Subgroups — C₂×#20272792 #27298928

Order 13568.

726 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20272792	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₅₃	C₄	1	1	yes	characteristic
7	C₂²×#11719908	—	2	2	no	—
8	C₂²×#11719908	—	2	2	no	—
9	C₂²×#11719908	C₄	1	4	yes	complemented
10	C₂²×#11719908	C₂²	1	2	yes	—
11	C₂×C₈×Dic₅₃	—	2	2	no	—
12	C₂×C₈×Dic₅₃	—	2	2	no	—
13	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
14	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
15	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₅₃	—	2	2	no	—
17	C₂²×C₁₆×C₅₃	—	2	2	no	—
18	C₂²×C₁₆×C₅₃	C₄	1	4	yes	complemented
19	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
20	C₂³×Dic₅₃	C₈	1	2	yes	—
21	C₂²×#88322	—	2	2	no	—
22	C₂²×#88322	—	2	2	no	—
23	C₂²×#88322	C₈	1	4	yes	—
24	C₂²×#88322	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₅₃	—	2	2	no	—
26	C₂×C₄×Dic₅₃	—	2	2	no	—
27	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
28	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
29	C₂×#11719908	—	2	8	no	—
30	C₂×#11719908	—	2	2	no	—
31	C₂×#11719908	—	2	2	no	—
32	C₂×#11719908	C₂×C₄	1	4	yes	complemented
33	C₈×Dic₅₃	—	2	4	no	—
34	C₈×Dic₅₃	—	2	2	no	—
35	C₈×Dic₅₃	—	2	2	no	—
36	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	D₄	1	2	yes	—
41	C₂²×C₈×C₅₃	D₄	1	2	yes	—
42	C₂²×C₈×C₅₃	D₄	1	2	yes	—
43	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×C₁₆×C₅₃	—	2	8	no	—
46	C₂×C₁₆×C₅₃	—	2	2	no	—
47	C₂×C₁₆×C₅₃	—	2	2	no	—
48	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
49	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
56	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×Dic₅₃	—	2	4	no	—
59	C₂²×Dic₅₃	—	2	4	no	—
60	C₂²×Dic₅₃	C₁₆	1	8	yes	complemented
61	C₂²×Dic₅₃	C₂×C₈	1	4	yes	—
62	C₂×C₈×C₅₃	C₄²	1	2	yes	—
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
66	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
68	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
75	C₄×Dic₅₃	—	2	4	no	—
76	C₄×Dic₅₃	—	2	2	no	—
77	C₄×Dic₅₃	—	2	2	no	—
78	C₂×#88322	—	2	8	no	—
79	C₂×#88322	—	2	2	no	—
80	C₂×#88322	—	2	2	no	—
81	C₂×#88322	C₂×C₈	1	4	yes	—
82	C₁₆×C₅₃	—	2	4	no	—
83	#11719908	—	2	4	no	—
84	C₂³×C₅₃	C₄×C₈	1	2	yes	—
85	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
86	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
87	C₂³×C₅₃	C₂×C₁₆	1	4	yes	—
88	C₂³×C₅₃	M₃₂	1	4	yes	—
89	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#95	1	4	yes	—
91	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
92	C₂×C₄×C₅₃	#97	1	4	yes	—
93	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
94	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
95	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
96	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
97	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
103	C₂×Dic₅₃	—	2	16	no	—
104	C₂×Dic₅₃	—	2	4	no	—
105	C₂×Dic₅₃	—	2	4	no	—
106	C₂×Dic₅₃	C₂×C₁₆	1	8	yes	complemented
107	C₈×C₅₃	#95	1	4	yes	—
108	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	#88322	—	2	4	no	—
113	C₂×#1374	—	53	1	no	maximal, Sylow
114	C₂²×C₅₃	#336	1	4	yes	—
115	C₂²×C₅₃	C₄×C₁₆	1	4	yes	—
116	C₂²×C₅₃	#346	1	4	yes	—
117	C₂²×C₅₃	C₂²⋊C₁₆	1	8	yes	—
118	C₂²×C₅₃	C₄⋊C₁₆	1	8	yes	—
119	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
