—
113	C₂×#88322	—	2	12	no	—
114	C₂×#88322	—	2	32	no	—
115	C₂×#88322	C₂×D₄	1	24	yes	complemented
116	C₂×#88322	C₂⁴	1	4	yes	—
117	C₂×#88322	C₂⁴	1	4	yes	—
118	C₈×D₅₃	—	2	32	no	—
119	C₂³×C₅₃	—	2	12	no	—
120	C₂³×C₅₃	—	2	12	no	—
121	C₂³×C₅₃	—	2	32	no	—
122	C₂³×C₅₃	C₄×D₄	1	4	yes	—
123	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
124	C₂³×C₅₃	C₂²×C₈	1	24	yes	complemented
125	C₂³×C₅₃	#131	1	4	yes	—
126	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
127	C₂³×C₅₃	C₂³×C₄	1	3	yes	—
128	C₂²×D₅₃	—	2	12	no	—
129	C₂²×D₅₃	—	2	12	no	—
130	C₂²×D₅₃	—	2	32	no	—
131	C₂²×D₅₃	C₂²×C₈	1	24	yes	complemented
132	C₂²×D₅₃	C₂³×C₄	1	4	yes	—
133	C₂×C₄×C₅₃	—	2	12	no	—
134	C₂×C₄×C₅₃	—	2	24	no	—
135	C₂×C₄×C₅₃	—	2	4	no	—
136	C₂×C₄×C₅₃	—	2	4	no	—
137	C₂×C₄×C₅₃	C₄×D₄	1	12	yes	—
138	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
139	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
140	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
141	C₂×C₄×C₅₃	C₂×#41	1	6	yes	—
142	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
143	C₂²⋊D₅₃	C₂²×C₈	1	64	yes	complemented
144	C₂×Dic₅₃	—	2	12	no	—
145	C₂×Dic₅₃	—	2	12	no	—
146	C₂×Dic₅₃	C₂²×C₈	1	24	yes	complemented
147	C₂×Dic₅₃	C₂³×C₄	1	4	yes	—
148	C₂×Dic₅₃	C₂³×C₄	1	4	yes	—
149	C₂×Dic₅₃	C₂³×C₄	1	4	yes	—
150	C₄×D₅₃	—	2	24	no	—
151	C₄×D₅₃	—	2	4	no	—
152	C₄×D₅₃	—	2	4	no	—
153	C₈×C₅₃	—	2	4	no	—
154	C₈×C₅₃	—	2	4	no	—
155	#88322	—	2	4	no	—
156	#88322	C₂²×D₄	1	8	yes	complemented
157	C₂²×C₈×D₄	—	53	1	no	maximal, Sylow
158	C₂²×C₅₃	—	2	4	no	—
159	C₂²×C₅₃	—	2	48	no	—
160	C₂²×C₅₃	—	2	4	no	—
161	C₂²×C₅₃	C₈×D₄	1	16	yes	complemented
162	C₂²×C₅₃	C₂×C₄×D₄	1	6	yes	—
163	C₂²×C₅₃	C₂³×C₈	1	6	yes	—
164	C₂²×C₅₃	C₂³×C₈	1	8	yes	complemented
165	C₂²×C₅₃	C₂×#131	1	6	yes	—
166	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
167	C₂×D₅₃	—	2	4	no	—
168	C₂×D₅₃	—	2	48	no	—
169	C₂×D₅₃	—	2	4	no	—
170	C₂×D₅₃	C₂³×C₈	1	8	yes	complemented
171	C₄×C₅₃	—	2	4	no	—
172	C₄×C₅₃	C₂×C₄×D₄	1	6	yes	—
173	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
174	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
175	Dic₅₃	—	2	4	no	—
176	Dic₅₃	—	2	4	no	—
177	Dic₅₃	C₂³×C₈	1	8	yes	complemented
178	C₂⁴×C₈	—	53	1	no	—
179	C₂⁴×C₈	—	53	1	no	—
180	C₂²×C₄×D₄	—	53	1	no	—
181	C₂×C₈×D₄	—	53	24	no	—
182	C₂²×C₄⋊C₈	—	53	1	no	—
183	C₂²×C₂²⋊C₈	—	53	1	no	—
184	C₂²×C₂²⋊C₈	D₅₃	1	1	yes	characteristic, complemented
185	C₂²×C₄×C₈	—	53	1	no	—
