Subgroups — C₂×C₅₃×#2138 #27240888

Order 13568.

1190 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2138	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅₃×#358	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#357	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₅₃	—	2	1	no	—
8	C₂×#135×C₅₃	—	2	4	no	—
9	C₂×#135×C₅₃	—	2	2	no	—
10	C₂×#135×C₅₃	C₄	1	8	yes	complemented
11	C₂×#135×C₅₃	C₂²	1	4	yes	—
12	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₅₃	—	2	1	no	—
14	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#117×C₅₃	—	2	1	no	—
17	C₂²×C₁₆×C₅₃	—	2	4	no	—
18	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
19	C₂²×C₁₆×C₅₃	C₂²	1	4	yes	complemented
20	C₅₃×#427	—	2	2	no	—
21	C₅₃×#427	C₂²	1	4	yes	complemented
22	C₂×#107×C₅₃	C₂²	1	2	yes	—
23	C₂×#106×C₅₃	—	2	1	no	—
24	C₅₃×#358	C₂²	1	8	yes	complemented
25	C₅₃×#357	C₂²	1	8	yes	complemented
26	C₂×#41×C₅₃	—	2	8	no	—
27	C₂×#41×C₅₃	—	2	4	no	—
28	C₂×#41×C₅₃	—	4	2	no	—
29	C₂×#41×C₅₃	—	4	4	no	—
30	C₂²×Q₈×C₅₃	—	2	1	no	—
31	C₂²×D₄×C₅₃	—	2	1	no	—
32	C₂²×D₄×C₅₃	—	4	1	no	—
33	C₂³×C₄×C₅₃	—	4	1	no	—
34	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
35	#135×C₅₃	—	2	16	no	—
36	#135×C₅₃	—	2	8	no	—
37	#135×C₅₃	C₂×C₄	1	16	yes	complemented
38	C₂×Q₁₆×C₅₃	—	2	2	no	—
39	C₂×Q₁₆×C₅₃	—	2	1	no	—
40	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	—
41	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
42	C₂×SD₁₆×C₅₃	—	2	8	no	—
43	C₂×SD₁₆×C₅₃	—	2	4	no	—
44	C₂×D₈×C₅₃	—	2	2	no	—
45	C₂×D₈×C₅₃	—	2	1	no	—
46	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
47	C₂×D₈×C₅₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₅₃	D₄	1	4	yes	—
50	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	D₄	1	2	yes	—
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
54	#117×C₅₃	—	2	4	no	—
55	#117×C₅₃	—	4	2	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
58	C₂×C₄²×C₅₃	—	4	1	no	—
59	C₂×C₁₆×C₅₃	—	2	4	no	—
60	C₂×C₁₆×C₅₃	—	2	8	no	—
61	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
62	#107×C₅₃	—	2	2	no	—
63	#107×C₅₃	C₂³	1	4	yes	—
64	#106×C₅₃	—	2	4	no	—
65	C₂⁴×C₅₃	—	4	1	no	—
66	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
67	#41×C₅₃	—	2	16	no	—
68	#41×C₅₃	—	4	16	no	—
69	#41×C₅₃	—	4	8	no	—
70	C₂×Q₈×C₅₃	—	2	4	no	—
71	C₂×Q₈×C₅₃	—	2	2	no	—
72	C₂×Q₈×C₅₃	—	4	1	no	—
73	C₂×Q₈×C₅₃	—	4	2	no	—
74	C₂×D₄×C₅₃	—	2	4	no	—
75	C₂×D₄×C₅₃	—	2	2	no	—
76	C₂×D₄×C₅₃	—	4	8	no	—
77	C₂×D₄×C₅₃	—	4	4	no	—
78	C₂×D₄×C₅₃	—	4	1	no	—
79	C₂×D₄×C₅₃	—	4	2	no	—
80	C₂²×C₄×C₅₃	—	4	1	no	—
81	C₂²×C₄×C₅₃	—	4	8	no	—
82	C₂²×C₄×C₅₃	—	4	4	no	—
83	C₂²×C₄×C₅₃	—	4	2	no	—
84	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
85	C₂²×C₄×C₅₃	D₈	1	2	yes	—
86	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	SD₁₆	1	4	yes	—
88	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
90	Q₁₆×C₅₃	—	2	4	no	—
91	Q₁₆×C₅₃	—	2	2	no	—
92	Q₁₆×C₅₃	C₂²×C₄	1	4	yes	—
93	SD₁₆×C₅₃	—	2	16	no	—
94	D₈×C₅₃	—	2	4	no	—
