Subgroups — C₈×#13133176 #27382295

Order 13568.

1174 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133176	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊Dic₅₃	C₂	1	4	yes	maximal, complemented
4	C₈×C₂²⋊Dic₅₃	C₂	1	4	yes	maximal
5	#20274460	C₂	1	2	yes	maximal, complemented
6	C₄×C₈×Dic₅₃	C₂	1	2	yes	maximal, complemented
7	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133176	C₄	1	2	yes	—
9	#16810676	C₄	1	8	yes	complemented
10	C₂×C₄⋊#88322	C₂²	1	2	yes	—
11	C₂×C₂²⋊#88322	C₂²	1	2	yes	—
12	C₂×C₄×#88322	C₂²	1	2	yes	—
13	C₂×C₈×Dic₅₃	C₂²	1	8	yes	—
14	C₄×C₄⋊Dic₅₃	C₄	1	2	yes	—
15	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
16	C₄×C₂²⋊Dic₅₃	C₄	1	2	yes	—
17	C₄×C₂²⋊Dic₅₃	C₂²	1	2	yes	—
18	#16810469	C₄	1	2	yes	—
19	C₄²×Dic₅₃	C₂²	1	2	yes	—
20	#16810377	C₂²	1	8	yes	—
21	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
22	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₃	C₄	1	8	yes	complemented
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×#88322	C₂³	1	4	yes	—
28	#13133176	C₈	1	16	yes	direct product factor
29	C₂×C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
30	C₂×C₂²⋊Dic₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
32	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₅₃	C₂³	1	4	yes	—
34	C₄⋊#88322	C₂×C₄	1	8	yes	—
35	C₂²⋊#88322	C₂×C₄	1	8	yes	—
36	C₄×#88322	C₂×C₄	1	8	yes	—
37	C₈×Dic₅₃	—	2	16	no	—
38	#13133124	C₂×C₄	1	4	yes	—
39	#13133124	C₂×C₄	1	4	yes	—
40	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	—	2	8	no	—
44	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
45	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
46	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
47	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
54	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
55	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
56	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	—	2	4	no	—
58	C₂²×C₄×C₅₃	—	2	4	no	—
59	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
60	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
61	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₅₃	C₂²×C₄	1	8	yes	—
71	C₂×C₈×C₅₃	—	2	4	no	—
72	C₂×C₈×C₅₃	—	2	16	no	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₅₃	#41	1	8	yes	—
76	C₄²×C₅₃	C₄²	1	4	yes	—
77	C₄²×C₅₃	C₄²	1	4	yes	—
78	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
79	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
80	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
81	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂²⋊Dic₅₃	C₂×C₈	1	16	yes	—
83	C₄⋊Dic₅₃	C₂×C₈	1	16	yes	complemented
84	C₄×Dic₅₃	—	2	8	no	—
85	C₄×Dic₅₃	—	2	8	no	—
86	C₄×Dic₅₃	C₂×C₈	1	16	yes	complemented
87	C₂×#88322	—	2	8	no	—
88	C₂×#88322	C₂²×C₄	1	8	yes	—
89	C₂³×C₅₃	—	2	2	no	—
90	C₂³×C₅₃	—	2	2	no	—
91	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
92	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
93	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	4	no	—
97	C₂×C₄×C₅₃	—	2	4	no	—
98	C₂×C₄×C₅₃	—	2	8	no	—
99	C₂×C₄×C₅₃	C₄×C₈	1	16	yes	complemented
100	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
101	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
106	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
107	C₂×C₄×C₅₃	#131	1	4	yes	—
108	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
111	C₂×Dic₅₃	—	2	8	no	—
112	C₂×Dic₅₃	—	2	8	no	—
113	C₂×Dic₅₃	C₂²×C₈	1	16	yes	—
114	C₈×C₅₃	—	2	4	no	—
115	C₈×C₅₃	#117	1	8	yes	complemented
116	#88322	—	2	8	no	—
117	C₈×#117	—	53	1	no	maximal, Sylow
118	C₂²×C₅₃	—	2	2	no	—
119	C₂²×C₅₃	—	2	8	no	—
120	C₂²×C₅₃	—	2	2	no	—
121	C₂²×C₅₃	C₂×C₄×C₈	1	4	yes	—
122	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
123	C₂²×C₅₃	C₄×#41	1	2	yes	—
124	C₂²×C₅₃	C₂³×C₈	1	2	yes	—
125	C₂²×C₅₃	C₂×#131	1	2	yes	—
126	C₄×C₅₃	—	2	2	no	—
127	C₄×C₅₃	—	2	2	no	—
128	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
129	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
130	C₄×C₅₃	#405	1	4	yes	—
131	C₄×C₅₃	C₂×#117	1	2	yes	—
132	C₄×C₅₃	C₄×#41	1	2	yes	—
133	Dic₅₃	—	2	16	no	—
134	C₂×#431	—	53	1	no	—
135	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
136	C₄×#117	—	53	1	no	—
137	C₈×C₄⋊C₄	—	53	4	no	—
138	C₈×C₂²⋊C₄	—	53	4	no	—
139	#1744	—	53	2	no	—
140	C₄²×C₈	—	53	2	no	—
141	C₂×C₅₃	—	2	4	no	—
142	C₂×C₅₃	C₄×#117	1	1	yes	characteristic
143	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
144	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
145	C₂×C₅₃	C₈×#41	1	4	yes	—
146	C₂³×C₈	C₂×D₅₃	1	2	yes	—
147	C₂×#117	—	53	2	no	—
148	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
149	#431	—	53	8	no	—
150	C₂×C₄⋊C₈	—	53	2	no	—
151	C₂×C₂²⋊C₈	—	53	2	no	—
152	C₂×C₄×C₈	—	53	2	no	—
153	C₂×C₄×C₈	—	53	8	no	—
154	C₂×C₄×C₈	Dic₅₃	1	8	yes	complemented
