Subgroups — C₂×#20274873 #27384492

Order 13568.

1358 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133151	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13133150	C₂	1	2	yes	maximal
3	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810492	C₂	1	2	yes	maximal
5	#20274873	C₂	1	8	yes	maximal, direct product factor
6	C₂²×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#88322	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
9	C₂²×#11719919	—	2	4	no	—
10	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×#13133151	—	2	4	no	—
12	C₂×#13133151	C₂²	1	4	yes	—
13	C₂×#13133150	—	2	8	no	—
14	C₂×#13133150	C₂²	1	8	yes	—
15	C₂×C₈×Dic₅₃	C₄	1	8	yes	complemented
16	C₂×C₈×Dic₅₃	C₂²	1	4	yes	—
17	#16810492	—	2	4	no	—
18	#16810492	C₂²	1	8	yes	—
19	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₂×#107×C₅₃	—	2	4	no	—
22	C₂×#107×C₅₃	C₂²	1	4	yes	—
23	C₂³×Dic₅₃	—	2	2	no	—
24	C₂³×Dic₅₃	D₄	1	2	yes	—
25	C₂²×#88322	C₂×C₄	1	4	yes	—
26	C₂²×#88322	D₄	1	4	yes	—
27	C₂²×#88322	Q₈	1	4	yes	—
28	C₂²×#88322	C₂³	1	2	yes	—
29	C₂×C₄⋊Dic₅₃	—	2	2	no	—
30	C₂×C₄⋊Dic₅₃	—	2	4	no	—
31	C₂×C₄⋊Dic₅₃	—	2	4	no	—
32	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
33	C₂×#11719919	—	2	16	no	—
34	C₂×#11719919	—	4	16	no	—
35	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
37	C₂×C₄×Dic₅₃	Q₈	1	4	yes	—
38	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
39	#13133151	—	2	8	no	—
40	#13133150	—	2	16	no	—
41	C₈×Dic₅₃	C₂×C₄	1	16	yes	complemented
42	C₂³×C₄×C₅₃	—	2	2	no	—
43	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₅₃	D₄	1	4	yes	—
46	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
47	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
50	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
51	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
52	#107×C₅₃	—	2	8	no	—
53	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	—	2	4	no	—
55	C₂²×C₄×C₅₃	—	4	4	no	—
56	C₂²×C₄×C₅₃	—	4	8	no	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₅₃	—	2	4	no	—
66	C₂²×Dic₅₃	—	4	4	no	—
67	C₂²×Dic₅₃	—	4	8	no	—
68	C₂²×Dic₅₃	C₄⋊C₄	1	8	yes	—
69	C₂²×Dic₅₃	D₈	1	8	yes	—
70	C₂²×Dic₅₃	Q₁₆	1	8	yes	—
71	C₂²×Dic₅₃	C₂×D₄	1	4	yes	—
72	C₂×C₈×C₅₃	C₄⋊C₄	1	16	yes	complemented
73	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
75	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
76	C₄⋊C₄×C₅₃	—	2	4	no	—
77	C₄⋊C₄×C₅₃	—	2	4	no	—
78	C₄⋊C₄×C₅₃	—	4	4	no	—
79	C₄⋊Dic₅₃	—	2	4	no	—
80	C₄⋊Dic₅₃	—	2	4	no	—
81	C₄⋊Dic₅₃	—	4	4	no	—
82	#11719919	—	4	32	no	—
83	C₄×Dic₅₃	C₄⋊C₄	1	8	yes	—
84	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
85	C₄×Dic₅₃	C₂×Q₈	1	4	yes	—
86	C₂×#88322	C₄⋊C₄	1	16	yes	complemented
87	C₂×#88322	C₂²×C₄	1	4	yes	—
88	C₂×#88322	C₂×D₄	1	4	yes	—
89	C₂×#88322	C₂×Q₈	1	4	yes	—
90	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
91	C₂³×C₅₃	#127	1	2	yes	—
92	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
93	C₂³×C₅₃	C₂×D₈	1	4	yes	—
94	C₂³×C₅₃	C₂×Q₁₆	1	4	yes	—
95	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	—	4	4	no	—
97	C₂×C₄×C₅₃	—	4	8	no	—
98	C₂×C₄×C₅₃	—	8	8	no	—
99	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
100	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×C₄×C₅₃	#127	1	2	yes	—
103	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
104	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
105	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
106	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