95	D₈×C₅₃	—	2	2	no	—
96	D₈×C₅₃	C₂²×C₄	1	4	yes	—
97	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
98	C₂×C₈×C₅₃	C₂²⋊C₄	1	8	yes	—
99	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
105	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₅₃	—	2	4	no	—
107	C₄⋊C₄×C₅₃	—	4	2	no	—
108	C₂²⋊C₄×C₅₃	—	4	4	no	—
109	C₄²×C₅₃	—	4	4	no	—
110	C₁₆×C₅₃	—	2	4	no	—
111	C₂³×C₅₃	—	2	4	no	—
112	C₂³×C₅₃	—	4	2	no	—
113	C₂³×C₅₃	—	4	4	no	—
114	C₂³×C₅₃	—	8	4	no	—
115	C₂³×C₅₃	#102	1	4	yes	—
116	C₂³×C₅₃	C₂×D₈	1	2	yes	—
117	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
118	Q₈×C₅₃	—	2	4	no	—
119	Q₈×C₅₃	—	4	2	no	—
120	Q₈×C₅₃	—	4	4	no	—
121	D₄×C₅₃	—	2	4	no	—
122	D₄×C₅₃	—	4	16	no	—
123	D₄×C₅₃	—	4	2	no	—
124	D₄×C₅₃	—	4	4	no	—
125	C₂×C₄×C₅₃	—	4	2	no	—
126	C₂×C₄×C₅₃	—	4	4	no	—
127	C₂×C₄×C₅₃	—	4	16	no	—
128	C₂×C₄×C₅₃	—	4	4	no	—
129	C₂×C₄×C₅₃	—	8	4	no	—
130	C₂×C₄×C₅₃	#102	1	4	yes	—
131	C₂×C₄×C₅₃	#102	1	8	yes	—
132	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
133	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
134	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
137	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
138	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
140	C₈×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
141	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
142	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×#2138	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₅₃	—	2	4	no	—
145	C₂²×C₅₃	—	2	8	no	—
146	C₂²×C₅₃	—	4	4	no	—
147	C₂²×C₅₃	—	8	4	no	—
148	C₂²×C₅₃	—	8	8	no	—
149	C₂²×C₅₃	C₂×#102	1	4	yes	—
150	C₂²×C₅₃	C₂×#102	1	2	yes	—
151	C₂²×C₅₃	#508	1	4	yes	—
152	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
153	C₄×C₅₃	—	4	4	no	—
154	C₄×C₅₃	—	8	4	no	—
155	C₄×C₅₃	C₂×#102	1	4	yes	—
156	C₄×C₅₃	C₂×#102	1	2	yes	—
157	C₄×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
158	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
159	C₂²×#135	C₂×C₅₃	1	1	yes	characteristic
160	C₂³×C₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
161	C₂×#427	C₂×C₅₃	1	1	yes	characteristic, complemented
162	#2138	C₂×C₅₃	1	8	yes	direct product factor
163	C₂×#358	C₂×C₅₃	1	2	yes	complemented
164	C₂×#357	C₂×C₅₃	1	2	yes	complemented
165	C₂×C₅₃	—	2	4	no	—
166	C₂×C₅₃	—	8	4	no	—
167	C₂×C₅₃	#2138	1	4	yes	direct product factor
168	C₂×C₅₃	C₂²×#102	1	1	yes	characteristic
169	C₂×C₅₃	C₂×#508	1	2	yes	—
170	C₂²×#41	—	2	1	no	—
171	C₂×#135	—	2	4	no	—
172	C₂×#135	—	2	2	no	—
173	C₂×#135	C₄×C₅₃	1	8	yes	complemented
174	C₂×#135	C₂²×C₅₃	1	4	yes	—
175	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
176	C₂²×SD₁₆	—	2	1	no	—
177	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
178	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
179	C₂×#117	—	2	1	no	—
180	C₂²×C₁₆	—	2	4	no	—
181	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	—
182	C₂²×C₁₆	C₂²×C₅₃	1	4	yes	complemented
