Subgroups — C₂×C₅₃×#3169 #27268183

Order 13568.

2566 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#532	C₂	1	2	yes	maximal, complemented
2	C₅₃×#3169	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₃×#486	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#466	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×D₈×C₅₃	C₂	1	4	yes	maximal, complemented
8	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₅₃	—	2	2	no	—
10	C₂²×D₈×C₅₃	C₂²	1	4	yes	—
11	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#532	—	2	8	no	—
13	C₅₃×#532	C₂²	1	16	yes	complemented
14	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
15	C₂×#127×C₅₃	C₂²	1	2	yes	—
16	C₂×#124×C₅₃	—	2	2	no	—
17	C₂×#121×C₅₃	—	2	2	no	—
18	C₂×#121×C₅₃	C₂²	1	4	yes	—
19	C₂×C₄×D₄×C₅₃	—	2	2	no	—
20	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
21	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#500	C₂²	1	8	yes	complemented
23	C₅₃×#493	C₂²	1	8	yes	complemented
24	C₅₃×#486	C₂²	1	16	yes	complemented
25	C₅₃×#466	C₂²	1	32	yes	complemented
26	C₄×D₈×C₅₃	C₂²	1	32	yes	complemented
27	C₂×#107×C₅₃	C₂²	1	4	yes	—
28	C₂×#102×C₅₃	C₂²	1	8	yes	—
29	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₅₃	C₂²	1	8	yes	complemented
33	C₂×#41×C₅₃	—	2	16	no	—
34	C₂×#41×C₅₃	—	4	16	no	—
35	C₂²×Q₈×C₅₃	—	2	2	no	—
36	C₂²×D₄×C₅₃	—	2	4	no	—
37	C₂²×D₄×C₅₃	—	2	2	no	—
38	C₂²×D₄×C₅₃	—	2	2	no	—
39	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
40	C₂³×C₄×C₅₃	—	2	4	no	—
41	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×D₈×C₅₃	—	2	16	no	—
45	C₂×D₈×C₅₃	C₂³	1	16	yes	—
46	C₂²×C₈×C₅₃	—	2	8	no	—
47	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₅₃	C₂³	1	8	yes	—
49	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₅₃	—	2	4	no	—
51	C₄⋊Q₈×C₅₃	C₂³	1	8	yes	—
52	#127×C₅₃	—	2	4	no	—
53	#127×C₅₃	C₂³	1	8	yes	—
54	#124×C₅₃	—	2	16	no	—
55	#121×C₅₃	—	2	16	no	—
56	#121×C₅₃	—	2	8	no	—
57	#121×C₅₃	C₂³	1	16	yes	—
58	C₄×D₄×C₅₃	—	2	16	no	—
59	C₄×D₄×C₅₃	—	2	8	no	—
60	C₄×D₄×C₅₃	C₂³	1	16	yes	—
61	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
62	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
63	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
64	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
65	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	D₄	1	4	yes	—
67	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
70	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
71	#107×C₅₃	C₂³	1	16	yes	—
72	#102×C₅₃	C₂³	1	32	yes	—
73	C₄×C₈×C₅₃	C₂³	1	4	yes	—
74	C₄×C₈×C₅₃	C₂³	1	4	yes	—
75	C₄×C₈×C₅₃	C₂³	1	8	yes	—
76	C₂⁴×C₅₃	—	2	4	no	—
77	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
78	#41×C₅₃	—	4	64	no	—
79	C₂×Q₈×C₅₃	—	2	8	no	—
80	C₂×Q₈×C₅₃	—	4	8	no	—
81	C₂×D₄×C₅₃	—	2	16	no	—
82	C₂×D₄×C₅₃	—	2	4	no	—
83	C₂×D₄×C₅₃	—	2	8	no	—
84	C₂×D₄×C₅₃	—	2	8	no	—
85	C₂×D₄×C₅₃	—	2	16	no	—
