Subgroups — C₄×#16810744 #27382665

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×#16810744	C₂	1	2	yes	maximal
2	C₄×#13133239	C₂	1	1	yes	maximal, characteristic
3	C₄×#13133225	C₂	1	2	yes	maximal, complemented
4	#20274602	C₂	1	2	yes	maximal, complemented
5	#20274558	C₂	1	4	yes	maximal, complemented
6	C₄×#13133143	C₂	1	2	yes	maximal, complemented
7	C₄×#13133127	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#108×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133239	C₂²	1	1	yes	characteristic
10	C₂×#13133225	C₂²	1	4	yes	—
11	C₄×#11719945	C₂²	1	2	yes	—
12	#16810744	C₄	1	16	yes	direct product factor
13	C₄×Dic₂₁₂	C₄	1	2	yes	complemented
14	C₄×#11719913	C₄	1	4	yes	complemented
15	C₄×D₄₂₄	C₄	1	2	yes	complemented
16	#16810661	C₂²	1	1	yes	characteristic
17	C₂×#13133143	C₂²	1	4	yes	—
18	#16810620	C₂²	1	2	yes	—
19	#16810581	C₂²	1	4	yes	—
20	#16810580	—	2	4	no	—
21	C₄×#11719912	—	2	2	no	—
22	C₂×#13133127	C₂²	1	2	yes	—
23	C₂×#13133127	C₂²	1	2	yes	—
24	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
25	C₄×C₈×D₅₃	—	2	2	no	—
26	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
27	#16810384	C₂²	1	4	yes	—
28	C₂×#108×C₅₃	C₂²	1	2	yes	—
29	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#341	C₂²	1	2	yes	—
32	C₂×#11719945	C₂³	1	2	yes	—
33	#13133239	C₂×C₄	1	8	yes	—
34	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
35	C₂×#11719913	C₂×C₄	1	2	yes	—
36	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
37	#13133225	—	2	8	no	—
38	#13133225	C₂×C₄	1	16	yes	complemented
39	C₂×#11719912	—	2	4	no	—
40	C₂²×#88322	C₂³	1	2	yes	—
41	C₂×C₈×D₅₃	—	2	4	no	—
42	C₄×Dic₁₀₆	C₂×C₄	1	4	yes	—
43	C₄×D₂₁₂	C₂×C₄	1	4	yes	—
44	C₄×C₂²⋊D₅₃	—	2	4	no	—
45	#13133176	C₂³	1	2	yes	—
46	#13133174	—	2	2	no	—
47	C₂×C₄×Dic₅₃	—	2	2	no	—
48	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
49	C₄²×D₅₃	—	2	2	no	—
50	#13133151	C₂×C₄	1	2	yes	—
51	#13133149	C₂×C₄	1	2	yes	—
52	#13133143	C₂×C₄	1	16	yes	complemented
53	#13133142	C₂×C₄	1	4	yes	—
54	#13133139	C₂×C₄	1	4	yes	—
55	#13133127	—	2	4	no	—
56	#13133127	—	2	4	no	—
57	#13133127	C₂×C₄	1	8	yes	complemented
58	C₄×#88322	D₄	1	2	yes	—
59	C₄×#88322	D₄	1	2	yes	—
60	C₈×Dic₅₃	—	2	4	no	—
61	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
62	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	#108×C₅₃	C₂×C₄	1	8	yes	complemented
66	#97×C₅₃	—	2	2	no	—
67	C₄×C₈×C₅₃	—	2	2	no	—
68	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
70	C₂²×C₄×C₅₃	#41	1	2	yes	—
71	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
72	C₂×C₂²⋊D₅₃	—	2	2	no	—
73	C₂²×Dic₅₃	—	2	2	no	—
74	#11719945	—	2	4	no	—
75	#11719945	C₂²×C₄	1	8	yes	—
76	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
77	C₂×C₄×D₅₃	—	2	2	no	—
78	C₂×C₄×D₅₃	—	2	4	no	—
79	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
80	M₁₆×C₅₃	—	2	4	no	—
81	M₁₆×C₅₃	C₂²×C₄	1	8	yes	—
82	C₂×C₈×C₅₃	—	2	4	no	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
88	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂²⋊Dic₅₃	—	2	2	no	—
92	#11719921	—	2	4	no	—
93	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
94	#11719919	—	2	4	no	—
95	C₄×Dic₅₃	—	2	4	no	—
96	C₄×Dic₅₃	—	2	2	no	—
97	C₄×Dic₅₃	—	2	2	no	—
98	C₄×Dic₅₃	—	4	4	no	—
99	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
