Subgroups — C₂×#20290932 #28327920

Order 13568.

782 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅₃⋊M₃₂	C₂	1	2	yes	maximal
2	C₂×#16811516	C₂	1	2	yes	maximal, complemented
3	#20290932	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16811515	C₂	1	2	yes	maximal, complemented
5	C₂×#97×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133292	C₄	1	4	yes	complemented
7	D₅₃⋊M₃₂	C₄	1	8	yes	complemented
8	D₅₃⋊M₃₂	C₂²	1	8	yes	—
9	C₂×#13133291	C₂²	1	4	yes	—
10	C₂×D₅₃⋊C₁₆	C₄	1	4	yes	complemented
11	#16811516	C₂²	1	8	yes	complemented
12	#16811515	C₂²	1	8	yes	complemented
13	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
14	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×D₅₃	C₄	1	1	yes	characteristic
16	C₂×#13133128	C₂²	1	1	yes	characteristic
17	C₂×#13133127	C₄	1	1	yes	characteristic
18	C₂×#13133127	C₂²	1	1	yes	characteristic
19	#97×D₅₃	C₄	1	4	yes	—
20	#97×D₅₃	C₂²	1	4	yes	—
21	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
22	#13133292	C₂×C₄	1	16	yes	complemented
23	#13133291	C₂×C₄	1	8	yes	complemented
24	#13133291	C₂³	1	8	yes	—
25	D₅₃⋊C₁₆	C₂×C₄	1	16	yes	complemented
26	C₂×#11719910	C₂×C₄	1	8	yes	complemented
27	C₂²×C₄×D₅₃	—	2	1	no	—
28	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
30	C₂²×#88322	C₂³	1	1	yes	characteristic
31	C₂×C₈×D₅₃	—	2	1	no	—
32	C₂×C₈×D₅₃	—	2	2	no	—
33	C₂×C₈×D₅₃	—	2	1	no	—
34	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
36	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
38	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₅₃	C₂³	1	2	yes	—
40	C₂×C₄×Dic₅₃	—	2	1	no	—
41	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄²×D₅₃	—	2	2	no	—
43	C₄²×D₅₃	C₂×C₄	1	4	yes	—
44	#13133128	C₂×C₄	1	2	yes	—
45	#13133128	C₂³	1	2	yes	—
46	#13133127	C₂×C₄	1	2	yes	—
47	#13133127	C₂×C₄	1	4	yes	—
48	#13133127	C₂³	1	2	yes	—
49	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
52	#97×C₅₃	C₂×C₄	1	4	yes	—
53	C₂³×D₅₃	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	—	2	1	no	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₅₃	—	2	1	no	—
57	C₂²×Dic₅₃	M₁₆	1	1	yes	characteristic
58	C₂×C₄×D₅₃	—	2	4	no	—
59	C₂×C₄×D₅₃	—	4	4	no	—
60	C₂×C₄×D₅₃	C₄²	1	2	yes	—
61	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
62	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
63	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
64	C₂×C₄×D₅₃	M₁₆	1	1	yes	characteristic
65	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
66	C₂×C₄×D₅₃	M₁₆	1	1	yes	characteristic
67	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₅₃	—	2	1	no	—
69	C₂×C₈×C₅₃	—	2	1	no	—
70	C₂×C₈×C₅₃	C₄²	1	2	yes	—
71	C₂×C₈×C₅₃	C₄²	1	2	yes	—
72	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
74	C₄²×C₅₃	—	2	1	no	—
75	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
76	C₄×Dic₅₃	—	2	1	no	—
77	C₄×Dic₅₃	—	2	4	no	—
78	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
79	C₂×#88322	—	2	1	no	—
80	C₂×#88322	—	2	1	no	—
81	C₂×#88322	C₂²×C₄	1	2	yes	—
82	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
83	C₂×#88322	C₂²×C₄	1	2	yes	—
84	C₂×#88322	C₂²×C₄	1	2	yes	—
85	C₂×#88322	C₂⁴	1	1	yes	characteristic
86	C₈×D₅₃	—	2	4	no	—
87	C₈×D₅₃	—	2	4	no	—
88	C₈×D₅₃	C₄²	1	4	yes	—
89	C₈×D₅₃	C₄²	1	4	yes	—
90	C₈×D₅₃	C₂²×C₄	1	4	yes	—
91	C₈×D₅₃	C₂²×C₄	1	4	yes	—
92	#11719910	C₂²×C₄	1	16	yes	complemented
93	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
94	C₂²×D₅₃	—	2	4	no	—
