Subgroups — C₂×#20272817 #27299001

Order 13568.

586 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810661	C₂	1	2	yes	maximal
2	#20272817	C₂	1	4	yes	maximal, direct product factor
3	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133176	—	2	2	no	—
5	#16810661	—	2	2	no	—
6	#16810661	—	2	2	no	—
7	#16810661	C₄	1	4	yes	complemented
8	#16810661	C₂²	1	4	yes	—
9	C₂×#13133151	—	2	2	no	—
10	C₂×#13133149	C₄	1	4	yes	complemented
11	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×M₃₂×C₅₃	—	2	2	no	—
13	C₂×M₃₂×C₅₃	—	2	2	no	—
14	C₂×M₃₂×C₅₃	C₄	1	2	yes	complemented
15	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
16	C₂²×C₁₆×C₅₃	C₄	1	2	yes	complemented
17	#13133176	—	2	4	no	—
18	#13133176	—	2	4	no	—
19	#13133176	—	4	2	no	—
20	#13133176	—	4	2	no	—
21	C₂×C₄⋊Dic₅₃	—	2	4	no	—
22	C₂×C₂²⋊Dic₅₃	—	4	2	no	—
23	C₂×C₄×Dic₅₃	—	4	2	no	—
24	#13133151	—	2	4	no	—
25	#13133151	—	2	4	no	—
26	#13133149	—	2	4	no	—
27	#13133149	C₂×C₄	1	8	yes	complemented
28	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	D₄	1	2	yes	—
34	C₂²×C₈×C₅₃	D₄	1	2	yes	—
35	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	M₃₂×C₅₃	—	2	4	no	—
40	M₃₂×C₅₃	—	2	4	no	—
41	C₂×C₁₆×C₅₃	—	2	2	no	—
42	C₂×C₁₆×C₅₃	—	2	2	no	—
43	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
44	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
51	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
52	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₅₃	—	4	4	no	—
56	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₄²	1	2	yes	—
58	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
68	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
74	C₂²⋊Dic₅₃	—	4	4	no	—
75	C₂²⋊Dic₅₃	—	4	4	no	—
76	C₄⋊Dic₅₃	—	2	8	no	—
77	C₄⋊Dic₅₃	—	4	4	no	—
78	C₄×Dic₅₃	—	4	4	no	—
79	C₄×Dic₅₃	—	4	4	no	—
80	C₁₆×C₅₃	—	2	4	no	—
81	C₂³×C₅₃	—	4	2	no	—
82	C₂³×C₅₃	#102	1	2	yes	—
83	C₂³×C₅₃	#102	1	2	yes	—
84	C₂³×C₅₃	#107	1	1	yes	characteristic
85	C₂³×C₅₃	#107	1	1	yes	characteristic
86	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#95	1	2	yes	—
88	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#102	1	2	yes	—
91	C₂×C₄×C₅₃	#102	1	2	yes	—
92	C₂×C₄×C₅₃	#103	1	2	yes	—
93	C₂×C₄×C₅₃	#103	1	2	yes	—
94	C₂×C₄×C₅₃	#103	1	2	yes	—
95	C₂×C₄×C₅₃	#103	1	2	yes	—
96	C₂×C₄×C₅₃	#106	1	2	yes	—
97	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
102	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
104	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
105	C₂×Dic₅₃	—	4	8	no	—
106	C₂×Dic₅₃	—	8	8	no	—
107	C₈×C₅₃	#95	1	1	yes	characteristic
108	C₈×C₅₃	#95	1	2	yes	—
109	C₈×C₅₃	#95	1	1	yes	characteristic
110	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
111	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×#1384	—	53	1	no	maximal, Sylow
114	C₂²×C₅₃	—	2	4	no	—
115	C₂²×C₅₃	—	2	2	no	—
116	C₂²×C₅₃	—	4	2	no	—
117	C₂²×C₅₃	—	4	1	no	—
118	C₂²×C₅₃	—	4	1	no	—
119	C₂²×C₅₃	#340	1	2	yes	—
120	C₂²×C₅₃	#340	1	1	yes	characteristic
121	C₂²×C₅₃	#340	1	1	yes	characteristic
122	C₂²×C₅₃	C₂×#102	1	2	yes	—
123	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
124	C₄×C₅₃	#340	1	2	yes	—
125	C₄×C₅₃	#340	1	1	yes	characteristic
126	C₄×C₅₃	#340	1	1	yes	characteristic
127	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
128	C₄×C₅₃	C₂×#103	1	2	yes	—
129	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
130	Dic₅₃	—	8	8	no	—
131	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
132	C₂×#427	—	53	2	no	—
133	#1384	—	53	4	no	—
134	C₂×C₅₃	—	2	2	no	—
135	C₂×C₅₃	—	2	2	no	—
136	C₂×C₅₃	—	2	1	no	—
137	C₂×C₅₃	—	2	1	no	—
138	C₂×C₅₃	#1384	1	2	yes	complemented
139	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
140	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
141	C₂×#117	—	106	2	no	—
142	C₂×M₃₂	—	2	2	no	—
143	C₂×M₃₂	—	2	2	no	—
144	C₂×M₃₂	Dic₅₃	1	2	yes	complemented
145	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
146	C₂²×C₁₆	Dic₅₃	1	2	yes	complemented
147	#427	—	53	4	no	—
148	#427	—	53	4	no	—
149	#427	—	106	2	no	—
150	#427	—	106	2	no	—
151	C₂×#107	—	106	2	no	—
152	C₂×#106	—	53	4	no	—
153	C₅₃	C₂×#1384	1	1	yes	characteristic, complemented, Sylow
154	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
155	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
156	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
157	C₂²×C₈	C₄×D₅₃	1	2	yes	—
158	C₂²×C₈	C₄×D₅₃	1	2	yes	—
159	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
160	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
161	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
162	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
163	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
