Subgroups — C₂×#20275464 #27387502

Order 13568.

974 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133153	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810650	C₂	1	2	yes	maximal
3	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	#20275464	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16810382	C₂	1	2	yes	maximal
6	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₂²	1	2	yes	—
8	C₂³×#88322	C₂²	1	1	yes	characteristic
9	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂×#13133153	—	2	4	no	—
11	C₂×#13133153	C₂²	1	4	yes	—
12	#16810650	—	2	4	no	—
13	#16810650	C₂²	1	8	yes	—
14	C₂×C₄⋊#88322	—	2	4	no	—
15	C₂×C₈×Dic₅₃	—	2	4	no	—
16	C₂×C₈×Dic₅₃	C₂²	1	4	yes	—
17	#16810382	C₂²	1	8	yes	—
18	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#117×C₅₃	C₂²	1	2	yes	—
20	C₅₃×#427	C₄	1	4	yes	complemented
21	C₅₃×#427	C₂²	1	4	yes	—
22	C₂×#107×C₅₃	C₄	1	2	yes	complemented
23	C₂×#106×C₅₃	C₄	1	2	yes	complemented
24	C₂³×Dic₅₃	—	2	1	no	—
25	C₂×#11719917	—	2	2	no	—
26	C₂×#11719917	—	2	4	no	—
27	C₂×#11719917	—	2	4	no	—
28	C₂×#11719917	C₂³	1	4	yes	—
29	C₂²×#88322	—	2	2	no	—
30	C₂²×#88322	—	2	2	no	—
31	C₂²×#88322	—	2	2	no	—
32	C₂²×#88322	D₄	1	2	yes	—
33	C₂²×#88322	D₄	1	2	yes	—
34	C₂²×#88322	C₂³	1	2	yes	—
35	C₂×C₄×Dic₅₃	—	2	2	no	—
36	C₂×C₄×Dic₅₃	—	2	1	no	—
37	C₂×C₄×Dic₅₃	Q₈	1	4	yes	—
38	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
39	#13133153	—	2	8	no	—
40	C₄⋊#88322	—	2	16	no	—
41	C₈×Dic₅₃	—	2	8	no	—
42	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₅₃	D₄	1	4	yes	—
47	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
48	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
49	#117×C₅₃	—	2	2	no	—
50	#117×C₅₃	—	2	2	no	—
51	#117×C₅₃	C₂×C₄	1	4	yes	—
52	#117×C₅₃	C₂³	1	4	yes	—
53	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
54	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
55	C₂×C₄²×C₅₃	—	2	2	no	—
56	#107×C₅₃	C₂×C₄	1	8	yes	complemented
57	#106×C₅₃	C₂×C₄	1	8	yes	complemented
58	C₂⁴×C₅₃	#41	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	—	2	4	no	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	#41	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₅₃	—	2	8	no	—
71	C₂²×Dic₅₃	—	2	4	no	—
72	C₂²×Dic₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
74	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
75	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
79	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
80	C₄⋊C₄×C₅₃	—	2	4	no	—
81	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
82	C₂²⋊C₄×C₅₃	—	2	4	no	—
83	C₂²⋊C₄×C₅₃	—	2	4	no	—
84	C₄²×C₅₃	—	2	4	no	—
85	C₄²×C₅₃	—	2	4	no	—
86	C₄×Dic₅₃	—	2	4	no	—
87	C₄×Dic₅₃	—	2	2	no	—
88	C₄×Dic₅₃	C₂×Q₈	1	4	yes	—
89	#11719917	—	2	4	no	—
90	#11719917	—	2	4	no	—
91	#11719917	—	4	4	no	—
92	C₂×#88322	—	2	4	no	—
93	C₂×#88322	—	2	4	no	—
94	C₂×#88322	—	2	2	no	—
95	C₂×#88322	—	2	2	no	—
96	C₂×#88322	—	2	4	no	—
97	C₂×#88322	—	4	4	no	—
98	C₂×#88322	C₂×D₄	1	4	yes	—
99	C₂³×C₅₃	—	2	4	no	—
100	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
101	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
102	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
103	C₂³×C₅₃	#125	1	2	yes	—
104	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₅₃	—	2	8	no	—
106	C₂×C₄×C₅₃	—	4	8	no	—
107	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
113	C₂×C₄×C₅₃	#121	1	2	yes	—
114	C₂×C₄×C₅₃	#121	1	2	yes	—
115	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₅₃	#125	1	2	yes	—
117	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
118	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
