Subgroups — C₅₃×#60086 #27267598

Order 13568.

4874 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#585	C₂	1	1	yes	maximal, characteristic
2	C₂²×#131×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3428	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2898	C₂	1	8	yes	maximal, complemented
5	C₅₃×#2897	C₂	1	8	yes	maximal, complemented
6	C₅₃×#2896	C₂	1	6	yes	maximal, complemented
7	C₂×C₅₃×#416	C₂	1	6	yes	maximal, complemented
8	C₅₃×#585	—	2	8	no	—
9	C₅₃×#585	C₄	1	16	yes	complemented
10	C₂×2₋¹⁺⁴×C₅₃	—	2	1	no	—
11	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	4	yes	—
12	C₂×H(5,𝔽₂)×C₅₃	—	2	3	no	—
13	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	4	yes	—
14	C₂²×#41×C₅₃	—	2	4	no	—
15	C₂²×#41×C₅₃	C₂²	1	6	yes	—
16	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#131×C₅₃	—	2	8	no	—
18	C₂×#131×C₅₃	C₂²	1	8	yes	complemented
19	C₂²×M₁₆×C₅₃	C₂²	1	3	yes	complemented
20	C₂³×C₈×C₅₃	C₂²	1	3	yes	complemented
21	C₅₃×#520	C₂²	1	8	yes	complemented
22	C₅₃×#519	—	2	1	no	—
23	C₅₃×#518	—	2	3	no	—
24	C₄×#41×C₅₃	—	2	4	no	—
25	C₂×#117×C₅₃	C₂²	1	6	yes	complemented
26	C₅₃×#419	C₂²	1	12	yes	complemented
27	C₅₃×#418	C₂²	1	12	yes	complemented
28	C₅₃×#417	C₂²	1	24	yes	complemented
29	C₅₃×#416	C₂²	1	4	yes	complemented
30	C₅₃×#416	C₂²	1	12	yes	complemented
31	C₅₃×#416	C₂²	1	24	yes	complemented
32	C₂×#103×C₅₃	C₂²	1	12	yes	complemented
33	C₂×#102×C₅₃	C₂²	1	12	yes	complemented
34	2₋¹⁺⁴×C₅₃	—	2	16	no	—
35	2₋¹⁺⁴×C₅₃	—	2	16	no	—
36	2₋¹⁺⁴×C₅₃	C₂×C₄	1	32	yes	complemented
37	H(5,𝔽₂)×C₅₃	—	2	16	no	—
38	H(5,𝔽₂)×C₅₃	—	2	48	no	—
39	H(5,𝔽₂)×C₅₃	C₂×C₄	1	32	yes	complemented
40	C₂×#41×C₅₃	—	2	64	no	—
41	C₂×#41×C₅₃	—	2	4	no	—
42	C₂×#41×C₅₃	—	2	12	no	—
43	C₂×#41×C₅₃	—	2	6	no	—
44	C₂×#41×C₅₃	—	2	6	no	—
45	C₂×#41×C₅₃	—	2	24	no	—
46	C₂×#41×C₅₃	C₂×C₄	1	8	yes	—
47	C₂×#41×C₅₃	C₂×C₄	1	48	yes	complemented
48	C₂×#41×C₅₃	D₄	1	8	yes	—
49	C₂×#41×C₅₃	D₄	1	4	yes	—
50	C₂×#41×C₅₃	C₂³	1	4	yes	—
51	C₂×#41×C₅₃	C₂³	1	12	yes	—
52	C₂×#41×C₅₃	C₂³	1	4	yes	—
53	C₂²×Q₈×C₅₃	—	2	4	no	—
54	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
55	C₂²×Q₈×C₅₃	C₂³	1	6	yes	—
56	C₂²×D₄×C₅₃	—	2	6	no	—
57	C₂²×D₄×C₅₃	—	2	12	no	—
58	C₂²×D₄×C₅₃	D₄	1	3	yes	—
59	C₂²×D₄×C₅₃	C₂³	1	6	yes	—
60	C₂³×C₄×C₅₃	—	2	6	no	—
61	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
62	#131×C₅₃	—	2	32	no	—
63	C₂×M₁₆×C₅₃	—	2	12	no	—
64	C₂×M₁₆×C₅₃	C₂³	1	12	yes	—
65	C₂²×C₈×C₅₃	—	2	12	no	—
