Subgroups — C₂×#20272590 #27274036

Order 13568.

690 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810647	C₂	1	2	yes	maximal
2	#20272590	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719917	—	2	2	no	—
5	#16810647	—	2	2	no	—
6	#16810647	—	2	2	no	—
7	#16810647	C₄	1	4	yes	—
8	#16810647	C₂²	1	4	yes	—
9	C₂×C₄⋊#88322	—	2	2	no	—
10	C₂×C₄⋊#88322	C₄	1	4	yes	—
11	C₂³×C₈×C₅₃	C₄	1	2	yes	—
12	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
14	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
15	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×#11719917	—	2	4	no	—
18	C₂×#11719917	—	2	4	no	—
19	C₂×#11719917	—	4	4	no	—
20	C₂×#11719917	—	4	4	no	—
21	C₂²×#88322	—	2	4	no	—
22	C₄⋊#88322	—	2	4	no	—
23	C₄⋊#88322	—	2	4	no	—
24	C₄⋊#88322	—	2	4	no	—
25	C₄⋊#88322	C₂×C₄	1	8	yes	—
26	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	—	2	2	no	—
31	C₂²×C₈×C₅₃	—	4	4	no	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	D₄	1	2	yes	—
37	C₂×C₄²×C₅₃	D₄	1	2	yes	—
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₅₃	—	2	4	no	—
42	C₄⋊C₈×C₅₃	—	2	4	no	—
43	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	—	2	2	no	—
45	C₂²×C₄×C₅₃	—	2	2	no	—
46	C₂²×C₄×C₅₃	—	4	2	no	—
47	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	D₈	1	2	yes	—
59	C₂²×C₄×C₅₃	SD₁₆	1	4	yes	—
60	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₅₃	—	2	4	no	—
65	C₂×C₈×C₅₃	—	4	4	no	—
66	C₂×C₈×C₅₃	—	4	4	no	—
67	C₂×C₈×C₅₃	—	4	4	no	—
68	C₄²×C₅₃	C₄²	1	2	yes	—
69	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
73	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
74	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
75	C₄²×C₅₃	C₂×D₄	1	2	yes	—
76	#11719917	—	4	8	no	—
77	#11719917	—	4	4	no	—
78	#11719917	—	4	4	no	—
79	C₂×#88322	—	2	8	no	—
80	C₂×#88322	—	4	8	no	—
81	C₂³×C₅₃	—	2	2	no	—
82	C₂³×C₅₃	—	2	2	no	—
83	C₂³×C₅₃	#95	1	1	yes	characteristic
84	C₂³×C₅₃	#95	1	1	yes	characteristic
85	C₂³×C₅₃	#106	1	2	yes	—
86	C₂³×C₅₃	#107	1	2	yes	—
87	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	1	no	—
98	C₂×C₄×C₅₃	—	4	1	no	—
99	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#95	1	2	yes	—
101	C₂×C₄×C₅₃	#95	1	2	yes	—
102	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#100	1	2	yes	—
104	C₂×C₄×C₅₃	#101	1	2	yes	—
105	C₂×C₄×C₅₃	#102	1	4	yes	—
106	C₂×C₄×C₅₃	#102	1	4	yes	—
107	C₂×C₄×C₅₃	#103	1	4	yes	—
108	C₂×C₄×C₅₃	#103	1	4	yes	—
109	C₂×C₄×C₅₃	#106	1	2	yes	—
110	C₂×C₄×C₅₃	#107	1	2	yes	—
111	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
117	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
118	C₈×C₅₃	—	4	4	no	—
119	#88322	—	4	8	no	—
120	C₂×#1292	—	53	1	no	maximal, Sylow
121	C₂²×C₅₃	—	2	4	no	—
122	C₂²×C₅₃	—	2	4	no	—
123	C₂²×C₅₃	—	2	1	no	—
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	—	2	1	no	—
126	C₂²×C₅₃	—	2	1	no	—
127	C₂²×C₅₃	#338	1	2	yes	—
128	C₂²×C₅₃	#340	1	2	yes	—
129	C₂²×C₅₃	#340	1	2	yes	—
130	C₂²×C₅₃	#343	1	2	yes	—
131	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
132	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
133	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
134	C₄×C₅₃	—	2	2	no	—
135	C₄×C₅₃	—	2	2	no	—
136	C₄×C₅₃	—	2	1	no	—
137	C₄×C₅₃	—	2	1	no	—
138	C₄×C₅₃	#340	1	4	yes	—
139	C₄×C₅₃	#343	1	2	yes	—
140	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
141	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
142	C₄×C₅₃	C₂×#102	1	2	yes	—
143	C₄×C₅₃	C₂×#103	1	2	yes	—
144	C₂×#423	—	53	2	no	—
145	C₂²×C₄⋊C₈	D₅₃	1	1	yes	characteristic
146	#1292	—	53	4	no	—
147	C₂×C₅₃	—	2	2	no	—
148	C₂×C₅₃	—	2	1	no	—
149	C₂×C₅₃	—	2	1	no	—
150	C₂×C₅₃	#1292	1	4	yes	complemented
151	C₂×C₅₃	C₂×#338	1	1	yes	characteristic
152	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
153	C₂×C₅₃	C₂×#343	1	1	yes	characteristic
154	C₂²×M₁₆	—	106	2	no	—
155	C₂³×C₈	Dic₅₃	1	2	yes	—
156	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
157	#423	—	53	4	no	—
158	#423	—	53	4	no	—
159	#423	—	106	2	no	—
160	#423	—	106	2	no	—
161	C₂×C₄⋊C₈	—	2	2	no	—
162	C₂×C₄⋊C₈	—	2	2	no	—
163	C₂×C₄⋊C₈	—	53	4	no	—
164	C₂×C₄⋊C₈	—	106	2	no	—
165	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
166	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
167	C₅₃	C₂×#1292	1	1	yes	characteristic, complemented, Sylow
168	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
169	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
