Subgroups — C₄×#16810402 #27299120

Order 13568.

614 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	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810402	C₂	1	2	yes	maximal
3	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272792	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₅₃	C₄	1	1	yes	characteristic
9	C₂²×#11719908	C₂²	1	2	yes	—
10	C₂×C₄×#88322	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₄×C₈×D₅₃	—	2	2	no	—
14	C₄×C₈×D₅₃	—	2	2	no	—
15	#16810402	C₄	1	8	yes	direct product factor
16	C₄×#11719908	—	2	2	no	—
17	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₄×C₁₆×C₅₃	—	2	2	no	—
20	C₂²×C₄×D₅₃	C₈	1	2	yes	—
21	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
23	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₅₃	—	2	2	no	—
25	C₂×C₈×D₅₃	—	2	2	no	—
26	C₂×C₈×D₅₃	—	2	1	no	—
27	C₂×C₈×D₅₃	—	2	1	no	—
28	C₂×C₈×D₅₃	—	2	2	no	—
29	C₂×C₈×D₅₃	C₈	1	4	yes	—
30	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
32	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄²×D₅₃	—	2	2	no	—
34	C₄²×D₅₃	—	2	2	no	—
35	C₂×#11719908	—	2	2	no	—
36	C₂×#11719908	—	2	2	no	—
37	C₂×#11719908	C₂×C₄	1	4	yes	complemented
38	C₄×#88322	—	2	2	no	—
39	C₈×Dic₅₃	—	2	2	no	—
40	C₈×Dic₅₃	—	2	1	no	—
41	C₈×Dic₅₃	—	2	1	no	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₆×C₅₃	—	2	2	no	—
46	C₂×C₁₆×C₅₃	—	2	2	no	—
47	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
48	C₄×C₈×C₅₃	D₄	1	2	yes	—
49	C₄×C₈×C₅₃	D₄	1	2	yes	—
50	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
55	C₂×C₄×D₅₃	—	2	4	no	—
56	C₂×C₄×D₅₃	—	2	4	no	—
57	C₂×C₄×D₅₃	—	2	1	no	—
58	C₂×C₄×D₅₃	—	2	1	no	—
59	C₂×C₄×D₅₃	—	2	1	no	—
60	C₂×C₄×D₅₃	—	2	1	no	—
61	C₂×C₄×D₅₃	C₁₆	1	8	yes	complemented
62	C₂×C₄×D₅₃	C₄²	1	2	yes	—
63	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
66	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	#41	1	2	yes	—
71	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₅₃	M₁₆	1	2	yes	—
73	C₄×Dic₅₃	—	2	4	no	—
74	C₄×Dic₅₃	—	2	1	no	—
75	C₄×Dic₅₃	—	2	1	no	—
76	C₂×#88322	—	2	1	no	—
77	C₂×#88322	—	2	1	no	—
78	C₂×#88322	—	2	1	no	—
79	C₂×#88322	—	2	1	no	—
80	C₂×#88322	C₄²	1	2	yes	—
81	C₂×#88322	C₂×C₈	1	2	yes	—
82	C₈×D₅₃	—	2	8	no	—
83	C₈×D₅₃	—	2	4	no	—
84	C₈×D₅₃	—	2	4	no	—
85	C₁₆×C₅₃	—	2	4	no	—
86	#11719908	—	2	4	no	—
87	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
88	C₂²×D₅₃	—	2	1	no	—
89	C₂²×D₅₃	—	2	1	no	—
90	C₂²×D₅₃	—	2	2	no	—
91	C₂²×D₅₃	C₄×C₈	1	2	yes	—
92	C₂²×D₅₃	C₂×C₁₆	1	4	yes	—
93	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
94	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
95	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
96	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
102	C₂×Dic₅₃	—	2	2	no	—
103	C₂×Dic₅₃	—	2	1	no	—
104	C₂×Dic₅₃	—	2	1	no	—
105	C₂×Dic₅₃	—	2	2	no	—
106	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
107	C₂×Dic₅₃	C₂×C₁₆	1	4	yes	—
108	C₄×D₅₃	—	2	16	no	—
109	C₄×D₅₃	—	2	2	no	—
110	C₄×D₅₃	—	2	2	no	—
111	C₄×D₅₃	—	2	2	no	—
112	C₄×D₅₃	—	2	2	no	—
