Subgroups — C₂×C₅₃×#3342 #27268745

Order 13568.

1854 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#556	C₂	1	2	yes	maximal
2	C₅₃×#3342	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#485	C₂	1	4	yes	maximal, complemented
5	C₂×C₅₃×#475	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#467	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₅₃×#427	C₂	1	2	yes	maximal, complemented
10	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#556	—	2	8	no	—
13	C₅₃×#556	C₂²	1	16	yes	—
14	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
15	C₂×#126×C₅₃	—	2	4	no	—
16	C₂×#124×C₅₃	—	2	2	no	—
17	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	—
18	C₂×C₄×Q₈×C₅₃	—	2	2	no	—
19	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#117×C₅₃	C₂²	1	4	yes	—
22	C₅₃×#500	C₂²	1	8	yes	complemented
23	C₅₃×#485	C₂²	1	32	yes	complemented
24	C₅₃×#475	C₂²	1	16	yes	complemented
25	C₅₃×#467	C₂²	1	16	yes	complemented
26	C₄×Q₁₆×C₅₃	C₂²	1	8	yes	complemented
27	C₈×D₄×C₅₃	C₂²	1	8	yes	—
28	C₅₃×#427	C₂²	1	16	yes	complemented
29	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×#107×C₅₃	C₂²	1	2	yes	—
31	C₂×#107×C₅₃	C₂²	1	2	yes	—
32	C₂×#106×C₅₃	C₂²	1	4	yes	—
33	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×#103×C₅₃	C₂²	1	2	yes	—
35	C₂×#103×C₅₃	C₂²	1	4	yes	—
36	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₂²×Q₈×C₅₃	—	2	2	no	—
39	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
40	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
42	C₂×Q₁₆×C₅₃	—	2	4	no	—
43	C₂×Q₁₆×C₅₃	C₂³	1	4	yes	—
44	C₂²×C₈×C₅₃	—	2	8	no	—
45	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
46	C₂²×C₈×C₅₃	C₂³	1	8	yes	—
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₅₃	—	2	4	no	—
50	C₄⋊Q₈×C₅₃	C₂³	1	8	yes	—
51	#126×C₅₃	—	2	32	no	—
52	#124×C₅₃	—	2	16	no	—
53	C₂²⋊Q₈×C₅₃	—	2	8	no	—
54	C₂²⋊Q₈×C₅₃	C₂³	1	16	yes	—
55	C₄×Q₈×C₅₃	—	2	16	no	—
56	C₄×Q₈×C₅₃	—	2	4	no	—
57	C₄×Q₈×C₅₃	C₂³	1	8	yes	—
58	C₄×D₄×C₅₃	D₄	1	4	yes	—
59	C₄×D₄×C₅₃	C₂³	1	4	yes	—
60	#117×C₅₃	—	2	8	no	—
61	#117×C₅₃	C₂³	1	16	yes	—
62	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
63	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
65	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
68	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
69	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
70	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
71	C₂×C₄²×C₅₃	—	2	4	no	—
72	C₂×C₄²×C₅₃	—	2	2	no	—
73	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
74	#107×C₅₃	C₂³	1	8	yes	—
75	#107×C₅₃	C₂³	1	8	yes	—
76	#107×C₅₃	C₂³	1	4	yes	—
77	#106×C₅₃	C₂³	1	16	yes	—
78	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
79	#103×C₅₃	C₂³	1	8	yes	—
80	#103×C₅₃	C₂³	1	16	yes	—
81	C₂²⋊C₈×C₅₃	C₂³	1	8	yes	—
82	C₄×C₈×C₅₃	C₂³	1	4	yes	—
83	C₂⁴×C₅₃	C₂×D₄	1	2	yes	—
84	C₂×Q₈×C₅₃	—	2	2	no	—
85	C₂×Q₈×C₅₃	—	2	8	no	—
86	C₂×Q₈×C₅₃	—	2	4	no	—
87	C₂×Q₈×C₅₃	—	2	4	no	—
88	C₂×Q₈×C₅₃	—	4	8	no	—
89	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
90	C₂×D₄×C₅₃	—	2	4	no	—
91	C₂×D₄×C₅₃	C₂×D₄	1	4	yes	—
92	C₂²×C₄×C₅₃	—	2	4	no	—
93	C₂²×C₄×C₅₃	—	2	4	no	—
94	C₂²×C₄×C₅₃	—	2	2	no	—
95	C₂²×C₄×C₅₃	—	2	2	no	—
96	C₂²×C₄×C₅₃	—	2	4	no	—
97	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
98	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
102	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
103	Q₁₆×C₅₃	—	2	8	no	—
104	C₂×C₈×C₅₃	—	2	4	no	—
