Subgroups — C₂×C₅₃×#1998 #27228848

Order 13568.

1126 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1998	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅₃×#343	C₂	1	4	yes	maximal
6	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
8	C₂×#125×C₅₃	—	2	1	no	—
9	C₂×#125×C₅₃	—	2	1	no	—
10	C₂×#125×C₅₃	—	2	2	no	—
11	C₂×#125×C₅₃	—	2	2	no	—
12	C₂×#125×C₅₃	—	2	2	no	—
13	C₂×#125×C₅₃	C₄	1	4	yes	—
14	C₂×#125×C₅₃	C₂²	1	4	yes	—
15	C₂²×C₄⋊C₄×C₅₃	C₄	1	4	yes	—
16	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#423	—	2	2	no	—
19	C₅₃×#423	C₂²	1	4	yes	—
20	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
21	C₄×M₁₆×C₅₃	—	2	2	no	—
22	C₄×M₁₆×C₅₃	C₂²	1	4	yes	—
23	C₂×#97×C₅₃	—	2	1	no	—
24	C₂×C₄×C₈×C₅₃	—	2	1	no	—
25	C₅₃×#343	C₂²	1	16	yes	—
26	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
27	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₅₃	—	2	2	no	—
30	C₂×M₁₆×C₅₃	—	2	2	no	—
31	C₂×M₁₆×C₅₃	—	2	4	no	—
32	C₂×M₁₆×C₅₃	—	2	8	no	—
33	C₂×M₁₆×C₅₃	Q₈	1	8	yes	—
34	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
35	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
36	C₂²×C₈×C₅₃	—	2	2	no	—
37	C₂²×C₈×C₅₃	—	2	2	no	—
38	#125×C₅₃	—	2	4	no	—
39	#125×C₅₃	—	2	4	no	—
40	#125×C₅₃	—	2	8	no	—
41	#125×C₅₃	—	2	4	no	—
42	#125×C₅₃	—	2	4	no	—
43	#125×C₅₃	—	2	8	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
45	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
46	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
47	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
48	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
50	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
51	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	16	yes	—
52	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
53	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
54	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	D₄	1	2	yes	—
58	C₂×C₄²×C₅₃	D₄	1	2	yes	—
59	C₂×C₄²×C₅₃	D₄	1	2	yes	—
60	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
61	C₄⋊C₈×C₅₃	—	2	8	no	—
62	#97×C₅₃	—	2	4	no	—
63	C₄×C₈×C₅₃	—	2	4	no	—
64	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	—	2	4	no	—
66	C₂²×C₄×C₅₃	—	2	4	no	—
67	C₂²×C₄×C₅₃	—	4	4	no	—
68	C₂²×C₄×C₅₃	—	4	8	no	—
69	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
72	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
73	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
80	C₂²×C₄×C₅₃	#41	1	4	yes	—
81	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
82	M₁₆×C₅₃	—	2	8	no	—
83	M₁₆×C₅₃	—	2	4	no	—
84	M₁₆×C₅₃	—	2	8	no	—
85	M₁₆×C₅₃	—	4	4	no	—
86	M₁₆×C₅₃	C₂×Q₈	1	8	yes	—
87	C₂×C₈×C₅₃	—	2	8	no	—
88	C₂×C₈×C₅₃	—	2	4	no	—
89	C₂×C₈×C₅₃	—	2	4	no	—
90	C₂×C₈×C₅₃	—	4	4	no	—
91	C₄⋊C₄×C₅₃	—	2	16	no	—
92	C₄⋊C₄×C₅₃	—	2	16	no	—
93	C₄⋊C₄×C₅₃	—	2	4	no	—
94	C₄⋊C₄×C₅₃	—	2	4	no	—
95	C₄⋊C₄×C₅₃	—	2	4	no	—
96	C₄⋊C₄×C₅₃	—	2	4	no	—
97	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
98	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
99	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
100	C₄²×C₅₃	C₂×D₄	1	2	yes	—
101	C₄²×C₅₃	C₂×D₄	1	2	yes	—
102	C₄²×C₅₃	C₂×D₄	1	4	yes	—
103	C₂³×C₅₃	—	2	4	no	—
104	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂³×C₅₃	C₄×Q₈	1	4	yes	—
107	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	—	2	4	no	—
109	C₂×C₄×C₅₃	—	2	4	no	—
110	C₂×C₄×C₅₃	—	2	8	no	—
111	C₂×C₄×C₅₃	—	4	4	no	—
112	C₂×C₄×C₅₃	—	4	4	no	—
113	C₂×C₄×C₅₃	—	4	8	no	—