186	C₂×C₅₃	—	2	8	no	—
187	C₂×C₅₃	C₂×C₈×D₄	1	12	yes	complemented
188	C₂×C₅₃	C₂²×C₄×D₄	1	1	yes	characteristic
189	C₂×C₅₃	C₂⁴×C₈	1	1	yes	characteristic
190	C₂×C₅₃	C₂²×#131	1	1	yes	characteristic
191	D₅₃	—	2	8	no	—
192	C₂³×D₄	—	53	1	no	—
193	C₂⁴×C₄	—	53	1	no	—
194	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
195	C₂³×C₈	—	53	1	no	—
196	C₂³×C₈	—	53	12	no	—
197	C₂³×C₈	—	53	1	no	—
198	C₂³×C₈	—	53	12	no	—
199	C₂³×C₈	—	106	8	no	—
200	C₂³×C₈	—	106	8	no	—
201	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
202	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
203	C₂×C₄×D₄	—	53	12	no	—
204	C₂×C₄×D₄	—	53	12	no	—
205	C₂²×C₄⋊C₄	—	53	1	no	—
206	C₂²×C₂²⋊C₄	—	53	1	no	—
207	C₂²×C₂²⋊C₄	—	53	1	no	—
208	C₂²×C₄²	—	53	1	no	—
209	C₈×D₄	—	53	64	no	—
210	C₂×C₄⋊C₈	—	53	12	no	—
211	C₂×C₂²⋊C₈	—	53	12	no	—
212	C₂×C₂²⋊C₈	C₂×D₅₃	1	12	yes	complemented
213	C₂×C₄×C₈	—	53	12	no	—
214	C₅₃	C₂²×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
215	C₂⁵	—	53	1	no	—
216	C₂⁵	C₄×D₅₃	1	1	yes	characteristic
217	C₂²×D₄	—	53	12	no	—
218	C₂²×D₄	—	53	16	no	—
219	C₂³×C₄	—	2	4	no	—
220	C₂³×C₄	—	2	4	no	—
221	C₂³×C₄	—	53	1	no	—
222	C₂³×C₄	—	53	1	no	—
223	C₂³×C₄	—	53	1	no	—
224	C₂³×C₄	—	53	6	no	—
225	C₂³×C₄	—	53	6	no	—
226	C₂³×C₄	—	106	4	no	—
227	C₂³×C₄	—	106	4	no	—
228	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
229	C₂³×C₄	C₄×D₅₃	1	6	yes	—
230	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
231	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
232	C₂²×C₈	—	2	4	no	—
233	C₂²×C₈	—	2	4	no	—
234	C₂²×C₈	—	53	8	no	—
235	C₂²×C₈	—	53	6	no	—
236	C₂²×C₈	—	53	6	no	—
237	C₂²×C₈	—	53	16	no	—
238	C₂²×C₈	—	53	16	no	—
239	C₂²×C₈	—	106	48	no	—
240	C₂²×C₈	—	106	48	no	—
241	C₂²×C₈	—	106	4	no	—
242	C₂²×C₈	C₂²×D₅₃	1	6	yes	—
243	C₂²×C₈	C₂²×D₅₃	1	6	yes	—
244	C₄×D₄	—	53	48	no	—
245	C₄×D₄	—	53	16	no	—
246	C₂×C₄⋊C₄	—	53	12	no	—
247	C₂×C₂²⋊C₄	—	53	12	no	—
248	C₂×C₂²⋊C₄	—	53	12	no	—
249	C₂×C₄²	—	53	6	no	—
250	C₂×C₄²	—	53	6	no	—
251	C₄⋊C₈	—	53	16	no	—
252	C₂²⋊C₈	—	53	16	no	—
253	C₂²⋊C₈	C₂²×D₅₃	1	16	yes	complemented
254	C₄×C₈	—	53	16	no	—
255	C₂⁴	—	2	4	no	—
256	C₂⁴	—	2	4	no	—
257	C₂⁴	—	53	8	no	—
258	C₂⁴	—	53	6	no	—
259	C₂⁴	—	106	4	no	—
260	C₂⁴	—	106	4	no	—
261	C₂⁴	C₈×D₅₃	1	8	yes	complemented
262	C₂⁴	C₂×C₄×D₅₃	1	6	yes	—
263	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
264	C₂×D₄	—	53	96	no	—
265	C₂×D₄	—	53	16	no	—
266	C₂²×C₄	—	2	4	no	—
