Subgroups — C₄×#16810745 #27382667

Order 13568.

1022 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810745	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	#20274601	C₂	1	2	yes	maximal, complemented
5	#20274562	C₂	1	4	yes	maximal, complemented
6	C₄×#13133144	C₂	1	2	yes	maximal, complemented
7	C₄×#13133127	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	#16810745	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×#13133127	C₂²	1	2	yes	—
18	C₂×#13133127	C₂²	1	2	yes	—
19	C₂×C₄×#88322	—	2	2	no	—
20	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
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	#13133127	—	2	4	no	—
45	#13133127	C₂×C₄	1	8	yes	complemented
46	C₄×#88322	—	2	2	no	—
47	C₄×#88322	—	2	4	no	—
48	C₄×#88322	D₄	1	2	yes	—
49	C₄×#88322	D₄	1	2	yes	—
50	C₂×#41×C₅₃	C₂³	1	2	yes	—
51	#135×C₅₃	C₂×C₄	1	8	yes	—
52	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
54	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
58	C₄×D₄×C₅₃	—	2	4	no	—
59	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
60	#117×C₅₃	—	2	2	no	—
61	#117×C₅₃	C₂³	1	2	yes	—
62	C₂×C₄²×C₅₃	—	2	2	no	—
63	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
64	#107×C₅₃	C₂×C₄	1	2	yes	—
65	#106×C₅₃	C₂×C₄	1	2	yes	—
66	#103×C₅₃	C₂×C₄	1	4	yes	—
67	#102×C₅₃	C₂×C₄	1	4	yes	—
68	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
70	#41×C₅₃	—	2	4	no	—
71	#41×C₅₃	C₂²×C₄	1	8	yes	—
72	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
73	C₂×D₄×C₅₃	—	2	2	no	—
74	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₅₃	—	2	2	no	—
76	C₂²×C₄×C₅₃	—	2	4	no	—
77	C₂²×C₄×C₅₃	#41	1	2	yes	—
78	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
79	C₂²×Dic₅₃	—	2	2	no	—
80	Q₁₆×C₅₃	C₄²	1	4	yes	complemented
81	SD₁₆×C₅₃	C₄²	1	8	yes	complemented
82	D₈×C₅₃	C₄²	1	4	yes	complemented
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₄⋊C₄×C₅₃	—	2	4	no	—
89	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
90	C₂²⋊C₄×C₅₃	—	2	2	no	—
91	C₂²⋊C₄×C₅₃	—	2	4	no	—
92	C₄²×C₅₃	—	2	4	no	—
93	C₄²×C₅₃	—	2	2	no	—
94	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
95	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₄×Dic₅₃	—	2	2	no	—
98	C₄×Dic₅₃	—	4	4	no	—
99	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
100	#11719917	—	2	4	no	—
101	#11719917	—	2	16	no	—
102	#11719917	C₂²×C₄	1	8	yes	—
103	C₂×#88322	—	2	4	no	—
104	C₂×#88322	—	2	16	no	—
105	C₂×#88322	C₂²×C₄	1	4	yes	—
106	C₂×#88322	C₂×D₄	1	4	yes	—
107	C₂×#88322	#41	1	4	yes	—
108	C₂³×C₅₃	—	2	2	no	—
109	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
110	Q₈×C₅₃	—	2	2	no	—
111	Q₈×C₅₃	C₂×C₄²	1	4	yes	—
112	D₄×C₅₃	—	2	8	no	—
113	D₄×C₅₃	—	2	2	no	—
114	D₄×C₅₃	C₂×C₄²	1	4	yes	—
115	C₂×C₄×C₅₃	—	2	2	no	—
116	C₂×C₄×C₅₃	—	2	8	no	—
117	C₂×C₄×C₅₃	—	2	4	no	—
118	C₂×C₄×C₅₃	—	2	4	no	—
119	C₂×C₄×C₅₃	—	4	8	no	—
120	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
123	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
126	C₂×Dic₅₃	—	2	4	no	—
127	C₂×Dic₅₃	—	4	4	no	—
128	C₈×C₅₃	—	2	2	no	—
