Subgroups — C₄×#16810377 #27289238

Order 13568.

1318 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄²×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810377	C₂	1	4	yes	maximal
4	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	C₂²	1	2	yes	—
6	C₂²×C₄×Dic₅₃	C₄	1	4	yes	—
7	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	—
8	C₂×C₄×#88322	—	2	2	no	—
9	C₂×C₄×#88322	—	2	2	no	—
10	C₂×C₄×#88322	C₄	1	4	yes	complemented
11	C₄²×Dic₅₃	—	2	2	no	—
12	C₄²×Dic₅₃	—	2	2	no	—
13	#16810377	C₄	1	16	yes	direct product factor
14	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	—	2	2	no	—
17	C₂×C₄×C₈×C₅₃	—	2	2	no	—
18	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	complemented
19	C₂³×Dic₅₃	C₂×C₄	1	4	yes	—
20	C₂²×#88322	—	2	2	no	—
21	C₂²×#88322	—	2	2	no	—
22	C₂²×#88322	—	2	4	no	—
23	C₂²×#88322	C₂×C₄	1	4	yes	—
24	C₂²×#88322	C₂×C₄	1	8	yes	—
25	C₂×C₄×Dic₅₃	—	2	8	no	—
26	C₂×C₄×Dic₅₃	—	2	8	no	—
27	C₂×C₄×Dic₅₃	—	2	2	no	—
28	C₂×C₄×Dic₅₃	—	2	2	no	—
29	C₂×C₄×Dic₅₃	—	2	4	no	—
30	C₂×C₄×Dic₅₃	C₈	1	16	yes	complemented
31	C₂×C₄×Dic₅₃	C₂×C₄	1	8	yes	—
32	C₄×#88322	—	2	8	no	—
33	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	—	2	2	no	—
36	C₂²×C₈×C₅₃	—	2	2	no	—
37	C₂²×C₈×C₅₃	—	2	4	no	—
38	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
39	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅₃	D₄	1	2	yes	—
43	C₂×C₄²×C₅₃	D₄	1	2	yes	—
44	C₂×C₄²×C₅₃	D₄	1	2	yes	—
45	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
46	C₄×C₈×C₅₃	—	2	8	no	—
47	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
49	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	—
52	C₂²×C₄×C₅₃	M₁₆	1	8	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	#41	1	4	yes	—
61	C₂²×Dic₅₃	—	2	4	no	—
62	C₂²×Dic₅₃	—	2	4	no	—
63	C₂²×Dic₅₃	—	2	8	no	—
64	C₂²×Dic₅₃	C₄²	1	8	yes	—
65	C₂²×Dic₅₃	C₂×C₈	1	16	yes	—
66	C₂×C₈×C₅₃	—	2	4	no	—
67	C₂×C₈×C₅₃	—	2	4	no	—
68	C₂×C₈×C₅₃	—	2	8	no	—
69	C₂×C₈×C₅₃	—	2	8	no	—
70	C₂×C₈×C₅₃	C₄²	1	8	yes	complemented
71	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
72	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
73	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
74	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
75	C₄×Dic₅₃	—	2	32	no	—
76	C₄×Dic₅₃	—	2	8	no	—
77	C₄×Dic₅₃	—	2	8	no	—
78	C₂×#88322	—	2	4	no	—
79	C₂×#88322	—	2	4	no	—
80	C₂×#88322	—	2	8	no	—
81	C₂×#88322	—	2	8	no	—
82	C₂×#88322	C₄²	1	8	yes	complemented
83	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
84	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
85	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂³×C₅₃	#117	1	2	yes	—
88	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
89	C₂³×C₅₃	C₂×M₁₆	1	4	yes	—
90	C₂×C₄×C₅₃	#95	1	8	yes	—
91	C₂×C₄×C₅₃	C₄×C₈	1	8	yes	—
92	C₂×C₄×C₅₃	#97	1	8	yes	—
93	C₂×C₄×C₅₃	C₂²⋊C₈	1	16	yes	—
94	C₂×C₄×C₅₃	C₄⋊C₈	1	16	yes	—
95	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#117	1	2	yes	—
102	C₂×C₄×C₅₃	#117	1	2	yes	—
103	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
104	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
105	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
106	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
107	C₂×Dic₅₃	—	2	8	no	—
