Subgroups — C₄×#16810740 #27382648

Order 13568.

1046 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810740	C₂	1	2	yes	maximal
2	C₄×#13133224	C₂	1	2	yes	maximal, complemented
3	C₄×#13133153	C₂	1	1	yes	maximal, characteristic, complemented
4	#20274596	C₂	1	2	yes	maximal, complemented
5	#20274555	C₂	1	4	yes	maximal, complemented
6	C₄×#13133144	C₂	1	2	yes	maximal, complemented
7	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#135×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133224	C₂²	1	4	yes	—
10	#16810740	C₄	1	16	yes	direct product factor
11	C₂×#13133153	C₂²	1	2	yes	—
12	C₂×#13133144	C₂²	1	4	yes	—
13	#16810579	—	2	4	no	—
14	#16810579	C₂²	1	4	yes	—
15	C₄×#11719917	—	2	2	no	—
16	C₄×#11719917	C₂²	1	2	yes	—
17	C₂×C₄×#88322	—	2	2	no	—
18	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
20	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
21	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
22	#16810388	C₂²	1	2	yes	—
23	#16810382	C₂²	1	4	yes	—
24	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
25	C₄×#41×C₅₃	C₂²	1	2	yes	—
26	C₄×Q₁₆×C₅₃	C₄	1	2	yes	complemented
27	C₄×SD₁₆×C₅₃	C₄	1	4	yes	complemented
28	C₄×D₈×C₅₃	C₄	1	2	yes	complemented
29	C₅₃×#427	C₂²	1	1	yes	characteristic
30	C₅₃×#416	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	#13133224	—	2	8	no	—
33	#13133224	C₂×C₄	1	16	yes	complemented
34	C₂×#11719917	—	2	4	no	—
35	C₂×#11719917	C₂³	1	4	yes	—
36	C₂²×#88322	—	2	4	no	—
37	C₂²×#88322	C₂³	1	2	yes	—
38	C₂×C₄×Dic₅₃	—	2	2	no	—
39	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
40	#13133153	C₂×C₄	1	8	yes	complemented
41	#13133144	C₂×C₄	1	16	yes	complemented
42	#13133128	—	2	2	no	—
43	#13133128	—	2	4	no	—
44	C₄×#88322	—	2	2	no	—
45	C₄×#88322	—	2	4	no	—
46	C₄×#88322	D₄	1	1	yes	characteristic
47	C₄×#88322	D₄	1	1	yes	characteristic
48	C₄×#88322	D₄	1	2	yes	—
49	C₈×Dic₅₃	—	2	4	no	—
50	C₈×Dic₅₃	C₂×C₄	1	8	yes	complemented
51	C₂×#41×C₅₃	C₂³	1	2	yes	—
52	#135×C₅₃	C₂×C₄	1	8	yes	—
53	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
55	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
58	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
59	C₄×D₄×C₅₃	—	2	4	no	—
60	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
61	#117×C₅₃	—	2	2	no	—
62	#117×C₅₃	C₂³	1	2	yes	—
63	C₂×C₄²×C₅₃	—	2	2	no	—
64	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	#107×C₅₃	C₂×C₄	1	2	yes	—
66	#106×C₅₃	C₂×C₄	1	2	yes	—
67	#103×C₅₃	C₂×C₄	1	4	yes	—
68	#102×C₅₃	C₂×C₄	1	4	yes	—
69	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
71	#41×C₅₃	—	2	4	no	—
72	#41×C₅₃	C₂²×C₄	1	8	yes	—
73	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂×D₄×C₅₃	—	2	2	no	—
75	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₅₃	—	2	2	no	—
