Subgroups — C₂×C₅₃×#2308 #27241552

Order 13568.

8142 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₂²⋊C₄×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₅₃×#2308	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#386	C₂	1	8	yes	maximal, complemented
5	C₂×C₅₃×#381	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#379	C₂	1	2	yes	maximal, complemented
7	C₂⁶×C₅₃	C₂²	1	1	yes	characteristic
8	C₂⁶×C₅₃	C₂²	1	1	yes	characteristic
9	C₂³×D₄×C₅₃	—	2	16	no	—
10	C₂³×D₄×C₅₃	C₄	1	16	yes	complemented
11	C₂³×D₄×C₅₃	C₂²	1	4	yes	—
12	C₂³×D₄×C₅₃	C₂²	1	8	yes	—
13	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₅₃	C₂²	1	4	yes	complemented
15	C₂²×C₂²⋊C₄×C₅₃	—	2	16	no	—
16	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
17	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	16	yes	complemented
18	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
19	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	complemented
20	C₅₃×#386	C₂²	1	64	yes	complemented
21	C₅₃×#381	C₂²	1	16	yes	complemented
22	C₅₃×#379	C₂²	1	16	yes	complemented
23	C₂×#95×C₅₃	C₂²	1	8	yes	complemented
24	C₂⁵×C₅₃	—	2	8	no	—
25	C₂⁵×C₅₃	—	4	8	no	—
26	C₂⁵×C₅₃	—	4	8	no	—
27	C₂⁵×C₅₃	C₂×C₄	1	8	yes	—
28	C₂⁵×C₅₃	C₂×C₄	1	8	yes	complemented
29	C₂⁵×C₅₃	D₄	1	16	yes	—
30	C₂⁵×C₅₃	C₂³	1	2	yes	—
31	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
32	C₂⁵×C₅₃	C₂³	1	2	yes	—
33	C₂⁵×C₅₃	C₂³	1	4	yes	—
34	C₂⁵×C₅₃	C₂³	1	4	yes	—
35	C₂²×D₄×C₅₃	—	2	128	no	—
36	C₂²×D₄×C₅₃	—	2	16	no	—
37	C₂²×D₄×C₅₃	—	2	16	no	—
38	C₂²×D₄×C₅₃	—	2	64	no	—
39	C₂²×D₄×C₅₃	C₂×C₄	1	32	yes	complemented
40	C₂²×D₄×C₅₃	C₂×C₄	1	64	yes	complemented
41	C₂²×D₄×C₅₃	C₂³	1	16	yes	—
42	C₂³×C₄×C₅₃	—	2	8	no	—
43	C₂³×C₄×C₅₃	—	2	16	no	—
44	C₂³×C₄×C₅₃	—	4	16	no	—
45	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	complemented
46	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
47	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
48	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
49	C₂³×C₄×C₅₃	C₂³	1	16	yes	complemented
50	C₂×C₂²⋊C₄×C₅₃	—	2	16	no	—
51	C₂×C₂²⋊C₄×C₅₃	—	2	16	no	—
52	C₂×C₂²⋊C₄×C₅₃	—	2	64	no	—
53	C₂×C₂²⋊C₄×C₅₃	—	4	16	no	—
54	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
55	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	complemented
56	C₂×C₂²⋊C₄×C₅₃	C₂³	1	32	yes	complemented
57	#95×C₅₃	C₂³	1	32	yes	complemented
58	C₂⁴×C₅₃	—	2	16	no	—
59	C₂⁴×C₅₃	—	2	8	no	—
