Subgroups — C₂×#20276595 #27537491

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811055	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810739	C₂	1	2	yes	maximal, complemented
3	#20276595	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810429	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×#368	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13133239	C₂²	1	2	yes	—
8	C₂×#13133225	—	2	2	no	—
9	#16811055	C₄	1	4	yes	complemented
10	#16811055	C₂²	1	4	yes	—
11	C₂×Dic₄₂₄	C₄	1	1	yes	characteristic, complemented
12	C₂×#13133162	C₄	1	2	yes	complemented
13	C₂×D₈₄₈	C₄	1	1	yes	characteristic, complemented
14	C₂²×#11719908	C₂²	1	1	yes	characteristic
15	#16810739	—	2	4	no	—
16	#16810739	C₂²	1	8	yes	complemented
17	C₂×#13133143	—	2	2	no	—
18	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
19	#16810429	C₂²	1	8	yes	complemented
20	C₁₆×Dic₅₃	C₂²	1	4	yes	complemented
21	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#108×C₅₃	C₂²	1	2	yes	—
23	C₅₃×#368	C₂²	1	4	yes	complemented
24	C₂×#11719945	—	2	2	no	—
25	#13133239	—	2	2	no	—
26	#13133239	—	2	2	no	—
27	#13133239	C₂×C₄	1	4	yes	—
28	#13133239	C₂³	1	4	yes	—
29	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
30	C₂×#11719913	—	2	2	no	—
31	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
32	#13133225	—	2	8	no	—
33	#13133225	—	4	4	no	—
34	C₂×#11719912	—	4	2	no	—
35	C₂²×#88322	D₄	1	1	yes	characteristic
36	C₂×C₈×D₅₃	—	4	2	no	—
37	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
38	Dic₄₂₄	C₂×C₄	1	4	yes	complemented
39	#13133162	C₂×C₄	1	8	yes	complemented
40	D₈₄₈	C₂×C₄	1	4	yes	complemented
41	C₂×#11719908	—	2	1	no	—
42	C₂×#11719908	—	2	1	no	—
43	C₂×#11719908	C₂³	1	2	yes	—
44	#13133143	—	2	8	no	—
45	C₈×Dic₅₃	D₄	1	2	yes	—
46	C₈×Dic₅₃	C₂³	1	2	yes	—
47	C₂×M₁₆×C₅₃	—	2	2	no	—
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
53	#108×C₅₃	—	2	2	no	—
54	#108×C₅₃	C₂³	1	4	yes	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₂²⋊D₅₃	—	4	2	no	—
57	C₂²×Dic₅₃	—	2	1	no	—
58	#11719945	—	2	4	no	—
59	#11719945	—	2	4	no	—
60	#11719945	—	4	2	no	—
61	#11719945	—	4	2	no	—
62	C₂×D₂₁₂	—	2	2	no	—
63	C₂×C₄×D₅₃	—	4	2	no	—
64	C₂×Dic₁₀₆	—	2	2	no	—
65	M₁₆×C₅₃	—	2	4	no	—
66	M₁₆×C₅₃	—	4	2	no	—
67	C₂×C₈×C₅₃	—	4	2	no	—
68	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
72	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
74	C₄×Dic₅₃	—	2	1	no	—
75	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
76	C₂×#88322	D₈	1	1	yes	characteristic
77	C₂×#88322	D₈	1	1	yes	characteristic
78	C₂×#88322	D₈	1	2	yes	—
79	C₂×#88322	C₂×D₄	1	2	yes	—
80	Dic₂₁₂	—	2	2	no	—
81	Dic₂₁₂	C₂²×C₄	1	4	yes	—
82	D₄₂₄	—	2	2	no	—
83	D₄₂₄	C₂²×C₄	1	4	yes	—
84	#11719913	—	2	4	no	—
85	#11719913	—	2	4	no	—
86	#11719912	—	4	8	no	—
87	C₈×D₅₃	—	4	8	no	—
88	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
89	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
90	#11719908	—	2	2	no	—
91	C₂³×C₅₃	#135	1	1	yes	characteristic
92	C₂²×D₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
99	C₂²⋊D₅₃	—	4	4	no	—
100	C₂²⋊D₅₃	—	4	4	no	—
101	C₂×Dic₅₃	—	2	4	no	—
102	C₂×Dic₅₃	—	2	2	no	—
103	C₂×Dic₅₃	—	4	2	no	—
104	D₂₁₂	—	2	4	no	—