77	C₂²×C₄×C₅₃	—	2	4	no	—
78	C₂²×C₄×C₅₃	#41	1	2	yes	—
79	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
80	C₂²×Dic₅₃	—	2	2	no	—
81	Q₁₆×C₅₃	C₄²	1	4	yes	complemented
82	SD₁₆×C₅₃	C₄²	1	8	yes	complemented
83	D₈×C₅₃	C₄²	1	4	yes	complemented
84	C₂×C₈×C₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₄⋊C₄×C₅₃	—	2	4	no	—
90	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
91	C₂²⋊C₄×C₅₃	—	2	2	no	—
92	C₂²⋊C₄×C₅₃	—	2	4	no	—
93	C₄²×C₅₃	—	2	4	no	—
94	C₄²×C₅₃	—	2	2	no	—
95	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
96	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₄×Dic₅₃	—	2	8	no	—
99	C₄×Dic₅₃	—	2	2	no	—
100	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
101	#11719917	—	2	4	no	—
102	#11719917	—	2	16	no	—
103	#11719917	C₂²×C₄	1	8	yes	—
104	C₂×#88322	—	2	4	no	—
105	C₂×#88322	—	2	16	no	—
106	C₂×#88322	C₂²×C₄	1	4	yes	—
107	C₂×#88322	C₂×D₄	1	2	yes	—
108	C₂×#88322	C₂×D₄	1	2	yes	—
109	C₂×#88322	#41	1	4	yes	—
110	C₂³×C₅₃	—	2	2	no	—
111	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
112	Q₈×C₅₃	—	2	2	no	—
113	Q₈×C₅₃	C₂×C₄²	1	4	yes	—
114	D₄×C₅₃	—	2	8	no	—
115	D₄×C₅₃	—	2	2	no	—
116	D₄×C₅₃	C₂×C₄²	1	4	yes	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	8	no	—
119	C₂×C₄×C₅₃	—	2	4	no	—
120	C₂×C₄×C₅₃	—	2	4	no	—
121	C₂×C₄×C₅₃	—	4	8	no	—
122	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
125	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
128	C₂×Dic₅₃	—	2	8	no	—
129	C₂×Dic₅₃	—	2	4	no	—
130	C₈×C₅₃	—	2	2	no	—
131	C₈×C₅₃	C₂×C₄²	1	2	yes	—
132	C₈×C₅₃	C₂×C₄²	1	2	yes	—
133	#88322	—	2	8	no	—
134	#88322	C₄×D₄	1	4	yes	—
135	#88322	C₄×D₄	1	4	yes	—
136	C₄×#443	—	53	1	no	maximal, Sylow
137	C₂²×C₅₃	—	2	2	no	—
138	C₂²×C₅₃	—	2	4	no	—
139	C₂²×C₅₃	—	4	4	no	—
140	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂²×C₅₃	C₄×#41	1	2	yes	—
142	C₄×C₅₃	—	2	2	no	—
143	C₄×C₅₃	—	2	4	no	—
144	C₄×C₅₃	—	4	4	no	—
145	C₄×C₅₃	#443	1	4	yes	complemented
146	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
147	C₄×C₅₃	C₂×C₄×D₄	1	2	yes	—
148	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
149	Dic₅₃	—	2	8	no	—
150	C₂×#443	—	53	2	no	—
151	C₄×#135	D₅₃	1	1	yes	characteristic
152	C₄×#131	—	53	2	no	—
153	C₄×#108	—	53	1	no	—
154	#1772	—	53	2	no	—
155	#1764	—	53	4	no	—
156	C₄×#104	—	53	2	no	—
157	C₄²×C₈	—	53	1	no	—
158	C₂×C₅₃	—	2	2	no	—
159	C₂×C₅₃	—	4	4	no	—
160	C₂×C₅₃	C₄²×D₄	1	1	yes	characteristic
161	C₂×C₅₃	C₂×#443	1	2	yes	—
162	C₂×#135	C₂×D₅₃	1	1	yes	characteristic
163	C₂×#131	—	53	4	no	—
164	C₄×#41	C₂×D₅₃	1	2	yes	—
165	#443	—	53	16	no	—
166	C₄×Q₁₆	Dic₅₃	1	2	yes	complemented
167	C₄×SD₁₆	Dic₅₃	1	4	yes	complemented
168	C₄×D₈	Dic₅₃	1	2	yes	complemented
169	C₂×#108	—	53	2	no	—
