Subgroups — C₂×#20276469 #27537069

Order 13568.

798 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	C₂²×#13133156	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20276469	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810441	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄²×D₅₃	C₄	1	1	yes	characteristic
10	C₂×#13133156	—	2	8	no	—
11	C₂×#13133156	C₂²	1	8	yes	—
12	C₂²×#11719908	C₂²	1	2	yes	—
13	C₂×C₄×#88322	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₅₃	C₄	1	4	yes	—
17	C₄×C₈×D₅₃	C₂²	1	4	yes	—
18	C₄⋊#11719908	C₂²	1	4	yes	complemented
19	#16810441	C₂²	1	8	yes	complemented
20	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
23	C₂²×C₄×D₅₃	C₈	1	1	yes	characteristic
24	C₂²×C₄×D₅₃	C₈	1	1	yes	characteristic
25	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
27	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₅₃	—	2	4	no	—
29	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
30	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₅₃	D₄	1	4	yes	—
32	C₂×C₈×D₅₃	Q₈	1	4	yes	—
33	C₂×C₈×D₅₃	C₂³	1	2	yes	—
34	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
35	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
36	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄²×D₅₃	C₈	1	4	yes	—
38	C₄²×D₅₃	C₂×C₄	1	4	yes	—
39	#13133156	—	2	16	no	—
40	C₂×#11719908	—	2	4	no	—
41	C₂×#11719908	C₂³	1	4	yes	—
42	C₄×#88322	C₂×C₄	1	4	yes	—
43	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
44	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
45	C₈×Dic₅₃	C₂³	1	2	yes	—
46	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₁₆×C₅₃	—	2	4	no	—
50	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
51	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₅₃	C₂³	1	2	yes	—
53	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
58	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
60	C₂×C₄×D₅₃	—	2	4	no	—
61	C₂×C₄×D₅₃	—	2	4	no	—
62	C₂×C₄×D₅₃	C₄⋊C₄	1	4	yes	—
63	C₂×C₄×D₅₃	C₂×C₈	1	4	yes	—
64	C₂×C₄×D₅₃	C₂×C₈	1	4	yes	—
65	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
66	C₂×C₄×D₅₃	M₁₆	1	4	yes	—
67	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₅₃	—	2	2	no	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
73	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
74	C₄²×C₅₃	C₂×C₈	1	2	yes	—
75	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
76	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
77	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
78	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
79	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
80	C₂×#88322	—	2	2	no	—
81	C₂×#88322	C₄⋊C₄	1	4	yes	—
82	C₂×#88322	C₂²×C₄	1	2	yes	—
83	C₂×#88322	C₂²×C₄	1	2	yes	—
84	C₈×D₅₃	—	2	8	no	—
85	C₈×D₅₃	C₄⋊C₄	1	8	yes	—
86	C₈×D₅₃	C₂×D₄	1	4	yes	—
87	C₈×D₅₃	C₂×Q₈	1	4	yes	—
88	C₁₆×C₅₃	—	2	4	no	—
89	#11719908	—	2	4	no	—
90	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
91	C₂²×D₅₃	—	2	2	no	—
92	C₂²×D₅₃	—	2	2	no	—
93	C₂²×D₅₃	C₄⋊C₈	1	4	yes	—
94	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	M₃₂	1	4	yes	complemented
97	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
99	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
101	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
102	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×Dic₅₃	—	2	2	no	—