105	C₂×C₈×C₅₃	—	2	2	no	—
106	C₂×C₈×C₅₃	—	2	16	no	—
107	C₂×C₈×C₅₃	#41	1	4	yes	—
108	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
109	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
110	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
111	C₄⋊C₄×C₅₃	—	2	16	no	—
112	C₄⋊C₄×C₅₃	—	2	8	no	—
113	C₄⋊C₄×C₅₃	—	2	8	no	—
114	C₄⋊C₄×C₅₃	—	2	2	no	—
115	C₄⋊C₄×C₅₃	—	2	4	no	—
116	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
117	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
118	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
119	C₂²⋊C₄×C₅₃	—	2	16	no	—
120	C₂²⋊C₄×C₅₃	—	2	16	no	—
121	C₂²⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
122	C₄²×C₅₃	—	2	16	no	—
123	C₄²×C₅₃	—	2	8	no	—
124	C₄²×C₅₃	C₂×D₄	1	2	yes	—
125	C₄²×C₅₃	C₂⁴	1	2	yes	—
126	C₂³×C₅₃	—	2	4	no	—
127	C₂³×C₅₃	—	2	4	no	—
128	C₂³×C₅₃	—	2	4	no	—
129	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
130	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
131	Q₈×C₅₃	—	2	4	no	—
132	Q₈×C₅₃	—	2	4	no	—
133	Q₈×C₅₃	—	4	4	no	—
134	Q₈×C₅₃	—	4	16	no	—
135	D₄×C₅₃	—	2	8	no	—
136	C₂×C₄×C₅₃	—	2	4	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	8	no	—
139	C₂×C₄×C₅₃	—	2	4	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	4	no	—
144	C₂×C₄×C₅₃	—	4	4	no	—
145	C₂×C₄×C₅₃	—	4	8	no	—
146	C₂×C₄×C₅₃	—	4	4	no	—
147	C₂×C₄×C₅₃	#135	1	4	yes	—
148	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
149	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
150	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
152	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
153	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
154	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
155	C₈×C₅₃	—	2	4	no	—
156	C₈×C₅₃	—	2	2	no	—
157	C₈×C₅₃	C₂×#41	1	4	yes	—
158	C₂×#3342	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂²×C₅₃	—	2	4	no	—
160	C₂²×C₅₃	—	2	4	no	—
161	C₂²×C₅₃	—	2	4	no	—
162	C₂²×C₅₃	—	4	8	no	—
163	C₂²×C₅₃	#547	1	2	yes	—
164	C₂²×C₅₃	C₂×#135	1	2	yes	—
165	C₂²×C₅₃	#578	1	2	yes	—
166	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
167	C₄×C₅₃	—	2	4	no	—
168	C₄×C₅₃	—	2	2	no	—
169	C₄×C₅₃	—	4	4	no	—
170	C₄×C₅₃	—	4	8	no	—
171	C₄×C₅₃	—	4	4	no	—
172	C₄×C₅₃	#547	1	2	yes	—
173	C₄×C₅₃	C₂×#135	1	4	yes	—
174	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
175	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
176	C₂×#556	C₂×C₅₃	1	2	yes	—
177	#3342	C₂×C₅₃	1	16	yes	direct product factor
178	C₂×#500	C₂×C₅₃	1	1	yes	characteristic, complemented
179	C₂×#485	C₂×C₅₃	1	4	yes	complemented
180	C₂×#475	C₂×C₅₃	1	2	yes	complemented
181	C₂×#467	C₂×C₅₃	1	2	yes	complemented
182	C₂×C₄×Q₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
183	C₂×C₈×D₄	C₂×C₅₃	1	1	yes	characteristic
184	C₂×#427	C₂×C₅₃	1	2	yes	complemented
185	C₂×C₅₃	—	4	4	no	—
186	C₂×C₅₃	#3342	1	4	yes	direct product factor
187	C₂×C₅₃	C₂×#547	1	1	yes	characteristic
188	C₂×C₅₃	C₂²×#135	1	1	yes	characteristic
189	C₂×C₅₃	C₂×#578	1	1	yes	characteristic
190	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
191	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
192	#556	—	2	8	no	—
193	#556	C₂²×C₅₃	1	16	yes	—
194	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
195	C₂×#126	—	2	4	no	—
196	C₂×#124	—	2	2	no	—
197	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	—
198	C₂×C₄×Q₈	—	2	2	no	—
199	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