108	C₂×Dic₅₃	—	2	8	no	—
109	C₂×Dic₅₃	—	2	16	no	—
110	C₂×Dic₅₃	—	2	16	no	—
111	C₂×Dic₅₃	C₄×C₈	1	16	yes	complemented
112	C₈×C₅₃	—	2	8	no	—
113	#88322	—	2	8	no	—
114	C₄×#336	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	C₄³	1	1	yes	characteristic
116	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
117	C₂²×C₅₃	#376	1	1	yes	characteristic
118	C₂²×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
119	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
120	C₂²×C₅₃	C₂×C₄×C₈	1	4	yes	—
121	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
122	C₂²×C₅₃	C₂×#97	1	2	yes	—
123	C₂²×C₅₃	C₄×M₁₆	1	4	yes	—
124	C₂²×C₅₃	C₂×C₂²⋊C₈	1	4	yes	—
125	C₂²×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
126	C₂²×C₅₃	#431	1	8	yes	—
127	C₄×C₅₃	#336	1	16	yes	complemented
128	C₄×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
129	C₄×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
130	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
131	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
132	Dic₅₃	—	2	16	no	—
133	C₂²×C₄×C₈	—	53	1	no	—
134	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
135	C₂×C₄³	—	53	1	no	—
136	C₂×#336	—	53	4	no	—
137	C₂×C₅₃	C₄×#95	1	1	yes	characteristic
138	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
139	C₂×C₅₃	C₄×#97	1	1	yes	characteristic
140	C₂×C₅₃	C₂×#336	1	2	yes	—
141	C₂×C₅₃	#1744	1	2	yes	—
142	C₂×C₅₃	C₄×C₂²⋊C₈	1	4	yes	—
143	C₂×C₅₃	C₄×C₄⋊C₈	1	4	yes	—
144	C₂³×C₈	—	53	2	no	—
145	C₂³×C₈	C₂×D₅₃	1	2	yes	—
146	C₂²×C₄²	—	53	4	no	—
147	C₂²×C₄²	—	53	2	no	—
148	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
149	C₂×C₄×C₈	—	2	2	no	—
150	C₂×C₄×C₈	—	2	2	no	—
151	C₂×C₄×C₈	—	53	4	no	—
152	C₂×C₄×C₈	—	106	2	no	—
153	C₂×C₄×C₈	—	106	2	no	—
154	C₂×C₄×C₈	Dic₅₃	1	4	yes	complemented
155	C₄³	—	106	2	no	—
156	C₄³	—	106	2	no	—
157	#336	—	53	16	no	—
158	C₅₃	C₄×#336	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₄	—	53	4	no	—
160	C₂³×C₄	C₄×D₅₃	1	2	yes	—
161	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
162	C₂²×C₈	—	2	4	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	53	8	no	—
166	C₂²×C₈	—	53	4	no	—
167	C₂²×C₈	—	106	2	no	—
168	C₂²×C₈	—	106	4	no	—
169	C₂²×C₈	—	106	2	no	—
170	C₂²×C₈	C₄×D₅₃	1	8	yes	—
171	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
172	C₂×C₄²	—	53	8	no	—
173	C₂×C₄²	—	53	16	no	—
174	C₂×C₄²	—	106	2	no	—
175	C₂×C₄²	—	106	4	no	—
176	C₂×C₄²	—	106	2	no	—
177	C₂×C₄²	—	106	8	no	—
178	C₂×C₄²	—	106	8	no	—
179	C₂×C₄²	Dic₁₀₆	1	2	yes	—
180	C₂×C₄²	C₄×D₅₃	1	2	yes	—
181	C₂×C₄²	D₂₁₂	1	2	yes	—
182	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
183	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
184	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
185	C₄×C₈	—	2	8	no	—
186	C₄×C₈	—	106	8	no	—
187	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
188	C₂²×C₄	—	53	8	no	—
189	C₂²×C₄	—	53	16	no	—
190	C₂²×C₄	—	106	4	no	—
191	C₂²×C₄	—	106	4	no	—
192	C₂²×C₄	—	106	8	no	—
193	C₂²×C₄	C₈×D₅₃	1	8	yes	—
194	C₂²×C₄	#11719912	1	8	yes	—
195	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
196	C₂²×C₄	#11719919	1	4	yes	—
197	C₂²×C₄	#11719921	1	4	yes	—
198	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