183	#427	—	2	2	no	—
184	#427	C₂²×C₅₃	1	4	yes	complemented
185	C₂×#107	C₂²×C₅₃	1	2	yes	—
186	C₂×#106	—	2	1	no	—
187	#358	C₂²×C₅₃	1	8	yes	complemented
188	#357	C₂²×C₅₃	1	8	yes	complemented
189	C₅₃	C₂×#2138	1	1	yes	characteristic, direct product factor, Sylow
190	C₂×#41	—	2	4	no	—
191	C₂×#41	—	2	8	no	—
192	C₂×#41	—	4	2	no	—
193	C₂×#41	—	4	4	no	—
194	C₂²×Q₈	—	2	1	no	—
195	C₂²×D₄	—	2	1	no	—
196	C₂²×D₄	—	4	1	no	—
197	C₂³×C₄	—	4	1	no	—
198	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
199	#135	—	2	8	no	—
200	#135	—	2	16	no	—
201	#135	C₂×C₄×C₅₃	1	16	yes	complemented
202	C₂×Q₁₆	—	2	1	no	—
203	C₂×Q₁₆	—	2	2	no	—
204	C₂×Q₁₆	C₂×C₄×C₅₃	1	4	yes	—
205	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
206	C₂×SD₁₆	—	2	4	no	—
207	C₂×SD₁₆	—	2	8	no	—
208	C₂×D₈	—	2	1	no	—
209	C₂×D₈	—	2	2	no	—
210	C₂×D₈	C₂×C₄×C₅₃	1	4	yes	—
211	C₂×D₈	C₂³×C₅₃	1	2	yes	—
212	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
213	C₂²×C₈	D₄×C₅₃	1	4	yes	—
214	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
215	C₂²×C₈	D₄×C₅₃	1	2	yes	—
216	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
217	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
218	#117	—	2	4	no	—
219	#117	—	4	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₂²⋊C₄	—	4	1	no	—
222	C₂×C₄²	—	4	1	no	—
223	C₂×C₁₆	—	2	4	no	—
224	C₂×C₁₆	—	2	8	no	—
225	C₂×C₁₆	C₂³×C₅₃	1	4	yes	—
226	#107	—	2	2	no	—
227	#107	C₂³×C₅₃	1	4	yes	—
228	#106	—	2	4	no	—
229	C₂⁴	—	4	1	no	—
230	C₂⁴	D₈×C₅₃	1	1	yes	characteristic
231	#41	—	2	16	no	—
232	#41	—	4	16	no	—
233	#41	—	4	8	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂×D₄	—	2	4	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	4	8	no	—
241	C₂×D₄	—	4	4	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂²×C₄	—	4	1	no	—
245	C₂²×C₄	—	4	8	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
249	C₂²×C₄	D₈×C₅₃	1	2	yes	—
250	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
251	C₂²×C₄	SD₁₆×C₅₃	1	4	yes	—
252	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
254	Q₁₆	—	2	4	no	—
255	Q₁₆	—	2	2	no	—
256	Q₁₆	C₂²×C₄×C₅₃	1	4	yes	—
257	SD₁₆	—	2	16	no	—
258	D₈	—	2	4	no	—
259	D₈	—	2	2	no	—
260	D₈	C₂²×C₄×C₅₃	1	4	yes	—
261	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
262	C₂×C₈	C₂²⋊C₄×C₅₃	1	8	yes	—
263	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
264	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
265	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
266	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
267	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
268	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
269	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	4	no	—
273	C₄²	—	4	4	no	—
274	C₁₆	—	2	4	no	—
275	C₂³	—	2	4	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	4	no	—
278	C₂³	—	8	4	no	—
279	C₂³	#102×C₅₃	1	4	yes	—
280	C₂³	C₂×D₈×C₅₃	1	2	yes	—
281	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