155	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
156	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
157	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
158	C₄×C₄⋊C₄	—	53	2	no	—
159	C₄×C₄⋊C₄	—	53	2	no	—
160	C₄×C₂²⋊C₄	—	53	2	no	—
161	C₄×C₂²⋊C₄	—	53	2	no	—
162	#376	—	53	2	no	—
163	C₄³	—	53	2	no	—
164	#336	—	53	8	no	—
165	C₅₃	C₈×#117	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
167	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
168	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
169	C₂²×C₈	—	2	8	no	—
170	C₂²×C₈	—	53	4	no	—
171	C₂²×C₈	C₂×Dic₅₃	1	8	yes	—
172	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
173	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
174	#117	—	53	16	no	—
175	C₂×C₄⋊C₄	—	53	4	no	—
176	C₂×C₂²⋊C₄	—	53	4	no	—
177	C₂×C₄²	—	53	4	no	—
178	C₂×C₄²	—	53	4	no	—
179	C₂×C₄²	—	53	4	no	—
180	C₂×C₄²	C₄×D₅₃	1	4	yes	—
181	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
182	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
183	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
184	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
185	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
186	C₄⋊C₈	—	53	8	no	—
187	C₂²⋊C₈	—	53	8	no	—
188	C₄×C₈	—	53	8	no	—
189	C₄×C₈	—	106	16	no	—
190	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
191	C₄×C₈	C₂×Dic₅₃	1	8	yes	—
192	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
193	#95	—	53	4	no	—
194	#95	—	53	4	no	—
195	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	53	8	no	—
199	C₂²×C₄	C₈×D₅₃	1	8	yes	—
200	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
201	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
202	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
205	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
208	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
209	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	16	no	—
212	C₂×C₈	—	53	8	no	—
213	C₂×C₈	—	106	8	no	—
214	C₂×C₈	#11719945	1	8	yes	—
215	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
216	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
217	C₄⋊C₄	—	53	16	no	—
218	C₂²⋊C₄	—	53	16	no	—
219	C₄²	—	53	16	no	—
220	C₄²	—	106	8	no	—
221	C₄²	—	106	8	no	—
222	C₄²	C₄×Dic₅₃	1	4	yes	—
223	C₄²	C₄×Dic₅₃	1	4	yes	—
224	C₄²	C₂×C₄×D₅₃	1	2	yes	—
225	C₄²	C₂×C₄×D₅₃	1	2	yes	—
226	C₄²	C₂²×Dic₅₃	1	2	yes	—
227	C₄²	C₂²×Dic₅₃	1	2	yes	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
231	C₂³	C₂×C₈×D₅₃	1	4	yes	—
232	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	53	16	no	—
239	C₂×C₄	—	106	8	no	—
240	C₂×C₄	—	106	8	no	—
241	C₂×C₄	C₈×Dic₅₃	1	16	yes	complemented
242	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
243	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
244	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
245	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
246	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
247	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
248	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
249	C₂×C₄	#13133223	1	4	yes	—
250	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
251	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
252	C₂×C₄	C₂×#11719945	1	4	yes	—
253	C₈	—	2	4	no	—
254	C₈	—	106	8	no	—
255	C₈	#13133176	1	8	yes	direct product factor
256	C₂²	—	2	8	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	C₂×C₈×Dic₅₃	1	4	yes	—
260	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
261	C₂²	C₄×#11719945	1	2	yes	—
262	C₂²	C₂²×C₈×D₅₃	1	2	yes	—
263	C₂²	C₂×#13133223	1	2	yes	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	106	16	no	—
267	C₄	C₂×C₈×Dic₅₃	1	4	yes	—
268	C₄	C₂×C₈×Dic₅₃	1	4	yes	—
269	C₄	#16810578	1	4	yes	—
270	C₄	C₂×#13133176	1	2	yes	—
271	C₄	C₄×#11719945	1	2	yes	—
272	C₂	—	2	4	no	—
273	C₂	C₄×#13133176	1	1	yes	characteristic
274	C₂	C₂²×C₈×Dic₅₃	1	1	yes	characteristic
275	C₂	C₂×#16810578	1	1	yes	characteristic
276	C₂	C₈×#11719945	1	4	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	C₈×#117
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₈×C₄⋊Dic₅₃	C₂	semidirect
5. #20274460	C₂	semidirect
6. C₄×C₈×Dic₅₃	C₂	semidirect
9. #16810676	C₄	semidirect
23. C₂×C₄×C₈×C₅₃	C₄	semidirect
28. #13133176	C₈	direct (N × H)
83. C₄⋊Dic₅₃	C₂×C₈	semidirect
86. C₄×Dic₅₃	C₂×C₈	semidirect
99. C₂×C₄×C₅₃	C₄×C₈	semidirect
115. C₈×C₅₃	#117	semidirect
154. C₂×C₄×C₈	Dic₅₃	semidirect
165. C₅₃	C₈×#117	semidirect
241. C₂×C₄	C₈×Dic₅₃	semidirect
255. C₈	#13133176	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×#16810676	2
C₄×#13133176	2
C₈×C₄⋊Dic₅₃	2
C₈×C₂²⋊Dic₅₃	2
#20274460	2
C₄×C₈×Dic₅₃	2
C₂²×C₄×C₈×C₅₃	2
C₈×#117	53

Subgroups — C8×#13133176 #27382295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#13133176 #27382295