86	C₂×D₄×C₅₃	—	4	16	no	—
87	C₂×D₄×C₅₃	—	4	8	no	—
88	C₂×D₄×C₅₃	—	4	8	no	—
89	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
90	C₂²×C₄×C₅₃	—	2	4	no	—
91	C₂²×C₄×C₅₃	—	2	4	no	—
92	C₂²×C₄×C₅₃	—	2	4	no	—
93	C₂²×C₄×C₅₃	—	2	16	no	—
94	C₂²×C₄×C₅₃	—	4	16	no	—
95	C₂²×C₄×C₅₃	D₈	1	8	yes	complemented
96	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
100	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
102	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
105	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
106	D₈×C₅₃	—	2	32	no	—
107	C₂×C₈×C₅₃	—	2	4	no	—
108	C₂×C₈×C₅₃	—	2	16	no	—
109	C₂×C₈×C₅₃	#41	1	8	yes	complemented
110	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
111	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
112	C₄⋊C₄×C₅₃	—	2	8	no	—
113	C₄⋊C₄×C₅₃	—	2	4	no	—
114	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
115	C₂²⋊C₄×C₅₃	—	2	16	no	—
116	C₂²⋊C₄×C₅₃	—	2	16	no	—
117	C₄²×C₅₃	C₂×D₄	1	2	yes	—
118	C₄²×C₅₃	C₂×D₄	1	4	yes	—
119	C₄²×C₅₃	C₂×D₄	1	4	yes	—
120	C₄²×C₅₃	C₂×D₄	1	2	yes	—
121	C₄²×C₅₃	C₂⁴	1	2	yes	—
122	C₄²×C₅₃	C₂⁴	1	2	yes	—
123	C₂³×C₅₃	—	2	2	no	—
124	C₂³×C₅₃	—	2	2	no	—
125	C₂³×C₅₃	—	2	8	no	—
126	C₂³×C₅₃	—	4	8	no	—
127	C₂³×C₅₃	—	4	8	no	—
128	C₂³×C₅₃	—	4	8	no	—
129	C₂³×C₅₃	C₂×D₈	1	4	yes	—
130	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
131	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
132	Q₈×C₅₃	—	4	16	no	—
133	D₄×C₅₃	—	2	16	no	—
134	D₄×C₅₃	—	4	32	no	—
135	D₄×C₅₃	—	4	8	no	—
136	D₄×C₅₃	—	4	16	no	—
137	D₄×C₅₃	—	4	16	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	2	4	no	—
141	C₂×C₄×C₅₃	—	2	8	no	—
142	C₂×C₄×C₅₃	—	2	4	no	—
143	C₂×C₄×C₅₃	—	2	8	no	—
144	C₂×C₄×C₅₃	—	4	8	no	—
145	C₂×C₄×C₅₃	—	4	32	no	—
146	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
147	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
148	C₂×C₄×C₅₃	C₂×D₈	1	16	yes	complemented
149	C₂×C₄×C₅₃	#135	1	4	yes	—
150	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
151	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
152	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
154	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
155	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
156	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
157	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
158	C₈×C₅₃	—	2	4	no	—
159	C₈×C₅₃	C₂×#41	1	8	yes	complemented
160	C₂×#3169	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂²×C₅₃	—	2	8	no	—
162	C₂²×C₅₃	—	2	2	no	—
163	C₂²×C₅₃	—	2	2	no	—
164	C₂²×C₅₃	—	4	8	no	—
165	C₂²×C₅₃	—	4	8	no	—
166	C₂²×C₅₃	—	4	8	no	—
167	C₂²×C₅₃	—	8	16	no	—
168	C₂²×C₅₃	#532	1	2	yes	—
169	C₂²×C₅₃	C₂²×D₈	1	4	yes	—
170	C₂²×C₅₃	C₂²×D₈	1	2	yes	—
171	C₂²×C₅₃	C₂×#135	1	2	yes	—
172	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
173	C₄×C₅₃	—	2	2	no	—
