Subgroups — C₅₃×#60639 #27268151

Order 13568.

3314 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#575	C₂	1	4	yes	maximal, complemented
2	C₂×C₅₃×#532	C₂	1	2	yes	maximal, complemented
3	C₅₃×#3153	C₂	1	8	yes	maximal, complemented
4	C₅₃×#3152	C₂	1	4	yes	maximal, complemented
5	C₅₃×#3151	C₂	1	4	yes	maximal, complemented
6	C₅₃×#3150	C₂	1	4	yes	maximal, complemented
7	C₅₃×#3147	C₂	1	4	yes	maximal, complemented
8	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₅₃	C₂²	1	4	yes	—
10	C₅₃×#575	—	2	32	no	—
11	C₂×#137×C₅₃	C₂²	1	16	yes	complemented
12	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	16	yes	complemented
13	C₂×#135×C₅₃	C₂²	1	16	yes	complemented
14	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
15	C₅₃×#532	—	2	4	no	—
16	C₅₃×#532	—	2	4	no	—
17	C₅₃×#532	C₂²	1	4	yes	—
18	C₅₃×#532	C₂²	1	4	yes	—
19	C₅₃×#532	C₂²	1	4	yes	complemented
20	C₅₃×#532	C₂²	1	4	yes	complemented
21	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
22	C₂×#127×C₅₃	C₂²	1	2	yes	complemented
23	C₂×#124×C₅₃	C₂²	1	4	yes	complemented
24	C₂×#121×C₅₃	—	2	4	no	—
25	C₂×C₄×D₄×C₅₃	—	2	4	no	—
26	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#497	C₂²	1	16	yes	complemented
28	C₅₃×#496	C₂²	1	16	yes	complemented
29	C₅₃×#495	C₂²	1	16	yes	complemented
30	C₅₃×#494	C₂²	1	4	yes	complemented
31	C₅₃×#493	C₂²	1	4	yes	complemented
32	C₅₃×#492	C₂²	1	8	yes	complemented
33	C₄×M₁₆×C₅₃	C₂²	1	4	yes	complemented
34	C₄×M₁₆×C₅₃	C₂²	1	4	yes	complemented
35	C₂×#97×C₅₃	C₂²	1	2	yes	complemented
36	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	complemented
37	C₂×#41×C₅₃	—	2	4	no	—
38	C₂×#41×C₅₃	—	2	4	no	—
39	C₂×#41×C₅₃	—	2	32	no	—
40	C₂×#41×C₅₃	C₂³	1	8	yes	—
41	C₂×#41×C₅₃	C₂³	1	8	yes	—
42	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
43	C₂²×D₄×C₅₃	—	2	4	no	—
44	C₂²×D₄×C₅₃	—	2	4	no	—
45	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
46	C₂³×C₄×C₅₃	—	2	4	no	—
47	C₂³×C₄×C₅₃	D₄	1	2	yes	—
48	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
49	#137×C₅₃	—	2	64	no	—
50	AGL(1,ℤ/8)×C₅₃	—	2	64	no	—
51	#135×C₅₃	—	2	64	no	—
52	C₂×Q₁₆×C₅₃	C₂³	1	16	yes	—
53	C₂×SD₁₆×C₅₃	C₂³	1	32	yes	—
54	C₂×D₈×C₅₃	C₂³	1	16	yes	—
55	C₂×M₁₆×C₅₃	D₄	1	16	yes	complemented
56	C₂×M₁₆×C₅₃	C₂³	1	8	yes	—
57	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
58	C₄⋊Q₈×C₅₃	—	2	4	no	—
59	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
60	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
61	#127×C₅₃	—	2	4	no	—
62	#127×C₅₃	C₂³	1	4	yes	—
63	#127×C₅₃	C₂³	1	4	yes	—
64	#124×C₅₃	—	2	8	no	—
65	#124×C₅₃	C₂³	1	8	yes	—
66	#124×C₅₃	C₂³	1	8	yes	—
67	#121×C₅₃	—	2	8	no	—
68	#121×C₅₃	—	2	8	no	—
69	#121×C₅₃	—	2	8	no	—
70	#121×C₅₃	—	2	8	no	—
71	C₄×D₄×C₅₃	—	2	8	no	—
72	C₄×D₄×C₅₃	—	2	8	no	—
73	C₄×D₄×C₅₃	—	2	8	no	—
74	C₄×D₄×C₅₃	—	2	8	no	—
75	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
76	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
77	C₂×C₄²×C₅₃	D₄	1	4	yes	—
78	C₂×C₄²×C₅₃	D₄	1	2	yes	—
79	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
80	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