100	C₂×#88322	C₂²×C₄	1	4	yes	—
101	C₂×#88322	C₂×D₄	1	4	yes	—
102	C₂×#88322	#41	1	4	yes	—
103	Dic₂₁₂	C₄²	1	4	yes	complemented
104	D₄₂₄	C₄²	1	4	yes	complemented
105	#11719913	C₄²	1	8	yes	complemented
106	#11719912	—	2	16	no	—
107	C₈×D₅₃	—	2	16	no	—
108	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
109	C₂²×D₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	2	4	no	—
111	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
114	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
117	C₂²⋊D₅₃	—	2	8	no	—
118	C₂×Dic₅₃	—	2	2	no	—
119	C₂×Dic₅₃	—	2	4	no	—
120	C₂×Dic₅₃	—	2	4	no	—
121	C₂×Dic₅₃	—	4	4	no	—
122	D₂₁₂	—	2	2	no	—
123	D₂₁₂	C₂×C₄²	1	4	yes	—
124	C₄×D₅₃	—	2	8	no	—
125	C₄×D₅₃	—	4	8	no	—
126	Dic₁₀₆	—	2	2	no	—
127	Dic₁₀₆	C₂×C₄²	1	4	yes	—
128	C₈×C₅₃	—	2	8	no	—
129	C₈×C₅₃	—	2	2	no	—
130	C₈×C₅₃	C₂×C₄²	1	2	yes	—
131	C₈×C₅₃	C₂×C₄²	1	2	yes	—
132	#88322	C₄×D₄	1	8	yes	—
133	C₄×#444	—	53	1	no	maximal, Sylow
134	C₂²×C₅₃	—	2	2	no	—
135	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂²×C₅₃	C₄×#41	1	2	yes	—
137	C₂×D₅₃	—	2	4	no	—
138	C₂×D₅₃	—	4	4	no	—
139	C₄×C₅₃	—	2	2	no	—
140	C₄×C₅₃	#444	1	4	yes	complemented
141	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
142	C₄×C₅₃	C₂×C₄×D₄	1	2	yes	—
143	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
144	Dic₅₃	—	2	4	no	—
145	Dic₅₃	—	4	4	no	—
146	Dic₅₃	—	4	4	no	—
147	C₂×#444	—	53	2	no	—
148	C₄×#135	—	53	1	no	—
149	C₄×#131	—	53	2	no	—
150	C₄×#108	D₅₃	1	1	yes	characteristic, complemented
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	C₄²×D₄	1	1	yes	characteristic
157	C₂×C₅₃	C₂×#444	1	2	yes	—
158	D₅₃	—	4	4	no	—
159	C₂×#135	—	53	1	no	—
160	C₂×#131	—	53	4	no	—
161	C₄×#41	—	53	2	no	—
162	#444	—	53	16	no	—
163	C₄×Q₁₆	—	53	2	no	—
164	C₄×SD₁₆	—	53	4	no	—
165	C₄×D₈	—	53	2	no	—
166	C₂×#108	C₂×D₅₃	1	2	yes	—
167	#427	—	53	1	no	—
168	C₂×#104	—	53	4	no	—
169	#416	—	53	2	no	—
170	#405	—	53	4	no	—
171	#405	—	106	4	no	—
172	C₄×M₁₆	—	106	2	no	—
173	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
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	C₂×D₅₃	1	2	yes	—
181	#339	—	53	4	no	—
182	C₅₃	C₄×#444	1	1	yes	characteristic, complemented, Sylow
183	C₂×#41	—	53	2	no	—
184	#135	—	53	8	no	—
185	C₂×Q₁₆	—	53	1	no	—
186	C₂×SD₁₆	—	53	2	no	—
187	C₂×D₈	—	53	1	no	—
188	#131	—	53	16	no	—
189	#131	—	106	8	no	—
190	C₂×M₁₆	—	106	4	no	—
191	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
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	—	53	4	no	—
197	C₄×D₄	—	53	4	no	—
198	C₄×D₄	—	106	4	no	—
199	#117	—	53	2	no	—
200	#117	—	106	2	no	—
201	C₂×C₄²	—	53	2	no	—
202	C₂×C₄²	—	106	2	no	—
203	C₂×C₄²	—	106	2	no	—
204	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
205	#108	C₄×D₅₃	1	8	yes	complemented
206	#107	—	53	2	no	—
207	#106	—	53	2	no	—
208	#104	—	53	16	no	—
209	#103	—	53	4	no	—
210	#102	—	53	4	no	—
211	#97	—	2	2	no	—
212	#97	—	53	8	no	—
213	#97	—	106	4	no	—
214	#97	—	106	4	no	—
215	C₄×C₈	—	2	2	no	—
216	C₄×C₈	—	53	2	no	—
217	C₄×C₈	—	53	2	no	—
218	C₄×C₈	—	106	4	no	—
219	C₄×C₈	C₄×D₅₃	1	2	yes	—
220	C₄×C₈	C₄×D₅₃	1	2	yes	—
221	#41	—	53	8	no	—
222	#41	—	106	4	no	—
223	C₂×Q₈	—	53	2	no	—
224	C₂×D₄	—	53	2	no	—
225	C₂×D₄	—	106	2	no	—
226	C₂²×C₄	—	106	2	no	—
227	C₂²×C₄	—	106	2	no	—