66	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
68	C₂²×C₈×C₅₃	C₂³	1	12	yes	—
69	C₄×Q₈×C₅₃	—	2	8	no	—
70	C₄×D₄×C₅₃	—	2	24	no	—
71	#117×C₅₃	—	2	6	no	—
72	#117×C₅₃	—	2	6	no	—
73	#117×C₅₃	C₂³	1	4	yes	—
74	#117×C₅₃	C₂³	1	12	yes	—
75	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
76	C₂×C₂²⋊C₄×C₅₃	—	2	6	no	—
77	C₂×C₄²×C₅₃	—	2	6	no	—
78	#103×C₅₃	C₂³	1	8	yes	—
79	#103×C₅₃	C₂³	1	24	yes	—
80	#102×C₅₃	C₂³	1	8	yes	—
81	#102×C₅₃	C₂³	1	24	yes	—
82	C₂⁴×C₅₃	—	2	6	no	—
83	C₂⁴×C₅₃	C₂×D₄	1	3	yes	—
84	#41×C₅₃	—	2	32	no	—
85	#41×C₅₃	—	2	96	no	—
86	#41×C₅₃	—	2	48	no	—
87	#41×C₅₃	—	2	48	no	—
88	#41×C₅₃	—	2	8	no	—
89	#41×C₅₃	—	2	24	no	—
90	#41×C₅₃	C₂²⋊C₄	1	32	yes	—
91	#41×C₅₃	C₂²×C₄	1	16	yes	—
92	#41×C₅₃	C₂²×C₄	1	48	yes	—
93	#41×C₅₃	C₂²×C₄	1	16	yes	—
94	#41×C₅₃	C₂×D₄	1	16	yes	—
95	C₂×Q₈×C₅₃	—	2	32	no	—
96	C₂×Q₈×C₅₃	—	2	6	no	—
97	C₂×Q₈×C₅₃	—	2	12	no	—
98	C₂×Q₈×C₅₃	C₂²⋊C₄	1	8	yes	—
99	C₂×Q₈×C₅₃	C₂²×C₄	1	24	yes	—
100	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
101	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
102	C₂×D₄×C₅₃	—	2	48	no	—
103	C₂×D₄×C₅₃	—	2	12	no	—
104	C₂×D₄×C₅₃	—	2	4	no	—
105	C₂×D₄×C₅₃	—	2	96	no	—
106	C₂×D₄×C₅₃	—	2	6	no	—
107	C₂×D₄×C₅₃	—	2	12	no	—
108	C₂×D₄×C₅₃	C₂²⋊C₄	1	24	yes	—
109	C₂×D₄×C₅₃	C₂²×C₄	1	24	yes	—
110	C₂×D₄×C₅₃	C₂×D₄	1	12	yes	—
111	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
112	C₂²×C₄×C₅₃	—	2	6	no	—
113	C₂²×C₄×C₅₃	—	2	48	no	—
114	C₂²×C₄×C₅₃	—	2	12	no	—
115	C₂²×C₄×C₅₃	—	2	4	no	—
116	C₂²×C₄×C₅₃	—	2	12	no	—
117	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
118	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
119	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
120	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
121	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
122	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
123	M₁₆×C₅₃	—	2	24	no	—
124	C₂×C₈×C₅₃	—	2	2	no	—
125	C₂×C₈×C₅₃	—	2	6	no	—
126	C₂×C₈×C₅₃	—	2	24	no	—
127	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
128	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
129	C₄⋊C₄×C₅₃	—	2	12	no	—
130	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
131	C₂²⋊C₄×C₅₃	—	2	4	no	—
132	C₂²⋊C₄×C₅₃	—	2	12	no	—
133	C₄²×C₅₃	—	2	12	no	—
134	C₄²×C₅₃	—	2	4	no	—
135	C₂³×C₅₃	—	2	4	no	—
136	C₂³×C₅₃	—	2	24	no	—
137	C₂³×C₅₃	—	4	6	no	—
138	C₂³×C₅₃	—	8	12	no	—
139	C₂³×C₅₃	C₂×C₂²⋊C₄	1	12	yes	—
140	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