81	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
83	#97×C₅₃	C₂³	1	8	yes	—
84	C₄×C₈×C₅₃	C₂³	1	4	yes	—
85	C₄×C₈×C₅₃	C₂³	1	4	yes	—
86	C₂⁴×C₅₃	—	2	4	no	—
87	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
88	#41×C₅₃	—	2	32	no	—
89	#41×C₅₃	—	2	32	no	—
90	#41×C₅₃	—	4	16	no	—
91	#41×C₅₃	—	4	16	no	—
92	C₂×Q₈×C₅₃	—	2	4	no	—
93	C₂×Q₈×C₅₃	—	2	16	no	—
94	C₂×Q₈×C₅₃	C₂⁴	1	8	yes	—
95	C₂×D₄×C₅₃	—	2	8	no	—
96	C₂×D₄×C₅₃	—	2	8	no	—
97	C₂×D₄×C₅₃	—	2	16	no	—
98	C₂×D₄×C₅₃	—	2	8	no	—
99	C₂×D₄×C₅₃	—	2	8	no	—
100	C₂×D₄×C₅₃	—	2	4	no	—
101	C₂×D₄×C₅₃	—	4	16	no	—
102	C₂×D₄×C₅₃	—	4	16	no	—
103	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
104	C₂²×C₄×C₅₃	—	2	4	no	—
105	C₂²×C₄×C₅₃	—	2	8	no	—
106	C₂²×C₄×C₅₃	—	2	8	no	—
107	C₂²×C₄×C₅₃	—	4	4	no	—
108	C₂²×C₄×C₅₃	—	4	16	no	—
109	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
110	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
112	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
116	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
117	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
118	Q₁₆×C₅₃	—	2	32	no	—
119	SD₁₆×C₅₃	—	2	64	no	—
120	D₈×C₅₃	—	2	32	no	—
121	M₁₆×C₅₃	C₂×D₄	1	32	yes	complemented
122	C₂×C₈×C₅₃	C₂×D₄	1	16	yes	complemented
123	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
124	C₄⋊C₄×C₅₃	—	2	8	no	—
125	C₄⋊C₄×C₅₃	—	2	8	no	—
126	C₂²⋊C₄×C₅₃	—	2	16	no	—
127	C₂²⋊C₄×C₅₃	—	2	16	no	—
128	C₄²×C₅₃	C₂×D₄	1	4	yes	—
129	C₄²×C₅₃	C₂×D₄	1	4	yes	—
130	C₄²×C₅₃	C₂×D₄	1	2	yes	—
131	C₄²×C₅₃	C₂×D₄	1	2	yes	—
132	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
133	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
134	C₄²×C₅₃	C₂⁴	1	2	yes	—
135	C₂³×C₅₃	—	2	8	no	—
136	C₂³×C₅₃	—	4	2	no	—
137	C₂³×C₅₃	—	4	16	no	—
138	C₂³×C₅₃	—	4	8	no	—
139	C₂³×C₅₃	#127	1	4	yes	—
140	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
141	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
142	Q₈×C₅₃	—	2	16	no	—
143	Q₈×C₅₃	—	4	8	no	—
144	D₄×C₅₃	—	2	16	no	—
145	D₄×C₅₃	—	4	16	no	—
146	D₄×C₅₃	—	4	16	no	—
147	D₄×C₅₃	—	4	16	no	—
148	D₄×C₅₃	—	4	8	no	—
149	D₄×C₅₃	—	4	16	no	—
150	C₂×C₄×C₅₃	—	2	4	no	—
151	C₂×C₄×C₅₃	—	2	8	no	—
152	C₂×C₄×C₅₃	—	2	8	no	—
153	C₂×C₄×C₅₃	—	4	4	no	—
154	C₂×C₄×C₅₃	—	4	2	no	—
155	C₂×C₄×C₅₃	—	4	8	no	—
156	C₂×C₄×C₅₃	—	4	16	no	—
157	C₂×C₄×C₅₃	—	4	16	no	—
158	C₂×C₄×C₅₃	#127	1	8	yes	—
159	C₂×C₄×C₅₃	#127	1	4	yes	—
160	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
161	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
162	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
163	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
164	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
165	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
166	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
168	C₈×C₅₃	C₂²×D₄	1	16	yes	—
169	#60639	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂²×C₅₃	—	2	4	no	—
171	C₂²×C₅₃	—	2	2	no	—
172	C₂²×C₅₃	—	4	4	no	—
173	C₂²×C₅₃	—	4	8	no	—
174	C₂²×C₅₃	—	4	16	no	—
175	C₂²×C₅₃	—	8	16	no	—
176	C₂²×C₅₃	C₂×#127	1	4	yes	—
177	C₂²×C₅₃	C₂×#127	1	2	yes	—