105	D₂₁₂	—	4	2	no	—
106	C₄×D₅₃	—	4	4	no	—
107	C₄×D₅₃	—	4	4	no	—
108	Dic₁₀₆	—	2	4	no	—
109	Dic₁₀₆	—	4	2	no	—
110	C₈×C₅₃	—	4	4	no	—
111	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
112	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
113	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
114	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
115	#88322	C₂×D₈	1	2	yes	—
116	#88322	C₂×D₈	1	2	yes	—
117	C₂×#2178	—	53	1	no	maximal, Sylow
118	C₂²×C₅₃	—	2	2	no	—
119	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
120	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
121	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
122	C₂×D₅₃	—	4	4	no	—
123	C₂×D₅₃	—	8	4	no	—
124	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
125	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
126	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
127	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
128	Dic₅₃	—	2	4	no	—
129	Dic₅₃	—	4	4	no	—
130	C₂×#508	—	53	1	no	—
131	C₂×#443	—	53	2	no	—
132	#2178	—	53	8	no	—
133	C₂×#368	D₅₃	1	1	yes	characteristic, complemented
134	C₂×#359	—	53	2	no	—
135	C₂×C₄×C₁₆	—	53	1	no	—
136	C₂×C₅₃	—	2	1	no	—
137	C₂×C₅₃	—	2	1	no	—
138	C₂×C₅₃	#2178	1	2	yes	complemented
139	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
140	D₅₃	—	8	4	no	—
141	C₂×#135	—	53	2	no	—
142	C₂×#131	—	106	2	no	—
143	#508	—	53	4	no	—
144	#508	—	53	4	no	—
145	C₂×Q₃₂	—	53	1	no	—
146	C₂×SD₃₂	—	53	2	no	—
147	C₂×D₁₆	—	53	1	no	—
148	C₂²×C₁₆	—	53	1	no	—
149	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
150	#443	—	53	8	no	—
151	#443	—	106	4	no	—
152	C₂×#108	C₂×D₅₃	1	2	yes	—
153	C₂×#104	—	106	2	no	—
154	C₂×C₄×C₈	—	53	1	no	—
155	#368	C₂×D₅₃	1	4	yes	complemented
156	#359	—	53	8	no	—
157	C₄×C₁₆	—	53	4	no	—
158	C₅₃	C₂×#2178	1	1	yes	characteristic, complemented, Sylow
159	C₂×#41	—	106	2	no	—
160	#135	—	53	4	no	—
161	#135	—	53	4	no	—
162	#135	—	106	2	no	—
163	#135	—	106	2	no	—
164	C₂×Q₁₆	—	53	2	no	—
165	C₂×SD₁₆	—	106	2	no	—
166	C₂×D₈	—	53	2	no	—
167	#131	—	106	8	no	—
168	#131	—	212	4	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	212	2	no	—
171	C₂²×C₈	—	53	1	no	—
172	C₂²×C₈	—	212	2	no	—
173	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
174	C₂×C₄²	—	53	1	no	—
175	Q₃₂	—	53	4	no	—
176	SD₃₂	—	53	8	no	—
177	D₁₆	—	53	4	no	—
178	C₂×C₁₆	—	53	2	no	—
179	C₂×C₁₆	—	106	1	no	—
180	C₂×C₁₆	—	106	1	no	—
181	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
182	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
183	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
184	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
185	#108	—	2	2	no	—
186	#108	C₂²×D₅₃	1	4	yes	—
187	#104	—	106	8	no	—
188	C₄×C₈	—	53	2	no	—
189	C₄×C₈	—	53	2	no	—
190	#41	—	106	4	no	—
191	#41	—	106	4	no	—
192	#41	—	212	2	no	—
193	#41	—	212	2	no	—
194	C₂×Q₈	—	106	2	no	—
195	C₂×D₄	—	106	2	no	—
196	C₂×D₄	—	212	2	no	—
197	C₂²×C₄	—	106	1	no	—
198	C₂²×C₄	—	212	2	no	—
199	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
200	Q₁₆	—	53	4	no	—
201	Q₁₆	—	106	2	no	—
202	SD₁₆	—	106	4	no	—
203	SD₁₆	—	106	4	no	—
204	D₈	—	53	4	no	—
205	D₈	—	106	2	no	—
206	M₁₆	—	2	4	no	—