129	C₈×C₅₃	C₂×C₄²	1	2	yes	—
130	C₈×C₅₃	C₂×C₄²	1	2	yes	—
131	#88322	—	2	8	no	—
132	#88322	C₄×D₄	1	8	yes	—
133	C₄×#444	—	53	1	no	maximal, Sylow
134	C₂²×C₅₃	—	2	2	no	—
135	C₂²×C₅₃	—	2	4	no	—
136	C₂²×C₅₃	—	4	4	no	—
137	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂²×C₅₃	C₄×#41	1	2	yes	—
139	C₄×C₅₃	—	2	2	no	—
140	C₄×C₅₃	—	2	4	no	—
141	C₄×C₅₃	—	4	4	no	—
142	C₄×C₅₃	#444	1	4	yes	complemented
143	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
144	C₄×C₅₃	C₂×C₄×D₄	1	2	yes	—
145	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
146	Dic₅₃	—	4	4	no	—
147	C₂×#444	—	53	2	no	—
148	C₄×#135	D₅₃	1	1	yes	characteristic
149	C₄×#131	—	53	2	no	—
150	C₄×#108	—	53	1	no	—
151	#1773	—	53	2	no	—
152	#1765	—	53	4	no	—
153	C₄×#104	—	53	2	no	—
154	C₄×#97	—	53	1	no	—
155	C₂×C₅₃	—	2	2	no	—
156	C₂×C₅₃	—	4	4	no	—
157	C₂×C₅₃	C₄²×D₄	1	1	yes	characteristic
158	C₂×C₅₃	C₂×#444	1	2	yes	—
159	C₂×#135	C₂×D₅₃	1	1	yes	characteristic
160	C₂×#131	—	53	4	no	—
161	C₄×#41	C₂×D₅₃	1	2	yes	—
162	#444	—	53	16	no	—
163	C₄×Q₁₆	Dic₅₃	1	2	yes	complemented
164	C₄×SD₁₆	Dic₅₃	1	4	yes	complemented
165	C₄×D₈	Dic₅₃	1	2	yes	complemented
166	C₂×#108	—	53	2	no	—
167	#427	C₂×D₅₃	1	1	yes	characteristic
168	C₂×#104	—	53	4	no	—
169	#416	C₂×D₅₃	1	2	yes	—
170	#405	—	53	4	no	—
171	#405	—	106	4	no	—
172	C₄×M₁₆	—	53	2	no	—
173	C₄×M₁₆	—	106	2	no	—
174	C₂×#97	—	53	2	no	—
175	C₂×#97	—	53	2	no	—
176	C₂×C₄×C₈	—	53	1	no	—
177	C₂×C₄×C₈	—	106	2	no	—
178	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
179	C₄³	—	53	1	no	—
180	#341	—	53	2	no	—
181	#339	—	53	4	no	—
182	C₅₃	C₄×#444	1	1	yes	characteristic, complemented, Sylow
183	C₂×#41	C₂²×D₅₃	1	2	yes	—
184	#135	C₄×D₅₃	1	8	yes	—
185	C₂×Q₁₆	C₂×Dic₅₃	1	1	yes	characteristic
186	C₂×SD₁₆	C₂×Dic₅₃	1	2	yes	—
187	C₂×D₈	C₂×Dic₅₃	1	1	yes	characteristic
188	#131	—	53	16	no	—
189	#131	—	106	8	no	—
190	C₂×M₁₆	—	53	4	no	—
191	C₂×M₁₆	—	106	4	no	—
192	C₂²×C₈	—	53	2	no	—
193	C₂²×C₈	—	106	4	no	—
194	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
196	C₄×Q₈	C₂×Dic₅₃	1	4	yes	—
197	C₄×D₄	—	2	4	no	—
198	C₄×D₄	C₂×Dic₅₃	1	4	yes	—
199	#117	—	2	2	no	—
200	#117	C₂²×D₅₃	1	2	yes	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	—	53	2	no	—
203	C₂×C₄²	—	106	2	no	—
204	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
205	#108	—	53	8	no	—
206	#107	C₂×Dic₅₃	1	2	yes	—
207	#106	C₂×Dic₅₃	1	2	yes	—
208	#104	—	53	16	no	—
209	#103	C₂×Dic₅₃	1	4	yes	—
210	#102	C₂×Dic₅₃	1	4	yes	—
211	#97	—	53	8	no	—
212	#97	—	106	4	no	—
213	#97	—	106	4	no	—
214	#97	—	106	2	no	—
215	C₄×C₈	—	53	2	no	—
216	C₄×C₈	—	53	2	no	—
217	C₄×C₈	—	106	4	no	—
218	C₄×C₈	—	106	2	no	—
219	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
220	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
221	#41	—	2	4	no	—
222	#41	C₂×C₄×D₅₃	1	8	yes	—
223	C₂×Q₈	C₂²×Dic₅₃	1	2	yes	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	C₂²×Dic₅₃	1	2	yes	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	106	2	no	—
229	C₂²×C₄	#11719947	1	2	yes	—
230	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