141	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
142	Q₈×C₅₃	—	2	24	no	—
143	Q₈×C₅₃	—	2	4	no	—
144	Q₈×C₅₃	C₂×C₂²⋊C₄	1	16	yes	—
145	Q₈×C₅₃	C₂³×C₄	1	8	yes	—
146	D₄×C₅₃	—	2	48	no	—
147	D₄×C₅₃	—	2	16	no	—
148	D₄×C₅₃	—	2	24	no	—
149	D₄×C₅₃	—	2	4	no	—
150	D₄×C₅₃	C₂×C₂²⋊C₄	1	48	yes	—
151	D₄×C₅₃	C₂³×C₄	1	8	yes	—
152	C₂×C₄×C₅₃	—	2	24	no	—
153	C₂×C₄×C₅₃	—	2	4	no	—
154	C₂×C₄×C₅₃	—	2	16	no	—
155	C₂×C₄×C₅₃	—	2	48	no	—
156	C₂×C₄×C₅₃	—	2	8	no	—
157	C₂×C₄×C₅₃	—	8	12	no	—
158	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	24	yes	—
159	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	12	yes	—
160	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
161	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
162	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
163	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
164	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
165	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
167	C₈×C₅₃	—	2	2	no	—
168	C₈×C₅₃	—	2	6	no	—
169	#60086	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂²×C₅₃	—	2	12	no	—
171	C₂²×C₅₃	—	2	2	no	—
172	C₂²×C₅₃	—	2	8	no	—
173	C₂²×C₅₃	—	4	12	no	—
174	C₂²×C₅₃	—	4	8	no	—
175	C₂²×C₅₃	—	8	24	no	—
176	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	12	yes	—
177	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	2	yes	—
178	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
179	C₄×C₅₃	—	2	8	no	—
180	C₄×C₅₃	—	4	8	no	—
181	C₄×C₅₃	C₂²×C₂²⋊C₄	1	2	yes	—
182	C₄×C₅₃	C₂²×C₂²⋊C₄	1	12	yes	—
183	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
184	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
185	C₂×#585	C₂×C₅₃	1	1	yes	characteristic
186	C₂²×#131	C₂×C₅₃	1	1	yes	characteristic, complemented
187	#3428	C₂×C₅₃	1	1	yes	characteristic, complemented
188	#2898	C₂×C₅₃	1	8	yes	complemented
189	#2897	C₂×C₅₃	1	8	yes	complemented
190	#2896	C₂×C₅₃	1	6	yes	complemented
191	C₂×#416	C₂×C₅₃	1	6	yes	complemented
192	C₂×C₅₃	—	2	2	no	—
193	C₂×C₅₃	—	2	12	no	—
194	C₂×C₅₃	—	4	8	no	—
195	C₂×C₅₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
196	C₂×C₅₃	#3584	1	2	yes	—
197	#585	—	2	8	no	—
198	#585	C₄×C₅₃	1	16	yes	complemented
199	C₂×2₋¹⁺⁴	—	2	1	no	—
200	C₂×2₋¹⁺⁴	C₂²×C₅₃	1	4	yes	—
201	C₂×H(5,𝔽₂)	—	2	3	no	—
202	C₂×H(5,𝔽₂)	C₂²×C₅₃	1	4	yes	—
203	C₂²×#41	—	2	4	no	—
204	C₂²×#41	C₂²×C₅₃	1	6	yes	—
205	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
206	C₂×#131	—	2	8	no	—
