Subgroups — C₂×#20273509 #27353448

Order 13568.

910 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810688	C₂	1	2	yes	maximal, complemented
3	#20273509	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810350	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₄₂₄	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₅₃	—	2	2	no	—
9	C₂×C₄×D₂₁₂	C₂²	1	2	yes	—
10	C₄×D₄₂₄	C₄	1	4	yes	—
11	C₄×D₄₂₄	C₂²	1	4	yes	—
12	#16810688	—	2	4	no	—
13	#16810688	C₂²	1	8	yes	complemented
14	C₂×#13133151	C₄	1	1	yes	characteristic
15	C₂×#13133139	C₄	1	2	yes	—
16	C₂×#13133129	—	2	2	no	—
17	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
18	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
19	#16810350	C₂²	1	8	yes	complemented
20	C₈×#88322	C₂²	1	4	yes	complemented
21	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#335	C₂²	1	4	yes	complemented
24	C₂²×D₂₁₂	C₂×C₄	1	2	yes	—
25	C₂²×C₄×D₅₃	—	2	2	no	—
26	C₂×D₄₂₄	C₈	1	8	yes	complemented
27	C₂×D₄₂₄	C₂×C₄	1	4	yes	—
28	C₂²×#88322	—	2	1	no	—
29	C₂²×#88322	D₄	1	1	yes	characteristic
30	C₂²×#88322	D₄	1	1	yes	characteristic
31	C₂×C₈×D₅₃	—	2	8	no	—
32	C₂×C₈×D₅₃	—	4	8	no	—
33	C₄×D₂₁₂	—	2	2	no	—
34	C₄×D₂₁₂	—	2	2	no	—
35	C₄×D₂₁₂	C₂×C₄	1	4	yes	—
36	C₄×D₂₁₂	C₂³	1	4	yes	—
37	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
38	C₂×#11719921	—	2	2	no	—
39	#13133151	C₂×C₄	1	4	yes	—
40	#13133139	C₂×C₄	1	8	yes	—
41	#13133129	—	2	8	no	—
42	C₄×#88322	—	2	1	no	—
43	C₄×#88322	D₄	1	2	yes	—
44	C₄×#88322	C₂³	1	2	yes	—
45	C₄×#88322	C₂³	1	2	yes	—
46	C₂²×C₈×C₅₃	—	2	2	no	—
47	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₅₃	—	2	2	no	—
51	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
52	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₅₃	C₂³	1	2	yes	—
54	C₂³×D₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
58	C₂²×Dic₅₃	—	2	2	no	—
59	C₂×D₂₁₂	—	2	2	no	—
60	C₂×D₂₁₂	—	2	4	no	—
61	C₂×D₂₁₂	C₂×C₈	1	8	yes	complemented
62	C₂×D₂₁₂	C₂²×C₄	1	4	yes	—
63	C₂×C₄×D₅₃	—	2	4	no	—
64	C₂×C₄×D₅₃	—	2	4	no	—
65	C₂×C₄×D₅₃	—	4	4	no	—
66	C₂×C₄×D₅₃	—	4	4	no	—
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	—	2	4	no	—
69	C₂×C₈×C₅₃	—	4	4	no	—
70	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	complemented
71	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
73	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅₃	#41	1	1	yes	characteristic
76	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
77	#11719921	—	2	4	no	—
78	#11719921	—	2	4	no	—
79	C₄⋊Dic₅₃	—	2	2	no	—
80	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
81	C₂×#88322	—	2	4	no	—
82	C₂×#88322	—	2	2	no	—
83	C₂×#88322	—	2	2	no	—
84	C₂×#88322	D₈	1	4	yes	complemented
85	C₂×#88322	C₂×D₄	1	2	yes	—
86	C₂×#88322	C₂×D₄	1	2	yes	—
87	D₄₂₄	C₂×C₈	1	16	yes	complemented
88	C₈×D₅₃	—	4	16	no	—
89	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
90	C₂²×D₅₃	—	2	4	no	—
91	C₂²×D₅₃	—	2	8	no	—
92	C₂²×D₅₃	—	4	4	no	—
93	C₂²×D₅₃	—	4	4	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
99	C₂×C₄×C₅₃	#131	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
101	C₂×C₄×C₅₃	#135	1	2	yes	—
102	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
105	C₂×Dic₅₃	—	2	4	no	—
106	C₂×Dic₅₃	—	4	4	no	—
107	D₂₁₂	—	2	4	no	—
108	D₂₁₂	—	2	8	no	—
109	D₂₁₂	C₂²×C₈	1	8	yes	complemented
110	C₄×D₅₃	—	4	4	no	—