200	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
201	C₂×#117	C₂²×C₅₃	1	4	yes	—
202	#500	C₂²×C₅₃	1	8	yes	complemented
203	#485	C₂²×C₅₃	1	32	yes	complemented
204	#475	C₂²×C₅₃	1	16	yes	complemented
205	#467	C₂²×C₅₃	1	16	yes	complemented
206	C₄×Q₁₆	C₂²×C₅₃	1	8	yes	complemented
207	C₈×D₄	C₂²×C₅₃	1	8	yes	—
208	#427	C₂²×C₅₃	1	16	yes	complemented
209	C₂×#107	C₂²×C₅₃	1	2	yes	—
210	C₂×#107	C₂²×C₅₃	1	2	yes	—
211	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
212	C₂×#106	C₂²×C₅₃	1	4	yes	—
213	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
214	C₂×#103	C₂²×C₅₃	1	4	yes	—
215	C₂×#103	C₂²×C₅₃	1	2	yes	—
216	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
217	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
218	C₅₃	C₂×#3342	1	1	yes	characteristic, direct product factor, Sylow
219	C₂²×Q₈	—	2	2	no	—
220	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
221	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
222	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
223	C₂×Q₁₆	—	2	4	no	—
224	C₂×Q₁₆	C₂³×C₅₃	1	4	yes	—
225	C₂²×C₈	—	2	8	no	—
226	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
227	C₂²×C₈	C₂³×C₅₃	1	8	yes	—
228	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
229	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
230	C₄⋊Q₈	—	2	4	no	—
231	C₄⋊Q₈	C₂³×C₅₃	1	8	yes	—
232	#126	—	2	32	no	—
233	#124	—	2	16	no	—
234	C₂²⋊Q₈	—	2	8	no	—
235	C₂²⋊Q₈	C₂³×C₅₃	1	16	yes	—
236	C₄×Q₈	—	2	16	no	—
237	C₄×Q₈	—	2	4	no	—
238	C₄×Q₈	C₂³×C₅₃	1	8	yes	—
239	C₄×D₄	D₄×C₅₃	1	4	yes	—
240	C₄×D₄	C₂³×C₅₃	1	4	yes	—
241	#117	—	2	8	no	—
242	#117	C₂³×C₅₃	1	16	yes	—
243	C₂×C₄⋊C₄	—	2	4	no	—
244	C₂×C₄⋊C₄	—	2	2	no	—
245	C₂×C₄⋊C₄	—	2	2	no	—
246	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
247	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
248	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
249	C₂×C₂²⋊C₄	—	2	4	no	—
250	C₂×C₂²⋊C₄	—	2	4	no	—
251	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
252	C₂×C₄²	—	2	4	no	—
253	C₂×C₄²	—	2	2	no	—
254	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
255	#107	C₂³×C₅₃	1	4	yes	—
256	#107	C₂³×C₅₃	1	8	yes	—
257	#107	C₂³×C₅₃	1	8	yes	—
258	#106	C₂³×C₅₃	1	16	yes	—
259	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
260	#103	C₂³×C₅₃	1	8	yes	—
261	#103	C₂³×C₅₃	1	16	yes	—
262	C₂²⋊C₈	C₂³×C₅₃	1	8	yes	—
263	C₄×C₈	C₂³×C₅₃	1	4	yes	—
264	C₂⁴	C₂×D₄×C₅₃	1	2	yes	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	2	8	no	—
267	C₂×Q₈	—	2	4	no	—
268	C₂×Q₈	—	2	4	no	—
269	C₂×Q₈	—	4	8	no	—
270	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
271	C₂×D₄	—	2	4	no	—
272	C₂×D₄	C₂×D₄×C₅₃	1	4	yes	—
273	C₂²×C₄	—	2	4	no	—
274	C₂²×C₄	—	2	4	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	2	4	no	—
278	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
279	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
280	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
281	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
282	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
283	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
284	Q₁₆	—	2	8	no	—
285	C₂×C₈	—	2	4	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	16	no	—
288	C₂×C₈	#41×C₅₃	1	4	yes	—
289	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
290	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
291	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
