Subgroups — C₄×#16810408 #27299143

Order 13568.

490 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133223	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810408	C₂	1	2	yes	maximal
3	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272805	C₂	1	2	yes	maximal, complemented
5	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13133223	C₄	1	1	yes	characteristic
7	C₂×#13133223	C₂²	1	1	yes	characteristic
8	C₄×#11719945	C₄	1	1	yes	characteristic
9	C₂²×#11719908	C₂²	1	2	yes	—
10	#16810578	C₄	1	1	yes	characteristic
11	#16810578	C₂²	1	1	yes	characteristic
12	#16810577	—	2	1	no	—
13	#16810577	—	2	1	no	—
14	C₄×#11719917	C₄	1	1	yes	characteristic
15	C₄×#11719912	—	2	1	no	—
16	C₄×C₈×D₅₃	—	2	1	no	—
17	#16810408	C₄	1	8	yes	direct product factor
18	C₄×#11719908	—	2	1	no	—
19	C₄×#11719908	—	2	1	no	—
20	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#346	—	2	1	no	—
23	C₄×C₁₆×C₅₃	—	2	1	no	—
24	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
25	#13133223	—	2	2	no	—
26	#13133223	—	2	2	no	—
27	#13133223	C₈	1	4	yes	—
28	#13133223	C₂×C₄	1	4	yes	—
29	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
30	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
31	C₂×#11719912	—	2	1	no	—
32	C₂×#11719912	—	2	1	no	—
33	C₂×C₈×D₅₃	—	2	1	no	—
34	C₂×C₈×D₅₃	—	2	1	no	—
35	C₄×Dic₁₀₆	C₈	1	2	yes	—
36	C₄×D₂₁₂	C₈	1	2	yes	—
37	C₄×C₂²⋊D₅₃	—	2	2	no	—
38	#13133176	C₂×C₄	1	1	yes	characteristic
39	#13133174	—	2	1	no	—
40	C₄²×D₅₃	—	2	1	no	—
41	C₂×#11719908	—	2	2	no	—
42	C₂×#11719908	—	2	2	no	—
43	C₂×#11719908	C₂×C₄	1	4	yes	complemented
44	#13133128	—	2	1	no	—
45	#13133127	—	2	1	no	—
46	#13133127	—	2	1	no	—
47	C₄×#88322	—	2	1	no	—
48	C₈×Dic₅₃	—	2	1	no	—
49	C₈×Dic₅₃	—	2	1	no	—
50	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
53	M₃₂×C₅₃	—	2	2	no	—
54	M₃₂×C₅₃	C₂×C₄	1	4	yes	complemented
55	C₂×C₁₆×C₅₃	—	2	2	no	—
56	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
57	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
58	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
59	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₅₃	—	2	1	no	—
63	#11719945	—	2	1	no	—
64	#11719945	—	2	1	no	—
65	#11719945	C₄²	1	2	yes	—
66	#11719945	C₂×C₈	1	2	yes	—
67	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
68	C₂×C₄×D₅₃	—	2	1	no	—
69	C₂×C₄×D₅₃	—	2	2	no	—
70	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
74	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
75	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	#41	1	2	yes	—
79	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₄²×C₅₃	C₂×C₈	1	2	yes	—
82	C₂²⋊Dic₅₃	—	2	1	no	—
83	#11719921	—	2	2	no	—
84	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
85	#11719919	—	2	2	no	—
86	C₄×Dic₅₃	—	2	2	no	—
87	C₄×Dic₅₃	—	2	1	no	—
88	#11719917	—	2	1	no	—
89	#11719917	—	2	1	no	—
90	#11719917	C₄²	1	2	yes	—
91	#11719917	C₂×C₈	1	2	yes	—
92	C₂×#88322	—	2	1	no	—
93	C₂×#88322	—	2	1	no	—
94	#11719912	—	2	4	no	—
95	#11719912	—	2	4	no	—
96	C₈×D₅₃	—	2	4	no	—
97	C₈×D₅₃	—	2	4	no	—
98	C₁₆×C₅₃	—	2	4	no	—
99	#11719908	—	2	4	no	—