164	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
165	#117	—	106	4	no	—
166	#117	—	106	4	no	—
167	#117	—	212	2	no	—
168	#117	—	212	2	no	—
169	C₂×C₄⋊C₄	—	106	4	no	—
170	C₂×C₂²⋊C₄	—	212	2	no	—
171	C₂×C₄²	—	212	2	no	—
172	M₃₂	—	2	4	no	—
173	M₃₂	—	2	4	no	—
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	complemented
177	#107	—	106	4	no	—
178	#107	—	106	4	no	—
179	#106	—	53	8	no	—
180	#106	—	106	4	no	—
181	C₂⁴	D₄₂₄	1	1	yes	characteristic
182	C₂²×C₄	—	212	4	no	—
183	C₂²×C₄	#11719913	1	2	yes	—
184	C₂²×C₄	#11719913	1	2	yes	—
185	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
186	C₂²×C₄	Dic₂₁₂	1	1	yes	characteristic
187	C₂²×C₄	Dic₂₁₂	1	1	yes	characteristic
188	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
189	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
190	C₂²×C₄	#11719921	1	2	yes	—
191	C₂²×C₄	#11719921	1	2	yes	—
192	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
193	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
194	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
195	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
196	C₂×C₈	#11719919	1	2	yes	—
197	C₂×C₈	#11719919	1	2	yes	—
198	C₂×C₈	#11719919	1	2	yes	—
199	C₂×C₈	#11719919	1	2	yes	—
200	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
201	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
202	C₂×C₈	#11719921	1	2	yes	—
203	C₂×C₈	#11719921	1	2	yes	—
204	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
205	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
207	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
208	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
209	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
210	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
211	C₄⋊C₄	—	106	8	no	—
212	C₄⋊C₄	—	212	4	no	—
213	C₂²⋊C₄	—	212	4	no	—
214	C₂²⋊C₄	—	212	4	no	—
215	C₄²	—	212	4	no	—
216	C₄²	—	212	4	no	—
217	C₁₆	—	2	4	no	—
218	C₂³	—	4	2	no	—
219	C₂³	#13133139	1	2	yes	—
220	C₂³	#13133139	1	2	yes	—
221	C₂³	#13133151	1	1	yes	characteristic
222	C₂³	#13133151	1	1	yes	characteristic
223	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
224	C₂×C₄	—	424	8	no	—
225	C₂×C₄	—	212	8	no	—
226	C₂×C₄	#13133124	1	2	yes	—
227	C₂×C₄	#13133124	1	1	yes	characteristic
228	C₂×C₄	#13133124	1	1	yes	characteristic
229	C₂×C₄	#13133139	1	2	yes	—
230	C₂×C₄	#13133139	1	2	yes	—
231	C₂×C₄	#13133142	1	2	yes	—
232	C₂×C₄	#13133142	1	2	yes	—
233	C₂×C₄	#13133142	1	2	yes	—
234	C₂×C₄	#13133142	1	2	yes	—
235	C₂×C₄	#13133149	1	1	yes	characteristic
236	C₂×C₄	#13133149	1	2	yes	—
237	C₂×C₄	#13133149	1	1	yes	characteristic
238	C₂×C₄	#13133151	1	1	yes	characteristic
239	C₂×C₄	#13133151	1	1	yes	characteristic
240	C₂×C₄	C₂×#11719921	1	2	yes	—
241	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
242	C₂×C₄	C₂×#11719913	1	2	yes	—
243	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
244	C₈	#13133124	1	1	yes	characteristic
245	C₈	#13133124	1	2	yes	—
246	C₈	#13133124	1	1	yes	characteristic
247	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
248	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
249	C₈	C₂×#11719919	1	2	yes	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	1	no	—
254	C₂²	—	4	1	no	—
255	C₂²	#16810386	1	2	yes	—
256	C₂²	#16810386	1	1	yes	characteristic
257	C₂²	#16810386	1	1	yes	characteristic
258	C₂²	C₂×#13133139	1	2	yes	—
259	C₂²	C₂×#13133151	1	1	yes	characteristic
260	C₄	—	424	8	no	—
261	C₄	#16810386	1	2	yes	—
262	C₄	#16810386	1	1	yes	characteristic
263	C₄	#16810386	1	1	yes	characteristic
264	C₄	C₂×#13133124	1	1	yes	characteristic
265	C₄	C₂×#13133142	1	2	yes	—
266	C₄	C₂×#13133149	1	1	yes	characteristic
267	C₂	—	2	2	no	—
268	C₂	—	2	2	no	—
269	C₂	—	2	1	no	—
270	C₂	—	2	1	no	—
271	C₂	#20272817	1	2	yes	direct product factor
272	C₂	C₂×#16810386	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133149
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	#340
prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	C₂×#1384
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
2. #20272817	C₂	direct (N × H)
7. #16810661	C₄	semidirect
10. C₂×#13133149	C₄	semidirect
14. C₂×M₃₂×C₅₃	C₄	semidirect
16. C₂²×C₁₆×C₅₃	C₄	semidirect
27. #13133149	C₂×C₄	semidirect
43. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
138. C₂×C₅₃	#1384	semidirect
144. C₂×M₃₂	Dic₅₃	semidirect
146. C₂²×C₁₆	Dic₅₃	semidirect
153. C₅₃	C₂×#1384	semidirect
176. C₂×C₁₆	C₂×Dic₅₃	semidirect
271. C₂	#20272817	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×#16810661	2
#20272817	2
C₂²×M₃₂×C₅₃	2
C₂×#1384	53

Subgroups — C2×#20272817 #27299001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272817 #27299001