170	#427	C₂×D₅₃	1	1	yes	characteristic
171	C₂×#104	—	53	4	no	—
172	#416	C₂×D₅₃	1	2	yes	—
173	#405	—	53	4	no	—
174	#405	—	106	4	no	—
175	C₄×M₁₆	—	53	2	no	—
176	C₄×M₁₆	—	106	2	no	—
177	C₂×C₄×C₈	—	53	2	no	—
178	C₂×C₄×C₈	—	53	1	no	—
179	C₂×C₄×C₈	—	53	2	no	—
180	C₂×C₄×C₈	—	106	2	no	—
181	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
182	C₄³	—	53	1	no	—
183	#341	—	53	2	no	—
184	#339	—	53	4	no	—
185	C₅₃	C₄×#443	1	1	yes	characteristic, complemented, Sylow
186	C₂×#41	C₂²×D₅₃	1	2	yes	—
187	#135	C₄×D₅₃	1	8	yes	—
188	C₂×Q₁₆	C₂×Dic₅₃	1	1	yes	characteristic
189	C₂×SD₁₆	C₂×Dic₅₃	1	2	yes	—
190	C₂×D₈	C₂×Dic₅₃	1	1	yes	characteristic
191	#131	—	53	16	no	—
192	#131	—	106	8	no	—
193	C₂×M₁₆	—	53	4	no	—
194	C₂×M₁₆	—	106	4	no	—
195	C₂²×C₈	—	53	2	no	—
196	C₂²×C₈	—	106	4	no	—
197	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
199	C₄×Q₈	C₂×Dic₅₃	1	4	yes	—
200	C₄×D₄	—	2	4	no	—
201	C₄×D₄	C₂×Dic₅₃	1	4	yes	—
202	#117	—	2	2	no	—
203	#117	C₂²×D₅₃	1	2	yes	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	53	2	no	—
206	C₂×C₄²	—	106	2	no	—
207	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
208	#108	—	53	8	no	—
209	#107	C₂×Dic₅₃	1	2	yes	—
210	#106	C₂×Dic₅₃	1	2	yes	—
211	#104	—	53	16	no	—
212	#103	C₂×Dic₅₃	1	4	yes	—
213	#102	C₂×Dic₅₃	1	4	yes	—
214	#97	—	106	4	no	—
215	#97	—	106	2	no	—
216	C₄×C₈	—	53	8	no	—
217	C₄×C₈	—	53	2	no	—
218	C₄×C₈	—	53	1	no	—
219	C₄×C₈	—	53	1	no	—
220	C₄×C₈	—	106	4	no	—
221	C₄×C₈	—	106	4	no	—
222	C₄×C₈	—	106	2	no	—
223	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
224	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
225	#41	—	2	4	no	—
226	#41	C₂×C₄×D₅₃	1	8	yes	—
227	C₂×Q₈	C₂²×Dic₅₃	1	2	yes	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂²×Dic₅₃	1	2	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	106	2	no	—
233	C₂²×C₄	#11719947	1	2	yes	—
234	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
235	Q₁₆	C₄×Dic₅₃	1	4	yes	complemented
236	SD₁₆	C₄×Dic₅₃	1	8	yes	complemented
237	D₈	C₄×Dic₅₃	1	4	yes	complemented
238	M₁₆	—	53	8	no	—
239	M₁₆	—	106	16	no	—
240	M₁₆	—	106	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	53	4	no	—
243	C₂×C₈	—	53	2	no	—
244	C₂×C₈	—	53	2	no	—
245	C₂×C₈	—	53	4	no	—
246	C₂×C₈	—	106	16	no	—
247	C₂×C₈	—	106	4	no	—
248	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
249	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
250	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
251	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	C₂²×Dic₅₃	1	4	yes	—
254	C₂²⋊C₄	—	2	2	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₄²	—	2	4	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	53	4	no	—
259	C₄²	—	106	8	no	—
260	C₄²	—	106	2	no	—