207	C₂×#131	C₂²×C₅₃	1	8	yes	complemented
208	C₂²×M₁₆	C₂²×C₅₃	1	3	yes	complemented
209	C₂³×C₈	C₂²×C₅₃	1	3	yes	complemented
210	#520	C₂²×C₅₃	1	8	yes	complemented
211	#519	—	2	1	no	—
212	#518	—	2	3	no	—
213	C₄×#41	—	2	4	no	—
214	C₂×#117	C₂²×C₅₃	1	6	yes	complemented
215	#419	C₂²×C₅₃	1	12	yes	complemented
216	#418	C₂²×C₅₃	1	12	yes	complemented
217	#417	C₂²×C₅₃	1	24	yes	complemented
218	#416	C₂²×C₅₃	1	4	yes	complemented
219	#416	C₂²×C₅₃	1	12	yes	complemented
220	#416	C₂²×C₅₃	1	24	yes	complemented
221	C₂×#103	C₂²×C₅₃	1	12	yes	complemented
222	C₂×#102	C₂²×C₅₃	1	12	yes	complemented
223	C₅₃	#60086	1	1	yes	characteristic, direct product factor, Sylow
224	2₋¹⁺⁴	—	2	16	no	—
225	2₋¹⁺⁴	—	2	16	no	—
226	2₋¹⁺⁴	C₂×C₄×C₅₃	1	32	yes	complemented
227	H(5,𝔽₂)	—	2	16	no	—
228	H(5,𝔽₂)	—	2	48	no	—
229	H(5,𝔽₂)	C₂×C₄×C₅₃	1	32	yes	complemented
230	C₂×#41	—	2	64	no	—
231	C₂×#41	—	2	4	no	—
232	C₂×#41	—	2	12	no	—
233	C₂×#41	—	2	6	no	—
234	C₂×#41	—	2	6	no	—
235	C₂×#41	—	2	24	no	—
236	C₂×#41	C₂×C₄×C₅₃	1	8	yes	—
237	C₂×#41	C₂×C₄×C₅₃	1	48	yes	complemented
238	C₂×#41	D₄×C₅₃	1	4	yes	—
239	C₂×#41	D₄×C₅₃	1	8	yes	—
240	C₂×#41	C₂³×C₅₃	1	4	yes	—
241	C₂×#41	C₂³×C₅₃	1	12	yes	—
242	C₂×#41	C₂³×C₅₃	1	4	yes	—
243	C₂²×Q₈	—	2	4	no	—
244	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
245	C₂²×Q₈	C₂³×C₅₃	1	6	yes	—
246	C₂²×D₄	—	2	6	no	—
247	C₂²×D₄	—	2	12	no	—
248	C₂²×D₄	D₄×C₅₃	1	3	yes	—
249	C₂²×D₄	C₂³×C₅₃	1	6	yes	—
250	C₂³×C₄	—	2	6	no	—
251	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
252	#131	—	2	32	no	—
253	C₂×M₁₆	—	2	12	no	—
254	C₂×M₁₆	C₂³×C₅₃	1	12	yes	—
255	C₂²×C₈	—	2	12	no	—
256	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
257	C₂²×C₈	C₂³×C₅₃	1	3	yes	—
258	C₂²×C₈	C₂³×C₅₃	1	12	yes	—
259	C₄×Q₈	—	2	8	no	—
260	C₄×D₄	—	2	24	no	—
261	#117	—	2	6	no	—
262	#117	—	2	6	no	—
263	#117	C₂³×C₅₃	1	4	yes	—
264	#117	C₂³×C₅₃	1	12	yes	—
265	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
266	C₂×C₂²⋊C₄	—	2	6	no	—
267	C₂×C₄²	—	2	6	no	—
268	#103	C₂³×C₅₃	1	8	yes	—
269	#103	C₂³×C₅₃	1	24	yes	—
270	#102	C₂³×C₅₃	1	8	yes	—
271	#102	C₂³×C₅₃	1	24	yes	—
272	C₂⁴	—	2	6	no	—
273	C₂⁴	C₂×D₄×C₅₃	1	3	yes	—
274	#41	—	2	32	no	—
275	#41	—	2	96	no	—
276	#41	—	2	48	no	—
277	#41	—	2	48	no	—
278	#41	—	2	8	no	—
279	#41	—	2	24	no	—
280	#41	C₂²⋊C₄×C₅₃	1	32	yes	—
281	#41	C₂²×C₄×C₅₃	1	16	yes	—
282	#41	C₂²×C₄×C₅₃	1	48	yes	—
283	#41	C₂²×C₄×C₅₃	1	16	yes	—
284	#41	C₂×D₄×C₅₃	1	16	yes	—
285	C₂×Q₈	—	2	32	no	—
