Subgroups — C₂×#20273468 #27353072

Order 13568.

834 conjugacy classes of subgroups.

The subgroup lattice has 324 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×C₄×#11719917	C₂	1	1	yes	maximal, characteristic
3	#20273468	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810375	C₂	1	2	yes	maximal, complemented
5	C₂×#16810373	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₄	1	1	yes	characteristic
8	C₂²×#11719917	C₄	1	1	yes	characteristic
9	C₂³×#88322	—	2	1	no	—
10	C₂×C₄⋊#88322	—	2	1	no	—
11	C₂×C₄⋊#88322	—	2	2	no	—
12	C₂×C₄⋊#88322	C₂²	1	2	yes	—
13	C₂×C₄⋊#88322	C₂²	1	4	yes	—
14	C₄×#11719917	C₄	1	4	yes	—
15	C₄×#11719917	C₂²	1	4	yes	—
16	C₂×#13133128	C₄	1	2	yes	—
17	C₂×C₄×#88322	C₂²	1	2	yes	—
18	#16810375	C₂²	1	8	yes	complemented
19	#16810373	C₂²	1	8	yes	complemented
20	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#431	—	2	2	no	—
22	C₅₃×#431	C₂²	1	4	yes	—
23	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
26	C₂×C₄×C₈×C₅₃	—	2	1	no	—
27	C₂×#11719917	—	2	4	no	—
28	C₂×#11719917	C₈	1	8	yes	complemented
29	C₂×#11719917	C₂×C₄	1	4	yes	—
30	C₂×#11719917	C₂×C₄	1	4	yes	—
31	C₂²×#88322	—	2	4	no	—
32	C₂²×#88322	—	2	2	no	—
33	C₂²×#88322	—	4	4	no	—
34	C₂²×#88322	C₂×C₄	1	2	yes	—
35	C₂²×#88322	C₂×C₄	1	2	yes	—
36	C₄⋊#88322	—	2	4	no	—
37	C₄⋊#88322	—	2	2	no	—
38	C₄⋊#88322	C₂³	1	4	yes	—
39	#13133128	C₂×C₄	1	8	yes	—
40	C₄×#88322	C₂×C₄	1	4	yes	—
41	C₄×#88322	C₂³	1	4	yes	—
42	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	—	2	1	no	—
46	C₂²×C₈×C₅₃	—	2	1	no	—
47	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	D₄	1	2	yes	—
51	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
55	C₄⋊C₈×C₅₃	—	2	1	no	—
56	C₄⋊C₈×C₅₃	—	2	1	no	—
57	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
58	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
59	C₂²⋊C₈×C₅₃	—	2	4	no	—
60	C₄×C₈×C₅₃	—	2	4	no	—
61	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	—	2	2	no	—
64	C₂²×C₄×C₅₃	—	4	2	no	—
65	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
72	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
73	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
79	C₂×C₈×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	—	2	2	no	—
81	C₂×C₈×C₅₃	—	4	2	no	—
82	C₂×C₈×C₅₃	—	4	2	no	—
83	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
84	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
85	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
86	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
88	C₄²×C₅₃	C₂×D₄	1	2	yes	—
89	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
91	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
92	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
93	#11719917	—	2	8	no	—
94	#11719917	C₂×C₈	1	16	yes	complemented
95	C₂×#88322	—	2	4	no	—
96	C₂×#88322	—	2	4	no	—
97	C₂×#88322	—	4	8	no	—
98	C₂×#88322	—	4	4	no	—
99	C₂×#88322	C₂×C₈	1	8	yes	complemented
100	C₂×#88322	C₂²×C₄	1	4	yes	—
101	C₂×#88322	C₂²×C₄	1	4	yes	—
102	C₂³×C₅₃	—	2	2	no	—
103	C₂³×C₅₃	—	2	2	no	—
104	C₂³×C₅₃	C₄⋊C₈	1	4	yes	—
105	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	2	4	no	—
113	C₂×C₄×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	—	4	2	no	—
115	C₂×C₄×C₅₃	C₄⋊C₈	1	8	yes	—
116	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
117	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
124	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
125	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