60	C₂⁴×C₅₃	—	2	32	no	—
61	C₂⁴×C₅₃	—	2	8	no	—
62	C₂⁴×C₅₃	—	2	16	no	—
63	C₂⁴×C₅₃	—	2	16	no	—
64	C₂⁴×C₅₃	—	4	32	no	—
65	C₂⁴×C₅₃	—	4	32	no	—
66	C₂⁴×C₅₃	—	4	64	no	—
67	C₂⁴×C₅₃	—	4	16	no	—
68	C₂⁴×C₅₃	—	4	16	no	—
69	C₂⁴×C₅₃	—	4	64	no	—
70	C₂⁴×C₅₃	C₂²⋊C₄	1	64	yes	complemented
71	C₂⁴×C₅₃	C₂²×C₄	1	8	yes	—
72	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
73	C₂⁴×C₅₃	C₂²×C₄	1	8	yes	—
74	C₂⁴×C₅₃	C₂²×C₄	1	16	yes	—
75	C₂⁴×C₅₃	C₂²×C₄	1	16	yes	complemented
76	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
77	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
78	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
79	C₂⁴×C₅₃	C₂×D₄	1	32	yes	—
80	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
81	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
82	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
83	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
84	C₂×D₄×C₅₃	—	2	128	no	—
85	C₂×D₄×C₅₃	—	2	256	no	—
86	C₂×D₄×C₅₃	—	2	32	no	—
87	C₂×D₄×C₅₃	—	2	32	no	—
88	C₂×D₄×C₅₃	C₂²×C₄	1	64	yes	complemented
89	C₂²×C₄×C₅₃	—	2	32	no	—
90	C₂²×C₄×C₅₃	—	2	8	no	—
91	C₂²×C₄×C₅₃	—	2	16	no	—
92	C₂²×C₄×C₅₃	—	2	16	no	—
93	C₂²×C₄×C₅₃	—	2	32	no	—
94	C₂²×C₄×C₅₃	—	2	32	no	—
95	C₂²×C₄×C₅₃	—	4	32	no	—
96	C₂²×C₄×C₅₃	—	4	64	no	—
97	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
98	C₂²×C₄×C₅₃	C₂²×C₄	1	16	yes	complemented
99	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
100	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
101	C₂²⋊C₄×C₅₃	—	2	32	no	—
102	C₂²⋊C₄×C₅₃	—	2	32	no	—
103	C₂²⋊C₄×C₅₃	—	4	32	no	—
104	C₂³×C₅₃	—	2	16	no	—
105	C₂³×C₅₃	—	2	32	no	—
106	C₂³×C₅₃	—	2	32	no	—
107	C₂³×C₅₃	—	2	32	no	—
108	C₂³×C₅₃	—	2	32	no	—
109	C₂³×C₅₃	—	2	16	no	—
110	C₂³×C₅₃	—	2	8	no	—
111	C₂³×C₅₃	—	2	16	no	—
112	C₂³×C₅₃	—	2	32	no	—
113	C₂³×C₅₃	—	2	32	no	—
114	C₂³×C₅₃	—	2	8	no	—
115	C₂³×C₅₃	—	2	16	no	—
116	C₂³×C₅₃	—	4	128	no	—
117	C₂³×C₅₃	—	4	32	no	—
118	C₂³×C₅₃	—	4	32	no	—
119	C₂³×C₅₃	—	4	64	no	—
120	C₂³×C₅₃	—	4	128	no	—
121	C₂³×C₅₃	—	4	64	no	—
122	C₂³×C₅₃	—	4	32	no	—
123	C₂³×C₅₃	C₂×C₂²⋊C₄	1	16	yes	—
124	C₂³×C₅₃	C₂×C₂²⋊C₄	1	32	yes	—
125	C₂³×C₅₃	C₂×C₂²⋊C₄	1	64	yes	complemented
126	C₂³×C₅₃	C₂³×C₄	1	8	yes	—
127	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
128	C₂³×C₅₃	C₂³×C₄	1	8	yes	—
129	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
130	C₂³×C₅₃	C₂²×D₄	1	8	yes	—
131	C₂³×C₅₃	C₂²×D₄	1	8	yes	—
132	C₂³×C₅₃	C₂²×D₄	1	8	yes	—