95	C₂²×D₅₃	#97	1	1	yes	characteristic
96	C₂²×D₅₃	#97	1	2	yes	—
97	C₂²×D₅₃	#97	1	1	yes	characteristic
98	C₂²×D₅₃	C₂×M₁₆	1	2	yes	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×Dic₅₃	—	2	2	no	—
108	C₂×Dic₅₃	—	4	2	no	—
109	C₂×Dic₅₃	#97	1	2	yes	—
110	C₂×Dic₅₃	#97	1	1	yes	characteristic
111	C₂×Dic₅₃	#97	1	1	yes	characteristic
112	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
113	C₄×D₅₃	—	4	8	no	—
114	C₄×D₅₃	#97	1	2	yes	—
115	C₄×D₅₃	#97	1	2	yes	—
116	C₄×D₅₃	#97	1	2	yes	—
117	C₄×D₅₃	#97	1	2	yes	—
118	C₄×D₅₃	C₂×C₄²	1	2	yes	—
119	C₄×D₅₃	C₂×C₄²	1	2	yes	—
120	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
121	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
122	C₈×C₅₃	—	2	2	no	—
123	C₈×C₅₃	C₂×C₄²	1	4	yes	—
124	#88322	—	2	2	no	—
125	#88322	C₂×C₄²	1	2	yes	—
126	#88322	C₂³×C₄	1	2	yes	—
127	C₂²×#347	—	53	1	no	maximal, Sylow
128	C₂²×C₅₃	—	2	2	no	—
129	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
130	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
131	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
132	C₂×D₅₃	—	2	2	no	—
133	C₂×D₅₃	—	2	4	no	—
134	C₂×D₅₃	—	2	4	no	—
135	C₂×D₅₃	#347	1	4	yes	complemented
136	C₂×D₅₃	C₂×#97	1	2	yes	—
137	C₂×D₅₃	C₂×#97	1	2	yes	—
138	C₄×C₅₃	—	4	2	no	—
139	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
140	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
141	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
142	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
143	Dic₅₃	—	4	2	no	—
144	Dic₅₃	C₂×#97	1	2	yes	—
145	Dic₅₃	C₂×#97	1	2	yes	—
146	C₂²×M₃₂	—	53	2	no	—
147	C₂²×#97	—	53	1	no	—
148	C₂×#347	—	53	2	no	—
149	C₂×#347	—	53	8	no	—
150	C₂×#347	—	53	2	no	—
151	C₂×C₅₃	—	2	1	no	—
152	C₂×C₅₃	—	2	1	no	—
153	C₂×C₅₃	C₂×#347	1	2	yes	complemented
154	C₂×C₅₃	C₂²×#97	1	1	yes	characteristic
155	D₅₃	—	2	2	no	—
156	D₅₃	C₂×#347	1	4	yes	complemented
157	C₂³×C₈	—	53	1	no	—
158	C₂³×C₈	—	53	1	no	—
159	C₂²×C₄²	—	53	1	no	—
160	C₂×M₃₂	—	53	4	no	—
161	C₂×M₃₂	—	53	8	no	—
162	C₂×M₃₂	—	53	8	no	—
163	C₂×M₃₂	—	53	4	no	—
164	C₂²×C₁₆	—	53	4	no	—
165	C₂×#97	—	53	1	no	—
166	C₂×#97	—	53	1	no	—
167	C₂×#97	—	53	1	no	—
168	C₂×#97	—	53	4	no	—
169	C₂×#97	—	53	4	no	—
170	C₂×#97	#6157	1	1	yes	characteristic
171	#347	—	53	8	no	—
172	#347	—	53	8	no	—
173	C₅₃	C₂²×#347	1	1	yes	characteristic, complemented, Sylow
174	C₂³×C₄	—	53	1	no	—
175	C₂³×C₄	—	106	1	no	—
176	C₂²×C₈	—	53	1	no	—
177	C₂²×C₈	—	53	2	no	—
178	C₂²×C₈	—	53	4	no	—
179	C₂²×C₈	—	53	2	no	—
180	C₂²×C₈	—	53	2	no	—
181	C₂²×C₈	—	53	2	no	—
182	C₂²×C₈	—	53	1	no	—
183	C₂²×C₈	—	53	4	no	—
184	C₂²×C₈	—	106	1	no	—
185	C₂²×C₈	—	106	1	no	—
186	C₂²×C₈	—	106	2	no	—
187	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
188	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
189	C₂×C₄²	—	53	1	no	—
190	C₂×C₄²	—	53	4	no	—
191	C₂×C₄²	—	106	2	no	—
192	C₂×C₄²	—	106	1	no	—
193	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
194	M₃₂	—	53	8	no	—
195	M₃₂	—	53	8	no	—
196	M₃₂	—	53	16	no	—
197	C₂×C₁₆	—	53	8	no	—
198	C₂×C₁₆	—	53	16	no	—
199	#97	—	53	2	no	—
200	#97	—	53	4	no	—
201	#97	—	53	2	no	—
202	#97	—	53	2	no	—
203	#97	—	53	2	no	—
204	#97	C₂×#6157	1	4	yes	—
205	C₂⁴	—	53	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	53	1	no	—
208	C₂²×C₄	—	53	2	no	—
209	C₂²×C₄	—	53	1	no	—
210	C₂²×C₄	—	53	1	no	—
211	C₂²×C₄	—	53	2	no	—
212	C₂²×C₄	—	53	1	no	—
213	C₂²×C₄	—	53	2	no	—
214	C₂²×C₄	—	53	1	no	—
215	C₂²×C₄	—	53	2	no	—