127	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
128	C₂×C₄×C₅₃	#137	1	2	yes	—
129	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
132	C₈×C₅₃	—	4	2	no	—
133	C₈×C₅₃	—	4	2	no	—
134	#88322	—	2	4	no	—
135	#88322	—	4	4	no	—
136	#88322	C₂²×C₈	1	8	yes	complemented
137	C₂×#1565	—	53	1	no	maximal, Sylow
138	C₂²×C₅₃	—	2	8	no	—
139	C₂²×C₅₃	—	2	2	no	—
140	C₂²×C₅₃	—	2	2	no	—
141	C₂²×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
142	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
143	C₂²×C₅₃	#428	1	2	yes	—
144	C₂²×C₅₃	#430	1	2	yes	—
145	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₅₃	—	2	2	no	—
147	C₄×C₅₃	—	2	2	no	—
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	2	1	no	—
150	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
151	C₄×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
152	C₄×C₅₃	#428	1	2	yes	—
153	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
154	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
155	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
157	C₂×#431	D₅₃	1	1	yes	characteristic
158	C₂²×C₄⋊C₈	—	53	1	no	—
159	C₂×C₄×M₁₆	—	53	1	no	—
160	#1565	—	53	8	no	—
161	C₂×#335	—	53	2	no	—
162	C₂×#334	—	53	2	no	—
163	C₂×C₅₃	—	2	4	no	—
164	C₂×C₅₃	#1565	1	4	yes	complemented
165	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
166	C₂×C₅₃	C₂×#428	1	1	yes	characteristic
167	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
168	C₂²×M₁₆	—	53	1	no	—
169	C₂²×M₁₆	—	53	1	no	—
170	C₂³×C₈	—	106	1	no	—
171	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
172	#431	—	2	2	no	—
173	#431	C₂×D₅₃	1	4	yes	—
174	C₂×C₄⋊C₈	—	53	2	no	—
175	C₂×C₄⋊C₈	—	53	4	no	—
176	C₂×C₄⋊C₈	—	106	2	no	—
177	C₂×C₄⋊C₈	—	106	1	no	—
178	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
179	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
180	C₂×C₂²⋊C₈	—	2	1	no	—
181	C₄×M₁₆	—	53	4	no	—
182	C₄×M₁₆	—	53	4	no	—
183	C₂×#97	—	53	2	no	—
184	C₂×C₄×C₈	—	2	1	no	—
185	C₂×C₄×C₈	—	53	2	no	—
186	#335	—	53	8	no	—
187	#334	—	53	8	no	—
188	C₅₃	C₂×#1565	1	1	yes	characteristic, complemented, Sylow
189	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
190	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
191	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
192	C₂×M₁₆	—	53	4	no	—
193	C₂×M₁₆	—	53	8	no	—
194	C₂×M₁₆	—	53	4	no	—
195	C₂×M₁₆	—	106	4	no	—
196	C₂²×C₈	—	2	1	no	—
197	C₂²×C₈	—	2	1	no	—
198	C₂²×C₈	—	53	2	no	—
199	C₂²×C₈	—	53	2	no	—
200	C₂²×C₈	—	106	4	no	—
201	C₂²×C₈	—	106	2	no	—
202	C₂²×C₈	—	212	4	no	—
203	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
204	C₂×C₄²	Dic₁₀₆	1	2	yes	—
205	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
206	C₂×C₄²	C₄×D₅₃	1	2	yes	—
207	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
208	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
209	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
210	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	—	53	4	no	—
214	C₄⋊C₈	—	106	4	no	—
215	C₄⋊C₈	—	106	2	no	—
216	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
217	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
218	C₂²⋊C₈	—	2	4	no	—
219	#97	—	53	8	no	—
220	C₄×C₈	—	2	4	no	—
221	C₄×C₈	—	53	4	no	—
222	C₄×C₈	—	53	4	no	—
223	C₂⁴	#11719919	1	1	yes	characteristic
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	C₈×D₅₃	1	4	yes	—
228	C₂²×C₄	#11719912	1	4	yes	—
229	C₂²×C₄	#11719919	1	1	yes	characteristic
230	C₂²×C₄	#11719919	1	2	yes	—
231	C₂²×C₄	#11719919	1	1	yes	characteristic
232	C₂²×C₄	#11719919	1	2	yes	—
