Subgroups — C₂×C₈×#11719917 #27349666

Order 13568.

1046 conjugacy classes of subgroups.

The subgroup lattice has 292 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	C₂×C₄×#11719917	C₂	1	1	yes	maximal, characteristic
3	C₈×#11719917	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810345	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×#88322	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₄	1	2	yes	—
8	#16810676	C₄	1	4	yes	—
9	#16810676	C₂²	1	4	yes	—
10	C₂×C₄⋊#88322	C₄	1	2	yes	—
11	C₂×C₂²⋊#88322	C₄	1	2	yes	—
12	C₄×#11719917	C₄	1	4	yes	—
13	C₄×#11719917	C₂²	1	4	yes	—
14	C₂×#13133128	C₄	1	2	yes	—
15	C₂×C₄×#88322	C₂²	1	2	yes	—
16	C₂×C₄×#88322	C₂²	1	2	yes	—
17	#16810345	C₂²	1	8	yes	complemented
18	C₈×#88322	C₂²	1	8	yes	complemented
19	C₂³×C₈×C₅₃	C₄	1	2	yes	—
20	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	—
22	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#11719917	C₈	1	16	yes	direct product factor
27	C₂×#11719917	C₂×C₄	1	8	yes	—
28	C₂²×#88322	C₂×C₄	1	4	yes	—
29	C₂²×#88322	C₂×C₄	1	4	yes	—
30	C₄⋊#88322	C₂×C₄	1	8	yes	—
31	C₂²⋊#88322	C₂×C₄	1	8	yes	—
32	#13133128	C₂×C₄	1	8	yes	—
33	C₄×#88322	C₂×C₄	1	4	yes	—
34	C₄×#88322	C₂×C₄	1	4	yes	—
35	C₄×#88322	C₂³	1	4	yes	—
36	C₄×#88322	C₂³	1	4	yes	—
37	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	—	2	8	no	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
52	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
54	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₅₃	C₂³	1	2	yes	—
56	C₄×C₈×C₅₃	C₂³	1	2	yes	—
57	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	—	2	4	no	—
59	C₂²×C₄×C₅₃	—	2	4	no	—
60	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
63	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
64	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
65	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	—	2	4	no	—
74	C₂×C₈×C₅₃	—	2	16	no	—
75	C₂×C₈×C₅₃	M₁₆	1	8	yes	complemented
76	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
77	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
78	C₄²×C₅₃	C₄²	1	2	yes	—
79	C₄²×C₅₃	C₄²	1	2	yes	—
80	C₄²×C₅₃	C₄²	1	2	yes	—
81	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
84	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
85	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
86	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
88	#11719917	C₂×C₈	1	32	yes	direct product factor
89	C₂×#88322	—	2	8	no	—
90	C₂×#88322	C₂×C₈	1	16	yes	complemented
91	C₂×#88322	C₂²×C₄	1	8	yes	—
92	C₂×#88322	C₂²×C₄	1	8	yes	—
93	C₂³×C₅₃	—	2	2	no	—
94	C₂³×C₅₃	—	2	2	no	—
95	C₂³×C₅₃	C₄×C₈	1	4	yes	—
96	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
97	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	2	4	no	—
101	C₂×C₄×C₅₃	—	2	4	no	—
102	C₂×C₄×C₅₃	—	2	4	no	—
103	C₂×C₄×C₅₃	—	2	4	no	—
104	C₂×C₄×C₅₃	C₄×C₈	1	8	yes	—
105	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
106	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
110	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
113	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
114	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
115	C₂×C₄×C₅₃	#131	1	4	yes	—
116	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
119	C₈×C₅₃	—	2	4	no	—
120	C₈×C₅₃	C₂×M₁₆	1	8	yes	complemented
121	#88322	—	2	8	no	—
122	#88322	C₂²×C₈	1	16	yes	complemented
123	C₂×C₈×M₁₆	—	53	1	no	maximal, Sylow
124	C₂²×C₅₃	—	2	2	no	—
125	C₂²×C₅₃	—	2	2	no	—
126	C₂²×C₅₃	—	2	8	no	—
127	C₂²×C₅₃	C₂×C₄×C₈	1	4	yes	—
128	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
129	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
130	C₂²×C₅₃	#405	1	2	yes	—
131	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
132	C₄×C₅₃	—	2	2	no	—
133	C₄×C₅₃	—	2	2	no	—
134	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
135	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
136	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
137	C₄×C₅₃	#405	1	2	yes	—
138	C₄×C₅₃	C₂³×C₈	1	2	yes	—
139	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
140	C₄×C₅₃	C₂×#131	1	2	yes	—
141	C₂×#431	—	53	1	no	—
142	C₂×C₄×M₁₆	—	53	1	no	—
143	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
144	C₈×M₁₆	—	53	8	no	—
145	C₂×#322	—	53	2	no	—
146	C₂×C₈²	—	53	2	no	—
147	C₂×C₅₃	—	2	4	no	—
148	C₂×C₅₃	C₈×M₁₆	1	4	yes	complemented
149	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
150	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
151	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
152	C₂²×M₁₆	—	53	2	no	—
153	C₂³×C₈	Dic₅₃	1	2	yes	—
154	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
155	#431	—	53	4	no	—
156	#431	—	53	4	no	—
157	C₂×C₄⋊C₈	—	53	2	no	—
158	C₂×C₂²⋊C₈	—	53	2	no	—
159	C₄×M₁₆	—	53	4	no	—
160	C₄×M₁₆	—	53	4	no	—
161	C₂×#97	—	53	2	no	—
162	C₂×C₄×C₈	—	53	2	no	—
163	C₂×C₄×C₈	—	53	2	no	—
164	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
165	C₂×C₄×C₈	Dic₅₃	1	4	yes	—
166	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂×D₅₃	1	4	yes	—
168	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