178	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
179	C₄×C₅₃	—	2	2	no	—
180	C₄×C₅₃	—	2	4	no	—
181	C₄×C₅₃	—	4	8	no	—
182	C₄×C₅₃	C₂×#127	1	2	yes	—
183	C₄×C₅₃	C₂×#127	1	4	yes	—
184	C₄×C₅₃	#575	1	4	yes	complemented
185	C₄×C₅₃	C₂³×D₄	1	2	yes	—
186	C₂×#575	C₂×C₅₃	1	4	yes	complemented
187	C₂×#532	C₂×C₅₃	1	2	yes	complemented
188	#3153	C₂×C₅₃	1	8	yes	complemented
189	#3152	C₂×C₅₃	1	4	yes	complemented
190	#3151	C₂×C₅₃	1	4	yes	complemented
191	#3150	C₂×C₅₃	1	4	yes	complemented
192	#3147	C₂×C₅₃	1	4	yes	complemented
193	C₂×C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
194	C₂×C₅₃	—	2	4	no	—
195	C₂×C₅₃	—	2	2	no	—
196	C₂×C₅₃	—	8	8	no	—
197	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
198	C₂×C₅₃	C₂×#575	1	2	yes	—
199	C₂²×#41	C₂²×C₅₃	1	4	yes	—
200	#575	—	2	32	no	—
201	C₂×#137	C₂²×C₅₃	1	16	yes	complemented
202	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	16	yes	complemented
203	C₂×#135	C₂²×C₅₃	1	16	yes	complemented
204	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
205	#532	—	2	4	no	—
206	#532	—	2	4	no	—
207	#532	C₂²×C₅₃	1	4	yes	—
208	#532	C₂²×C₅₃	1	4	yes	—
209	#532	C₂²×C₅₃	1	4	yes	complemented
210	#532	C₂²×C₅₃	1	4	yes	complemented
211	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
212	C₂×#127	C₂²×C₅₃	1	2	yes	complemented
213	C₂×#124	C₂²×C₅₃	1	4	yes	complemented
214	C₂×#121	—	2	4	no	—
215	C₂×C₄×D₄	—	2	4	no	—
216	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
217	#497	C₂²×C₅₃	1	16	yes	complemented
218	#496	C₂²×C₅₃	1	16	yes	complemented
219	#495	C₂²×C₅₃	1	16	yes	complemented
220	#494	C₂²×C₅₃	1	4	yes	complemented
221	#493	C₂²×C₅₃	1	4	yes	complemented
222	#492	C₂²×C₅₃	1	8	yes	complemented
223	C₄×M₁₆	C₂²×C₅₃	1	4	yes	complemented
224	C₄×M₁₆	C₂²×C₅₃	1	4	yes	complemented
225	C₂×#97	C₂²×C₅₃	1	2	yes	complemented
226	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	complemented
227	C₅₃	#60639	1	1	yes	characteristic, direct product factor, Sylow
228	C₂×#41	—	2	4	no	—
229	C₂×#41	—	2	4	no	—
230	C₂×#41	—	2	32	no	—
231	C₂×#41	C₂³×C₅₃	1	8	yes	—
232	C₂×#41	C₂³×C₅₃	1	8	yes	—
233	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
234	C₂²×D₄	—	2	4	no	—
235	C₂²×D₄	—	2	4	no	—
236	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
237	C₂³×C₄	—	2	4	no	—
238	C₂³×C₄	D₄×C₅₃	1	2	yes	—
239	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
240	#137	—	2	64	no	—
241	AGL(1,ℤ/8)	—	2	64	no	—
242	#135	—	2	64	no	—
243	C₂×Q₁₆	C₂³×C₅₃	1	16	yes	—
244	C₂×SD₁₆	C₂³×C₅₃	1	32	yes	—
245	C₂×D₈	C₂³×C₅₃	1	16	yes	—
246	C₂×M₁₆	D₄×C₅₃	1	16	yes	complemented
247	C₂×M₁₆	C₂³×C₅₃	1	8	yes	—
248	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
249	C₄⋊Q₈	—	2	4	no	—
250	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
251	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
252	#127	—	2	4	no	—
253	#127	C₂³×C₅₃	1	4	yes	—
254	#127	C₂³×C₅₃	1	4	yes	—
255	#124	—	2	8	no	—
256	#124	C₂³×C₅₃	1	8	yes	—
257	#124	C₂³×C₅₃	1	8	yes	—
258	#121	—	2	8	no	—
259	#121	—	2	8	no	—
260	#121	—	2	8	no	—
261	#121	—	2	8	no	—
262	C₄×D₄	—	2	8	no	—