114	C₂×C₄×C₅₃	—	4	8	no	—
115	C₂×C₄×C₅₃	—	4	8	no	—
116	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
117	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
118	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
121	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
122	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
123	C₂×C₄×C₅₃	#124	1	2	yes	—
124	C₂×C₄×C₅₃	#124	1	2	yes	—
125	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
126	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
127	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
130	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
131	C₈×C₅₃	—	2	8	no	—
132	C₈×C₅₃	—	4	4	no	—
133	C₂×#1998	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂²×C₅₃	—	2	2	no	—
135	C₂²×C₅₃	—	2	2	no	—
136	C₂²×C₅₃	—	2	8	no	—
137	C₂²×C₅₃	#391	1	2	yes	—
138	C₂²×C₅₃	#391	1	2	yes	—
139	C₂²×C₅₃	#413	1	4	yes	—
140	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
141	C₂²×C₅₃	C₂×C₄×Q₈	1	2	yes	—
142	C₄×C₅₃	—	2	4	no	—
143	C₄×C₅₃	—	2	4	no	—
144	C₄×C₅₃	—	4	8	no	—
145	C₄×C₅₃	#391	1	2	yes	—
146	C₄×C₅₃	#391	1	2	yes	—
147	C₄×C₅₃	C₂×C₂²⋊Q₈	1	2	yes	—
148	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
149	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
150	C₂²×#125	C₂×C₅₃	1	1	yes	characteristic
151	C₂×#423	C₂×C₅₃	1	1	yes	characteristic
152	C₂×C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic
153	#1998	C₂×C₅₃	1	8	yes	direct product factor
154	C₂×#343	C₂×C₅₃	1	4	yes	—
155	C₂×C₅₃	—	2	2	no	—
156	C₂×C₅₃	—	2	2	no	—
157	C₂×C₅₃	#1998	1	4	yes	direct product factor
158	C₂×C₅₃	C₂×#391	1	1	yes	characteristic
159	C₂×C₅₃	C₂×#413	1	2	yes	—
160	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
161	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
162	C₂×#125	—	2	1	no	—
163	C₂×#125	—	2	1	no	—
164	C₂×#125	—	2	2	no	—
165	C₂×#125	—	2	2	no	—
166	C₂×#125	—	2	2	no	—
167	C₂×#125	C₄×C₅₃	1	4	yes	—
168	C₂×#125	C₂²×C₅₃	1	4	yes	—
169	C₂²×C₄⋊C₄	C₄×C₅₃	1	4	yes	—
170	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
171	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
172	#423	—	2	2	no	—
173	#423	C₂²×C₅₃	1	4	yes	—
174	C₂×C₄⋊C₈	—	2	2	no	—
175	C₄×M₁₆	—	2	2	no	—
176	C₄×M₁₆	C₂²×C₅₃	1	4	yes	—
177	C₂×#97	—	2	1	no	—
178	C₂×C₄×C₈	—	2	1	no	—
179	#343	C₂²×C₅₃	1	16	yes	—
180	C₅₃	C₂×#1998	1	1	yes	characteristic, direct product factor, Sylow
181	C₂³×C₄	C₂×C₄×C₅₃	1	4	yes	—
182	C₂³×C₄	C₂×C₄×C₅₃	1	2	yes	—
183	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	4	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	8	no	—
188	C₂×M₁₆	Q₈×C₅₃	1	8	yes	—
189	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
190	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	2	2	no	—
193	#125	—	2	4	no	—
194	#125	—	2	4	no	—
195	#125	—	2	8	no	—
196	#125	—	2	4	no	—
197	#125	—	2	4	no	—
198	#125	—	2	8	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	4	no	—
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	—	2	8	no	—
206	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	16	yes	—
207	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
208	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
209	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
210	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
211	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
212	C₂×C₄²	D₄×C₅₃	1	2	yes	—
213	C₂×C₄²	D₄×C₅₃	1	2	yes	—
214	C₂×C₄²	D₄×C₅₃	1	2	yes	—