170	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
171	C₂×M₁₆	—	106	4	no	—
172	C₂×M₁₆	—	106	4	no	—
173	C₂×M₁₆	—	212	4	no	—
174	C₂×M₁₆	—	212	4	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	4	4	no	—
178	C₂²×C₈	—	106	4	no	—
179	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
180	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
181	C₂×C₄²	C₄×D₅₃	1	2	yes	—
182	C₂×C₄²	C₄×D₅₃	1	2	yes	—
183	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
184	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
185	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
186	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
187	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
188	C₄⋊C₈	—	2	4	no	—
189	C₄⋊C₈	—	2	4	no	—
190	C₄⋊C₈	—	53	8	no	—
191	C₄⋊C₈	—	106	4	no	—
192	C₄⋊C₈	—	106	4	no	—
193	C₄⋊C₈	—	106	4	no	—
194	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	#11719913	1	4	yes	—
199	C₂²×C₄	D₄₂₄	1	2	yes	—
200	C₂²×C₄	Dic₂₁₂	1	2	yes	—
201	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
203	C₂²×C₄	#11719919	1	2	yes	—
204	C₂²×C₄	#11719919	1	2	yes	—
205	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
206	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
207	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
208	C₂²×C₄	#11719921	1	2	yes	—
209	C₂²×C₄	#11719921	1	2	yes	—
210	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
211	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
214	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
215	M₁₆	—	212	8	no	—
216	M₁₆	—	212	4	no	—
217	M₁₆	—	212	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	4	4	no	—
220	C₂×C₈	—	4	4	no	—
221	C₂×C₈	—	4	4	no	—
222	C₂×C₈	—	106	8	no	—
223	C₂×C₈	—	212	8	no	—
224	C₄²	C₄×Dic₅₃	1	2	yes	—
225	C₄²	#11719919	1	2	yes	—
226	C₄²	#11719919	1	2	yes	—
227	C₄²	#11719921	1	2	yes	—
228	C₄²	#11719921	1	2	yes	—
229	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
230	C₄²	C₂×C₄×D₅₃	1	2	yes	—
231	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	#13133124	1	1	yes	characteristic
235	C₂³	#13133124	1	1	yes	characteristic
236	C₂³	#13133149	1	2	yes	—
237	C₂³	#13133151	1	2	yes	—
238	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	#13133124	1	2	yes	—
251	C₂×C₄	#13133124	1	1	yes	characteristic
252	C₂×C₄	#13133124	1	2	yes	—
253	C₂×C₄	#13133124	1	1	yes	characteristic
254	C₂×C₄	#13133134	1	2	yes	—
255	C₂×C₄	#13133136	1	2	yes	—
256	C₂×C₄	#13133139	1	4	yes	—
257	C₂×C₄	#13133139	1	4	yes	—
258	C₂×C₄	#13133142	1	4	yes	—
259	C₂×C₄	#13133142	1	4	yes	—
260	C₂×C₄	#13133149	1	2	yes	—
261	C₂×C₄	#13133151	1	2	yes	—
262	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
263	C₂×C₄	C₂×#11719921	1	2	yes	—
264	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
265	C₂×C₄	C₂×#11719919	1	2	yes	—
266	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
267	C₂×C₄	C₂×#11719913	1	2	yes	—
268	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
269	C₈	—	4	4	no	—
270	C₈	—	212	8	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	#16810381	1	2	yes	—
278	C₂²	#16810386	1	2	yes	—
279	C₂²	#16810386	1	2	yes	—
280	C₂²	#16810392	1	2	yes	—
281	C₂²	C₂×#13133124	1	1	yes	characteristic
282	C₂²	C₂×#13133149	1	1	yes	characteristic
283	C₂²	C₂×#13133151	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	#16810386	1	4	yes	—
289	C₄	#16810392	1	2	yes	—
290	C₄	C₂×#13133134	1	1	yes	characteristic
291	C₄	C₂×#13133136	1	1	yes	characteristic
292	C₄	C₂×#13133139	1	2	yes	—
293	C₄	C₂×#13133142	1	2	yes	—
294	C₂	—	2	2	no	—
295	C₂	—	2	1	no	—
296	C₂	—	2	1	no	—
297	C₂	#20272590	1	4	yes	direct product factor
298	C₂	C₂×#16810381	1	1	yes	characteristic
299	C₂	C₂×#16810386	1	1	yes	characteristic
300	C₂	C₂×#16810392	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133124
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂²×C₄⋊C₈×C₅₃	C₂
socle	C₂³×C₅₃	#95
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂²×C₄⋊C₈	D₅₃
hypocenter	C₅₃	C₂×#1292
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #20272590	C₂	direct (N × H)
150. C₂×C₅₃	#1292	semidirect
167. C₅₃	C₂×#1292	semidirect
297. C₂	#20272590	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×#16810647	2
#20272590	2
C₂²×C₄⋊C₈×C₅₃	2
C₂×#1292	53

Subgroups — C2×#20272590 #27274036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272590 #27274036