Subgroups — C₂×#20276318 #27536748

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₅₃	C₂	1	2	yes	maximal
2	C₂×#16810580	C₂	1	1	yes	maximal, characteristic
3	#20276318	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810401	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810400	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#347	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11719912	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
9	C₂×#13133174	C₄	1	1	yes	characteristic
10	C₂×#13133157	C₂²	1	2	yes	—
11	C₂×#13133156	C₄	1	4	yes	complemented
12	M₃₂×D₅₃	C₄	1	8	yes	complemented
13	M₃₂×D₅₃	C₂²	1	8	yes	—
14	C₂×C₁₆×D₅₃	C₄	1	4	yes	complemented
15	#16810580	C₄	1	4	yes	—
16	#16810580	C₂²	1	4	yes	—
17	C₂×#13133127	C₂²	1	1	yes	characteristic
18	C₂×C₄×#88322	C₄	1	1	yes	characteristic
19	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
20	#16810401	C₂²	1	4	yes	complemented
21	#16810400	C₂²	1	8	yes	complemented
22	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
23	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#347	C₂²	1	4	yes	complemented
25	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×#11719912	—	2	2	no	—
27	C₂×#11719912	—	2	2	no	—
28	C₂×#11719912	C₂×C₄	1	4	yes	—
29	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
30	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
33	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₅₃	C₂³	1	2	yes	—
36	#13133174	—	2	2	no	—
37	#13133174	C₂×C₄	1	4	yes	—
38	C₂×#11719919	—	2	1	no	—
39	C₂×#11719921	—	2	1	no	—
40	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
41	#13133157	C₂×C₄	1	4	yes	—
42	#13133157	C₂³	1	4	yes	—
43	#13133156	C₂×C₄	1	16	yes	complemented
44	C₂×#11719908	C₂×C₄	1	4	yes	—
45	C₁₆×D₅₃	C₂×C₄	1	16	yes	complemented
46	#13133127	C₂×C₄	1	2	yes	—
47	#13133127	C₂³	1	2	yes	—
48	C₄×#88322	C₂×C₄	1	4	yes	—
49	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
53	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
54	M₃₂×C₅₃	C₂³	1	4	yes	—
55	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
56	#97×C₅₃	C₂×C₄	1	2	yes	—
57	#97×C₅₃	C₂³	1	2	yes	—
58	C₂³×D₅₃	M₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	—	2	1	no	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×Dic₅₃	—	2	1	no	—
62	C₂²×Dic₅₃	M₁₆	1	1	yes	characteristic
63	C₂×C₄×D₅₃	C₄²	1	2	yes	—
64	C₂×C₄×D₅₃	C₄²	1	2	yes	—
65	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
66	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
67	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
68	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₅₃	—	2	1	no	—
70	C₂×C₈×C₅₃	—	2	1	no	—
71	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
76	C₄²×C₅₃	—	2	1	no	—
77	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
78	#11719921	—	2	4	no	—
79	#11719919	—	2	4	no	—
80	C₄×Dic₅₃	—	2	1	no	—
81	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
82	C₂×#88322	—	2	1	no	—
83	C₂×#88322	—	2	1	no	—
84	C₂×#88322	C₄²	1	2	yes	—
85	C₂×#88322	C₄²	1	1	yes	characteristic
86	C₂×#88322	C₄²	1	1	yes	characteristic
87	C₂×#88322	C₂²×C₄	1	2	yes	—
88	C₂×#88322	C₂²×C₄	1	2	yes	—
89	#11719912	—	2	8	no	—
90	C₈×D₅₃	C₄²	1	4	yes	—
91	C₈×D₅₃	C₄²	1	4	yes	—
92	C₈×D₅₃	C₂²×C₄	1	4	yes	—
93	C₈×D₅₃	C₂²×C₄	1	4	yes	—