286	C₂×Q₈	—	2	6	no	—
287	C₂×Q₈	—	2	12	no	—
288	C₂×Q₈	C₂²⋊C₄×C₅₃	1	8	yes	—
289	C₂×Q₈	C₂²×C₄×C₅₃	1	24	yes	—
290	C₂×Q₈	C₂×D₄×C₅₃	1	4	yes	—
291	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
292	C₂×D₄	—	2	48	no	—
293	C₂×D₄	—	2	4	no	—
294	C₂×D₄	—	2	12	no	—
295	C₂×D₄	—	2	96	no	—
296	C₂×D₄	—	2	6	no	—
297	C₂×D₄	—	2	12	no	—
298	C₂×D₄	C₂²⋊C₄×C₅₃	1	24	yes	—
299	C₂×D₄	C₂²×C₄×C₅₃	1	24	yes	—
300	C₂×D₄	C₂×D₄×C₅₃	1	12	yes	—
301	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
302	C₂²×C₄	—	2	6	no	—
303	C₂²×C₄	—	2	48	no	—
304	C₂²×C₄	—	2	4	no	—
305	C₂²×C₄	—	2	12	no	—
306	C₂²×C₄	—	2	12	no	—
307	C₂²×C₄	C₂²×C₄×C₅₃	1	12	yes	—
308	C₂²×C₄	C₂×D₄×C₅₃	1	6	yes	—
309	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
310	C₂²×C₄	C₂×D₄×C₅₃	1	12	yes	—
311	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
312	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
313	M₁₆	—	2	24	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	6	no	—
316	C₂×C₈	—	2	24	no	—
317	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
318	C₂×C₈	C₂⁴×C₅₃	1	6	yes	—
319	C₄⋊C₄	—	2	12	no	—
320	C₄⋊C₄	C₂⁴×C₅₃	1	8	yes	—
321	C₂²⋊C₄	—	2	4	no	—
322	C₂²⋊C₄	—	2	12	no	—
323	C₄²	—	2	4	no	—
324	C₄²	—	2	12	no	—
325	C₂³	—	2	4	no	—
326	C₂³	—	2	24	no	—
327	C₂³	—	4	6	no	—
328	C₂³	—	8	12	no	—
329	C₂³	C₂×C₂²⋊C₄×C₅₃	1	12	yes	—
330	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
331	C₂³	C₂²×D₄×C₅₃	1	6	yes	—
332	Q₈	—	2	24	no	—
333	Q₈	—	2	4	no	—
334	Q₈	C₂×C₂²⋊C₄×C₅₃	1	16	yes	—
335	Q₈	C₂³×C₄×C₅₃	1	8	yes	—
336	D₄	—	2	48	no	—
337	D₄	—	2	16	no	—
338	D₄	—	2	24	no	—
339	D₄	—	2	4	no	—
340	D₄	C₂×C₂²⋊C₄×C₅₃	1	48	yes	—
341	D₄	C₂³×C₄×C₅₃	1	8	yes	—
342	C₂×C₄	—	2	24	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	2	16	no	—
345	C₂×C₄	—	2	48	no	—
346	C₂×C₄	—	2	8	no	—
347	C₂×C₄	—	8	12	no	—
348	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	24	yes	—
349	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	12	yes	—
350	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
351	C₂×C₄	C₂³×C₄×C₅₃	1	12	yes	—
352	C₂×C₄	C₂²×D₄×C₅₃	1	6	yes	—
353	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
354	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
355	C₂×C₄	C₂²×D₄×C₅₃	1	12	yes	—
356	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
357	C₈	—	2	2	no	—
358	C₈	—	2	6	no	—
359	C₂²	—	2	12	no	—
360	C₂²	—	2	2	no	—
361	C₂²	—	2	8	no	—
362	C₂²	—	4	12	no	—
363	C₂²	—	4	8	no	—
364	C₂²	—	8	24	no	—