112	C₂×Dic₅₃	—	4	4	no	—
113	C₂×Dic₅₃	—	4	8	no	—
114	C₂×Dic₅₃	—	8	8	no	—
115	C₂×Dic₅₃	#107	1	32	yes	complemented
116	C₂×Dic₅₃	C₂×C₄⋊C₄	1	8	yes	—
117	C₂×Dic₅₃	C₂×D₈	1	8	yes	—
118	C₂×Dic₅₃	C₂×Q₁₆	1	8	yes	—
119	C₈×C₅₃	C₂×C₄⋊C₄	1	8	yes	complemented
120	#88322	C₂×C₄⋊C₄	1	8	yes	complemented
121	C₂×#1845	—	53	1	no	maximal, Sylow
122	C₂²×C₅₃	#384	1	4	yes	—
123	C₂²×C₅₃	C₂×#107	1	8	yes	—
124	C₂²×C₅₃	#493	1	4	yes	—
125	C₂²×C₅₃	#494	1	4	yes	—
126	C₂²×C₅₃	#500	1	8	yes	—
127	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
129	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
130	C₂²×C₅₃	C₂²×D₈	1	2	yes	—
131	C₂²×C₅₃	C₂²×Q₁₆	1	2	yes	—
132	C₄×C₅₃	—	8	4	no	—
133	C₄×C₅₃	#384	1	4	yes	—
134	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
135	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
136	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
137	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
138	Dic₅₃	—	8	4	no	—
139	Dic₅₃	C₂×#107	1	16	yes	complemented
140	C₂²×#107	—	53	1	no	—
141	C₂²×#107	—	53	2	no	—
142	C₂²×#107	D₅₃	1	1	yes	characteristic
143	C₂²×C₄×C₈	—	53	1	no	—
144	C₂×#384	—	53	2	no	—
145	#1845	—	53	8	no	—
146	C₂×C₅₃	#1845	1	8	yes	complemented
147	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
148	C₂×C₅₃	C₂²×#107	1	2	yes	—
149	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
150	C₂×C₅₃	C₂×#494	1	1	yes	characteristic
151	C₂×C₅₃	C₂×#500	1	2	yes	—
152	C₂³×C₈	—	53	1	no	—
153	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
154	C₂²×C₄⋊C₄	—	53	2	no	—
155	C₂²×C₄⋊C₄	—	106	4	no	—
156	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
157	C₂²×C₄²	—	53	1	no	—
158	C₂×#107	—	2	4	no	—
159	C₂×#107	—	53	4	no	—
160	C₂×#107	—	53	8	no	—
161	C₂×#107	—	106	8	no	—
162	C₂×#107	—	106	4	no	—
163	C₂×#107	C₂×D₅₃	1	4	yes	—
164	C₂×C₄×C₈	—	53	8	no	—
165	C₂×C₄×C₈	—	53	4	no	—
166	#384	—	53	8	no	—
167	#384	—	106	4	no	—
168	C₅₃	C₂×#1845	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₄	—	2	2	no	—
170	C₂³×C₄	—	53	2	no	—
171	C₂³×C₄	—	106	2	no	—
172	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
173	C₂²×C₈	—	53	4	no	—
174	C₂²×C₈	—	53	4	no	—
175	C₂²×C₈	—	53	4	no	—
176	C₂²×C₈	—	53	2	no	—
177	C₂²×C₈	Dic₁₀₆	1	4	yes	—
178	C₂²×C₈	C₄×D₅₃	1	4	yes	—
179	C₂²×C₈	C₂²⋊D₅₃	1	4	yes	—
180	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
181	C₂×C₄⋊C₄	—	2	4	no	—
182	C₂×C₄⋊C₄	—	2	4	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	53	4	no	—
185	C₂×C₄⋊C₄	—	106	4	no	—
186	C₂×C₄⋊C₄	—	106	4	no	—
187	C₂×C₄⋊C₄	—	106	16	no	—
188	C₂×C₄⋊C₄	—	106	2	no	—
189	C₂×C₄⋊C₄	—	212	16	no	—
190	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
191	C₂×C₄²	—	53	4	no	—
192	C₂×C₄²	—	53	2	no	—
193	C₂×C₄²	—	53	4	no	—
194	C₂×C₄²	—	53	2	no	—
195	#107	—	2	8	no	—
196	#107	—	106	16	no	—
197	#107	—	106	8	no	—
198	C₄×C₈	—	53	16	no	—
199	C₂⁴	D₄×D₅₃	1	1	yes	characteristic
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	—	4	8	no	—
203	C₂²×C₄	—	53	8	no	—
204	C₂²×C₄	—	53	8	no	—
205	C₂²×C₄	—	53	8	no	—
206	C₂²×C₄	—	53	4	no	—
207	C₂²×C₄	—	106	4	no	—
208	C₂²×C₄	—	212	4	no	—
209	C₂²×C₄	—	212	8	no	—
210	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
211	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
212	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
213	C₂²×C₄	D₄×D₅₃	1	2	yes	—
214	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
215	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