282	Q₈	—	2	4	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	4	4	no	—
285	D₄	—	2	4	no	—
286	D₄	—	4	16	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	4	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	16	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	8	4	no	—
294	C₂×C₄	#102×C₅₃	1	4	yes	—
295	C₂×C₄	#102×C₅₃	1	8	yes	—
296	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
297	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
298	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
299	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
300	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
301	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
302	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
303	C₈	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
304	C₈	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
305	C₈	C₂³×C₄×C₅₃	1	1	yes	characteristic
306	C₈	C₂²×D₄×C₅₃	1	1	yes	characteristic
307	C₂²	—	2	4	no	—
308	C₂²	—	2	8	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	8	4	no	—
311	C₂²	—	8	8	no	—
312	C₂²	C₂×#102×C₅₃	1	4	yes	—
313	C₂²	C₂×#102×C₅₃	1	2	yes	—
314	C₂²	C₅₃×#508	1	4	yes	—
315	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
316	C₄	—	4	4	no	—
317	C₄	—	8	4	no	—
318	C₄	C₂×#102×C₅₃	1	4	yes	—
319	C₄	C₂×#102×C₅₃	1	2	yes	—
320	C₄	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
321	C₄	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
322	C₂	—	2	4	no	—
323	C₂	—	8	4	no	—
324	C₂	C₅₃×#2138	1	4	yes	direct product factor
325	C₂	C₂²×#102×C₅₃	1	1	yes	characteristic
326	C₂	C₂×C₅₃×#508	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	D₈
commutator	C₈	C₂³×C₄×C₅₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂²×#135×C₅₃	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ C₂³×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₁₆×C₅₃	C₂	semidirect
3. C₂×C₅₃×#427	C₂	semidirect
4. C₅₃×#2138	C₂	direct (N × H)
5. C₂×C₅₃×#358	C₂	semidirect
6. C₂×C₅₃×#357	C₂	semidirect
10. C₂×#135×C₅₃	C₄	semidirect
19. C₂²×C₁₆×C₅₃	C₂²	semidirect
21. C₅₃×#427	C₂²	semidirect
24. C₅₃×#358	C₂²	semidirect
25. C₅₃×#357	C₂²	semidirect
37. #135×C₅₃	C₂×C₄	semidirect
143. C₂×#2138	C₅₃	direct (N × H)
160. C₂³×C₁₆	C₂×C₅₃	semidirect
161. C₂×#427	C₂×C₅₃	semidirect
162. #2138	C₂×C₅₃	direct (N × H)
163. C₂×#358	C₂×C₅₃	semidirect
164. C₂×#357	C₂×C₅₃	semidirect
167. C₂×C₅₃	#2138	direct (N × H)
173. C₂×#135	C₄×C₅₃	semidirect
182. C₂²×C₁₆	C₂²×C₅₃	semidirect
184. #427	C₂²×C₅₃	semidirect
187. #358	C₂²×C₅₃	semidirect
188. #357	C₂²×C₅₃	semidirect
189. C₅₃	C₂×#2138	direct (N × H)
201. #135	C₂×C₄×C₅₃	semidirect
324. C₂	C₅₃×#2138	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#135×C₅₃	2
C₂³×C₁₆×C₅₃	2
C₂×C₅₃×#427	2
C₅₃×#2138	2
C₂×C₅₃×#358	2
C₂×C₅₃×#357	2
C₂×#2138	53

Subgroups — C2×C53×#2138 #27240888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2138 #27240888