216	C₂²×C₄	—	106	4	no	—
217	C₂²×C₄	—	106	1	no	—
218	C₂²×C₄	—	212	4	no	—
219	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	53	2	no	—
223	C₂×C₈	—	53	2	no	—
224	C₂×C₈	—	53	1	no	—
225	C₂×C₈	—	53	1	no	—
226	C₂×C₈	—	53	4	no	—
227	C₂×C₈	—	53	4	no	—
228	C₂×C₈	—	53	4	no	—
229	C₂×C₈	—	53	4	no	—
230	C₂×C₈	—	53	2	no	—
231	C₂×C₈	—	106	4	no	—
232	C₂×C₈	—	106	4	no	—
233	C₂×C₈	—	106	1	no	—
234	C₂×C₈	—	106	1	no	—
235	C₂×C₈	C₄×#6157	1	2	yes	—
236	C₂×C₈	C₄×#6157	1	2	yes	—
237	C₂×C₈	C₂²×#6157	1	2	yes	—
238	C₂×C₈	C₂²×#6157	1	2	yes	—
239	C₄²	—	2	1	no	—
240	C₄²	—	53	2	no	—
241	C₄²	—	106	4	no	—
242	C₄²	—	106	1	no	—
243	C₄²	C₂²×#6157	1	2	yes	—
244	C₁₆	—	53	16	no	—
245	C₂³	—	53	1	no	—
246	C₂³	—	53	2	no	—
247	C₂³	—	53	1	no	—
248	C₂³	—	53	2	no	—
249	C₂³	—	106	4	no	—
250	C₂³	D₅₃⋊M₁₆	1	1	yes	characteristic
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	53	2	no	—
254	C₂×C₄	—	53	1	no	—
255	C₂×C₄	—	53	1	no	—
256	C₂×C₄	—	53	2	no	—
257	C₂×C₄	—	53	2	no	—
258	C₂×C₄	—	53	2	no	—
259	C₂×C₄	—	53	2	no	—
260	C₂×C₄	—	53	2	no	—
261	C₂×C₄	—	53	2	no	—
262	C₂×C₄	—	53	2	no	—
263	C₂×C₄	—	53	2	no	—
264	C₂×C₄	—	53	2	no	—
265	C₂×C₄	—	106	2	no	—
266	C₂×C₄	—	212	8	no	—
267	C₂×C₄	—	212	2	no	—
268	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
269	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
270	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
271	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
272	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
273	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	—	53	2	no	—
276	C₈	—	53	2	no	—
277	C₈	—	106	2	no	—
278	C₈	C₂×C₄×#6157	1	4	yes	—
279	C₂²	—	2	2	no	—
280	C₂²	—	53	4	no	—
281	C₂²	—	53	2	no	—
282	C₂²	—	53	2	no	—
283	C₂²	—	106	4	no	—
284	C₂²	—	106	4	no	—
285	C₂²	—	106	2	no	—
286	C₂²	#16811582	1	1	yes	characteristic
287	C₂²	#16811582	1	1	yes	characteristic
288	C₂²	C₂×D₅₃⋊M₁₆	1	1	yes	characteristic
289	C₄	—	4	2	no	—
290	C₄	—	53	2	no	—
291	C₄	—	53	2	no	—
292	C₄	—	212	2	no	—
293	C₄	#16811582	1	1	yes	characteristic
294	C₄	#16811582	1	1	yes	characteristic
295	C₄	C₂²×C₄×#6157	1	1	yes	characteristic
296	C₄	C₂×D₅₃⋊M₁₆	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	53	4	no	—
300	C₂	—	106	2	no	—
301	C₂	#20290932	1	2	yes	direct product factor
302	C₂	C₂×#16811582	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#6157
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×#97×C₅₃	C₄
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×D₅₃	M₁₆
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂×#97	#6157
hypocenter	C₅₃	C₂²×#347
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. C₂×#16811516	C₂	semidirect
3. #20290932	C₂	direct (N × H)
4. C₂×#16811515	C₂	semidirect
6. C₂×#13133292	C₄	semidirect
7. D₅₃⋊M₃₂	C₄	semidirect
10. C₂×D₅₃⋊C₁₆	C₄	semidirect
11. #16811516	C₂²	semidirect
12. #16811515	C₂²	semidirect
22. #13133292	C₂×C₄	semidirect
23. #13133291	C₂×C₄	semidirect
25. D₅₃⋊C₁₆	C₂×C₄	semidirect
26. C₂×#11719910	C₂×C₄	semidirect
92. #11719910	C₂²×C₄	semidirect
135. C₂×D₅₃	#347	semidirect
153. C₂×C₅₃	C₂×#347	semidirect
156. D₅₃	C₂×#347	semidirect
173. C₅₃	C₂²×#347	semidirect
301. C₂	#20290932	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×D₅₃⋊M₃₂	2
C₂×#16811516	2
#20290932	2
C₂×#16811515	2
C₂×#97×D₅₃	2
C₂²×#347	53

Subgroups — C2×#20290932 #28327920

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20290932 #28327920