Subgroups — C₂×#20276517 #27537161

Order 13568.

686 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#20276517	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16810402	C₂	1	2	yes	maximal
5	C₂×#16810398	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	C₂²×#11719908	C₂²	1	2	yes	—
11	C₂×C₄×#88322	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×D₅₃	C₄	1	4	yes	—
15	C₄×C₈×D₅₃	C₂²	1	4	yes	—
16	C₄⋊#11719908	C₂²	1	4	yes	complemented
17	#16810402	C₂²	1	8	yes	—
18	#16810398	C₂²	1	8	yes	complemented
19	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
22	C₂²×C₄×D₅₃	C₈	1	1	yes	characteristic
23	C₂²×C₄×D₅₃	C₈	1	1	yes	characteristic
24	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
26	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₅₃	—	2	2	no	—
28	C₂×C₈×D₅₃	—	2	2	no	—
29	C₂×C₈×D₅₃	—	2	4	no	—
30	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
32	C₂×C₈×D₅₃	C₂³	1	2	yes	—
33	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
34	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
35	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄²×D₅₃	C₈	1	4	yes	—
37	C₄²×D₅₃	C₂×C₄	1	4	yes	—
38	C₂×#11719908	—	2	4	no	—
39	C₂×#11719908	C₂³	1	4	yes	—
40	C₄×#88322	C₂×C₄	1	4	yes	—
41	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
42	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
43	C₈×Dic₅₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₁₆×C₅₃	—	2	4	no	—
48	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
49	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₅₃	C₂³	1	2	yes	—
51	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
58	C₂×C₄×D₅₃	—	2	4	no	—
59	C₂×C₄×D₅₃	—	2	4	no	—
60	C₂×C₄×D₅₃	—	2	2	no	—
61	C₂×C₄×D₅₃	—	2	2	no	—
62	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
63	C₂×C₄×D₅₃	C₂×C₈	1	4	yes	—
64	C₂×C₄×D₅₃	C₂×C₈	1	4	yes	—
65	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	#41	1	2	yes	—
70	C₂×C₈×C₅₃	#41	1	2	yes	—
71	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
72	C₄²×C₅₃	C₂×C₈	1	2	yes	—
73	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
74	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
75	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
76	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
77	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
78	C₂×#88322	—	2	2	no	—
79	C₂×#88322	M₁₆	1	4	yes	—
80	C₂×#88322	C₂²×C₄	1	2	yes	—
81	C₂×#88322	C₂²×C₄	1	2	yes	—
82	C₈×D₅₃	—	2	4	no	—
83	C₈×D₅₃	—	2	2	no	—
84	C₈×D₅₃	—	2	2	no	—
85	C₈×D₅₃	—	2	8	no	—
86	C₁₆×C₅₃	—	2	4	no	—
87	#11719908	—	2	4	no	—
88	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
89	C₂²×D₅₃	—	2	2	no	—
90	C₂²×D₅₃	—	2	2	no	—
91	C₂²×D₅₃	M₃₂	1	4	yes	—
92	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
95	C₂×C₄×C₅₃	#117	1	2	yes	—
96	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
97	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
99	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
100	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
101	C₂×Dic₅₃	—	2	2	no	—