133	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
134	C₂³×C₅₃	H(5,𝔽₂)	1	8	yes	—
135	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
136	D₄×C₅₃	—	2	128	no	—
137	C₂×C₄×C₅₃	—	2	16	no	—
138	C₂×C₄×C₅₃	—	2	32	no	—
139	C₂×C₄×C₅₃	—	2	8	no	—
140	C₂×C₄×C₅₃	—	2	16	no	—
141	C₂×C₄×C₅₃	—	2	32	no	—
142	C₂×C₄×C₅₃	—	4	64	no	—
143	C₂×C₄×C₅₃	—	4	32	no	—
144	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
145	C₂×#2308	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂²×C₅₃	—	2	32	no	—
147	C₂²×C₅₃	—	2	64	no	—
148	C₂²×C₅₃	—	2	8	no	—
149	C₂²×C₅₃	—	2	16	no	—
150	C₂²×C₅₃	—	2	16	no	—
151	C₂²×C₅₃	—	2	16	no	—
152	C₂²×C₅₃	—	2	16	no	—
153	C₂²×C₅₃	—	2	8	no	—
154	C₂²×C₅₃	—	2	16	no	—
155	C₂²×C₅₃	—	4	64	no	—
156	C₂²×C₅₃	—	4	64	no	—
157	C₂²×C₅₃	—	4	32	no	—
158	C₂²×C₅₃	—	4	16	no	—
159	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	8	yes	—
160	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	16	yes	—
161	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	4	yes	—
162	C₂²×C₅₃	#518	1	8	yes	—
163	C₂²×C₅₃	#534	1	16	yes	—
164	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
165	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
166	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
167	C₄×C₅₃	—	2	8	no	—
168	C₄×C₅₃	—	4	16	no	—
169	C₂⁴×D₄	C₂×C₅₃	1	1	yes	characteristic
170	C₂³×C₂²⋊C₄	C₂×C₅₃	1	2	yes	complemented
171	#2308	C₂×C₅₃	1	16	yes	direct product factor
172	C₂×#386	C₂×C₅₃	1	8	yes	complemented
173	C₂×#381	C₂×C₅₃	1	2	yes	complemented
174	C₂×#379	C₂×C₅₃	1	2	yes	complemented
175	C₂×C₅₃	—	2	8	no	—
176	C₂×C₅₃	—	2	16	no	—
177	C₂×C₅₃	—	4	8	no	—
178	C₂×C₅₃	#2308	1	8	yes	direct product factor
179	C₂×C₅₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
180	C₂×C₅₃	C₂×#518	1	2	yes	—
181	C₂×C₅₃	C₂×#534	1	4	yes	—
182	C₂⁶	C₂²×C₅₃	1	1	yes	characteristic
183	C₂⁶	C₂²×C₅₃	1	1	yes	characteristic
184	C₂³×D₄	—	2	16	no	—
185	C₂³×D₄	C₄×C₅₃	1	16	yes	complemented
186	C₂³×D₄	C₂²×C₅₃	1	8	yes	—
187	C₂³×D₄	C₂²×C₅₃	1	4	yes	—
188	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
189	C₂⁴×C₄	C₂²×C₅₃	1	4	yes	complemented
190	C₂²×C₂²⋊C₄	—	2	16	no	—
191	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
192	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	16	yes	complemented
193	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
194	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	complemented
195	#386	C₂²×C₅₃	1	64	yes	complemented