263	C₄×D₄	—	2	8	no	—
264	C₄×D₄	—	2	8	no	—
265	C₄×D₄	—	2	8	no	—
266	C₂×C₄⋊C₄	—	2	4	no	—
267	C₂×C₂²⋊C₄	—	2	8	no	—
268	C₂×C₄²	D₄×C₅₃	1	4	yes	—
269	C₂×C₄²	D₄×C₅₃	1	2	yes	—
270	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
271	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
272	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
273	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
274	#97	C₂³×C₅₃	1	8	yes	—
275	C₄×C₈	C₂³×C₅₃	1	4	yes	—
276	C₄×C₈	C₂³×C₅₃	1	4	yes	—
277	C₂⁴	—	2	4	no	—
278	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
279	#41	—	2	32	no	—
280	#41	—	2	32	no	—
281	#41	—	4	16	no	—
282	#41	—	4	16	no	—
283	C₂×Q₈	—	2	4	no	—
284	C₂×Q₈	—	2	16	no	—
285	C₂×Q₈	C₂⁴×C₅₃	1	8	yes	—
286	C₂×D₄	—	2	8	no	—
287	C₂×D₄	—	2	8	no	—
288	C₂×D₄	—	2	16	no	—
289	C₂×D₄	—	2	8	no	—
290	C₂×D₄	—	2	8	no	—
291	C₂×D₄	—	2	4	no	—
292	C₂×D₄	—	4	16	no	—
293	C₂×D₄	—	4	16	no	—
294	C₂×D₄	C₂⁴×C₅₃	1	8	yes	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	8	no	—
297	C₂²×C₄	—	2	8	no	—
298	C₂²×C₄	—	4	4	no	—
299	C₂²×C₄	—	4	16	no	—
300	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
301	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
302	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
303	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
304	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
305	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
306	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
307	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
308	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
309	Q₁₆	—	2	32	no	—
310	SD₁₆	—	2	64	no	—
311	D₈	—	2	32	no	—
312	M₁₆	C₂×D₄×C₅₃	1	32	yes	complemented
313	C₂×C₈	C₂×D₄×C₅₃	1	16	yes	complemented
314	C₂×C₈	C₂⁴×C₅₃	1	8	yes	—
315	C₄⋊C₄	—	2	8	no	—
316	C₄⋊C₄	—	2	8	no	—
317	C₂²⋊C₄	—	2	16	no	—
318	C₂²⋊C₄	—	2	16	no	—
319	C₄²	C₂×D₄×C₅₃	1	2	yes	—
320	C₄²	C₂×D₄×C₅₃	1	2	yes	—
321	C₄²	C₂×D₄×C₅₃	1	4	yes	—
322	C₄²	C₂×D₄×C₅₃	1	4	yes	—
323	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
324	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
325	C₄²	C₂⁴×C₅₃	1	2	yes	—
326	C₂³	—	2	8	no	—
327	C₂³	—	4	2	no	—
328	C₂³	—	4	16	no	—
329	C₂³	—	4	8	no	—
330	C₂³	#127×C₅₃	1	4	yes	—
331	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
332	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
333	Q₈	—	2	16	no	—
334	Q₈	—	4	8	no	—
335	D₄	—	2	16	no	—
336	D₄	—	4	16	no	—
337	D₄	—	4	16	no	—
338	D₄	—	4	16	no	—
339	D₄	—	4	8	no	—
340	D₄	—	4	16	no	—
341	C₂×C₄	—	2	4	no	—
342	C₂×C₄	—	2	8	no	—
343	C₂×C₄	—	2	8	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	4	no	—
346	C₂×C₄	—	4	8	no	—
347	C₂×C₄	—	4	16	no	—
348	C₂×C₄	—	4	16	no	—
349	C₂×C₄	#127×C₅₃	1	8	yes	—
350	C₂×C₄	#127×C₅₃	1	4	yes	—
351	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
352	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
353	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
354	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
355	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
356	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