111	C₄×D₅₃	—	4	4	no	—
112	C₄×D₅₃	—	4	8	no	—
113	C₈×C₅₃	—	2	2	no	—
114	C₈×C₅₃	—	4	4	no	—
115	C₈×C₅₃	C₂²×C₈	1	4	yes	complemented
116	#88322	—	2	4	no	—
117	#88322	—	2	2	no	—
118	#88322	C₂×D₈	1	4	yes	complemented
119	C₂×C₈×D₈	—	53	1	no	maximal, Sylow
120	C₂²×C₅₃	C₈×D₄	1	2	yes	—
121	C₂²×C₅₃	C₄×D₈	1	2	yes	—
122	C₂²×C₅₃	#443	1	2	yes	—
123	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
124	C₂×D₅₃	—	2	8	no	—
125	C₂×D₅₃	—	4	16	no	—
126	C₂×D₅₃	—	4	4	no	—
127	C₂×D₅₃	—	4	4	no	—
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	C₈×D₄	1	2	yes	—
131	C₄×C₅₃	C₄×D₈	1	2	yes	—
132	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
133	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
134	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
135	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
136	Dic₅₃	—	4	4	no	—
137	C₂×C₄×D₈	—	53	1	no	—
138	C₂×C₈×D₄	—	53	2	no	—
139	C₈×D₈	—	53	8	no	—
140	C₂×#335	D₅₃	1	1	yes	characteristic, complemented
141	C₂×D₄⋊C₈	—	53	2	no	—
142	C₂×C₈²	—	53	1	no	—
143	C₂×C₅₃	C₈×D₈	1	4	yes	complemented
144	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
145	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
146	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
147	D₅₃	—	4	8	no	—
148	C₂²×D₈	—	53	1	no	—
149	C₂³×C₈	—	106	2	no	—
150	C₂×C₄×D₄	—	53	2	no	—
151	C₄×D₈	—	53	4	no	—
152	C₄×D₈	—	53	4	no	—
153	C₈×D₄	—	53	8	no	—
154	C₈×D₄	—	106	4	no	—
155	C₂×#107	—	53	1	no	—
156	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
157	C₂×#102	—	53	2	no	—
158	C₂×C₂²⋊C₈	—	106	2	no	—
159	C₂×C₄×C₈	—	53	1	no	—
160	C₂×C₄×C₈	—	53	1	no	—
161	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
162	#335	C₂×D₅₃	1	4	yes	complemented
163	D₄⋊C₈	—	53	8	no	—
164	C₈²	—	53	4	no	—
165	C₅₃	C₂×C₈×D₈	1	1	yes	characteristic, complemented, Sylow
166	C₂²×D₄	—	53	2	no	—
167	C₂³×C₄	—	106	2	no	—
168	C₂×D₈	—	53	4	no	—
169	C₂×D₈	—	53	8	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	53	1	no	—
172	C₂²×C₈	—	53	1	no	—
173	C₂²×C₈	—	106	8	no	—
174	C₂²×C₈	—	106	1	no	—
175	C₂²×C₈	—	212	8	no	—
176	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
177	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
178	C₄×D₄	—	53	4	no	—
179	C₄×D₄	—	53	4	no	—
180	C₄×D₄	—	106	2	no	—
181	C₄×D₄	—	106	2	no	—
182	C₂×C₄⋊C₄	—	53	2	no	—
183	C₂×C₂²⋊C₄	—	106	2	no	—
184	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
185	#107	—	53	4	no	—
186	C₄⋊C₈	—	2	2	no	—
187	C₄⋊C₈	C₂²×D₅₃	1	4	yes	—
188	#102	—	53	8	no	—
189	C₂²⋊C₈	—	106	8	no	—
190	C₄×C₈	—	53	2	no	—
191	C₄×C₈	—	53	2	no	—
192	C₄×C₈	—	53	2	no	—
193	C₄×C₈	—	106	1	no	—
194	C₄×C₈	C₄×D₅₃	1	2	yes	—
195	C₄×C₈	C₂²×D₅₃	1	2	yes	—
196	C₂⁴	—	106	2	no	—
197	C₂×D₄	—	53	8	no	—
198	C₂×D₄	—	53	4	no	—
199	C₂×D₄	—	106	2	no	—
200	C₂×D₄	—	106	4	no	—
201	C₂²×C₄	—	106	2	no	—
202	C₂²×C₄	—	106	4	no	—
203	C₂²×C₄	—	106	4	no	—
204	C₂²×C₄	—	212	4	no	—
205	C₂²×C₄	—	212	4	no	—
206	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
207	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
208	C₂²×C₄	#11719949	1	1	yes	characteristic
209	D₈	—	53	16	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	4	4	no	—
213	C₂×C₈	—	53	4	no	—
214	C₂×C₈	—	53	2	no	—
215	C₂×C₈	—	53	2	no	—
216	C₂×C₈	—	106	4	no	—
217	C₂×C₈	—	106	2	no	—
218	C₂×C₈	—	106	2	no	—