261	C₄²	D₄×D₅₃	1	1	yes	characteristic
262	C₄²	D₄×D₅₃	1	1	yes	characteristic
263	C₄²	C₂²×Dic₅₃	1	2	yes	—
264	C₂³	—	2	2	no	—
265	C₂³	C₂×#11719947	1	1	yes	characteristic
266	Q₈	—	2	2	no	—
267	Q₈	C₂×C₄×Dic₅₃	1	4	yes	—
268	D₄	—	2	8	no	—
269	D₄	—	2	2	no	—
270	D₄	C₂×C₄×Dic₅₃	1	4	yes	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	8	no	—
276	C₂×C₄	—	106	8	no	—
277	C₂×C₄	—	106	4	no	—
278	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
279	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
280	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
281	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
282	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
283	C₂×C₄	#41×D₅₃	1	2	yes	—
284	C₈	—	2	2	no	—
285	C₈	—	53	4	no	—
286	C₈	—	53	4	no	—
287	C₈	—	106	8	no	—
288	C₈	C₂×C₄×Dic₅₃	1	2	yes	—
289	C₈	C₂×C₄×Dic₅₃	1	2	yes	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	4	4	no	—
293	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
294	C₂²	C₄×#11719947	1	2	yes	—
295	C₄	—	2	4	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	4	no	—
298	C₄	—	106	8	no	—
299	C₄	#16810740	1	4	yes	direct product factor
300	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
301	C₄	C₄×D₄×D₅₃	1	2	yes	—
302	C₄	#41×Dic₅₃	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	4	4	no	—
305	C₂	C₄×D₄×Dic₅₃	1	1	yes	characteristic
306	C₂	C₂×#16810740	1	2	yes	—

	subgroup	G/H
center	C₄²	D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×#135×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#135×C₅₃	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#135	D₅₃
hypocenter	C₅₃	C₄×#443
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
2. C₄×#13133224	C₂	semidirect
3. C₄×#13133153	C₂	semidirect
4. #20274596	C₂	semidirect
5. #20274555	C₂	semidirect
6. C₄×#13133144	C₂	semidirect
7. C₄×C₈×Dic₅₃	C₂	semidirect
10. #16810740	C₄	direct (N × H)
26. C₄×Q₁₆×C₅₃	C₄	semidirect
27. C₄×SD₁₆×C₅₃	C₄	semidirect
28. C₄×D₈×C₅₃	C₄	semidirect
33. #13133224	C₂×C₄	semidirect
40. #13133153	C₂×C₄	semidirect
41. #13133144	C₂×C₄	semidirect
50. C₈×Dic₅₃	C₂×C₄	semidirect
81. Q₁₆×C₅₃	C₄²	semidirect
82. SD₁₆×C₅₃	C₄²	semidirect
83. D₈×C₅₃	C₄²	semidirect
145. C₄×C₅₃	#443	semidirect
166. C₄×Q₁₆	Dic₅₃	semidirect
167. C₄×SD₁₆	Dic₅₃	semidirect
168. C₄×D₈	Dic₅₃	semidirect
185. C₅₃	C₄×#443	semidirect
235. Q₁₆	C₄×Dic₅₃	semidirect
236. SD₁₆	C₄×Dic₅₃	semidirect
237. D₈	C₄×Dic₅₃	semidirect
299. C₄	#16810740	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×#16810740	2
C₄×#13133224	2
C₄×#13133153	2
#20274596	2
#20274555	2
C₄×#13133144	2
C₄×C₈×Dic₅₃	2
C₄×#135×C₅₃	2
C₄×#443	53

Subgroups — C4×#16810740 #27382648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810740 #27382648