357	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
358	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
359	C₈	C₂²×D₄×C₅₃	1	16	yes	—
360	C₂²	—	2	4	no	—
361	C₂²	—	2	2	no	—
362	C₂²	—	4	4	no	—
363	C₂²	—	4	16	no	—
364	C₂²	—	4	8	no	—
365	C₂²	—	8	16	no	—
366	C₂²	C₂×#127×C₅₃	1	4	yes	—
367	C₂²	C₂×#127×C₅₃	1	2	yes	—
368	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
369	C₄	—	2	4	no	—
370	C₄	—	2	2	no	—
371	C₄	—	4	8	no	—
372	C₄	C₂×#127×C₅₃	1	2	yes	—
373	C₄	C₂×#127×C₅₃	1	4	yes	—
374	C₄	C₅₃×#575	1	4	yes	complemented
375	C₄	C₂³×D₄×C₅₃	1	2	yes	—
376	C₂	—	2	4	no	—
377	C₂	—	2	2	no	—
378	C₂	—	8	8	no	—
379	C₂	C₂²×#127×C₅₃	1	1	yes	characteristic
380	C₂	C₂×C₅₃×#575	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
socle	C₂²×C₅₃	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#575	C₂	semidirect
2. C₂×C₅₃×#532	C₂	semidirect
3. C₅₃×#3153	C₂	semidirect
4. C₅₃×#3152	C₂	semidirect
5. C₅₃×#3151	C₂	semidirect
6. C₅₃×#3150	C₂	semidirect
7. C₅₃×#3147	C₂	semidirect
8. C₂×C₄×M₁₆×C₅₃	C₂	semidirect
11. C₂×#137×C₅₃	C₂²	semidirect
12. C₂×AGL(1,ℤ/8)×C₅₃	C₂²	semidirect
13. C₂×#135×C₅₃	C₂²	semidirect
19. C₅₃×#532	C₂²	semidirect
20. C₅₃×#532	C₂²	semidirect
21. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
22. C₂×#127×C₅₃	C₂²	semidirect
23. C₂×#124×C₅₃	C₂²	semidirect
27. C₅₃×#497	C₂²	semidirect
28. C₅₃×#496	C₂²	semidirect
29. C₅₃×#495	C₂²	semidirect
30. C₅₃×#494	C₂²	semidirect
31. C₅₃×#493	C₂²	semidirect
32. C₅₃×#492	C₂²	semidirect
33. C₄×M₁₆×C₅₃	C₂²	semidirect
34. C₄×M₁₆×C₅₃	C₂²	semidirect
35. C₂×#97×C₅₃	C₂²	semidirect
36. C₂×C₄×C₈×C₅₃	C₂²	semidirect
55. C₂×M₁₆×C₅₃	D₄	semidirect
121. M₁₆×C₅₃	C₂×D₄	semidirect
122. C₂×C₈×C₅₃	C₂×D₄	semidirect
169. #60639	C₅₃	direct (N × H)
184. C₄×C₅₃	#575	semidirect
186. C₂×#575	C₂×C₅₃	semidirect
187. C₂×#532	C₂×C₅₃	semidirect
188. #3153	C₂×C₅₃	semidirect
189. #3152	C₂×C₅₃	semidirect
190. #3151	C₂×C₅₃	semidirect
191. #3150	C₂×C₅₃	semidirect
192. #3147	C₂×C₅₃	semidirect
193. C₂×C₄×M₁₆	C₂×C₅₃	semidirect
201. C₂×#137	C₂²×C₅₃	semidirect
202. C₂×AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
203. C₂×#135	C₂²×C₅₃	semidirect
209. #532	C₂²×C₅₃	semidirect
210. #532	C₂²×C₅₃	semidirect
211. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
212. C₂×#127	C₂²×C₅₃	semidirect
213. C₂×#124	C₂²×C₅₃	semidirect
217. #497	C₂²×C₅₃	semidirect
218. #496	C₂²×C₅₃	semidirect
219. #495	C₂²×C₅₃	semidirect
220. #494	C₂²×C₅₃	semidirect
221. #493	C₂²×C₅₃	semidirect
222. #492	C₂²×C₅₃	semidirect
223. C₄×M₁₆	C₂²×C₅₃	semidirect
224. C₄×M₁₆	C₂²×C₅₃	semidirect
225. C₂×#97	C₂²×C₅₃	semidirect
226. C₂×C₄×C₈	C₂²×C₅₃	semidirect
227. C₅₃	#60639	direct (N × H)
246. C₂×M₁₆	D₄×C₅₃	semidirect
312. M₁₆	C₂×D₄×C₅₃	semidirect
313. C₂×C₈	C₂×D₄×C₅₃	semidirect
374. C₄	C₅₃×#575	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#575	2
C₂×C₅₃×#532	2
C₅₃×#3153	2
C₅₃×#3152	2
C₅₃×#3151	2
C₅₃×#3150	2
C₅₃×#3147	2
C₂×C₄×M₁₆×C₅₃	2
#60639	53

Subgroups — C53×#60639 #27268151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60639 #27268151