196	#381	C₂²×C₅₃	1	16	yes	complemented
197	#379	C₂²×C₅₃	1	16	yes	complemented
198	C₂×#95	C₂²×C₅₃	1	8	yes	complemented
199	C₅₃	C₂×#2308	1	1	yes	characteristic, direct product factor, Sylow
200	C₂⁵	—	2	8	no	—
201	C₂⁵	—	4	8	no	—
202	C₂⁵	—	4	8	no	—
203	C₂⁵	C₂×C₄×C₅₃	1	8	yes	—
204	C₂⁵	C₂×C₄×C₅₃	1	8	yes	complemented
205	C₂⁵	D₄×C₅₃	1	16	yes	—
206	C₂⁵	C₂³×C₅₃	1	2	yes	—
207	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
208	C₂⁵	C₂³×C₅₃	1	2	yes	—
209	C₂⁵	C₂³×C₅₃	1	4	yes	—
210	C₂⁵	C₂³×C₅₃	1	4	yes	—
211	C₂²×D₄	—	2	16	no	—
212	C₂²×D₄	—	2	16	no	—
213	C₂²×D₄	—	2	64	no	—
214	C₂²×D₄	—	2	128	no	—
215	C₂²×D₄	C₂×C₄×C₅₃	1	32	yes	complemented
216	C₂²×D₄	C₂×C₄×C₅₃	1	64	yes	complemented
217	C₂²×D₄	C₂³×C₅₃	1	16	yes	—
218	C₂³×C₄	—	2	8	no	—
219	C₂³×C₄	—	2	16	no	—
220	C₂³×C₄	—	4	16	no	—
221	C₂³×C₄	C₂×C₄×C₅₃	1	8	yes	complemented
222	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
223	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
224	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
225	C₂³×C₄	C₂³×C₅₃	1	16	yes	complemented
226	C₂×C₂²⋊C₄	—	2	16	no	—
227	C₂×C₂²⋊C₄	—	2	64	no	—
228	C₂×C₂²⋊C₄	—	2	16	no	—
229	C₂×C₂²⋊C₄	—	4	16	no	—
230	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
231	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	complemented
232	C₂×C₂²⋊C₄	C₂³×C₅₃	1	32	yes	complemented
233	#95	C₂³×C₅₃	1	32	yes	complemented
234	C₂⁴	—	2	16	no	—
235	C₂⁴	—	2	8	no	—
236	C₂⁴	—	2	32	no	—
237	C₂⁴	—	2	8	no	—
238	C₂⁴	—	2	16	no	—
239	C₂⁴	—	2	16	no	—
240	C₂⁴	—	4	32	no	—
241	C₂⁴	—	4	32	no	—
242	C₂⁴	—	4	64	no	—
243	C₂⁴	—	4	16	no	—
244	C₂⁴	—	4	16	no	—
245	C₂⁴	—	4	64	no	—
246	C₂⁴	C₂²⋊C₄×C₅₃	1	64	yes	complemented
247	C₂⁴	C₂²×C₄×C₅₃	1	8	yes	—
248	C₂⁴	C₂²×C₄×C₅₃	1	4	yes	—
249	C₂⁴	C₂²×C₄×C₅₃	1	8	yes	—
250	C₂⁴	C₂²×C₄×C₅₃	1	16	yes	—
251	C₂⁴	C₂²×C₄×C₅₃	1	16	yes	complemented
252	C₂⁴	C₂×D₄×C₅₃	1	8	yes	—
253	C₂⁴	C₂×D₄×C₅₃	1	8	yes	—
254	C₂⁴	C₂×D₄×C₅₃	1	8	yes	—
255	C₂⁴	C₂×D₄×C₅₃	1	32	yes	—
256	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
257	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
258	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
259	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
260	C₂×D₄	—	2	128	no	—
261	C₂×D₄	—	2	256	no	—
262	C₂×D₄	—	2	32	no	—
263	C₂×D₄	—	2	32	no	—
264	C₂×D₄	C₂²×C₄×C₅₃	1	64	yes	complemented
265	C₂²×C₄	—	2	32	no	—
266	C₂²×C₄	—	2	8	no	—
267	C₂²×C₄	—	2	16	no	—
268	C₂²×C₄	—	2	16	no	—