94	C₁₆×C₅₃	C₂²×C₄	1	8	yes	—
95	#11719908	C₂²×C₄	1	8	yes	—
96	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
97	C₂²×D₅₃	—	4	2	no	—
98	C₂²×D₅₃	#97	1	2	yes	—
99	C₂²×D₅₃	#97	1	2	yes	—
100	C₂²×D₅₃	C₂×M₁₆	1	2	yes	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×Dic₅₃	—	2	2	no	—
110	C₂×Dic₅₃	—	4	2	no	—
111	C₂×Dic₅₃	#97	1	2	yes	—
112	C₂×Dic₅₃	#97	1	2	yes	—
113	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
114	C₄×D₅₃	#97	1	4	yes	—
115	C₄×D₅₃	#97	1	4	yes	—
116	C₄×D₅₃	C₂×C₄²	1	4	yes	—
117	C₄×D₅₃	C₂×M₁₆	1	4	yes	—
118	C₈×C₅₃	—	2	2	no	—
119	C₈×C₅₃	C₂×C₄²	1	2	yes	—
120	C₈×C₅₃	C₂³×C₄	1	2	yes	—
121	#88322	—	2	2	no	—
122	#88322	C₂×C₄²	1	2	yes	—
123	#88322	C₂×C₄²	1	2	yes	—
124	C₂×#2110	—	53	1	no	maximal, Sylow
125	C₂²×C₅₃	—	2	2	no	—
126	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
127	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
128	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
129	C₂×D₅₃	—	2	4	no	—
130	C₂×D₅₃	—	4	2	no	—
131	C₂×D₅₃	—	4	2	no	—
132	C₂×D₅₃	C₂×#97	1	4	yes	—
133	C₄×C₅₃	—	4	2	no	—
134	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
135	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
136	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
137	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
138	Dic₅₃	—	4	2	no	—
139	Dic₅₃	C₂×#97	1	4	yes	—
140	C₂²×M₃₂	—	53	2	no	—
141	C₂×#405	—	53	1	no	—
142	#2110	—	53	8	no	—
143	C₂×#347	—	53	2	no	—
144	C₂×#347	—	53	1	no	—
145	C₂×#347	D₅₃	1	1	yes	characteristic, complemented
146	C₂×C₅₃	—	2	1	no	—
147	C₂×C₅₃	—	2	1	no	—
148	C₂×C₅₃	#2110	1	2	yes	complemented
149	C₂×C₅₃	C₂²×#97	1	1	yes	characteristic
150	D₅₃	—	2	4	no	—
151	C₂²×M₁₆	—	53	1	no	—
152	C₂³×C₈	—	53	1	no	—
153	C₂×#117	—	53	1	no	—
154	C₂×M₃₂	—	53	2	no	—
155	C₂×M₃₂	—	53	8	no	—
156	C₂×M₃₂	—	53	8	no	—
157	C₂×M₃₂	—	53	4	no	—
158	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
159	C₂²×C₁₆	—	53	4	no	—
160	#405	—	53	4	no	—
161	#405	—	53	4	no	—
162	C₂×#97	—	53	1	no	—
163	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
164	C₂×C₄×C₈	—	53	1	no	—
165	C₂×C₄×C₈	—	53	1	no	—
166	#347	—	53	8	no	—
167	#347	—	53	4	no	—
168	#347	C₂×D₅₃	1	4	yes	complemented
169	C₅₃	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₄	—	53	1	no	—
171	C₂×M₁₆	—	53	4	no	—
172	C₂×M₁₆	—	106	2	no	—
173	C₂×M₁₆	—	106	2	no	—
174	C₂²×C₈	—	53	1	no	—
175	C₂²×C₈	—	53	2	no	—
176	C₂²×C₈	—	53	2	no	—
177	C₂²×C₈	—	53	4	no	—
178	C₂²×C₈	—	53	2	no	—
179	C₂²×C₈	—	53	2	no	—
180	C₂²×C₈	—	53	1	no	—
181	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
183	#117	—	53	4	no	—
184	#117	—	106	2	no	—
185	C₂×C₄⋊C₄	—	106	1	no	—
186	C₂×C₂²⋊C₄	—	106	1	no	—
187	C₂×C₄²	—	53	1	no	—
188	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
189	M₃₂	—	53	4	no	—
190	M₃₂	—	53	4	no	—
191	M₃₂	—	53	16	no	—
192	M₃₂	C₄×D₅₃	1	4	yes	—
193	M₃₂	C₂²×D₅₃	1	4	yes	—
194	C₂×C₁₆	—	53	4	no	—
195	C₂×C₁₆	—	53	16	no	—
196	C₂×C₁₆	C₄×D₅₃	1	4	yes	—
197	#97	—	53	2	no	—
198	#97	—	53	2	no	—
199	#97	C₄×D₅₃	1	2	yes	—
200	#97	C₂²×D₅₃	1	2	yes	—
201	C₄×C₈	—	53	4	no	—
202	C₄×C₈	—	53	4	no	—
203	C₂⁴	—	53	1	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	53	1	no	—
206	C₂²×C₄	—	53	2	no	—
207	C₂²×C₄	—	53	2	no	—
208	C₂²×C₄	—	53	2	no	—
209	C₂²×C₄	—	53	2	no	—
210	C₂²×C₄	—	53	2	no	—