104	C₂×Dic₅₃	C₄⋊C₈	1	4	yes	—
105	C₂×Dic₅₃	M₃₂	1	4	yes	complemented
106	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
107	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
108	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
109	C₄×D₅₃	—	2	8	no	—
110	C₄×D₅₃	—	2	8	no	—
111	C₄×D₅₃	C₄⋊C₈	1	8	yes	—
112	C₄×D₅₃	C₂×C₄⋊C₄	1	4	yes	—
113	C₄×D₅₃	C₂×M₁₆	1	4	yes	—
114	C₈×C₅₃	—	2	2	no	—
115	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
116	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
117	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
118	#88322	—	2	2	no	—
119	#88322	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×#2150	—	53	1	no	maximal, Sylow
121	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
122	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
123	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
124	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
125	C₂×D₅₃	—	2	2	no	—
126	C₂×D₅₃	—	2	2	no	—
127	C₂×D₅₃	—	2	8	no	—
128	C₂×D₅₃	C₂×C₄⋊C₈	1	4	yes	—
129	C₄×C₅₃	—	2	2	no	—
130	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
131	C₄×C₅₃	C₂×M₃₂	1	4	yes	complemented
132	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
134	Dic₅₃	—	2	2	no	—
135	Dic₅₃	C₂×C₄⋊C₈	1	4	yes	—
136	Dic₅₃	C₂×M₃₂	1	4	yes	complemented
137	C₂²×M₃₂	—	53	2	no	—
138	C₂²×C₄×C₈	—	53	1	no	—
139	#2150	—	53	8	no	—
140	C₂×C₄⋊C₁₆	—	53	2	no	—
141	C₂×C₄⋊C₁₆	—	53	1	no	—
142	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
143	C₂×C₅₃	#2150	1	4	yes	complemented
144	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
145	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
146	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
147	D₅₃	—	2	4	no	—
148	C₂³×C₈	—	53	1	no	—
149	C₂³×C₈	—	53	1	no	—
150	C₂²×C₄²	—	53	1	no	—
151	C₂×M₃₂	—	53	8	no	—
152	C₂×M₃₂	—	106	8	no	—
153	C₂²×C₁₆	—	53	2	no	—
154	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
155	C₂×C₄×C₈	—	53	1	no	—
156	C₂×C₄×C₈	—	53	1	no	—
157	C₂×C₄×C₈	—	53	1	no	—
158	C₂×C₄×C₈	—	53	4	no	—
159	C₂×C₄×C₈	—	53	4	no	—
160	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
161	C₄⋊C₁₆	—	53	8	no	—
162	C₄⋊C₁₆	—	53	4	no	—
163	C₄⋊C₁₆	C₂×D₅₃	1	4	yes	complemented
164	C₅₃	C₂×#2150	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₄	—	53	1	no	—
166	C₂³×C₄	—	53	1	no	—
167	C₂³×C₄	—	53	1	no	—
168	C₂²×C₈	—	53	1	no	—
169	C₂²×C₈	—	53	2	no	—
170	C₂²×C₈	—	53	4	no	—
171	C₂²×C₈	—	53	4	no	—
172	C₂²×C₈	—	53	2	no	—
173	C₂²×C₈	—	53	1	no	—
174	C₂²×C₈	—	53	4	no	—
175	C₂²×C₈	—	106	4	no	—
176	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
177	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
178	C₂×C₄²	—	53	1	no	—
179	C₂×C₄²	—	53	1	no	—
180	C₂×C₄²	—	53	1	no	—
181	C₂×C₄²	—	53	4	no	—
182	C₂×C₄²	—	53	4	no	—
183	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
184	M₃₂	—	106	16	no	—
185	C₂×C₁₆	—	2	4	no	—
186	C₂×C₁₆	—	53	4	no	—
187	C₂×C₁₆	—	106	4	no	—
188	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
189	C₄×C₈	—	53	4	no	—
190	C₄×C₈	—	53	2	no	—
191	C₄×C₈	—	53	2	no	—
192	C₄×C₈	—	53	4	no	—
193	C₄×C₈	C₄×D₅₃	1	2	yes	—
194	C₄×C₈	C₂²×D₅₃	1	2	yes	—
195	C₂⁴	—	53	1	no	—
196	C₂²×C₄	—	53	1	no	—
197	C₂²×C₄	—	53	1	no	—
198	C₂²×C₄	—	53	4	no	—
199	C₂²×C₄	—	53	2	no	—
200	C₂²×C₄	—	53	4	no	—
201	C₂²×C₄	—	53	2	no	—
202	C₂²×C₄	—	53	1	no	—
203	C₂²×C₄	—	53	4	no	—