120	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
121	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
122	C₂²×C₅₃	C₂²×C₁₆	1	2	yes	—
123	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
124	C₄×C₅₃	#336	1	4	yes	—
125	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
126	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
127	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
128	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
129	Dic₅₃	—	2	8	no	—
130	C₂³×C₁₆	—	53	1	no	—
131	C₂³×C₁₆	D₅₃	1	1	yes	characteristic
132	C₂²×C₄×C₈	—	53	1	no	—
133	#1374	—	53	4	no	—
134	C₂×C₅₃	#1374	1	8	yes	complemented
135	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
136	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
137	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
138	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	2	yes	—
139	C₂×C₅₃	C₂×C₄⋊C₁₆	1	2	yes	—
140	C₂³×C₈	—	53	1	no	—
141	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
142	C₂²×C₄²	—	53	1	no	—
143	C₂²×C₁₆	—	2	2	no	—
144	C₂²×C₁₆	—	2	2	no	—
145	C₂²×C₁₆	—	53	4	no	—
146	C₂²×C₁₆	—	53	2	no	—
147	C₂²×C₁₆	—	106	2	no	—
148	C₂²×C₁₆	—	106	2	no	—
149	C₂²×C₁₆	Dic₅₃	1	4	yes	complemented
150	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
151	C₂×C₄×C₈	—	53	2	no	—
152	C₂×C₄×C₈	—	53	2	no	—
153	C₂×C₄×C₈	—	106	2	no	—
154	C₂×C₄×C₈	—	106	2	no	—
155	C₅₃	C₂×#1374	1	1	yes	characteristic, complemented, Sylow
156	C₂³×C₄	—	53	2	no	—
157	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
158	C₂²×C₈	—	53	2	no	—
159	C₂²×C₈	—	53	4	no	—
160	C₂²×C₈	—	106	2	no	—
161	C₂²×C₈	—	106	2	no	—
162	C₂²×C₈	Dic₁₀₆	1	2	yes	—
163	C₂²×C₈	C₄×D₅₃	1	2	yes	—
164	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
165	C₂²×C₈	D₂₁₂	1	2	yes	—
166	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
167	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
168	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
169	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
170	C₂×C₄²	—	53	2	no	—
171	C₂×C₄²	—	53	2	no	—
172	C₂×C₄²	—	106	2	no	—
173	C₂×C₄²	—	106	2	no	—
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	—	2	8	no	—
177	C₂×C₁₆	—	53	4	no	—
178	C₂×C₁₆	—	106	2	no	—
179	C₂×C₁₆	—	106	2	no	—
180	C₂×C₁₆	—	106	8	no	—
181	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	complemented
182	C₄×C₈	—	106	2	no	—
183	C₄×C₈	—	106	2	no	—
184	C₄×C₈	—	106	4	no	—
185	C₂⁴	C₈×D₅₃	1	1	yes	characteristic
186	C₂²×C₄	—	53	8	no	—
187	C₂²×C₄	—	53	4	no	—
188	C₂²×C₄	—	106	4	no	—
189	C₂²×C₄	—	106	4	no	—
190	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
191	C₂²×C₄	C₈×D₅₃	1	2	yes	—
192	C₂²×C₄	#11719912	1	2	yes	—
193	C₂²×C₄	#11719912	1	2	yes	—
194	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
195	C₂²×C₄	#11719919	1	2	yes	—
196	C₂²×C₄	#11719921	1	2	yes	—
197	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
198	C₂×C₈	—	53	4	no	—
199	C₂×C₈	—	106	8	no	—
200	C₂×C₈	—	106	2	no	—
201	C₂×C₈	—	106	2	no	—
202	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
203	C₂×C₈	#11719919	1	2	yes	—
204	C₂×C₈	#11719919	1	4	yes	—
205	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
206	C₂×C₈	#11719921	1	2	yes	—
207	C₂×C₈	#11719921	1	4	yes	—
208	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
209	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