269	C₂²×C₄	—	2	32	no	—
270	C₂²×C₄	—	2	32	no	—
271	C₂²×C₄	—	4	32	no	—
272	C₂²×C₄	—	4	64	no	—
273	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
274	C₂²×C₄	C₂²×C₄×C₅₃	1	16	yes	complemented
275	C₂²×C₄	C₂⁴×C₅₃	1	16	yes	—
276	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
277	C₂²⋊C₄	—	2	32	no	—
278	C₂²⋊C₄	—	2	32	no	—
279	C₂²⋊C₄	—	4	32	no	—
280	C₂³	—	2	16	no	—
281	C₂³	—	2	32	no	—
282	C₂³	—	2	32	no	—
283	C₂³	—	2	32	no	—
284	C₂³	—	2	32	no	—
285	C₂³	—	2	16	no	—
286	C₂³	—	2	8	no	—
287	C₂³	—	2	16	no	—
288	C₂³	—	2	32	no	—
289	C₂³	—	2	32	no	—
290	C₂³	—	2	8	no	—
291	C₂³	—	2	16	no	—
292	C₂³	—	4	32	no	—
293	C₂³	—	4	32	no	—
294	C₂³	—	4	64	no	—
295	C₂³	—	4	128	no	—
296	C₂³	—	4	128	no	—
297	C₂³	—	4	64	no	—
298	C₂³	—	4	32	no	—
299	C₂³	C₂×C₂²⋊C₄×C₅₃	1	16	yes	—
300	C₂³	C₂×C₂²⋊C₄×C₅₃	1	32	yes	—
301	C₂³	C₂×C₂²⋊C₄×C₅₃	1	64	yes	complemented
302	C₂³	C₂³×C₄×C₅₃	1	8	yes	—
303	C₂³	C₂³×C₄×C₅₃	1	4	yes	—
304	C₂³	C₂³×C₄×C₅₃	1	8	yes	—
305	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
306	C₂³	C₂²×D₄×C₅₃	1	8	yes	—
307	C₂³	C₂²×D₄×C₅₃	1	8	yes	—
308	C₂³	C₂²×D₄×C₅₃	1	8	yes	—
309	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
310	C₂³	H(5,𝔽₂)×C₅₃	1	8	yes	—
311	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
312	D₄	—	2	128	no	—
313	C₂×C₄	—	2	16	no	—
314	C₂×C₄	—	2	32	no	—
315	C₂×C₄	—	2	8	no	—
316	C₂×C₄	—	2	16	no	—
317	C₂×C₄	—	2	32	no	—
318	C₂×C₄	—	4	64	no	—
319	C₂×C₄	—	4	32	no	—
320	C₂×C₄	C₂³×C₄×C₅₃	1	8	yes	—
321	C₂²	—	2	32	no	—
322	C₂²	—	2	64	no	—
323	C₂²	—	2	8	no	—
324	C₂²	—	2	16	no	—
325	C₂²	—	2	16	no	—
326	C₂²	—	2	16	no	—
327	C₂²	—	2	16	no	—
328	C₂²	—	2	8	no	—
329	C₂²	—	2	16	no	—
330	C₂²	—	4	64	no	—
331	C₂²	—	4	64	no	—
332	C₂²	—	4	32	no	—
333	C₂²	—	4	16	no	—
334	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	8	yes	—
335	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	16	yes	—
336	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	4	yes	—
337	C₂²	C₅₃×#518	1	8	yes	—
338	C₂²	C₅₃×#534	1	16	yes	—
339	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
340	C₂²	C₂³×D₄×C₅₃	1	2	yes	—
341	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	4	yes	—
342	C₄	—	2	8	no	—
343	C₄	—	4	16	no	—
344	C₂	—	2	8	no	—
345	C₂	—	2	16	no	—
346	C₂	—	4	8	no	—
347	C₂	C₅₃×#2308	1	8	yes	direct product factor
348	C₂	C₂³×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
349	C₂	C₂×C₅₃×#518	1	2	yes	—