219	C₂×C₈	—	212	16	no	—
220	C₂×C₈	C₈×D₅₃	1	4	yes	complemented
221	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
222	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
223	C₄⋊C₄	—	53	4	no	—
224	C₄⋊C₄	—	106	2	no	—
225	C₂²⋊C₄	—	106	4	no	—
226	C₂²⋊C₄	—	106	4	no	—
227	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
228	C₄²	D₄×D₅₃	1	1	yes	characteristic
229	C₄²	#11719949	1	1	yes	characteristic
230	C₄²	C₂³×D₅₃	1	1	yes	characteristic
231	C₂³	—	106	4	no	—
232	C₂³	—	106	8	no	—
233	C₂³	—	212	4	no	—
234	C₂³	—	212	4	no	—
235	C₂³	#13133179	1	1	yes	characteristic
236	D₄	—	53	8	no	—
237	D₄	—	106	4	no	—
238	D₄	—	106	8	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	106	4	no	—
241	C₂×C₄	—	212	4	no	—
242	C₂×C₄	—	212	4	no	—
243	C₂×C₄	—	212	4	no	—
244	C₂×C₄	—	212	8	no	—
245	C₂×C₄	#13133179	1	1	yes	characteristic
246	C₂×C₄	#13133179	1	1	yes	characteristic
247	C₂×C₄	#13133179	1	1	yes	characteristic
248	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
249	C₂×C₄	#13133225	1	2	yes	—
250	C₂×C₄	D₈×D₅₃	1	2	yes	—
251	C₂×C₄	#13133240	1	2	yes	—
252	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
253	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
254	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	4	4	no	—
257	C₈	—	53	4	no	—
258	C₈	—	106	4	no	—
259	C₈	—	106	2	no	—
260	C₈	C₂×C₈×D₅₃	1	4	yes	complemented
261	C₂²	—	106	8	no	—
262	C₂²	—	212	4	no	—
263	C₂²	—	212	16	no	—
264	C₂²	—	212	4	no	—
265	C₂²	#16810688	1	2	yes	—
266	C₂²	#16810705	1	2	yes	—
267	C₂²	#16810739	1	2	yes	—
268	C₂²	C₂×#13133179	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	212	4	no	—
272	C₄	#16810688	1	2	yes	—
273	C₄	#16810705	1	2	yes	—
274	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
275	C₄	C₂×#13133225	1	1	yes	characteristic
276	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
277	C₄	C₂×#13133240	1	1	yes	characteristic
278	C₂	—	212	8	no	—
279	C₂	#20273509	1	4	yes	direct product factor
280	C₂	C₂×#16810688	1	1	yes	characteristic
281	C₂	C₂×#16810705	1	1	yes	characteristic
282	C₂	C₂×#16810739	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#335	C₂
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂²×D₄₂₄	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#335	D₅₃
hypocenter	C₅₃	C₂×C₈×D₈
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
2. C₂×#16810688	C₂	semidirect
3. #20273509	C₂	direct (N × H)
4. C₂×#16810350	C₂	semidirect
5. C₂×C₈×#88322	C₂	semidirect
6. C₂×C₅₃×#335	C₂	semidirect
13. #16810688	C₂²	semidirect
19. #16810350	C₂²	semidirect
20. C₈×#88322	C₂²	semidirect
23. C₅₃×#335	C₂²	semidirect
26. C₂×D₄₂₄	C₈	semidirect
61. C₂×D₂₁₂	C₂×C₈	semidirect
70. C₂×C₈×C₅₃	C₂×C₈	semidirect
84. C₂×#88322	D₈	semidirect
87. D₄₂₄	C₂×C₈	semidirect
109. D₂₁₂	C₂²×C₈	semidirect
115. C₈×C₅₃	C₂²×C₈	semidirect
118. #88322	C₂×D₈	semidirect
140. C₂×#335	D₅₃	semidirect
143. C₂×C₅₃	C₈×D₈	semidirect
162. #335	C₂×D₅₃	semidirect
165. C₅₃	C₂×C₈×D₈	semidirect
220. C₂×C₈	C₈×D₅₃	semidirect
260. C₈	C₂×C₈×D₅₃	semidirect
279. C₂	#20273509	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×C₄×D₄₂₄	2
C₂×#16810688	2
#20273509	2
C₂×#16810350	2
C₂×C₈×#88322	2
C₂×C₅₃×#335	2
C₂×C₈×D₈	53

Subgroups — C2×#20273509 #27353448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20273509 #27353448