216	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
217	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
218	C₂×C₈	—	53	16	no	—
219	C₂×C₈	—	53	4	no	—
220	C₂×C₈	—	53	4	no	—
221	C₂×C₈	—	53	4	no	—
222	C₂×C₈	#11719919	1	16	yes	complemented
223	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
224	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
225	C₂×C₈	C₂×C₂²⋊D₅₃	1	4	yes	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	106	4	no	—
230	C₄⋊C₄	—	106	4	no	—
231	C₄⋊C₄	—	212	32	no	—
232	C₄⋊C₄	—	212	4	no	—
233	C₄²	—	53	8	no	—
234	C₄²	—	53	4	no	—
235	C₄²	—	53	4	no	—
236	C₂³	C₄⋊C₄×D₅₃	1	2	yes	—
237	C₂³	#13133213	1	2	yes	—
238	C₂³	#13133216	1	2	yes	—
239	C₂³	D₈×D₅₃	1	4	yes	—
240	C₂³	Q₁₆×D₅₃	1	4	yes	—
241	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
242	C₂×C₄	—	4	8	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	8	8	no	—
245	C₂×C₄	—	53	8	no	—
246	C₂×C₄	—	53	8	no	—
247	C₂×C₄	—	53	8	no	—
248	C₂×C₄	—	53	32	no	—
249	C₂×C₄	—	424	8	no	—
250	C₂×C₄	—	212	8	no	—
251	C₂×C₄	—	212	4	no	—
252	C₂×C₄	C₄⋊C₄×D₅₃	1	4	yes	—
253	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
254	C₂×C₄	C₂×#11719919	1	4	yes	—
255	C₂×C₄	#13133213	1	2	yes	—
256	C₂×C₄	#13133215	1	4	yes	—
257	C₂×C₄	#13133216	1	4	yes	—
258	C₂×C₄	#13133216	1	2	yes	—
259	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
260	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
261	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
262	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
263	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
264	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
265	C₈	—	53	8	no	—
266	C₈	C₂×#11719919	1	8	yes	complemented
267	C₂²	#16810492	1	4	yes	—
268	C₂²	#107×D₅₃	1	8	yes	—
269	C₂²	#16810998	1	4	yes	—
270	C₂²	#16811001	1	4	yes	—
271	C₂²	#16811027	1	8	yes	—
272	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
273	C₂²	C₂×#13133213	1	1	yes	characteristic
274	C₂²	C₂×#13133216	1	1	yes	characteristic
275	C₂²	C₂×D₈×D₅₃	1	2	yes	—
276	C₂²	C₂×Q₁₆×D₅₃	1	2	yes	—
277	C₄	—	8	4	no	—
278	C₄	—	53	16	no	—
279	C₄	—	424	4	no	—
280	C₄	#16810492	1	4	yes	—
281	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
282	C₄	C₂²×#11719919	1	1	yes	characteristic
283	C₄	C₂×#13133215	1	1	yes	characteristic
284	C₄	C₂×#13133216	1	1	yes	characteristic
285	C₂	#20274873	1	8	yes	direct product factor
286	C₂	C₂×#16810492	1	1	yes	characteristic
287	C₂	C₂×#107×D₅₃	1	2	yes	—
288	C₂	C₂×#16810998	1	1	yes	characteristic
289	C₂	C₂×#16811001	1	1	yes	characteristic
290	C₂	C₂×#16811027	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂²×#107×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂²×#107	D₅₃
hypocenter	C₅₃	C₂×#1845
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
5. #20274873	C₂	direct (N × H)
15. C₂×C₈×Dic₅₃	C₄	semidirect
41. C₈×Dic₅₃	C₂×C₄	semidirect
72. C₂×C₈×C₅₃	C₄⋊C₄	semidirect
86. C₂×#88322	C₄⋊C₄	semidirect
115. C₂×Dic₅₃	#107	semidirect
119. C₈×C₅₃	C₂×C₄⋊C₄	semidirect
120. #88322	C₂×C₄⋊C₄	semidirect
139. Dic₅₃	C₂×#107	semidirect
146. C₂×C₅₃	#1845	semidirect
168. C₅₃	C₂×#1845	semidirect
222. C₂×C₈	#11719919	semidirect
266. C₈	C₂×#11719919	semidirect
285. C₂	#20274873	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133151	2
C₂²×#13133150	2
C₂²×C₈×Dic₅₃	2
C₂×#16810492	2
#20274873	2
C₂²×#107×C₅₃	2
C₂×#1845	53

Subgroups — C2×#20274873 #27384492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274873 #27384492