207	M₁₆	—	4	2	no	—
208	M₁₆	—	212	8	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	53	2	no	—
211	C₂×C₈	—	53	1	no	—
212	C₂×C₈	—	53	1	no	—
213	C₂×C₈	—	53	2	no	—
214	C₂×C₈	—	212	8	no	—
215	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
216	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
217	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
218	C₂×C₈	#11719949	1	1	yes	characteristic
219	C₂×C₈	#11719949	1	1	yes	characteristic
220	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
221	C₄²	—	53	2	no	—
222	C₄²	—	106	1	no	—
223	C₁₆	—	106	2	no	—
224	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
225	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
226	C₂³	—	212	2	no	—
227	C₂³	#13133240	1	1	yes	characteristic
228	Q₈	—	106	4	no	—
229	Q₈	—	212	2	no	—
230	D₄	—	106	4	no	—
231	D₄	—	212	4	no	—
232	D₄	—	212	4	no	—
233	D₄	—	212	2	no	—
234	C₂×C₄	—	106	4	no	—
235	C₂×C₄	—	106	2	no	—
236	C₂×C₄	—	212	2	no	—
237	C₂×C₄	—	212	4	no	—
238	C₂×C₄	—	212	4	no	—
239	C₂×C₄	#13133179	1	1	yes	characteristic
240	C₂×C₄	#13133179	1	1	yes	characteristic
241	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
242	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
243	C₂×C₄	#13133240	1	1	yes	characteristic
244	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
245	C₈	—	4	4	no	—
246	C₈	—	53	2	no	—
247	C₈	—	53	2	no	—
248	C₈	#13133179	1	1	yes	characteristic
249	C₈	#13133179	1	1	yes	characteristic
250	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
251	C₈	C₂×#11719949	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	424	4	no	—
254	C₂²	—	212	4	no	—
255	C₂²	#16810705	1	1	yes	characteristic
256	C₂²	#16810705	1	1	yes	characteristic
257	C₂²	C₂×#13133240	1	1	yes	characteristic
258	C₄	—	106	4	no	—
259	C₄	—	212	4	no	—
260	C₄	#16810705	1	1	yes	characteristic
261	C₄	#16810705	1	1	yes	characteristic
262	C₄	C₂×#13133179	1	1	yes	characteristic
263	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	424	4	no	—
267	C₂	#20276595	1	2	yes	direct product factor
268	C₂	C₂×#16810705	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#368	C₂
socle	C₂²×C₅₃	C₄×D₈
prime radical	C₂×#16811055	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#368	D₅₃
hypocenter	C₅₃	C₂×#2178
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
2. C₂×#16810739	C₂	semidirect
3. #20276595	C₂	direct (N × H)
4. C₂×#16810429	C₂	semidirect
5. C₂×C₁₆×Dic₅₃	C₂	semidirect
6. C₂×C₅₃×#368	C₂	semidirect
9. #16811055	C₄	semidirect
11. C₂×Dic₄₂₄	C₄	semidirect
12. C₂×#13133162	C₄	semidirect
13. C₂×D₈₄₈	C₄	semidirect
16. #16810739	C₂²	semidirect
19. #16810429	C₂²	semidirect
20. C₁₆×Dic₅₃	C₂²	semidirect
23. C₅₃×#368	C₂²	semidirect
38. Dic₄₂₄	C₂×C₄	semidirect
39. #13133162	C₂×C₄	semidirect
40. D₈₄₈	C₂×C₄	semidirect
133. C₂×#368	D₅₃	semidirect
138. C₂×C₅₃	#2178	semidirect
155. #368	C₂×D₅₃	semidirect
158. C₅₃	C₂×#2178	semidirect
267. C₂	#20276595	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×#16811055	2
C₂×#16810739	2
#20276595	2
C₂×#16810429	2
C₂×C₁₆×Dic₅₃	2
C₂×C₅₃×#368	2
C₂×#2178	53

Subgroups — C2×#20276595 #27537491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276595 #27537491