102	C₂×Dic₅₃	—	2	2	no	—
103	C₂×Dic₅₃	—	2	2	no	—
104	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
105	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
106	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
107	C₄×D₅₃	—	2	8	no	—
108	C₄×D₅₃	—	2	2	no	—
109	C₄×D₅₃	—	2	2	no	—
110	C₄×D₅₃	—	2	4	no	—
111	C₄×D₅₃	—	2	8	no	—
112	C₈×C₅₃	—	2	2	no	—
113	C₈×C₅₃	#117	1	2	yes	—
114	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
115	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
116	#88322	—	2	2	no	—
117	#88322	C₂×M₁₆	1	4	yes	—
118	C₂×#2163	—	53	1	no	maximal, Sylow
119	C₂²×C₅₃	#431	1	2	yes	—
120	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
121	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
122	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
123	C₂×D₅₃	—	2	2	no	—
124	C₂×D₅₃	—	2	2	no	—
125	C₂×D₅₃	—	2	8	no	—
126	C₂×D₅₃	C₂×M₃₂	1	4	yes	—
127	C₄×C₅₃	—	2	2	no	—
128	C₄×C₅₃	#431	1	2	yes	—
129	C₄×C₅₃	C₂×M₃₂	1	4	yes	—
130	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
131	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
132	Dic₅₃	—	2	2	no	—
133	Dic₅₃	—	2	2	no	—
134	Dic₅₃	—	2	2	no	—
135	C₂²×C₄×C₈	—	53	1	no	—
136	#2163	—	53	8	no	—
137	C₂×C₄⋊C₁₆	—	53	1	no	—
138	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
139	C₂×C₂²⋊C₁₆	—	53	2	no	—
140	C₂×#346	—	53	2	no	—
141	C₂×C₅₃	#2163	1	4	yes	complemented
142	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
143	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
144	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
145	D₅₃	—	2	4	no	—
146	C₂³×C₈	—	53	1	no	—
147	C₂³×C₈	—	53	1	no	—
148	C₂²×C₄²	—	53	1	no	—
149	C₂²×C₁₆	—	53	2	no	—
150	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
151	C₂×C₄×C₈	—	53	1	no	—
152	C₂×C₄×C₈	—	53	1	no	—
153	C₂×C₄×C₈	—	53	1	no	—
154	C₂×C₄×C₈	—	53	4	no	—
155	C₂×C₄×C₈	—	53	4	no	—
156	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
157	C₄⋊C₁₆	—	53	4	no	—
158	C₄⋊C₁₆	C₂×D₅₃	1	4	yes	complemented
159	C₂²⋊C₁₆	—	53	8	no	—
160	#346	—	53	8	no	—
161	C₅₃	C₂×#2163	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₄	—	53	1	no	—
163	C₂³×C₄	—	53	1	no	—
164	C₂³×C₄	—	53	1	no	—
165	C₂²×C₈	—	53	1	no	—
166	C₂²×C₈	—	53	1	no	—
167	C₂²×C₈	—	53	4	no	—
168	C₂²×C₈	—	53	2	no	—
169	C₂²×C₈	—	53	2	no	—
170	C₂²×C₈	—	106	4	no	—
171	C₂²×C₈	—	106	2	no	—
172	C₂²×C₈	—	106	2	no	—
173	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
175	C₂×C₄²	—	53	1	no	—
176	C₂×C₄²	—	53	1	no	—
177	C₂×C₄²	—	53	1	no	—
178	C₂×C₄²	—	53	4	no	—
179	C₂×C₄²	—	53	4	no	—
180	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
181	C₂×C₁₆	—	2	4	no	—
182	C₂×C₁₆	—	53	4	no	—
183	C₂×C₁₆	—	106	4	no	—
184	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
185	C₄×C₈	—	53	2	no	—
186	C₄×C₈	—	53	4	no	—
187	C₄×C₈	—	53	2	no	—
188	C₄×C₈	—	53	4	no	—
189	C₄×C₈	C₄×D₅₃	1	2	yes	—
190	C₄×C₈	C₂²×D₅₃	1	2	yes	—
191	C₂⁴	—	53	1	no	—
192	C₂²×C₄	—	53	1	no	—
193	C₂²×C₄	—	53	1	no	—
194	C₂²×C₄	—	53	1	no	—
195	C₂²×C₄	—	53	4	no	—
196	C₂²×C₄	—	53	4	no	—
197	C₂²×C₄	—	53	2	no	—
198	C₂²×C₄	—	53	2	no	—
