Subgroups — C₄×#16811041 #27536738

Order 13568.

834 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×#16811041	C₂	1	2	yes	maximal
2	C₄×#13133223	C₂	1	1	yes	maximal, characteristic
3	#20276314	C₂	1	2	yes	maximal, complemented
4	#20276313	C₂	1	4	yes	maximal, complemented
5	C₄×#13133156	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₁₆×D₅₃	C₂	1	2	yes	maximal, complemented
8	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133223	C₄	1	1	yes	characteristic
10	C₂×#13133223	C₂²	1	1	yes	characteristic
11	C₄×#11719945	C₄	1	1	yes	characteristic
12	#16811041	C₄	1	16	yes	direct product factor
13	C₂×#13133156	C₂²	1	4	yes	—
14	C₂²×#11719908	C₂²	1	2	yes	—
15	C₂×C₁₆×D₅₃	C₂²	1	4	yes	—
16	#16810578	C₄	1	1	yes	characteristic
17	#16810578	C₂²	1	1	yes	characteristic
18	#16810577	C₄	1	2	yes	—
19	#16810577	C₂²	1	2	yes	—
20	C₄×#11719917	C₄	1	1	yes	characteristic
21	C₄×#11719912	C₄	1	2	yes	—
22	C₄×C₈×D₅₃	C₂²	1	2	yes	—
23	#16810398	C₂²	1	4	yes	—
24	C₄×#11719908	C₂²	1	1	yes	characteristic
25	C₄×#11719908	C₂²	1	1	yes	characteristic
26	C₄×#11719908	C₂²	1	2	yes	—
27	C₁₆×Dic₅₃	C₂²	1	4	yes	—
28	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#346	C₂²	1	2	yes	—
31	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
32	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
33	#13133223	C₈	1	8	yes	—
34	#13133223	C₂×C₄	1	8	yes	—
35	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
36	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
37	C₂×#11719912	C₂×C₄	1	2	yes	—
38	C₂×#11719912	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₅₃	C₂³	1	2	yes	—
41	C₄×Dic₁₀₆	C₈	1	2	yes	—
42	C₄×D₂₁₂	C₈	1	2	yes	—
43	C₄×C₂²⋊D₅₃	C₈	1	4	yes	—
44	#13133176	C₂×C₄	1	1	yes	characteristic
45	#13133174	C₂×C₄	1	2	yes	—
46	C₄²×D₅₃	C₂×C₄	1	2	yes	—
47	#13133156	C₂×C₄	1	16	yes	complemented
48	C₂×#11719908	C₂×C₄	1	8	yes	complemented
49	C₂×#11719908	C₂³	1	2	yes	—
50	C₂×#11719908	C₂³	1	2	yes	—
51	C₁₆×D₅₃	C₂×C₄	1	16	yes	complemented
52	#13133128	C₂×C₄	1	2	yes	—
53	#13133127	C₂×C₄	1	2	yes	—
54	#13133127	C₂×C₄	1	2	yes	—
55	C₄×#88322	C₂×C₄	1	2	yes	—
56	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
57	C₈×Dic₅₃	C₂³	1	2	yes	—
58	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
61	M₃₂×C₅₃	C₂×C₄	1	8	yes	complemented
62	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
63	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
70	#11719945	C₄²	1	4	yes	—
71	#11719945	C₂×C₈	1	4	yes	—
72	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
73	C₂×C₄×D₅₃	C₂×C₈	1	4	yes	—
74	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
75	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
77	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
82	C₄²×C₅₃	C₂×C₈	1	2	yes	—
83	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²⋊Dic₅₃	C₂×C₈	1	2	yes	—
86	#11719921	C₂×C₈	1	4	yes	—
87	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
88	#11719919	C₂×C₈	1	4	yes	—
89	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
90	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
91	#11719917	C₄²	1	4	yes	—
92	#11719917	C₂×C₈	1	4	yes	—
93	C₂×#88322	C₂²×C₄	1	2	yes	—
94	C₂×#88322	C₂²×C₄	1	2	yes	—
95	#11719912	C₄²	1	8	yes	—