174	C₄×C₅₃	—	2	2	no	—
175	C₄×C₅₃	—	4	8	no	—
176	C₄×C₅₃	#532	1	2	yes	—
177	C₄×C₅₃	C₂²×D₈	1	4	yes	—
178	C₄×C₅₃	C₂²×D₈	1	4	yes	—
179	C₄×C₅₃	C₂×#135	1	4	yes	—
180	C₄×C₅₃	C₂²×#41	1	2	yes	—
181	C₂×#532	C₂×C₅₃	1	2	yes	complemented
182	#3169	C₂×C₅₃	1	16	yes	direct product factor
183	C₂×#500	C₂×C₅₃	1	1	yes	characteristic, complemented
184	C₂×#493	C₂×C₅₃	1	1	yes	characteristic, complemented
185	C₂×#486	C₂×C₅₃	1	2	yes	complemented
186	C₂×#466	C₂×C₅₃	1	4	yes	complemented
187	C₂×C₄×D₈	C₂×C₅₃	1	4	yes	complemented
188	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
189	C₂×C₅₃	—	2	4	no	—
190	C₂×C₅₃	—	8	8	no	—
191	C₂×C₅₃	#3169	1	4	yes	direct product factor
192	C₂×C₅₃	C₂×#532	1	1	yes	characteristic
193	C₂×C₅₃	C₂³×D₈	1	1	yes	characteristic
194	C₂×C₅₃	C₂²×#135	1	1	yes	characteristic
195	C₂²×#41	—	2	2	no	—
196	C₂²×D₈	C₂²×C₅₃	1	4	yes	—
197	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
198	#532	—	2	8	no	—
199	#532	C₂²×C₅₃	1	16	yes	complemented
200	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
201	C₂×#127	C₂²×C₅₃	1	2	yes	—
202	C₂×#124	—	2	2	no	—
203	C₂×#121	—	2	2	no	—
204	C₂×#121	C₂²×C₅₃	1	4	yes	—
205	C₂×C₄×D₄	—	2	2	no	—
206	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
207	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
208	#500	C₂²×C₅₃	1	8	yes	complemented
209	#493	C₂²×C₅₃	1	8	yes	complemented
210	#486	C₂²×C₅₃	1	16	yes	complemented
211	#466	C₂²×C₅₃	1	32	yes	complemented
212	C₄×D₈	C₂²×C₅₃	1	32	yes	complemented
213	C₂×#107	C₂²×C₅₃	1	4	yes	—
214	C₂×#102	C₂²×C₅₃	1	8	yes	—
215	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
216	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
217	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
218	C₂×C₄×C₈	C₂²×C₅₃	1	8	yes	complemented
219	C₅₃	C₂×#3169	1	1	yes	characteristic, direct product factor, Sylow
220	C₂×#41	—	2	16	no	—
221	C₂×#41	—	4	16	no	—
222	C₂²×Q₈	—	2	2	no	—
223	C₂²×D₄	—	2	4	no	—
224	C₂²×D₄	—	2	2	no	—
225	C₂²×D₄	—	2	2	no	—
226	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
227	C₂³×C₄	—	2	4	no	—
228	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
229	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
230	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
231	C₂×D₈	—	2	16	no	—
232	C₂×D₈	C₂³×C₅₃	1	16	yes	—
233	C₂²×C₈	—	2	8	no	—
234	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
235	C₂²×C₈	C₂³×C₅₃	1	8	yes	—
236	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
237	C₄⋊Q₈	—	2	4	no	—
238	C₄⋊Q₈	C₂³×C₅₃	1	8	yes	—
239	#127	—	2	4	no	—
240	#127	C₂³×C₅₃	1	8	yes	—
241	#124	—	2	16	no	—
242	#121	—	2	16	no	—
243	#121	—	2	8	no	—
244	#121	C₂³×C₅₃	1	16	yes	—
245	C₄×D₄	—	2	16	no	—
246	C₄×D₄	—	2	8	no	—
247	C₄×D₄	C₂³×C₅₃	1	16	yes	—
248	C₂×C₄⋊C₄	—	2	2	no	—
249	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
250	C₂×C₂²⋊C₄	—	2	4	no	—
251	C₂×C₂²⋊C₄	—	2	4	no	—