113	C₈×C₅₃	C₂²⋊C₈	1	4	yes	—
114	C₈×C₅₃	C₄×D₄	1	2	yes	—
115	C₈×C₅₃	C₄×D₄	1	2	yes	—
116	#88322	—	2	2	no	—
117	#88322	—	2	2	no	—
118	C₄×C₂²⋊C₁₆	—	53	1	no	maximal, Sylow
119	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
120	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
121	C₂²×C₅₃	#431	1	1	yes	characteristic
122	C₂²×C₅₃	C₂²×C₁₆	1	2	yes	—
123	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
124	C₂×D₅₃	—	2	2	no	—
125	C₂×D₅₃	—	2	4	no	—
126	C₂×D₅₃	—	2	4	no	—
127	C₂×D₅₃	—	2	2	no	—
128	C₂×D₅₃	C₄×C₁₆	1	4	yes	complemented
129	C₄×C₅₃	C₂²⋊C₁₆	1	8	yes	complemented
130	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
131	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
132	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
133	C₄×C₅₃	#431	1	1	yes	characteristic
134	Dic₅₃	—	2	4	no	—
135	Dic₅₃	—	2	2	no	—
136	C₂²×C₄×C₈	—	53	1	no	—
137	C₂×C₂²⋊C₁₆	—	53	2	no	—
138	C₂×C₄×C₁₆	—	53	1	no	—
139	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic, complemented
140	#1374	—	53	2	no	—
141	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
142	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
143	C₂×C₅₃	C₄×M₃₂	1	1	yes	characteristic
144	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	2	yes	—
145	C₂×C₅₃	#2162	1	2	yes	—
146	D₅₃	—	2	4	no	—
147	C₂³×C₈	—	53	1	no	—
148	C₂³×C₈	—	53	1	no	—
149	C₂²×C₄²	—	53	1	no	—
150	C₂²×C₁₆	—	53	2	no	—
151	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
152	C₂×C₄×C₈	—	53	1	no	—
153	C₂×C₄×C₈	—	53	1	no	—
154	C₂×C₄×C₈	—	53	1	no	—
155	C₂×C₄×C₈	—	106	2	no	—
156	C₂×C₄×C₈	—	106	2	no	—
157	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
158	C₂²⋊C₁₆	—	53	8	no	—
159	C₄×C₁₆	—	2	2	no	—
160	C₄×C₁₆	—	106	2	no	—
161	C₅₃	C₄×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₄	—	53	1	no	—
163	C₂³×C₄	—	53	2	no	—
164	C₂²×C₈	—	53	1	no	—
165	C₂²×C₈	—	53	1	no	—
166	C₂²×C₈	—	53	4	no	—
167	C₂²×C₈	—	53	4	no	—
168	C₂²×C₈	—	106	1	no	—
169	C₂²×C₈	—	106	2	no	—
170	C₂²×C₈	—	106	1	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	2	no	—
177	C₂×C₄²	—	106	2	no	—
178	C₂×C₄²	—	106	2	no	—
179	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
180	C₂×C₁₆	—	2	2	no	—
181	C₂×C₁₆	—	2	2	no	—
182	C₂×C₁₆	—	53	4	no	—
183	C₂×C₁₆	—	106	2	no	—
184	C₂×C₁₆	—	106	2	no	—
185	C₂×C₁₆	C₄×D₅₃	1	4	yes	complemented
186	C₄×C₈	—	106	1	no	—
187	C₄×C₈	—	106	1	no	—
188	C₄×C₈	—	106	2	no	—
189	C₄×C₈	—	106	2	no	—
190	C₄×C₈	D₂₁₂	1	2	yes	—
191	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
192	C₂⁴	—	53	1	no	—
193	C₂²×C₄	—	53	2	no	—
194	C₂²×C₄	—	53	1	no	—
195	C₂²×C₄	—	53	8	no	—
196	C₂²×C₄	—	53	2	no	—
197	C₂²×C₄	—	53	2	no	—
198	C₂²×C₄	—	106	4	no	—
199	C₂²×C₄	—	106	4	no	—
200	C₂²×C₄	—	106	1	no	—
201	C₂²×C₄	—	106	1	no	—
202	C₂²×C₄	—	106	1	no	—
203	C₂²×C₄	—	106	1	no	—
204	C₂²×C₄	C₈×D₅₃	1	2	yes	—
205	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
206	C₂×C₈	—	53	2	no	—
207	C₂×C₈	—	53	2	no	—
208	C₂×C₈	—	106	1	no	—
209	C₂×C₈	—	106	1	no	—
210	C₂×C₈	—	106	1	no	—
211	C₂×C₈	—	106	1	no	—