210	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
211	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
212	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
213	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
214	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
215	C₄²	—	106	4	no	—
216	C₄²	—	106	2	no	—
217	C₄²	—	106	2	no	—
218	C₁₆	—	2	4	no	—
219	C₁₆	—	106	4	no	—
220	C₂³	C₈×Dic₅₃	1	2	yes	—
221	C₂³	#13133129	1	2	yes	—
222	C₂³	#13133146	1	2	yes	—
223	C₂³	C₁₆×D₅₃	1	4	yes	—
224	C₂³	#13133156	1	4	yes	—
225	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
226	C₂×C₄	—	53	8	no	—
227	C₂×C₄	—	106	4	no	—
228	C₂×C₄	—	106	4	no	—
229	C₂×C₄	—	106	16	no	—
230	C₂×C₄	#13133124	1	4	yes	—
231	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
232	C₂×C₄	#13133127	1	4	yes	—
233	C₂×C₄	#13133129	1	4	yes	—
234	C₂×C₄	#13133129	1	2	yes	—
235	C₂×C₄	#13133146	1	2	yes	—
236	C₂×C₄	#13133146	1	4	yes	—
237	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
238	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
239	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
240	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
241	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
242	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
243	C₈	—	106	4	no	—
244	C₈	#13133124	1	4	yes	—
245	C₈	C₂×#11719921	1	1	yes	characteristic
246	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
247	C₈	C₂×#11719919	1	1	yes	characteristic
248	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
249	C₂²	#16810377	1	4	yes	—
250	C₂²	C₁₆×Dic₅₃	1	4	yes	—
251	C₂²	#16810398	1	4	yes	—
252	C₂²	#16810402	1	8	yes	—
253	C₂²	#16810441	1	8	yes	—
254	C₂²	C₂×C₈×Dic₅₃	1	1	yes	characteristic
255	C₂²	C₂×#13133129	1	1	yes	characteristic
256	C₂²	C₂×#13133146	1	1	yes	characteristic
257	C₂²	C₂×C₁₆×D₅₃	1	2	yes	—
258	C₂²	C₂×#13133156	1	2	yes	—
259	C₄	—	106	8	no	—
260	C₄	#16810377	1	4	yes	—
261	C₄	C₂×#13133124	1	1	yes	characteristic
262	C₄	C₂×#13133127	1	1	yes	characteristic
263	C₄	C₂×#13133129	1	1	yes	characteristic
264	C₄	C₂×#13133146	1	1	yes	characteristic
265	C₂	#20272792	1	8	yes	direct product factor
266	C₂	C₂×#16810377	1	1	yes	characteristic
267	C₂	C₂×C₁₆×Dic₅₃	1	1	yes	characteristic
268	C₂	C₂×#16810398	1	1	yes	characteristic
269	C₂	C₂×#16810402	1	2	yes	—
270	C₂	C₂×#16810441	1	2	yes	—

	subgroup	G/H
center	C₂³×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₂×C₄×C₁₆
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂³×C₁₆×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂³×C₁₆	D₅₃
hypocenter	C₅₃	C₂×#1374
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
3. #20272792	C₂	direct (N × H)
9. C₂²×#11719908	C₄	semidirect
18. C₂²×C₁₆×C₅₃	C₄	semidirect
32. C₂×#11719908	C₂×C₄	semidirect
48. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
60. C₂²×Dic₅₃	C₁₆	semidirect
106. C₂×Dic₅₃	C₂×C₁₆	semidirect
134. C₂×C₅₃	#1374	semidirect
149. C₂²×C₁₆	Dic₅₃	semidirect
155. C₅₃	C₂×#1374	semidirect
181. C₂×C₁₆	C₂×Dic₅₃	semidirect
265. C₂	#20272792	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11719908	2
C₂²×C₈×Dic₅₃	2
#20272792	2
C₂³×C₁₆×C₅₃	2
C₂×#1374	53

Subgroups — C2×#20272792 #27298928

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272792 #27298928