252	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
253	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
254	C₂×C₄²	D₄×C₅₃	1	4	yes	—
255	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
256	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
257	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
258	#107	C₂³×C₅₃	1	16	yes	—
259	#102	C₂³×C₅₃	1	32	yes	—
260	C₄×C₈	C₂³×C₅₃	1	8	yes	—
261	C₄×C₈	C₂³×C₅₃	1	4	yes	—
262	C₄×C₈	C₂³×C₅₃	1	4	yes	—
263	C₂⁴	—	2	4	no	—
264	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
265	#41	—	4	64	no	—
266	C₂×Q₈	—	2	8	no	—
267	C₂×Q₈	—	4	8	no	—
268	C₂×D₄	—	2	16	no	—
269	C₂×D₄	—	2	4	no	—
270	C₂×D₄	—	2	8	no	—
271	C₂×D₄	—	2	8	no	—
272	C₂×D₄	—	2	16	no	—
273	C₂×D₄	—	4	16	no	—
274	C₂×D₄	—	4	8	no	—
275	C₂×D₄	—	4	8	no	—
276	C₂×D₄	C₂⁴×C₅₃	1	8	yes	—
277	C₂²×C₄	—	2	4	no	—
278	C₂²×C₄	—	2	4	no	—
279	C₂²×C₄	—	2	4	no	—
280	C₂²×C₄	—	2	16	no	—
281	C₂²×C₄	—	4	16	no	—
282	C₂²×C₄	D₈×C₅₃	1	8	yes	complemented
283	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
284	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
285	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
287	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
288	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
289	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
290	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
291	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
292	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
293	D₈	—	2	32	no	—
294	C₂×C₈	—	2	4	no	—
295	C₂×C₈	—	2	16	no	—
296	C₂×C₈	#41×C₅₃	1	8	yes	complemented
297	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
298	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
299	C₄⋊C₄	—	2	8	no	—
300	C₄⋊C₄	—	2	4	no	—
301	C₄⋊C₄	C₂⁴×C₅₃	1	8	yes	—
302	C₂²⋊C₄	—	2	16	no	—
303	C₂²⋊C₄	—	2	16	no	—
304	C₄²	C₂×D₄×C₅₃	1	4	yes	—
305	C₄²	C₂×D₄×C₅₃	1	2	yes	—
306	C₄²	C₂×D₄×C₅₃	1	4	yes	—
307	C₄²	C₂×D₄×C₅₃	1	2	yes	—
308	C₄²	C₂⁴×C₅₃	1	2	yes	—
309	C₄²	C₂⁴×C₅₃	1	2	yes	—
310	C₂³	—	2	2	no	—
311	C₂³	—	2	2	no	—
312	C₂³	—	2	8	no	—
313	C₂³	—	4	8	no	—
314	C₂³	—	4	8	no	—
315	C₂³	—	4	8	no	—
316	C₂³	C₂×D₈×C₅₃	1	4	yes	—
317	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
318	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
319	Q₈	—	4	16	no	—
320	D₄	—	2	16	no	—
321	D₄	—	4	32	no	—
322	D₄	—	4	8	no	—
323	D₄	—	4	16	no	—
324	D₄	—	4	16	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	4	no	—
328	C₂×C₄	—	2	8	no	—
329	C₂×C₄	—	2	4	no	—
330	C₂×C₄	—	2	8	no	—
331	C₂×C₄	—	4	8	no	—
332	C₂×C₄	—	4	32	no	—
333	C₂×C₄	C₂×D₈×C₅₃	1	4	yes	—
334	C₂×C₄	C₂×D₈×C₅₃	1	4	yes	—
335	C₂×C₄	C₂×D₈×C₅₃	1	16	yes	complemented
336	C₂×C₄	#135×C₅₃	1	4	yes	—
337	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
338	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