350	C₂	C₂×C₅₃×#534	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂⁴×D₄×C₅₃	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₂²⋊C₄×C₅₃	C₂	semidirect
3. C₅₃×#2308	C₂	direct (N × H)
4. C₂×C₅₃×#386	C₂	semidirect
5. C₂×C₅₃×#381	C₂	semidirect
6. C₂×C₅₃×#379	C₂	semidirect
10. C₂³×D₄×C₅₃	C₄	semidirect
14. C₂⁴×C₄×C₅₃	C₂²	semidirect
17. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
18. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
19. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
20. C₅₃×#386	C₂²	semidirect
21. C₅₃×#381	C₂²	semidirect
22. C₅₃×#379	C₂²	semidirect
23. C₂×#95×C₅₃	C₂²	semidirect
28. C₂⁵×C₅₃	C₂×C₄	semidirect
39. C₂²×D₄×C₅₃	C₂×C₄	semidirect
40. C₂²×D₄×C₅₃	C₂×C₄	semidirect
45. C₂³×C₄×C₅₃	C₂×C₄	semidirect
49. C₂³×C₄×C₅₃	C₂³	semidirect
55. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
56. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
57. #95×C₅₃	C₂³	semidirect
70. C₂⁴×C₅₃	C₂²⋊C₄	semidirect
75. C₂⁴×C₅₃	C₂²×C₄	semidirect
88. C₂×D₄×C₅₃	C₂²×C₄	semidirect
98. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
125. C₂³×C₅₃	C₂×C₂²⋊C₄	semidirect
145. C₂×#2308	C₅₃	direct (N × H)
170. C₂³×C₂²⋊C₄	C₂×C₅₃	semidirect
171. #2308	C₂×C₅₃	direct (N × H)
172. C₂×#386	C₂×C₅₃	semidirect
173. C₂×#381	C₂×C₅₃	semidirect
174. C₂×#379	C₂×C₅₃	semidirect
178. C₂×C₅₃	#2308	direct (N × H)
185. C₂³×D₄	C₄×C₅₃	semidirect
189. C₂⁴×C₄	C₂²×C₅₃	semidirect
192. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
193. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
194. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
195. #386	C₂²×C₅₃	semidirect
196. #381	C₂²×C₅₃	semidirect
197. #379	C₂²×C₅₃	semidirect
198. C₂×#95	C₂²×C₅₃	semidirect
199. C₅₃	C₂×#2308	direct (N × H)
204. C₂⁵	C₂×C₄×C₅₃	semidirect
215. C₂²×D₄	C₂×C₄×C₅₃	semidirect
216. C₂²×D₄	C₂×C₄×C₅₃	semidirect
221. C₂³×C₄	C₂×C₄×C₅₃	semidirect
225. C₂³×C₄	C₂³×C₅₃	semidirect
231. C₂×C₂²⋊C₄	C₂³×C₅₃	semidirect
232. C₂×C₂²⋊C₄	C₂³×C₅₃	semidirect
233. #95	C₂³×C₅₃	semidirect
246. C₂⁴	C₂²⋊C₄×C₅₃	semidirect
251. C₂⁴	C₂²×C₄×C₅₃	semidirect
264. C₂×D₄	C₂²×C₄×C₅₃	semidirect
274. C₂²×C₄	C₂²×C₄×C₅₃	semidirect
301. C₂³	C₂×C₂²⋊C₄×C₅₃	semidirect
347. C₂	C₅₃×#2308	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×D₄×C₅₃	2
C₂³×C₂²⋊C₄×C₅₃	2
C₅₃×#2308	2
C₂×C₅₃×#386	2
C₂×C₅₃×#381	2
C₂×C₅₃×#379	2
C₂×#2308	53

Subgroups — C2×C53×#2308 #27241552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2308 #27241552