96	#11719912	C₂×C₈	1	8	yes	—
97	C₈×D₅₃	C₂×C₈	1	8	yes	—
98	C₈×D₅₃	C₂²×C₄	1	8	yes	—
99	C₁₆×C₅₃	C₂²×C₄	1	8	yes	—
100	#11719908	C₂²×C₄	1	4	yes	—
101	#11719908	C₂²×C₄	1	4	yes	—
102	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
103	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₅₃	—	2	4	no	—
105	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
111	C₂²⋊D₅₃	C₄×C₈	1	8	yes	—
112	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
113	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
114	D₂₁₂	C₄×C₈	1	4	yes	—
115	C₄×D₅₃	—	2	8	no	—
116	C₄×D₅₃	C₂×C₄²	1	4	yes	—
117	C₄×D₅₃	C₂²×C₈	1	4	yes	—
118	Dic₁₀₆	C₄×C₈	1	4	yes	—
119	C₈×C₅₃	C₂×C₄²	1	2	yes	—
120	C₈×C₅₃	C₂²×C₈	1	2	yes	—
121	C₈×C₅₃	C₂²×C₈	1	2	yes	—
122	C₈×C₅₃	C₂³×C₄	1	2	yes	—
123	#88322	C₂×C₄²	1	4	yes	—
124	#88322	C₂²×C₈	1	4	yes	—
125	C₄×#504	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	—	2	2	no	—
127	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
128	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
129	C₂×D₅₃	—	2	4	no	—
130	C₂×D₅₃	C₂×C₄×C₈	1	4	yes	—
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
133	C₄×C₅₃	#504	1	4	yes	complemented
134	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
135	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
136	Dic₅₃	—	2	4	no	—
137	Dic₅₃	C₂×C₄×C₈	1	4	yes	—
138	C₂×#504	—	53	2	no	—
139	C₄×#131	—	53	1	no	—
140	#2108	—	53	2	no	—
141	#2108	—	53	4	no	—
142	C₄×M₃₂	—	53	2	no	—
143	C₄×M₃₂	D₅₃	1	1	yes	characteristic, complemented
144	C₂×C₄×C₁₆	—	53	1	no	—
145	C₂×C₄×C₁₆	—	53	2	no	—
146	C₂×C₅₃	—	2	2	no	—
147	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
148	C₂×C₅₃	C₂×#504	1	2	yes	—
149	D₅₃	—	2	4	no	—
150	C₂×#131	—	53	1	no	—
151	C₂×#131	—	53	1	no	—
152	C₄×#41	—	53	1	no	—
153	#504	—	53	16	no	—
154	C₂×M₃₂	—	53	4	no	—
155	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
156	C₂²×C₁₆	—	53	2	no	—
157	C₂²×C₁₆	—	53	4	no	—
158	#405	—	53	1	no	—
159	#405	—	53	1	no	—
160	#405	—	53	2	no	—
161	#405	—	53	2	no	—
162	C₄×M₁₆	—	53	1	no	—
163	C₄×M₁₆	—	53	2	no	—
164	C₂×C₄×C₈	—	53	2	no	—
165	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
166	#346	—	53	4	no	—
167	#346	C₂×D₅₃	1	2	yes	—
168	C₄×C₁₆	—	53	4	no	—
169	C₄×C₁₆	—	53	2	no	—
170	C₄×C₁₆	—	53	1	no	—
171	C₄×C₁₆	—	53	1	no	—
172	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
173	C₅₃	C₄×#504	1	1	yes	characteristic, complemented, Sylow
174	C₂×#41	—	53	1	no	—
175	#131	—	53	8	no	—
176	#131	—	53	8	no	—
177	C₂×M₁₆	—	53	1	no	—
178	C₂×M₁₆	—	53	1	no	—
179	C₂×M₁₆	—	53	2	no	—
180	C₂×M₁₆	—	53	2	no	—
181	C₂²×C₈	—	53	2	no	—
182	C₂²×C₈	—	53	2	no	—
183	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
185	C₄×Q₈	—	53	2	no	—
186	C₄×D₄	—	53	4	no	—
187	C₄×D₄	—	53	2	no	—
188	#117	—	53	1	no	—
189	#117	—	53	2	no	—
190	C₂×C₄²	—	53	2	no	—
191	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
192	M₃₂	—	53	16	no	—
193	M₃₂	C₄×D₅₃	1	8	yes	complemented
194	C₂×C₁₆	—	53	8	no	—
195	C₂×C₁₆	—	53	2	no	—
196	C₂×C₁₆	—	53	2	no	—
197	C₂×C₁₆	—	53	16	no	—
198	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
199	#97	—	53	2	no	—
200	#97	—	53	2	no	—
201	#97	—	53	2	no	—
202	C₄×C₈	—	53	2	no	—
203	C₄×C₈	—	53	2	no	—
204	C₄×C₈	—	53	2	no	—