211	C₂²×C₄	—	53	2	no	—
212	C₂²×C₄	—	106	1	no	—
213	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
214	M₁₆	—	106	8	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	53	2	no	—
218	C₂×C₈	—	53	1	no	—
219	C₂×C₈	—	53	1	no	—
220	C₂×C₈	—	53	2	no	—
221	C₂×C₈	—	53	4	no	—
222	C₂×C₈	—	53	4	no	—
223	C₂×C₈	—	53	4	no	—
224	C₂×C₈	—	53	4	no	—
225	C₂×C₈	—	53	2	no	—
226	C₂×C₈	—	106	1	no	—
227	C₂×C₈	—	106	1	no	—
228	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
229	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
231	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
232	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
233	C₄⋊C₄	—	106	4	no	—
234	C₂²⋊C₄	—	106	4	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	53	2	no	—
237	C₄²	—	106	1	no	—
238	C₄²	C₂×C₄×D₅₃	1	2	yes	—
239	C₁₆	—	53	8	no	—
240	C₁₆	C₂×C₄×D₅₃	1	8	yes	—
241	C₂³	—	53	2	no	—
242	C₂³	—	53	2	no	—
243	C₂³	—	53	2	no	—
244	C₂³	—	212	2	no	—
245	C₂³	M₁₆×D₅₃	1	1	yes	characteristic
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	53	2	no	—
249	C₂×C₄	—	53	2	no	—
250	C₂×C₄	—	53	2	no	—
251	C₂×C₄	—	53	4	no	—
252	C₂×C₄	—	53	4	no	—
253	C₂×C₄	—	53	4	no	—
254	C₂×C₄	—	53	4	no	—
255	C₂×C₄	—	106	2	no	—
256	C₂×C₄	—	212	2	no	—
257	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
258	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
259	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
260	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
261	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	53	2	no	—
265	C₈	—	53	2	no	—
266	C₈	—	106	2	no	—
267	C₈	C₄²×D₅₃	1	2	yes	—
268	C₈	C₂²×C₄×D₅₃	1	2	yes	—
269	C₂²	—	2	2	no	—
270	C₂²	—	53	4	no	—
271	C₂²	—	106	4	no	—
272	C₂²	—	212	2	no	—
273	C₂²	—	212	2	no	—
274	C₂²	#97×D₅₃	1	1	yes	characteristic
275	C₂²	#97×D₅₃	1	1	yes	characteristic
276	C₂²	C₂×M₁₆×D₅₃	1	1	yes	characteristic
277	C₄	—	4	2	no	—
278	C₄	—	53	4	no	—
279	C₄	—	212	2	no	—
280	C₄	#97×D₅₃	1	1	yes	characteristic
281	C₄	#97×D₅₃	1	1	yes	characteristic
282	C₄	C₂×C₄²×D₅₃	1	1	yes	characteristic
283	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	106	4	no	—
287	C₂	#20276318	1	2	yes	direct product factor
288	C₂	C₂×#97×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄²×D₅₃
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#347	C₂
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×D₅₃	M₁₆
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#347	D₅₃
hypocenter	C₅₃	C₂×#2110
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
3. #20276318	C₂	direct (N × H)
4. C₂×#16810401	C₂	semidirect
5. C₂×#16810400	C₂	semidirect
6. C₂×C₅₃×#347	C₂	semidirect
11. C₂×#13133156	C₄	semidirect
12. M₃₂×D₅₃	C₄	semidirect
14. C₂×C₁₆×D₅₃	C₄	semidirect
20. #16810401	C₂²	semidirect
21. #16810400	C₂²	semidirect
24. C₅₃×#347	C₂²	semidirect
43. #13133156	C₂×C₄	semidirect
45. C₁₆×D₅₃	C₂×C₄	semidirect
145. C₂×#347	D₅₃	semidirect
148. C₂×C₅₃	#2110	semidirect
168. #347	C₂×D₅₃	semidirect
169. C₅₃	C₂×#2110	semidirect
287. C₂	#20276318	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×M₃₂×D₅₃	2
C₂×#16810580	2
#20276318	2
C₂×#16810401	2
C₂×#16810400	2
C₂×C₅₃×#347	2
C₂×#2110	53

Subgroups — C2×#20276318 #27536748

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276318 #27536748