228	C₂²×C₄	—	106	4	no	—
229	C₂²×C₄	#11719949	1	2	yes	—
230	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
231	Q₁₆	—	53	4	no	—
232	SD₁₆	—	53	8	no	—
233	D₈	—	53	4	no	—
234	M₁₆	—	2	4	no	—
235	M₁₆	—	106	16	no	—
236	M₁₆	C₂×C₄×D₅₃	1	8	yes	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	53	4	no	—
240	C₂×C₈	—	53	4	no	—
241	C₂×C₈	—	53	4	no	—
242	C₂×C₈	—	106	16	no	—
243	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
244	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
245	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
246	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
247	C₄⋊C₄	—	53	4	no	—
248	C₄⋊C₄	—	106	4	no	—
249	C₂²⋊C₄	—	106	2	no	—
250	C₂²⋊C₄	—	106	4	no	—
251	C₄²	—	53	4	no	—
252	C₄²	—	106	4	no	—
253	C₄²	—	106	2	no	—
254	C₄²	—	106	2	no	—
255	C₄²	—	212	4	no	—
256	C₄²	C₂×C₄×D₅₃	1	2	yes	—
257	C₄²	D₄×D₅₃	1	1	yes	characteristic
258	C₄²	D₄×D₅₃	1	1	yes	characteristic
259	C₂³	—	106	2	no	—
260	C₂³	C₂×#11719949	1	1	yes	characteristic
261	Q₈	—	53	4	no	—
262	Q₈	—	106	2	no	—
263	D₄	—	53	4	no	—
264	D₄	—	106	8	no	—
265	D₄	—	106	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	106	2	no	—
268	C₂×C₄	—	106	8	no	—
269	C₂×C₄	—	106	4	no	—
270	C₂×C₄	—	106	4	no	—
271	C₂×C₄	—	212	4	no	—
272	C₂×C₄	—	212	8	no	—
273	C₂×C₄	#13133179	1	2	yes	—
274	C₂×C₄	#13133179	1	2	yes	—
275	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
276	C₂×C₄	C₂²×C₄×D₅₃	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	8	no	—
280	C₈	—	2	2	no	—
281	C₈	—	53	8	no	—
282	C₈	C₄²×D₅₃	1	2	yes	—
283	C₈	C₄²×D₅₃	1	2	yes	—
284	C₂²	—	2	2	no	—
285	C₂²	—	106	4	no	—
286	C₂²	—	212	4	no	—
287	C₂²	C₂×#13133179	1	1	yes	characteristic
288	C₂²	C₄×#11719949	1	2	yes	—
289	C₄	—	2	2	no	—
290	C₄	—	106	4	no	—
291	C₄	—	212	4	no	—
292	C₄	—	212	4	no	—
293	C₄	#16810744	1	4	yes	direct product factor
294	C₄	C₂×C₄²×D₅₃	1	1	yes	characteristic
295	C₄	C₄×D₄×D₅₃	1	2	yes	—
296	C₄	#16811091	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	212	4	no	—
299	C₂	C₄×#13133179	1	1	yes	characteristic
300	C₂	C₂×#16810744	1	2	yes	—

	subgroup	G/H
center	C₄²	D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×#108×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133239	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#108	D₅₃
hypocenter	C₅₃	C₄×#444
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
3. C₄×#13133225	C₂	semidirect
4. #20274602	C₂	semidirect
5. #20274558	C₂	semidirect
6. C₄×#13133143	C₂	semidirect
7. C₄×#13133127	C₂	semidirect
8. C₄×#108×C₅₃	C₂	semidirect
12. #16810744	C₄	direct (N × H)
13. C₄×Dic₂₁₂	C₄	semidirect
14. C₄×#11719913	C₄	semidirect
15. C₄×D₄₂₄	C₄	semidirect
38. #13133225	C₂×C₄	semidirect
52. #13133143	C₂×C₄	semidirect
57. #13133127	C₂×C₄	semidirect
65. #108×C₅₃	C₂×C₄	semidirect
103. Dic₂₁₂	C₄²	semidirect
104. D₄₂₄	C₄²	semidirect
105. #11719913	C₄²	semidirect
140. C₄×C₅₃	#444	semidirect
150. C₄×#108	D₅₃	semidirect
182. C₅₃	C₄×#444	semidirect
205. #108	C₄×D₅₃	semidirect
293. C₄	#16810744	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×#16810744	2
C₄×#13133239	2
C₄×#13133225	2
#20274602	2
#20274558	2
C₄×#13133143	2
C₄×#13133127	2
C₄×#108×C₅₃	2
C₄×#444	53

Subgroups — C4×#16810744 #27382665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810744 #27382665