215	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
216	C₄⋊C₈	—	2	8	no	—
217	#97	—	2	4	no	—
218	C₄×C₈	—	2	4	no	—
219	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	4	8	no	—
224	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
225	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
226	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
227	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
228	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
229	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
234	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
235	C₂²×C₄	#41×C₅₃	1	4	yes	—
236	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
237	M₁₆	—	2	8	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	8	no	—
240	M₁₆	—	4	4	no	—
241	M₁₆	C₂×Q₈×C₅₃	1	8	yes	—
242	C₂×C₈	—	2	8	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	4	4	no	—
246	C₄⋊C₄	—	2	16	no	—
247	C₄⋊C₄	—	2	16	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
253	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
254	C₄²	C₂²⋊C₄×C₅₃	1	4	yes	—
255	C₄²	C₂×D₄×C₅₃	1	2	yes	—
256	C₄²	C₂×D₄×C₅₃	1	2	yes	—
257	C₄²	C₂×D₄×C₅₃	1	4	yes	—
258	C₂³	—	2	4	no	—
259	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
260	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
261	C₂³	C₄×Q₈×C₅₃	1	4	yes	—
262	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	8	no	—
269	C₂×C₄	—	4	8	no	—
270	C₂×C₄	—	4	8	no	—
271	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
272	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
273	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
274	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
275	C₂×C₄	C₄×Q₈×C₅₃	1	4	yes	—
276	C₂×C₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
277	C₂×C₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
278	C₂×C₄	#124×C₅₃	1	2	yes	—
279	C₂×C₄	#124×C₅₃	1	2	yes	—
280	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
281	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
282	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
283	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
284	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
285	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
286	C₈	—	2	8	no	—
287	C₈	—	4	4	no	—
288	C₂²	—	2	8	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	C₅₃×#391	1	2	yes	—
292	C₂²	C₅₃×#391	1	2	yes	—
293	C₂²	C₅₃×#413	1	4	yes	—
294	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
295	C₂²	C₂×C₄×Q₈×C₅₃	1	2	yes	—
296	C₄	—	2	4	no	—
297	C₄	—	2	4	no	—
298	C₄	—	4	8	no	—
299	C₄	C₅₃×#391	1	2	yes	—
300	C₄	C₅₃×#391	1	2	yes	—
301	C₄	C₂×C₂²⋊Q₈×C₅₃	1	2	yes	—
302	C₄	C₂×#124×C₅₃	1	1	yes	characteristic
303	C₄	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
304	C₂	—	2	2	no	—
305	C₂	—	2	2	no	—
306	C₂	C₅₃×#1998	1	4	yes	direct product factor
307	C₂	C₂×C₅₃×#391	1	1	yes	characteristic
308	C₂	C₂×C₅₃×#413	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂⁴×C₅₃	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1998	C₂	direct (N × H)
133. C₂×#1998	C₅₃	direct (N × H)
153. #1998	C₂×C₅₃	direct (N × H)
157. C₂×C₅₃	#1998	direct (N × H)
180. C₅₃	C₂×#1998	direct (N × H)
306. C₂	C₅₃×#1998	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×#125×C₅₃	2
C₂×C₅₃×#423	2
C₂×C₄×M₁₆×C₅₃	2
C₅₃×#1998	2
C₂×C₅₃×#343	2
C₂×#1998	53

Subgroups — C2×C53×#1998 #27228848

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1998 #27228848