231	Q₁₆	C₄×Dic₅₃	1	4	yes	complemented
232	SD₁₆	C₄×Dic₅₃	1	8	yes	complemented
233	D₈	C₄×Dic₅₃	1	4	yes	complemented
234	M₁₆	—	53	8	no	—
235	M₁₆	—	106	16	no	—
236	M₁₆	—	106	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	53	4	no	—
239	C₂×C₈	—	53	4	no	—
240	C₂×C₈	—	53	4	no	—
241	C₂×C₈	—	106	16	no	—
242	C₂×C₈	—	106	4	no	—
243	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
244	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
245	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
246	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	C₂²×Dic₅₃	1	4	yes	—
249	C₂²⋊C₄	—	2	2	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	53	4	no	—
254	C₄²	—	106	2	no	—
255	C₄²	—	212	4	no	—
256	C₄²	D₄×D₅₃	1	1	yes	characteristic
257	C₄²	D₄×D₅₃	1	1	yes	characteristic
258	C₄²	C₂²×Dic₅₃	1	2	yes	—
259	C₂³	—	2	2	no	—
260	C₂³	C₂×#11719947	1	1	yes	characteristic
261	Q₈	—	2	2	no	—
262	Q₈	C₂×C₄×Dic₅₃	1	4	yes	—
263	D₄	—	2	8	no	—
264	D₄	—	2	2	no	—
265	D₄	C₂×C₄×Dic₅₃	1	4	yes	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	4	8	no	—
271	C₂×C₄	—	106	4	no	—
272	C₂×C₄	—	212	4	no	—
273	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
274	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
275	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
276	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
277	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
278	C₂×C₄	#41×D₅₃	1	2	yes	—
279	C₈	—	2	2	no	—
280	C₈	—	53	8	no	—
281	C₈	—	106	8	no	—
282	C₈	C₂×C₄×Dic₅₃	1	2	yes	—
283	C₈	C₂×C₄×Dic₅₃	1	2	yes	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	4	4	no	—
287	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
288	C₂²	C₄×#11719947	1	2	yes	—
289	C₄	—	2	4	no	—
290	C₄	—	2	2	no	—
291	C₄	—	4	4	no	—
292	C₄	—	212	4	no	—
293	C₄	#16810745	1	4	yes	direct product factor
294	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
295	C₄	C₄×D₄×D₅₃	1	2	yes	—
296	C₄	#41×Dic₅₃	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	4	4	no	—
299	C₂	C₄×D₄×Dic₅₃	1	1	yes	characteristic
300	C₂	C₂×#16810745	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×#444
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. #20274601	C₂	semidirect
5. #20274562	C₂	semidirect
6. C₄×#13133144	C₂	semidirect
7. C₄×#13133127	C₂	semidirect
10. #16810745	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
45. #13133127	C₂×C₄	semidirect
80. Q₁₆×C₅₃	C₄²	semidirect
81. SD₁₆×C₅₃	C₄²	semidirect
82. D₈×C₅₃	C₄²	semidirect
142. C₄×C₅₃	#444	semidirect
163. C₄×Q₁₆	Dic₅₃	semidirect
164. C₄×SD₁₆	Dic₅₃	semidirect
165. C₄×D₈	Dic₅₃	semidirect
182. C₅₃	C₄×#444	semidirect
231. Q₁₆	C₄×Dic₅₃	semidirect
232. SD₁₆	C₄×Dic₅₃	semidirect
233. D₈	C₄×Dic₅₃	semidirect
293. C₄	#16810745	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×#16810745	2
C₄×#13133224	2
C₄×#13133153	2
#20274601	2
#20274562	2
C₄×#13133144	2
C₄×#13133127	2
C₄×#135×C₅₃	2
C₄×#444	53

Subgroups — C4×#16810745 #27382667

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810745 #27382667