169	#322	—	53	8	no	—
170	C₈²	—	53	8	no	—
171	C₅₃	C₂×C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
173	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
174	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
175	C₂×M₁₆	—	53	8	no	—
176	C₂×M₁₆	—	53	16	no	—
177	C₂²×C₈	—	2	8	no	—
178	C₂²×C₈	—	53	4	no	—
179	C₂²×C₈	—	53	4	no	—
180	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
181	C₂²×C₈	C₂×Dic₅₃	1	8	yes	—
182	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
183	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
184	C₂×C₄²	C₄×D₅₃	1	2	yes	—
185	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
186	C₂×C₄²	C₄×D₅₃	1	2	yes	—
187	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
188	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
189	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
190	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
191	C₄⋊C₈	—	53	8	no	—
192	C₂²⋊C₈	—	53	8	no	—
193	#97	—	53	8	no	—
194	C₄×C₈	—	53	4	no	—
195	C₄×C₈	—	53	4	no	—
196	C₄×C₈	—	53	4	no	—
197	C₄×C₈	—	53	4	no	—
198	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
199	C₄×C₈	C₂×Dic₅₃	1	8	yes	—
200	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
201	C₄×C₈	C₂²×D₅₃	1	2	yes	—
202	C₄×C₈	C₂²×D₅₃	1	2	yes	—
203	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	C₈×D₅₃	1	8	yes	—
207	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
208	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
209	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
210	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
211	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
212	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
214	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
216	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
218	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
219	M₁₆	—	53	32	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	16	no	—
222	C₂×C₈	—	53	16	no	—
223	C₂×C₈	—	53	8	no	—
224	C₂×C₈	—	53	8	no	—
225	C₂×C₈	—	106	8	no	—
226	C₂×C₈	#11719917	1	8	yes	direct product factor
227	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
228	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
229	C₄²	C₄×Dic₅₃	1	2	yes	—
230	C₄²	C₄×Dic₅₃	1	2	yes	—
231	C₄²	C₄×Dic₅₃	1	2	yes	—
232	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
233	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
234	C₄²	C₂×C₄×D₅₃	1	2	yes	—
235	C₄²	C₂×C₄×D₅₃	1	2	yes	—
236	C₄²	C₂²×Dic₅₃	1	2	yes	—
237	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
238	C₄²	C₂³×D₅₃	1	1	yes	characteristic
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	C₈×Dic₅₃	1	4	yes	—
242	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
243	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	C₈×Dic₅₃	1	8	yes	—
251	C₂×C₄	C₈×Dic₅₃	1	4	yes	—
252	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
253	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
254	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
256	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
258	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
259	C₂×C₄	C₂×C₈×D₅₃	1	8	yes	—
260	C₂×C₄	C₂×#11719917	1	4	yes	—
261	C₂×C₄	#13133223	1	4	yes	—
262	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
264	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
265	C₈	—	2	4	no	—
266	C₈	—	53	16	no	—
267	C₈	—	106	8	no	—
268	C₈	C₂×#11719917	1	8	yes	direct product factor
269	C₂²	—	2	8	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	C₂×C₈×Dic₅₃	1	4	yes	—
273	C₂²	C₂×C₈×Dic₅₃	1	2	yes	—
274	C₂²	C₄×#11719917	1	2	yes	—
275	C₂²	#16810578	1	2	yes	—
276	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	C₂×C₈×Dic₅₃	1	2	yes	—
280	C₄	C₂×C₈×Dic₅₃	1	4	yes	—
281	C₄	C₄×#11719917	1	2	yes	—
282	C₄	#16810578	1	2	yes	—
283	C₄	C₂²×C₈×D₅₃	1	2	yes	—
284	C₄	C₂²×#11719917	1	2	yes	—
285	C₄	C₂×#13133223	1	2	yes	—
286	C₂	—	2	4	no	—
287	C₂	C₈×#11719917	1	4	yes	direct product factor
288	C₂	C₂²×C₈×Dic₅₃	1	1	yes	characteristic
289	C₂	C₂×C₄×#11719917	1	1	yes	characteristic
290	C₂	C₂×#16810578	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄²
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂×C₈×M₁₆
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×#11719917	C₂	direct (N × H)
4. C₂×#16810345	C₂	semidirect
5. C₂×C₈×#88322	C₂	semidirect
17. #16810345	C₂²	semidirect
18. C₈×#88322	C₂²	semidirect
26. C₂×#11719917	C₈	direct (N × H)
75. C₂×C₈×C₅₃	M₁₆	semidirect
88. #11719917	C₂×C₈	direct (N × H)
90. C₂×#88322	C₂×C₈	semidirect
120. C₈×C₅₃	C₂×M₁₆	semidirect
122. #88322	C₂²×C₈	semidirect
148. C₂×C₅₃	C₈×M₁₆	semidirect
171. C₅₃	C₂×C₈×M₁₆	semidirect
226. C₂×C₈	#11719917	direct (N × H)
268. C₈	C₂×#11719917	direct (N × H)
287. C₂	C₈×#11719917	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
C₂×C₄×#11719917	2
C₈×#11719917	2
C₂×#16810345	2
C₂×C₈×#88322	2
C₂²×C₄×C₈×C₅₃	2
C₂×C₈×M₁₆	53

Subgroups — C2×C8×#11719917 #27349666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×#11719917 #27349666