267	C₂²×C₄	—	2	24	no	—
268	C₂²×C₄	—	2	12	no	—
269	C₂²×C₄	—	2	12	no	—
270	C₂²×C₄	—	53	8	no	—
271	C₂²×C₄	—	53	6	no	—
272	C₂²×C₄	—	53	6	no	—
273	C₂²×C₄	—	53	6	no	—
274	C₂²×C₄	—	53	12	no	—
275	C₂²×C₄	—	53	4	no	—
276	C₂²×C₄	—	106	4	no	—
277	C₂²×C₄	—	106	4	no	—
278	C₂²×C₄	—	106	24	no	—
279	C₂²×C₄	—	106	12	no	—
280	C₂²×C₄	—	106	12	no	—
281	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
282	C₂²×C₄	C₂×C₄×D₅₃	1	12	yes	—
283	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
284	C₂²×C₄	D₄×D₅₃	1	4	yes	—
285	C₂²×C₄	#11719947	1	4	yes	—
286	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
287	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
288	C₂×C₈	—	2	12	no	—
289	C₂×C₈	—	2	12	no	—
290	C₂×C₈	—	53	24	no	—
291	C₂×C₈	—	53	4	no	—
292	C₂×C₈	—	53	4	no	—
293	C₂×C₈	—	106	32	no	—
294	C₂×C₈	—	106	32	no	—
295	C₂×C₈	—	106	12	no	—
296	C₂×C₈	C₂³×D₅₃	1	4	yes	—
297	C₂×C₈	C₂³×D₅₃	1	4	yes	—
298	C₄⋊C₄	—	53	16	no	—
299	C₂²⋊C₄	—	53	16	no	—
300	C₂²⋊C₄	—	53	16	no	—
301	C₄²	—	53	12	no	—
302	C₄²	—	53	4	no	—
303	C₂³	—	2	12	no	—
304	C₂³	—	2	12	no	—
305	C₂³	—	2	32	no	—
306	C₂³	—	53	24	no	—
307	C₂³	—	53	4	no	—
308	C₂³	—	106	12	no	—
309	C₂³	—	106	12	no	—
310	C₂³	—	106	32	no	—
311	C₂³	#13133177	1	4	yes	—
312	C₂³	C₂×C₈×D₅₃	1	4	yes	—
313	C₂³	C₂×C₈×D₅₃	1	24	yes	complemented
314	C₂³	#13133225	1	4	yes	—
315	C₂³	C₂²×C₄×D₅₃	1	4	yes	—
316	C₂³	C₂²×C₄×D₅₃	1	3	yes	—
317	D₄	—	53	64	no	—
318	C₂×C₄	—	2	12	no	—
319	C₂×C₄	—	2	24	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	53	24	no	—
323	C₂×C₄	—	53	4	no	—
324	C₂×C₄	—	53	4	no	—
325	C₂×C₄	—	53	4	no	—
326	C₂×C₄	—	106	12	no	—
327	C₂×C₄	—	106	12	no	—
328	C₂×C₄	—	106	24	no	—
329	C₂×C₄	—	106	4	no	—
330	C₂×C₄	—	106	4	no	—
331	C₂×C₄	#13133177	1	12	yes	—
332	C₂×C₄	C₂²×C₄×D₅₃	1	4	yes	—
333	C₂×C₄	C₂²×C₄×D₅₃	1	3	yes	—
334	C₂×C₄	C₂×D₄×D₅₃	1	6	yes	—
335	C₂×C₄	C₂×#11719947	1	6	yes	—
336	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
337	C₈	—	2	4	no	—
338	C₈	—	2	4	no	—
339	C₈	—	53	8	no	—
340	C₈	—	106	4	no	—
341	C₂²	—	2	4	no	—
342	C₂²	—	2	4	no	—
343	C₂²	—	2	48	no	—
344	C₂²	—	53	8	no	—
345	C₂²	—	106	4	no	—
346	C₂²	—	106	4	no	—
347	C₂²	—	106	48	no	—
348	C₂²	#16810686	1	16	yes	direct product factor
349	C₂²	C₂×#13133177	1	6	yes	—
350	C₂²	C₂²×C₈×D₅₃	1	6	yes	—
351	C₂²	C₂²×C₈×D₅₃	1	8	yes	complemented
352	C₂²	C₂×#13133225	1	6	yes	—
353	C₂²	C₂³×C₄×D₅₃	1	1	yes	characteristic
354	C₄	—	2	4	no	—
