Subgroups — C₂×#20276601 #27537498

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810740	C₂	1	2	yes	maximal, complemented
2	#20276601	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810453	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16810430	C₂	1	2	yes	maximal, complemented
5	C₂×#16810398	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×#508	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133224	—	2	2	no	—
8	C₂²×#11719908	C₂²	1	1	yes	characteristic
9	#16810740	—	2	4	no	—
10	#16810740	C₂²	1	8	yes	complemented
11	C₂×#13133153	C₂²	1	2	yes	—
12	C₂×#13133144	—	2	2	no	—
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	#16810453	C₂²	1	4	yes	complemented
15	#16810430	C₂²	1	8	yes	complemented
16	#16810398	C₂²	1	4	yes	complemented
17	C₂×#135×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#508	C₄	1	4	yes	complemented
19	C₅₃×#508	C₂²	1	4	yes	—
20	C₂×Q₃₂×C₅₃	C₄	1	1	yes	characteristic, complemented
21	C₂×SD₃₂×C₅₃	C₄	1	2	yes	complemented
22	C₂×D₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
23	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
24	#13133224	—	2	8	no	—
25	#13133224	—	4	4	no	—
26	C₂×#11719917	—	2	2	no	—
27	C₂×#11719917	—	4	2	no	—
28	C₂²×#88322	—	4	2	no	—
29	C₂²×#88322	D₄	1	1	yes	characteristic
30	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
31	C₂×#11719908	—	2	1	no	—
32	C₂×#11719908	—	2	1	no	—
33	C₂×#11719908	C₂³	1	2	yes	—
34	#13133153	—	2	2	no	—
35	#13133153	C₂³	1	4	yes	—
36	#13133144	—	2	8	no	—
37	C₈×Dic₅₃	D₄	1	2	yes	—
38	C₈×Dic₅₃	C₂³	1	2	yes	—
39	C₂×#41×C₅₃	—	2	2	no	—
40	#135×C₅₃	—	2	2	no	—
41	#135×C₅₃	—	2	2	no	—
42	#135×C₅₃	C₂×C₄	1	4	yes	—
43	#135×C₅₃	C₂³	1	4	yes	—
44	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×SD₁₆×C₅₃	—	2	2	no	—
46	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	Q₃₂×C₅₃	C₂×C₄	1	4	yes	complemented
49	SD₃₂×C₅₃	C₂×C₄	1	8	yes	complemented
50	D₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
51	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
54	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
55	#41×C₅₃	—	2	4	no	—
56	#41×C₅₃	—	2	4	no	—
57	#41×C₅₃	—	4	2	no	—
58	#41×C₅₃	—	4	2	no	—
59	C₂×Q₈×C₅₃	—	2	2	no	—
60	C₂×D₄×C₅₃	—	2	2	no	—
61	C₂×D₄×C₅₃	—	4	2	no	—
62	C₂²×C₄×C₅₃	—	4	2	no	—
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×Dic₅₃	—	2	1	no	—
65	Q₁₆×C₅₃	—	2	2	no	—
66	Q₁₆×C₅₃	C₂²×C₄	1	4	yes	—
67	SD₁₆×C₅₃	—	2	4	no	—
68	SD₁₆×C₅₃	—	2	4	no	—
69	D₈×C₅₃	—	2	2	no	—
70	D₈×C₅₃	C₂²×C₄	1	4	yes	—
71	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
75	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
77	C₄×Dic₅₃	—	2	1	no	—
78	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
79	#11719917	—	2	4	no	—
80	#11719917	—	4	2	no	—
81	#11719917	—	4	8	no	—
82	C₂×#88322	—	4	2	no	—
83	C₂×#88322	—	4	8	no	—
84	C₂×#88322	D₈	1	2	yes	—
85	C₂×#88322	D₈	1	2	yes	—
86	C₂×#88322	C₂×D₄	1	2	yes	—
87	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
88	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
89	#11719908	—	2	2	no	—
90	C₂³×C₅₃	—	4	2	no	—
91	C₂³×C₅₃	#135	1	1	yes	characteristic
92	Q₈×C₅₃	—	2	4	no	—
93	Q₈×C₅₃	—	4	2	no	—
94	D₄×C₅₃	—	2	4	no	—
95	D₄×C₅₃	—	4	4	no	—
96	D₄×C₅₃	—	4	4	no	—
97	D₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	—	4	4	no	—