365	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	12	yes	—
366	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	2	yes	—
367	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
368	C₄	—	2	8	no	—
369	C₄	—	4	8	no	—
370	C₄	C₂²×C₂²⋊C₄×C₅₃	1	2	yes	—
371	C₄	C₂²×C₂²⋊C₄×C₅₃	1	12	yes	—
372	C₄	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
373	C₄	C₂³×D₄×C₅₃	1	1	yes	characteristic
374	C₂	—	2	2	no	—
375	C₂	—	2	12	no	—
376	C₂	—	4	8	no	—
377	C₂	C₂³×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
378	C₂	C₅₃×#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₄	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
socle	C₂²×C₅₃	C₂³×D₄
prime radical	C₂×C₅₃×#585	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#131×C₅₃	C₂	semidirect
3. C₅₃×#3428	C₂	semidirect
4. C₅₃×#2898	C₂	semidirect
5. C₅₃×#2897	C₂	semidirect
6. C₅₃×#2896	C₂	semidirect
7. C₂×C₅₃×#416	C₂	semidirect
9. C₅₃×#585	C₄	semidirect
18. C₂×#131×C₅₃	C₂²	semidirect
19. C₂²×M₁₆×C₅₃	C₂²	semidirect
20. C₂³×C₈×C₅₃	C₂²	semidirect
21. C₅₃×#520	C₂²	semidirect
25. C₂×#117×C₅₃	C₂²	semidirect
26. C₅₃×#419	C₂²	semidirect
27. C₅₃×#418	C₂²	semidirect
28. C₅₃×#417	C₂²	semidirect
29. C₅₃×#416	C₂²	semidirect
30. C₅₃×#416	C₂²	semidirect
31. C₅₃×#416	C₂²	semidirect
32. C₂×#103×C₅₃	C₂²	semidirect
33. C₂×#102×C₅₃	C₂²	semidirect
36. 2₋¹⁺⁴×C₅₃	C₂×C₄	semidirect
39. H(5,𝔽₂)×C₅₃	C₂×C₄	semidirect
47. C₂×#41×C₅₃	C₂×C₄	semidirect
169. #60086	C₅₃	direct (N × H)
186. C₂²×#131	C₂×C₅₃	semidirect
187. #3428	C₂×C₅₃	semidirect
188. #2898	C₂×C₅₃	semidirect
189. #2897	C₂×C₅₃	semidirect
190. #2896	C₂×C₅₃	semidirect
191. C₂×#416	C₂×C₅₃	semidirect
198. #585	C₄×C₅₃	semidirect
207. C₂×#131	C₂²×C₅₃	semidirect
208. C₂²×M₁₆	C₂²×C₅₃	semidirect
209. C₂³×C₈	C₂²×C₅₃	semidirect
210. #520	C₂²×C₅₃	semidirect
214. C₂×#117	C₂²×C₅₃	semidirect
215. #419	C₂²×C₅₃	semidirect
216. #418	C₂²×C₅₃	semidirect
217. #417	C₂²×C₅₃	semidirect
218. #416	C₂²×C₅₃	semidirect
219. #416	C₂²×C₅₃	semidirect
220. #416	C₂²×C₅₃	semidirect
221. C₂×#103	C₂²×C₅₃	semidirect
222. C₂×#102	C₂²×C₅₃	semidirect
223. C₅₃	#60086	direct (N × H)
226. 2₋¹⁺⁴	C₂×C₄×C₅₃	semidirect
229. H(5,𝔽₂)	C₂×C₄×C₅₃	semidirect
237. C₂×#41	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#585	2
C₂²×#131×C₅₃	2
C₅₃×#3428	2
C₅₃×#2898	2
C₅₃×#2897	2
C₅₃×#2896	2
C₂×C₅₃×#416	2
#60086	53

Subgroups — C53×#60086 #27267598

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60086 #27267598