119	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
125	C₂×Dic₅₃	—	2	8	no	—
126	C₂×Dic₅₃	—	2	16	no	—
127	C₂×Dic₅₃	—	2	4	no	—
128	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
129	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
130	#88322	—	2	4	no	—
131	#88322	—	4	4	no	—
132	C₂×#1945	—	53	1	no	maximal, Sylow
133	C₂²×C₅₃	—	2	2	no	—
134	C₂²×C₅₃	—	2	2	no	—
135	C₂²×C₅₃	—	2	4	no	—
136	C₂²×C₅₃	—	2	4	no	—
137	C₂²×C₅₃	#389	1	2	yes	—
138	C₂²×C₅₃	#389	1	2	yes	—
139	C₂²×C₅₃	#443	1	4	yes	—
140	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
142	C₂²×C₅₃	C₂×#125	1	1	yes	characteristic
143	C₄×C₅₃	—	4	8	no	—
144	C₄×C₅₃	#389	1	2	yes	—
145	C₄×C₅₃	#389	1	2	yes	—
146	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
148	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
149	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
150	Dic₅₃	—	2	8	no	—
151	C₂²×#108	—	53	1	no	—
152	C₂×#427	D₅₃	1	1	yes	characteristic
153	C₂×#424	—	53	2	no	—
154	C₂²×C₄×C₈	—	53	1	no	—
155	#1945	—	53	8	no	—
156	C₂×#339	—	53	2	no	—
157	C₂×C₅₃	—	2	1	no	—
158	C₂×C₅₃	—	2	1	no	—
159	C₂×C₅₃	—	2	2	no	—
160	C₂×C₅₃	#1945	1	4	yes	complemented
161	C₂×C₅₃	C₂×#389	1	1	yes	characteristic
162	C₂×C₅₃	C₂×#443	1	2	yes	—
163	C₂²×M₁₆	—	53	2	no	—
164	C₂³×C₈	—	53	1	no	—
165	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
166	C₂×#117	C₂×D₅₃	1	2	yes	—
167	C₂²×C₄²	—	53	1	no	—
168	C₂×#108	—	53	4	no	—
169	C₂×#108	—	106	4	no	—
170	#427	Dic₅₃	1	4	yes	complemented
171	#427	C₂×D₅₃	1	4	yes	—
172	C₂×#107	Dic₅₃	1	2	yes	complemented
173	C₂×#106	Dic₅₃	1	2	yes	complemented
174	#424	—	53	8	no	—
175	#424	—	106	4	no	—
176	C₂×C₄⋊C₈	—	106	4	no	—
177	C₂×C₄×C₈	—	53	4	no	—
178	C₂×C₄×C₈	—	106	4	no	—
179	#339	—	53	8	no	—
180	C₅₃	C₂×#1945	1	1	yes	characteristic, complemented, Sylow
181	C₂³×C₄	—	106	1	no	—
182	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
183	C₂×M₁₆	—	53	4	no	—
184	C₂×M₁₆	—	106	2	no	—
185	C₂×M₁₆	—	106	4	no	—
186	C₂×M₁₆	—	106	4	no	—
187	C₂²×C₈	—	53	2	no	—
188	C₂²×C₈	—	53	2	no	—
189	C₂²×C₈	—	53	2	no	—
190	C₂²×C₈	—	106	2	no	—
191	C₂²×C₈	—	106	2	no	—
192	C₂²×C₈	—	106	2	no	—
193	C₂²×C₈	Dic₁₀₆	1	4	yes	—
194	C₂²×C₈	D₂₁₂	1	4	yes	—
195	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
196	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
197	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
198	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
199	#117	—	2	2	no	—
200	#117	—	2	2	no	—
201	#117	C₂×Dic₅₃	1	4	yes	—
202	#117	C₂²×D₅₃	1	4	yes	—
203	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	53	4	no	—
207	C₂×C₄²	—	53	2	no	—
208	C₂×C₄²	—	106	2	no	—
209	C₂×C₄²	—	106	1	no	—
210	#108	—	106	8	no	—
211	#107	C₂×Dic₅₃	1	8	yes	complemented
212	#106	C₂×Dic₅₃	1	8	yes	complemented
213	C₄⋊C₈	—	106	16	no	—
214	C₄×C₈	—	106	8	no	—
215	C₂⁴	#11719947	1	1	yes	characteristic
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	106	8	no	—
218	C₂²×C₄	—	106	4	no	—
219	C₂²×C₄	—	106	2	no	—
220	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
221	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
222	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
223	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
224	C₂²×C₄	#11719947	1	1	yes	characteristic
225	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
226	C₂²×C₄	#11719949	1	2	yes	—
227	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
228	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
229	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
230	M₁₆	—	106	4	no	—
231	M₁₆	—	106	4	no	—
232	M₁₆	—	212	4	no	—
233	C₂×C₈	—	53	4	no	—