199	C₂²×C₄	—	106	4	no	—
200	C₂²×C₄	—	106	4	no	—
201	C₂²×C₄	—	106	2	no	—
202	C₂²×C₄	—	106	2	no	—
203	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
204	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	53	4	no	—
208	C₂×C₈	—	53	2	no	—
209	C₂×C₈	—	53	2	no	—
210	C₂×C₈	—	106	8	no	—
211	C₂×C₈	—	106	2	no	—
212	C₂×C₈	—	106	4	no	—
213	C₂×C₈	—	106	2	no	—
214	C₂×C₈	—	106	2	no	—
215	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
216	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
217	C₂×C₈	#11719947	1	2	yes	—
218	C₂×C₈	#11719949	1	2	yes	—
219	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
220	C₄²	—	53	4	no	—
221	C₄²	—	53	4	no	—
222	C₄²	—	53	2	no	—
223	C₄²	—	53	2	no	—
224	C₄²	C₈×D₅₃	1	2	yes	—
225	C₄²	C₂×C₄×D₅₃	1	2	yes	—
226	C₁₆	—	2	4	no	—
227	C₁₆	—	106	4	no	—
228	C₂³	—	53	4	no	—
229	C₂³	—	53	2	no	—
230	C₂³	—	106	2	no	—
231	C₂³	—	106	2	no	—
232	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	53	2	no	—
235	C₂×C₄	—	53	2	no	—
236	C₂×C₄	—	53	2	no	—
237	C₂×C₄	—	106	8	no	—
238	C₂×C₄	—	106	2	no	—
239	C₂×C₄	—	106	2	no	—
240	C₂×C₄	—	106	2	no	—
241	C₂×C₄	—	106	8	no	—
242	C₂×C₄	—	106	2	no	—
243	C₂×C₄	—	106	2	no	—
244	C₂×C₄	—	106	4	no	—
245	C₂×C₄	#13133156	1	4	yes	—
246	C₂×C₄	#13133175	1	2	yes	—
247	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
248	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
249	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
250	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
251	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	—	53	4	no	—
254	C₈	—	106	2	no	—
255	C₈	#13133175	1	2	yes	—
256	C₈	C₂×#11719947	1	1	yes	characteristic
257	C₈	C₂×#11719949	1	1	yes	characteristic
258	C₂²	—	53	4	no	—
259	C₂²	—	106	8	no	—
260	C₂²	—	106	2	no	—
261	C₂²	—	106	2	no	—
262	C₂²	#16810675	1	2	yes	—
263	C₂²	M₃₂×D₅₃	1	2	yes	—
264	C₂²	C₂×#13133156	1	2	yes	—
265	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	106	2	no	—
268	C₄	—	106	2	no	—
269	C₄	—	106	2	no	—
270	C₄	#16810675	1	2	yes	—
271	C₄	C₂×#13133156	1	4	yes	—
272	C₄	C₂×#13133175	1	1	yes	characteristic
273	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
274	C₂	—	106	4	no	—
275	C₂	#20276517	1	4	yes	direct product factor
276	C₂	C₂×#16810675	1	1	yes	characteristic
277	C₂	C₂×M₃₂×D₅₃	1	1	yes	characteristic
278	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×#2163
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
2. #20276517	C₂	direct (N × H)
3. C₂×C₄⋊#11719908	C₂	semidirect
5. C₂×#16810398	C₂	semidirect
6. C₂×C₅₃×C₄⋊C₁₆	C₂	semidirect
16. C₄⋊#11719908	C₂²	semidirect
18. #16810398	C₂²	semidirect
21. C₅₃×C₄⋊C₁₆	C₂²	semidirect
138. C₂×C₄⋊C₁₆	D₅₃	semidirect
141. C₂×C₅₃	#2163	semidirect
158. C₄⋊C₁₆	C₂×D₅₃	semidirect
161. C₅₃	C₂×#2163	semidirect
275. C₂	#20276517	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×C₈×D₅₃	2
#20276517	2
C₂×C₄⋊#11719908	2
C₂×#16810402	2
C₂×#16810398	2
C₂×C₅₃×C₄⋊C₁₆	2
C₂×#2163	53

Subgroups — C2×#20276517 #27537161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276517 #27537161