212	C₂×C₈	—	106	8	no	—
213	C₂×C₈	—	106	4	no	—
214	C₂×C₈	—	106	4	no	—
215	C₂×C₈	C₈×D₅₃	1	2	yes	—
216	C₂×C₈	#11719912	1	2	yes	—
217	C₂×C₈	#11719921	1	2	yes	—
218	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
219	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
220	C₂×C₈	#11719945	1	2	yes	—
221	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
222	C₄²	—	106	4	no	—
223	C₄²	—	106	1	no	—
224	C₄²	—	106	1	no	—
225	C₄²	#11719912	1	2	yes	—
226	C₄²	#11719921	1	2	yes	—
227	C₁₆	—	2	4	no	—
228	C₁₆	—	106	4	no	—
229	C₂³	—	53	4	no	—
230	C₂³	—	53	2	no	—
231	C₂³	—	106	2	no	—
232	C₂³	—	106	1	no	—
233	C₂³	—	106	1	no	—
234	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
235	C₂×C₄	—	53	4	no	—
236	C₂×C₄	—	53	2	no	—
237	C₂×C₄	—	106	2	no	—
238	C₂×C₄	—	106	1	no	—
239	C₂×C₄	—	106	2	no	—
240	C₂×C₄	—	106	1	no	—
241	C₂×C₄	—	106	2	no	—
242	C₂×C₄	—	106	2	no	—
243	C₂×C₄	—	106	16	no	—
244	C₂×C₄	—	106	2	no	—
245	C₂×C₄	—	106	2	no	—
246	C₂×C₄	#13133129	1	4	yes	—
247	C₂×C₄	C₁₆×D₅₃	1	4	yes	—
248	C₂×C₄	#13133156	1	4	yes	—
249	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
250	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
251	C₂×C₄	#13133174	1	1	yes	characteristic
252	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
253	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
254	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
255	C₈	—	106	2	no	—
256	C₈	—	106	2	no	—
257	C₈	#13133129	1	4	yes	—
258	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
259	C₈	C₄×D₂₁₂	1	2	yes	—
260	C₂²	—	53	4	no	—
261	C₂²	—	106	4	no	—
262	C₂²	—	106	2	no	—
263	C₂²	—	106	2	no	—
264	C₂²	—	106	4	no	—
265	C₂²	C₄×C₈×D₅₃	1	1	yes	characteristic
266	C₂²	C₂×#13133129	1	1	yes	characteristic
267	C₂²	#16810674	1	1	yes	characteristic
268	C₂²	C₂×C₁₆×D₅₃	1	2	yes	—
269	C₂²	C₂×#13133156	1	2	yes	—
270	C₄	—	106	2	no	—
271	C₄	—	106	4	no	—
272	C₄	#16810402	1	8	yes	direct product factor
273	C₄	C₄×#11719921	1	1	yes	characteristic
274	C₄	C₄×#11719912	1	1	yes	characteristic
275	C₄	C₂×#13133129	1	1	yes	characteristic
276	C₄	#16810674	1	1	yes	characteristic
277	C₂	—	106	4	no	—
278	C₂	C₄×#13133129	1	1	yes	characteristic
279	C₂	C₄×C₁₆×D₅₃	1	1	yes	characteristic
280	C₂	C₄×#13133156	1	1	yes	characteristic
281	C₂	C₂×#16810402	1	2	yes	—
282	C₂	#20276512	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×#11719908	C₂	semidirect
4. #20272792	C₂	semidirect
5. C₂×C₄×C₁₆×C₅₃	C₂	semidirect
15. #16810402	C₄	direct (N × H)
37. C₂×#11719908	C₂×C₄	semidirect
47. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
61. C₂×C₄×D₅₃	C₁₆	semidirect
128. C₂×D₅₃	C₄×C₁₆	semidirect
129. C₄×C₅₃	C₂²⋊C₁₆	semidirect
139. C₂×C₄×C₁₆	D₅₃	semidirect
161. C₅₃	C₄×C₂²⋊C₁₆	semidirect
185. C₂×C₁₆	C₄×D₅₃	semidirect
272. C₄	#16810402	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
C₂×#16810402	2
C₂×C₄×#11719908	2
#20272792	2
C₂×C₄×C₁₆×C₅₃	2
C₄×C₂²⋊C₁₆	53

Subgroups — C4×#16810402 #27299120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810402 #27299120