100	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂²×D₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	4	no	—
103	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
111	C₂²⋊D₅₃	—	2	2	no	—
112	C₂²⋊D₅₃	—	2	2	no	—
113	C₂×Dic₅₃	—	2	1	no	—
114	C₂×Dic₅₃	—	2	2	no	—
115	D₂₁₂	—	2	1	no	—
116	D₂₁₂	C₄×C₈	1	2	yes	—
117	C₄×D₅₃	—	2	2	no	—
118	C₄×D₅₃	—	2	2	no	—
119	C₄×D₅₃	—	4	4	no	—
120	Dic₁₀₆	—	2	1	no	—
121	Dic₁₀₆	C₄×C₈	1	2	yes	—
122	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
123	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
124	C₈×C₅₃	C₄×D₄	1	2	yes	—
125	C₈×C₅₃	C₄×D₄	1	2	yes	—
126	#88322	—	2	2	no	—
127	#88322	—	2	2	no	—
128	C₄×#350	—	53	1	no	maximal, Sylow
129	C₂²×C₅₃	—	2	2	no	—
130	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
131	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
132	C₂²×C₅₃	#431	1	1	yes	characteristic
133	C₂×D₅₃	—	2	2	no	—
134	C₂×D₅₃	—	4	2	no	—
135	C₄×C₅₃	—	2	2	no	—
136	C₄×C₅₃	#350	1	4	yes	complemented
137	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
139	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
140	C₄×C₅₃	#431	1	1	yes	characteristic
141	Dic₅₃	—	2	2	no	—
142	Dic₅₃	—	4	2	no	—
143	C₄×#131	—	53	1	no	—
144	C₂×#350	—	53	2	no	—
145	C₄×M₃₂	D₅₃	1	1	yes	characteristic, complemented
146	C₂×C₄×C₁₆	—	53	1	no	—
147	#1379	—	53	2	no	—
148	C₂×C₅₃	—	2	1	no	—
149	C₂×C₅₃	—	2	1	no	—
150	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
151	C₂×C₅₃	C₂×#350	1	2	yes	—
152	D₅₃	—	4	2	no	—
153	C₂×#131	—	53	1	no	—
154	C₂×#131	—	53	1	no	—
155	C₄×#41	—	53	1	no	—
156	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
157	C₂²×C₁₆	—	53	2	no	—
158	#405	—	53	1	no	—
159	#405	—	53	1	no	—
160	#405	—	106	1	no	—
161	#405	—	106	1	no	—
162	C₄×M₁₆	—	53	1	no	—
163	C₄×M₁₆	—	106	1	no	—
164	C₂×C₄×C₈	—	106	1	no	—
165	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
166	#350	—	53	8	no	—
167	#346	—	2	1	no	—
168	C₄×C₁₆	—	2	1	no	—
169	C₄×C₁₆	—	106	1	no	—
170	C₄×C₁₆	—	106	1	no	—
171	C₅₃	C₄×#350	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	—	53	1	no	—
173	#131	—	53	4	no	—
174	#131	—	53	4	no	—
175	#131	—	106	2	no	—
176	#131	—	106	2	no	—
177	C₂×M₁₆	—	53	1	no	—
178	C₂×M₁₆	—	53	1	no	—
179	C₂×M₁₆	—	106	1	no	—
180	C₂×M₁₆	—	106	1	no	—
181	C₂²×C₈	—	106	1	no	—
182	C₂²×C₈	—	106	1	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	2	no	—
187	C₄×D₄	—	106	2	no	—
188	#117	—	53	1	no	—
189	#117	—	106	1	no	—
190	C₂×C₄²	—	106	1	no	—
191	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
192	M₃₂	—	2	2	no	—
193	M₃₂	C₄×D₅₃	1	4	yes	complemented
194	C₂×C₁₆	—	2	2	no	—
195	C₂×C₁₆	—	53	4	no	—
196	C₂×C₁₆	—	106	2	no	—
197	C₂×C₁₆	—	106	2	no	—
198	#97	—	106	1	no	—
199	#97	—	106	1	no	—
200	#97	—	106	1	no	—
201	C₄×C₈	—	106	1	no	—
202	C₄×C₈	—	106	1	no	—
203	C₄×C₈	—	106	1	no	—
204	C₄×C₈	D₂₁₂	1	1	yes	characteristic
205	C₄×C₈	D₂₁₂	1	1	yes	characteristic
206	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
207	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
208	#41	—	53	2	no	—
209	#41	—	53	2	no	—
210	#41	—	106	1	no	—
211	#41	—	106	1	no	—
212	C₂×Q₈	—	53	1	no	—
213	C₂×D₄	—	53	1	no	—
214	C₂×D₄	—	106	1	no	—
215	C₂²×C₄	—	106	1	no	—