204	C₂²×C₄	—	53	4	no	—
205	C₂²×C₄	—	106	4	no	—
206	C₂²×C₄	—	106	4	no	—
207	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
208	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	53	4	no	—
212	C₂×C₈	—	53	2	no	—
213	C₂×C₈	—	53	8	no	—
214	C₂×C₈	—	53	4	no	—
215	C₂×C₈	—	53	4	no	—
216	C₂×C₈	—	53	2	no	—
217	C₂×C₈	—	106	8	no	—
218	C₂×C₈	—	106	2	no	—
219	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
220	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
221	C₂×C₈	D₄×D₅₃	1	2	yes	—
222	C₂×C₈	Q₈×D₅₃	1	2	yes	—
223	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
224	C₄²	—	53	4	no	—
225	C₄²	—	53	4	no	—
226	C₄²	—	53	2	no	—
227	C₄²	—	53	2	no	—
228	C₄²	C₈×D₅₃	1	2	yes	—
229	C₄²	C₂×C₄×D₅₃	1	2	yes	—
230	C₁₆	—	2	4	no	—
231	C₁₆	—	106	4	no	—
232	C₂³	—	53	2	no	—
233	C₂³	—	53	4	no	—
234	C₂³	—	106	2	no	—
235	C₂³	—	106	2	no	—
236	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	53	4	no	—
239	C₂×C₄	—	53	2	no	—
240	C₂×C₄	—	53	2	no	—
241	C₂×C₄	—	53	4	no	—
242	C₂×C₄	—	53	4	no	—
243	C₂×C₄	—	53	4	no	—
244	C₂×C₄	—	53	8	no	—
245	C₂×C₄	—	53	2	no	—
246	C₂×C₄	—	106	8	no	—
247	C₂×C₄	—	106	2	no	—
248	C₂×C₄	—	106	8	no	—
249	C₂×C₄	#13133156	1	4	yes	complemented
250	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
251	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
252	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
253	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
254	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
255	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	53	4	no	—
258	C₈	—	106	2	no	—
259	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
260	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
261	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
262	C₂²	—	53	4	no	—
263	C₂²	—	106	2	no	—
264	C₂²	—	106	2	no	—
265	C₂²	—	106	8	no	—
266	C₂²	C₄⋊C₈×D₅₃	1	2	yes	—
267	C₂²	M₃₂×D₅₃	1	2	yes	—
268	C₂²	C₂×#13133156	1	2	yes	—
269	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	53	4	no	—
272	C₄	—	53	4	no	—
273	C₄	—	106	2	no	—
274	C₄	C₄⋊C₈×D₅₃	1	2	yes	—
275	C₄	C₂×#13133156	1	4	yes	complemented
276	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
277	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
278	C₂	—	106	4	no	—
279	C₂	#20276469	1	4	yes	direct product factor
280	C₂	C₂×C₄⋊C₈×D₅₃	1	1	yes	characteristic
281	C₂	C₂×M₃₂×D₅₃	1	1	yes	characteristic
282	C₂	C₂²×#13133156	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	C₂×#2150
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #20276469	C₂	direct (N × H)
4. C₂×C₄⋊#11719908	C₂	semidirect
5. C₂×#16810441	C₂	semidirect
6. C₂×C₅₃×C₄⋊C₁₆	C₂	semidirect
18. C₄⋊#11719908	C₂²	semidirect
19. #16810441	C₂²	semidirect
22. C₅₃×C₄⋊C₁₆	C₂²	semidirect
96. C₂×C₄×C₅₃	M₃₂	semidirect
105. C₂×Dic₅₃	M₃₂	semidirect
131. C₄×C₅₃	C₂×M₃₂	semidirect
136. Dic₅₃	C₂×M₃₂	semidirect
142. C₂×C₄⋊C₁₆	D₅₃	semidirect
143. C₂×C₅₃	#2150	semidirect
163. C₄⋊C₁₆	C₂×D₅₃	semidirect
164. C₅₃	C₂×#2150	semidirect
249. C₂×C₄	#13133156	semidirect
275. C₄	C₂×#13133156	semidirect
279. C₂	#20276469	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×#13133156	2
C₂×C₄×C₈×D₅₃	2
#20276469	2
C₂×C₄⋊#11719908	2
C₂×#16810441	2
C₂×C₅₃×C₄⋊C₁₆	2
C₂×#2150	53

Subgroups — C2×#20276469 #27537069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276469 #27537069