205	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
206	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
209	#41	—	53	4	no	—
210	#41	—	53	4	no	—
211	C₂×Q₈	—	53	1	no	—
212	C₂×D₄	—	53	2	no	—
213	C₂×D₄	—	53	1	no	—
214	C₂²×C₄	—	53	2	no	—
215	C₂²×C₄	—	53	4	no	—
216	C₂²×C₄	C₈×D₅₃	1	2	yes	—
217	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
218	M₁₆	—	53	4	no	—
219	M₁₆	—	53	4	no	—
220	M₁₆	—	53	8	no	—
221	M₁₆	—	53	8	no	—
222	C₂×C₈	—	53	2	no	—
223	C₂×C₈	—	53	2	no	—
224	C₂×C₈	—	53	8	no	—
225	C₂×C₈	—	53	8	no	—
226	C₂×C₈	C₈×D₅₃	1	4	yes	—
227	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
229	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
231	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
232	C₄⋊C₄	—	53	4	no	—
233	C₄⋊C₄	—	53	2	no	—
234	C₂²⋊C₄	—	53	2	no	—
235	C₂²⋊C₄	—	53	4	no	—
236	C₄²	—	53	4	no	—
237	C₄²	—	53	2	no	—
238	C₄²	C₈×D₅₃	1	2	yes	—
239	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
240	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
241	C₁₆	—	53	4	no	—
242	C₁₆	—	53	4	no	—
243	C₁₆	C₂×C₄×D₅₃	1	8	yes	—
244	C₂³	—	53	2	no	—
245	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
246	Q₈	—	53	4	no	—
247	D₄	—	53	8	no	—
248	D₄	—	53	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	53	2	no	—
251	C₂×C₄	—	53	4	no	—
252	C₂×C₄	—	53	4	no	—
253	C₂×C₄	—	53	4	no	—
254	C₂×C₄	—	106	8	no	—
255	C₂×C₄	C₄²×D₅₃	1	2	yes	—
256	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
257	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
258	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
259	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
260	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
261	C₈	—	53	4	no	—
262	C₈	—	53	4	no	—
263	C₈	C₄²×D₅₃	1	2	yes	—
264	C₈	C₂×C₈×D₅₃	1	2	yes	—
265	C₈	C₂×C₈×D₅₃	1	2	yes	—
266	C₈	C₂²×C₄×D₅₃	1	2	yes	—
267	C₂²	—	2	2	no	—
268	C₂²	—	53	4	no	—
269	C₂²	—	106	4	no	—
270	C₂²	C₄×C₈×D₅₃	1	2	yes	—
271	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	53	4	no	—
274	C₄	—	106	4	no	—
275	C₄	C₄×C₈×D₅₃	1	2	yes	—
276	C₄	#16811041	1	4	yes	direct product factor
277	C₄	C₂×C₄²×D₅₃	1	1	yes	characteristic
278	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
279	C₂	—	2	2	no	—
280	C₂	—	106	4	no	—
281	C₂	C₂×C₄×C₈×D₅₃	1	1	yes	characteristic
282	C₂	C₂×#16811041	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂³×C₈
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	C₄×#504
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20276314	C₂	semidirect
4. #20276313	C₂	semidirect
5. C₄×#13133156	C₂	semidirect
6. C₂×C₄×#11719908	C₂	semidirect
7. C₄×C₁₆×D₅₃	C₂	semidirect
8. C₄×M₃₂×C₅₃	C₂	semidirect
12. #16811041	C₄	direct (N × H)
47. #13133156	C₂×C₄	semidirect
48. C₂×#11719908	C₂×C₄	semidirect
51. C₁₆×D₅₃	C₂×C₄	semidirect
61. M₃₂×C₅₃	C₂×C₄	semidirect
133. C₄×C₅₃	#504	semidirect
143. C₄×M₃₂	D₅₃	semidirect
173. C₅₃	C₄×#504	semidirect
193. M₃₂	C₄×D₅₃	semidirect
276. C₄	#16811041	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×#16811041	2
C₄×#13133223	2
#20276314	2
#20276313	2
C₄×#13133156	2
C₂×C₄×#11719908	2
C₄×C₁₆×D₅₃	2
C₄×M₃₂×C₅₃	2
C₄×#504	53

Subgroups — C4×#16811041 #27536738

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16811041 #27536738