216	C₂²×C₄	—	106	2	no	—
217	C₂²×C₄	#11719912	1	2	yes	—
218	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
219	M₁₆	—	53	2	no	—
220	M₁₆	—	53	2	no	—
221	M₁₆	—	106	1	no	—
222	M₁₆	—	106	1	no	—
223	M₁₆	—	106	4	no	—
224	M₁₆	—	106	4	no	—
225	C₂×C₈	—	106	1	no	—
226	C₂×C₈	—	106	1	no	—
227	C₂×C₈	—	106	4	no	—
228	C₂×C₈	—	106	4	no	—
229	C₂×C₈	C₈×D₅₃	1	2	yes	—
230	C₂×C₈	#11719912	1	2	yes	—
231	C₂×C₈	#11719921	1	1	yes	characteristic
232	C₂×C₈	#11719921	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
234	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
235	C₂×C₈	#11719945	1	2	yes	—
236	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
237	C₄⋊C₄	—	53	2	no	—
238	C₄⋊C₄	—	106	2	no	—
239	C₂²⋊C₄	—	106	1	no	—
240	C₂²⋊C₄	—	106	2	no	—
241	C₄²	—	106	2	no	—
242	C₄²	—	106	1	no	—
243	C₄²	C₈×D₅₃	1	2	yes	—
244	C₄²	#11719921	1	1	yes	characteristic
245	C₄²	#11719921	1	1	yes	characteristic
246	C₁₆	—	2	4	no	—
247	C₁₆	—	106	4	no	—
248	C₂³	—	106	1	no	—
249	C₂³	C₂×#11719912	1	1	yes	characteristic
250	Q₈	—	53	2	no	—
251	Q₈	—	106	1	no	—
252	D₄	—	53	2	no	—
253	D₄	—	106	2	no	—
254	D₄	—	106	2	no	—
255	D₄	—	106	1	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	106	1	no	—
258	C₂×C₄	—	106	2	no	—
259	C₂×C₄	—	106	2	no	—
260	C₂×C₄	—	106	2	no	—
261	C₂×C₄	—	212	4	no	—
262	C₂×C₄	#13133129	1	2	yes	—
263	C₂×C₄	#13133129	1	2	yes	—
264	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
265	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
266	C₂×C₄	#13133174	1	1	yes	characteristic
267	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
268	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
269	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
270	C₈	—	106	2	no	—
271	C₈	—	106	2	no	—
272	C₈	#13133129	1	2	yes	—
273	C₈	#13133129	1	2	yes	—
274	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
275	C₈	C₄×D₂₁₂	1	2	yes	—
276	C₂²	—	2	2	no	—
277	C₂²	—	106	2	no	—
278	C₂²	—	212	2	no	—
279	C₂²	C₄×#11719912	1	1	yes	characteristic
280	C₂²	C₂×#13133129	1	1	yes	characteristic
281	C₂²	#16810674	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	106	2	no	—
284	C₄	—	212	2	no	—
285	C₄	#16810408	1	4	yes	direct product factor
286	C₄	C₄×#11719921	1	1	yes	characteristic
287	C₄	C₄×C₈×D₅₃	1	1	yes	characteristic
288	C₄	C₂×#13133129	1	1	yes	characteristic
289	C₄	#16810674	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	212	2	no	—
293	C₂	C₄×#13133129	1	1	yes	characteristic
294	C₂	C₂×#16810408	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	C₄×#350
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×#11719908	C₂	semidirect
4. #20272805	C₂	semidirect
5. C₄×M₃₂×C₅₃	C₂	semidirect
17. #16810408	C₄	direct (N × H)
43. C₂×#11719908	C₂×C₄	semidirect
54. M₃₂×C₅₃	C₂×C₄	semidirect
136. C₄×C₅₃	#350	semidirect
145. C₄×M₃₂	D₅₃	semidirect
171. C₅₃	C₄×#350	semidirect
193. M₃₂	C₄×D₅₃	semidirect
285. C₄	#16810408	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#13133223	2
C₂×#16810408	2
C₂×C₄×#11719908	2
#20272805	2
C₄×M₃₂×C₅₃	2
C₄×#350	53

Subgroups — C4×#16810408 #27299143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810408 #27299143