234	C₂×C₈	—	106	4	no	—
235	C₂×C₈	—	106	4	no	—
236	C₂×C₈	—	106	2	no	—
237	C₂×C₈	—	106	2	no	—
238	C₂×C₈	—	106	4	no	—
239	C₂×C₈	—	212	4	no	—
240	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
241	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
242	C₂×C₈	C₄⋊Dic₅₃	1	8	yes	—
243	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
244	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
245	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
246	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	C₂²×Dic₅₃	1	8	yes	—
249	C₂²⋊C₄	—	2	4	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	53	4	no	—
254	C₄²	—	106	4	no	—
255	C₄²	—	106	2	no	—
256	C₂³	—	2	4	no	—
257	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
258	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
259	C₂³	#13133197	1	2	yes	—
260	C₂³	#13133208	1	2	yes	—
261	C₂³	C₂×#11719947	1	1	yes	characteristic
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	4	8	no	—
264	C₂×C₄	—	106	16	no	—
265	C₂×C₄	—	106	8	no	—
266	C₂×C₄	—	106	4	no	—
267	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
268	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
269	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
270	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
271	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
272	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
273	C₂×C₄	#13133192	1	2	yes	—
274	C₂×C₄	#13133192	1	2	yes	—
275	C₂×C₄	#13133197	1	2	yes	—
276	C₂×C₄	#13133208	1	2	yes	—
277	C₂×C₄	#13133216	1	2	yes	—
278	C₂×C₄	#13133216	1	2	yes	—
279	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
280	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
282	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
283	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
284	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
285	C₈	—	106	4	no	—
286	C₈	—	212	4	no	—
287	C₈	C₂×C₄⋊Dic₅₃	1	4	yes	—
288	C₈	C₂×C₄⋊Dic₅₃	1	4	yes	—
289	C₂²	—	2	4	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	#16810522	1	2	yes	—
294	C₂²	#16810522	1	2	yes	—
295	C₂²	#16810740	1	4	yes	—
296	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
297	C₂²	C₂×#13133197	1	1	yes	characteristic
298	C₂²	C₂×#13133208	1	1	yes	characteristic
299	C₄	—	4	8	no	—
300	C₄	—	106	8	no	—
301	C₄	#16810522	1	2	yes	—
302	C₄	#16810522	1	2	yes	—
303	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
304	C₄	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
305	C₄	C₂×#13133192	1	1	yes	characteristic
306	C₄	C₂×#13133216	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	1	no	—
310	C₂	#20275464	1	4	yes	direct product factor
311	C₂	C₂×#16810522	1	1	yes	characteristic
312	C₂	C₂×#16810740	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#427	C₂
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#427	D₅₃
hypocenter	C₅₃	C₂×#1945
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
4. #20275464	C₂	direct (N × H)
20. C₅₃×#427	C₄	semidirect
22. C₂×#107×C₅₃	C₄	semidirect
23. C₂×#106×C₅₃	C₄	semidirect
56. #107×C₅₃	C₂×C₄	semidirect
57. #106×C₅₃	C₂×C₄	semidirect
160. C₂×C₅₃	#1945	semidirect
170. #427	Dic₅₃	semidirect
172. C₂×#107	Dic₅₃	semidirect
173. C₂×#106	Dic₅₃	semidirect
180. C₅₃	C₂×#1945	semidirect
211. #107	C₂×Dic₅₃	semidirect
212. #106	C₂×Dic₅₃	semidirect
310. C₂	#20275464	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×#13133153	2
C₂×#16810650	2
C₂²×C₈×Dic₅₃	2
#20275464	2
C₂×#16810382	2
C₂×C₅₃×#427	2
C₂×#1945	53

Subgroups — C2×#20275464 #27387502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20275464 #27387502