233	C₂²×C₄	#11719919	1	2	yes	—
234	C₂²×C₄	#11719919	1	1	yes	characteristic
235	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
236	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
237	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
239	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
240	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
241	M₁₆	—	53	16	no	—
242	M₁₆	—	106	8	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	53	8	no	—
248	C₂×C₈	—	53	4	no	—
249	C₂×C₈	—	53	4	no	—
250	C₂×C₈	—	106	4	no	—
251	C₂×C₈	—	106	4	no	—
252	C₂×C₈	—	212	8	no	—
253	C₂×C₈	—	212	4	no	—
254	C₄²	#11719919	1	2	yes	—
255	C₄²	#11719919	1	2	yes	—
256	C₄²	#11719919	1	2	yes	—
257	C₄²	C₂×Dic₁₀₆	1	2	yes	—
258	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
259	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
260	C₄²	C₂×C₄×D₅₃	1	2	yes	—
261	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
262	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
263	C₄²	C₂³×D₅₃	1	1	yes	characteristic
264	C₂³	—	2	2	no	—
265	C₂³	—	2	2	no	—
266	C₂³	#13133146	1	4	yes	—
267	C₂³	C₂×#11719919	1	2	yes	—
268	C₂³	C₂×#11719919	1	1	yes	characteristic
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	#13133146	1	8	yes	—
278	C₂×C₄	#13133146	1	4	yes	—
279	C₂×C₄	C₂×#11719919	1	2	yes	—
280	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
281	C₂×C₄	C₂×#11719919	1	2	yes	—
282	C₂×C₄	C₂×#11719919	1	2	yes	—
283	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
284	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
285	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
286	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
287	C₂×C₄	C₂×#11719912	1	2	yes	—
288	C₂×C₄	C₂×#11719912	1	4	yes	—
289	C₂×C₄	#13133242	1	2	yes	—
290	C₂×C₄	#13133247	1	2	yes	—
291	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
293	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
294	C₈	—	4	2	no	—
295	C₈	—	4	2	no	—
296	C₈	—	53	8	no	—
297	C₈	—	106	4	no	—
298	C₈	—	212	4	no	—
299	C₂²	—	2	8	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	C₂×#13133146	1	4	yes	—
303	C₂²	C₂×#13133146	1	2	yes	—
304	C₂²	#16810666	1	2	yes	—
305	C₂²	#16810671	1	2	yes	—
306	C₂²	C₂²×#11719919	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	1	no	—
311	C₄	C₂×#13133146	1	2	yes	—
312	C₄	C₂×#13133146	1	4	yes	—
313	C₄	#16810666	1	2	yes	—
314	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
315	C₄	C₂²×#11719912	1	1	yes	characteristic
316	C₄	C₂×#13133242	1	1	yes	characteristic
317	C₄	C₂×#13133247	1	1	yes	characteristic
318	C₂	—	2	4	no	—
319	C₂	#20273468	1	4	yes	direct product factor
320	C₂	C₂²×#13133146	1	1	yes	characteristic
321	C₂	C₂×#16810666	1	1	yes	characteristic
322	C₂	C₂×#16810671	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	C₂×#1565
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20273468	C₂	direct (N × H)
4. C₂×#16810375	C₂	semidirect
5. C₂×#16810373	C₂	semidirect
18. #16810375	C₂²	semidirect
19. #16810373	C₂²	semidirect
28. C₂×#11719917	C₈	semidirect
94. #11719917	C₂×C₈	semidirect
99. C₂×#88322	C₂×C₈	semidirect
136. #88322	C₂²×C₈	semidirect
164. C₂×C₅₃	#1565	semidirect
188. C₅₃	C₂×#1565	semidirect
319. C₂	#20273468	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×C₄×#11719917	2
#20273468	2
C₂×#16810375	2
C₂×#16810373	2
C₂×C₅₃×#431	2
C₂×#1565	53

Subgroups — C2×#20273468 #27353072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20273468 #27353072