355	C₄	—	53	8	no	—
356	C₄	—	106	4	no	—
357	C₄	—	106	4	no	—
358	C₄	C₂×#13133177	1	6	yes	—
359	C₄	C₂²×D₄×D₅₃	1	1	yes	characteristic
360	C₄	C₂²×#11719947	1	1	yes	characteristic
361	C₂	—	2	8	no	—
362	C₂	—	106	8	no	—
363	C₂	C₂×#16810686	1	12	yes	direct product factor
364	C₂	C₂²×#13133177	1	1	yes	characteristic
365	C₂	C₂³×C₈×D₅₃	1	1	yes	characteristic
366	C₂	C₂²×#13133225	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂²×C₂²⋊C₈×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂²×C₄
prime radical	C₂³×C₂²⋊D₅₃	C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂²×C₂²⋊C₈	D₅₃
hypocenter	C₅₃	C₂²×C₈×D₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂⁴×#88322	C₂	semidirect
2. C₂³×C₈×D₅₃	C₂	semidirect
4. C₂×#16810686	C₂	direct (N × H)
5. C₂²×#13133146	C₂	semidirect
6. C₂²×#13133129	C₂	semidirect
7. C₂²×C₈×Dic₅₃	C₂	semidirect
8. C₂²×C₂²⋊C₈×C₅₃	C₂	semidirect
14. C₂³×#88322	C₂²	semidirect
16. C₂²×C₈×D₅₃	C₂²	semidirect
23. #16810686	C₂²	direct (N × H)
24. C₂×#13133146	C₂²	semidirect
25. C₂×#13133129	C₂²	semidirect
26. C₂×C₈×Dic₅₃	C₂²	semidirect
30. C₂×C₂²⋊C₈×C₅₃	C₂²	semidirect
32. C₂²×C₂²⋊D₅₃	C₈	semidirect
43. C₂²×#88322	D₄	semidirect
46. C₂²×#88322	C₂³	semidirect
48. C₂×C₈×D₅₃	C₂³	semidirect
56. #13133146	C₂³	semidirect
57. #13133129	C₂³	semidirect
58. C₈×Dic₅₃	C₂³	semidirect
70. C₂²⋊C₈×C₅₃	C₂³	semidirect
73. C₂⁴×C₅₃	C₂×C₈	semidirect
78. C₂³×D₅₃	C₂×C₈	semidirect
91. C₂×C₂²⋊D₅₃	C₂×C₈	semidirect
95. C₂²×Dic₅₃	C₂×C₈	semidirect
115. C₂×#88322	C₂×D₄	semidirect
124. C₂³×C₅₃	C₂²×C₈	semidirect
131. C₂²×D₅₃	C₂²×C₈	semidirect
143. C₂²⋊D₅₃	C₂²×C₈	semidirect
146. C₂×Dic₅₃	C₂²×C₈	semidirect
156. #88322	C₂²×D₄	semidirect
161. C₂²×C₅₃	C₈×D₄	semidirect
164. C₂²×C₅₃	C₂³×C₈	semidirect
170. C₂×D₅₃	C₂³×C₈	semidirect
177. Dic₅₃	C₂³×C₈	semidirect
184. C₂²×C₂²⋊C₈	D₅₃	semidirect
187. C₂×C₅₃	C₂×C₈×D₄	semidirect
212. C₂×C₂²⋊C₈	C₂×D₅₃	semidirect
214. C₅₃	C₂²×C₈×D₄	semidirect
253. C₂²⋊C₈	C₂²×D₅₃	semidirect
261. C₂⁴	C₈×D₅₃	semidirect
313. C₂³	C₂×C₈×D₅₃	semidirect
348. C₂²	#16810686	direct (N × H)
351. C₂²	C₂²×C₈×D₅₃	semidirect
363. C₂	C₂×#16810686	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×#88322	2
C₂³×C₈×D₅₃	2
C₂²×C₄×C₂²⋊D₅₃	2
C₂×#16810686	2
C₂²×#13133146	2
C₂²×#13133129	2
C₂²×C₈×Dic₅₃	2
C₂²×C₂²⋊C₈×C₅₃	2
C₂²×C₈×D₄	53

Subgroups — C22×#16810686 #28247921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810686 #28247921