292	C₄⋊C₄	—	2	16	no	—
293	C₄⋊C₄	—	2	8	no	—
294	C₄⋊C₄	—	2	8	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	4	no	—
297	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
298	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
299	C₄⋊C₄	C₂⁴×C₅₃	1	8	yes	—
300	C₂²⋊C₄	—	2	16	no	—
301	C₂²⋊C₄	—	2	16	no	—
302	C₂²⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
303	C₄²	—	2	16	no	—
304	C₄²	—	2	8	no	—
305	C₄²	C₂×D₄×C₅₃	1	2	yes	—
306	C₄²	C₂⁴×C₅₃	1	2	yes	—
307	C₂³	—	2	4	no	—
308	C₂³	—	2	4	no	—
309	C₂³	—	2	4	no	—
310	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
311	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
312	Q₈	—	2	4	no	—
313	Q₈	—	2	4	no	—
314	Q₈	—	4	4	no	—
315	Q₈	—	4	16	no	—
316	D₄	—	2	8	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	8	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	2	4	no	—
323	C₂×C₄	—	2	4	no	—
324	C₂×C₄	—	2	8	no	—
325	C₂×C₄	—	4	4	no	—
326	C₂×C₄	—	4	8	no	—
327	C₂×C₄	—	4	4	no	—
328	C₂×C₄	#135×C₅₃	1	4	yes	—
329	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
330	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
331	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
332	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
333	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
334	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
335	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
336	C₈	—	2	4	no	—
337	C₈	—	2	2	no	—
338	C₈	C₂×#41×C₅₃	1	4	yes	—
339	C₂²	—	2	4	no	—
340	C₂²	—	2	4	no	—
341	C₂²	—	2	4	no	—
342	C₂²	—	4	8	no	—
343	C₂²	C₅₃×#547	1	2	yes	—
344	C₂²	C₂×#135×C₅₃	1	2	yes	—
345	C₂²	C₅₃×#578	1	2	yes	—
346	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
347	C₄	—	2	4	no	—
348	C₄	—	2	2	no	—
349	C₄	—	4	4	no	—
350	C₄	—	4	8	no	—
351	C₄	—	4	4	no	—
352	C₄	C₅₃×#547	1	2	yes	—
353	C₄	C₂×#135×C₅₃	1	4	yes	—
354	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
355	C₄	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
356	C₂	—	4	4	no	—
357	C₂	C₅₃×#3342	1	4	yes	direct product factor
358	C₂	C₂×C₅₃×#547	1	1	yes	characteristic
359	C₂	C₂²×#135×C₅₃	1	1	yes	characteristic
360	C₂	C₂×C₅₃×#578	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₅₃×#3342	C₂	direct (N × H)
3. C₂×C₅₃×#500	C₂	semidirect
4. C₂×C₅₃×#485	C₂	semidirect
5. C₂×C₅₃×#475	C₂	semidirect
6. C₂×C₅₃×#467	C₂	semidirect
7. C₂×C₄×Q₁₆×C₅₃	C₂	semidirect
9. C₂×C₅₃×#427	C₂	semidirect
22. C₅₃×#500	C₂²	semidirect
23. C₅₃×#485	C₂²	semidirect
24. C₅₃×#475	C₂²	semidirect
25. C₅₃×#467	C₂²	semidirect
26. C₄×Q₁₆×C₅₃	C₂²	semidirect
28. C₅₃×#427	C₂²	semidirect
158. C₂×#3342	C₅₃	direct (N × H)
177. #3342	C₂×C₅₃	direct (N × H)
178. C₂×#500	C₂×C₅₃	semidirect
179. C₂×#485	C₂×C₅₃	semidirect
180. C₂×#475	C₂×C₅₃	semidirect
181. C₂×#467	C₂×C₅₃	semidirect
182. C₂×C₄×Q₁₆	C₂×C₅₃	semidirect
184. C₂×#427	C₂×C₅₃	semidirect
186. C₂×C₅₃	#3342	direct (N × H)
202. #500	C₂²×C₅₃	semidirect
203. #485	C₂²×C₅₃	semidirect
204. #475	C₂²×C₅₃	semidirect
205. #467	C₂²×C₅₃	semidirect
206. C₄×Q₁₆	C₂²×C₅₃	semidirect
208. #427	C₂²×C₅₃	semidirect
218. C₅₃	C₂×#3342	direct (N × H)
357. C₂	C₅₃×#3342	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×#556	2
C₅₃×#3342	2
C₂×C₅₃×#500	2
C₂×C₅₃×#485	2
C₂×C₅₃×#475	2
C₂×C₅₃×#467	2
C₂×C₄×Q₁₆×C₅₃	2
C₂×C₈×D₄×C₅₃	2
C₂×C₅₃×#427	2
C₂×#3342	53

Subgroups — C2×C53×#3342 #27268745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#3342 #27268745