201	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
202	C₂²×C₄	#11719945	1	4	yes	—
203	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	8	no	—
209	C₂×C₈	—	2	8	no	—
210	C₂×C₈	—	53	8	no	—
211	C₂×C₈	—	106	4	no	—
212	C₂×C₈	—	106	4	no	—
213	C₂×C₈	—	106	8	no	—
214	C₂×C₈	—	106	8	no	—
215	C₂×C₈	C₄×Dic₅₃	1	8	yes	complemented
216	C₄²	—	106	32	no	—
217	C₄²	—	106	8	no	—
218	C₄²	—	106	8	no	—
219	C₄²	#11719919	1	4	yes	—
220	C₄²	C₄⋊Dic₅₃	1	4	yes	—
221	C₄²	#11719921	1	4	yes	—
222	C₄²	C₂²⋊Dic₅₃	1	4	yes	—
223	C₂³	C₄²×D₅₃	1	2	yes	—
224	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
225	C₂³	C₂×#11719921	1	1	yes	characteristic
226	C₂³	C₂×#11719919	1	1	yes	characteristic
227	C₂³	#13133174	1	2	yes	—
228	C₂³	C₂×C₈×D₅₃	1	4	yes	—
229	C₂³	C₂×#11719912	1	4	yes	—
230	C₂×C₄	—	53	16	no	—
231	C₂×C₄	—	106	8	no	—
232	C₂×C₄	—	106	16	no	—
233	C₂×C₄	—	106	16	no	—
234	C₂×C₄	—	106	8	no	—
235	C₂×C₄	#13133124	1	8	yes	—
236	C₂×C₄	C₈×Dic₅₃	1	8	yes	—
237	C₂×C₄	#13133127	1	8	yes	—
238	C₂×C₄	#13133129	1	16	yes	—
239	C₂×C₄	#13133146	1	16	yes	—
240	C₂×C₄	C₄²×D₅₃	1	2	yes	—
241	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
242	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
243	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
244	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
245	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
246	C₂×C₄	#13133174	1	2	yes	—
247	C₂×C₄	#13133176	1	2	yes	—
248	C₂×C₄	C₄×C₂²⋊D₅₃	1	4	yes	—
249	C₂×C₄	C₄×C₂²⋊D₅₃	1	4	yes	—
250	C₂×C₄	C₄×D₂₁₂	1	4	yes	—
251	C₂×C₄	C₄×Dic₁₀₆	1	4	yes	—
252	C₈	—	2	8	no	—
253	C₈	—	106	8	no	—
254	C₂²	C₄²×Dic₅₃	1	1	yes	characteristic
255	C₂²	C₂×#13133124	1	1	yes	characteristic
256	C₂²	#16810469	1	1	yes	characteristic
257	C₂²	C₄×#11719921	1	2	yes	—
258	C₂²	C₄×#11719919	1	2	yes	—
259	C₂²	C₄×C₈×D₅₃	1	4	yes	—
260	C₂²	C₂×C₈×Dic₅₃	1	2	yes	—
261	C₂²	C₂×#13133127	1	2	yes	—
262	C₂²	C₄×#11719912	1	4	yes	—
263	C₂²	C₂×#13133129	1	4	yes	—
264	C₂²	C₂×#13133146	1	4	yes	—
265	C₂²	#16810674	1	8	yes	—
266	C₄	—	106	16	no	—
267	C₄	#16810377	1	16	yes	direct product factor
268	C₄	C₄×C₂²⋊Dic₅₃	1	2	yes	—
269	C₄	C₄×#11719921	1	2	yes	—
270	C₄	C₄×C₄⋊Dic₅₃	1	2	yes	—
271	C₄	C₄×#11719919	1	2	yes	—
272	C₂	C₄×#13133124	1	1	yes	characteristic
273	C₂	C₄×C₈×Dic₅₃	1	1	yes	characteristic
274	C₂	C₄×#13133127	1	1	yes	characteristic
275	C₂	C₂×#16810377	1	2	yes	—
276	C₂	#20274460	1	2	yes	—
277	C₂	C₄×#13133129	1	4	yes	—
278	C₂	C₄×#13133146	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₅₃
commutator	C₂×C₅₃	C₄²×C₈
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	C₄×#336
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
10. C₂×C₄×#88322	C₄	semidirect
13. #16810377	C₄	direct (N × H)
18. C₂×C₄×C₈×C₅₃	C₄	semidirect
30. C₂×C₄×Dic₅₃	C₈	semidirect
70. C₂×C₈×C₅₃	C₄²	semidirect
82. C₂×#88322	C₄²	semidirect
111. C₂×Dic₅₃	C₄×C₈	semidirect
127. C₄×C₅₃	#336	semidirect
154. C₂×C₄×C₈	Dic₅₃	semidirect
158. C₅₃	C₄×#336	semidirect
215. C₂×C₈	C₄×Dic₅₃	semidirect
267. C₄	#16810377	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×#88322	2
C₂×C₄²×Dic₅₃	2
C₂×#16810377	2
C₂²×C₄×C₈×C₅₃	2
C₄×#336	53

Subgroups — C4×#16810377 #27289238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810377 #27289238