339	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
340	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
341	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
342	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
343	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
344	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
345	C₈	—	2	4	no	—
346	C₈	C₂×#41×C₅₃	1	8	yes	complemented
347	C₂²	—	2	2	no	—
348	C₂²	—	2	8	no	—
349	C₂²	—	2	2	no	—
350	C₂²	—	4	8	no	—
351	C₂²	—	4	8	no	—
352	C₂²	—	4	8	no	—
353	C₂²	—	8	16	no	—
354	C₂²	C₅₃×#532	1	2	yes	—
355	C₂²	C₂²×D₈×C₅₃	1	4	yes	—
356	C₂²	C₂²×D₈×C₅₃	1	2	yes	—
357	C₂²	C₂×#135×C₅₃	1	2	yes	—
358	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
359	C₄	—	2	2	no	—
360	C₄	—	2	2	no	—
361	C₄	—	4	8	no	—
362	C₄	C₅₃×#532	1	2	yes	—
363	C₄	C₂²×D₈×C₅₃	1	4	yes	—
364	C₄	C₂²×D₈×C₅₃	1	4	yes	—
365	C₄	C₂×#135×C₅₃	1	4	yes	—
366	C₄	C₂²×#41×C₅₃	1	2	yes	—
367	C₂	—	2	4	no	—
368	C₂	—	8	8	no	—
369	C₂	C₅₃×#3169	1	4	yes	direct product factor
370	C₂	C₂×C₅₃×#532	1	1	yes	characteristic
371	C₂	C₂³×D₈×C₅₃	1	1	yes	characteristic
372	C₂	C₂²×#135×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
socle	C₂³×C₅₃	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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×#532	C₂	semidirect
2. C₅₃×#3169	C₂	direct (N × H)
3. C₂×C₅₃×#500	C₂	semidirect
4. C₂×C₅₃×#493	C₂	semidirect
5. C₂×C₅₃×#486	C₂	semidirect
6. C₂×C₅₃×#466	C₂	semidirect
7. C₂×C₄×D₈×C₅₃	C₂	semidirect
8. C₂²×C₄×C₈×C₅₃	C₂	semidirect
13. C₅₃×#532	C₂²	semidirect
22. C₅₃×#500	C₂²	semidirect
23. C₅₃×#493	C₂²	semidirect
24. C₅₃×#486	C₂²	semidirect
25. C₅₃×#466	C₂²	semidirect
26. C₄×D₈×C₅₃	C₂²	semidirect
32. C₂×C₄×C₈×C₅₃	C₂²	semidirect
95. C₂²×C₄×C₅₃	D₈	semidirect
109. C₂×C₈×C₅₃	#41	semidirect
148. C₂×C₄×C₅₃	C₂×D₈	semidirect
159. C₈×C₅₃	C₂×#41	semidirect
160. C₂×#3169	C₅₃	direct (N × H)
181. C₂×#532	C₂×C₅₃	semidirect
182. #3169	C₂×C₅₃	direct (N × H)
183. C₂×#500	C₂×C₅₃	semidirect
184. C₂×#493	C₂×C₅₃	semidirect
185. C₂×#486	C₂×C₅₃	semidirect
186. C₂×#466	C₂×C₅₃	semidirect
187. C₂×C₄×D₈	C₂×C₅₃	semidirect
188. C₂²×C₄×C₈	C₂×C₅₃	semidirect
191. C₂×C₅₃	#3169	direct (N × H)
199. #532	C₂²×C₅₃	semidirect
208. #500	C₂²×C₅₃	semidirect
209. #493	C₂²×C₅₃	semidirect
210. #486	C₂²×C₅₃	semidirect
211. #466	C₂²×C₅₃	semidirect
212. C₄×D₈	C₂²×C₅₃	semidirect
218. C₂×C₄×C₈	C₂²×C₅₃	semidirect
219. C₅₃	C₂×#3169	direct (N × H)
282. C₂²×C₄	D₈×C₅₃	semidirect
296. C₂×C₈	#41×C₅₃	semidirect
335. C₂×C₄	C₂×D₈×C₅₃	semidirect
346. C₈	C₂×#41×C₅₃	semidirect
369. C₂	C₅₃×#3169	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#532	2
C₅₃×#3169	2
C₂×C₅₃×#500	2
C₂×C₅₃×#493	2
C₂×C₅₃×#486	2
C₂×C₅₃×#466	2
C₂×C₄×D₈×C₅₃	2
C₂²×C₄×C₈×C₅₃	2
C₂×#3169	53

Subgroups — C2×C53×#3169 #27268183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#3169 #27268183