100	C₂×C₄×C₅₃	—	4	4	no	—
101	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
107	C₂×Dic₅₃	—	2	2	no	—
108	C₂×Dic₅₃	—	4	2	no	—
109	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
110	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
111	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
112	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
113	#88322	—	4	4	no	—
114	#88322	C₂×D₈	1	4	yes	—
115	C₂×#2179	—	53	1	no	maximal, Sylow
116	C₂²×C₅₃	—	2	2	no	—
117	C₂²×C₅₃	—	4	4	no	—
118	C₂²×C₅₃	—	8	4	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	C₄×C₅₃	—	4	4	no	—
123	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
124	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
125	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
126	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
127	Dic₅₃	—	4	2	no	—
128	C₂×#508	D₅₃	1	1	yes	characteristic
129	C₂×#443	—	53	2	no	—
130	#2179	—	53	8	no	—
131	C₂×#368	—	53	1	no	—
132	C₂×#359	—	53	2	no	—
133	C₂×#346	—	53	1	no	—
134	C₂×C₅₃	—	2	1	no	—
135	C₂×C₅₃	—	2	1	no	—
136	C₂×C₅₃	—	8	4	no	—
137	C₂×C₅₃	#2179	1	2	yes	complemented
138	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
139	C₂×#135	C₂×D₅₃	1	2	yes	—
140	C₂×#131	—	106	2	no	—
141	#508	Dic₅₃	1	4	yes	complemented
142	#508	C₂×D₅₃	1	4	yes	—
143	C₂×Q₃₂	Dic₅₃	1	1	yes	characteristic, complemented
144	C₂×SD₃₂	Dic₅₃	1	2	yes	complemented
145	C₂×D₁₆	Dic₅₃	1	1	yes	characteristic, complemented
146	C₂²×C₁₆	—	53	1	no	—
147	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
148	#443	—	53	8	no	—
149	#443	—	106	4	no	—
150	C₂×#108	—	53	2	no	—
151	C₂×#104	—	106	2	no	—
152	C₂×C₄×C₈	—	53	1	no	—
153	#368	—	53	4	no	—
154	#359	—	53	8	no	—
155	#346	—	53	4	no	—
156	C₅₃	C₂×#2179	1	1	yes	characteristic, complemented, Sylow
157	C₂×#41	—	2	2	no	—
158	#135	—	2	2	no	—
159	#135	—	2	2	no	—
160	#135	C₂×Dic₅₃	1	4	yes	—
161	#135	C₂²×D₅₃	1	4	yes	—
162	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
163	C₂×SD₁₆	—	2	2	no	—
164	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
165	#131	—	106	8	no	—
166	#131	—	212	4	no	—
167	C₂×M₁₆	—	106	2	no	—
168	C₂×M₁₆	—	212	2	no	—
169	C₂²×C₈	—	53	1	no	—
170	C₂²×C₈	—	212	2	no	—
171	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
172	C₂×C₄²	—	53	1	no	—
173	Q₃₂	C₂×Dic₅₃	1	4	yes	complemented
174	SD₃₂	C₂×Dic₅₃	1	8	yes	complemented
175	D₁₆	C₂×Dic₅₃	1	4	yes	complemented
176	C₂×C₁₆	—	53	2	no	—
177	C₂×C₁₆	—	106	1	no	—
178	C₂×C₁₆	—	106	1	no	—
179	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
180	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
181	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
182	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
183	#108	—	53	4	no	—
184	#108	—	106	2	no	—
185	#104	—	106	8	no	—
186	C₄×C₈	—	53	2	no	—
187	C₄×C₈	—	53	2	no	—
188	#41	—	2	4	no	—
189	#41	—	2	4	no	—
190	#41	—	4	2	no	—
191	#41	—	4	2	no	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	106	1	no	—
197	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
198	Q₁₆	—	2	2	no	—
199	Q₁₆	C₂²×Dic₅₃	1	4	yes	—
200	SD₁₆	—	2	4	no	—
201	SD₁₆	—	2	4	no	—
202	D₈	—	2	2	no	—
203	D₈	C₂²×Dic₅₃	1	4	yes	—
204	M₁₆	—	106	4	no	—
205	M₁₆	—	212	8	no	—
206	M₁₆	—	212	2	no	—
207	C₂×C₈	—	53	2	no	—
208	C₂×C₈	—	53	2	no	—
209	C₂×C₈	—	53	2	no	—
210	C₂×C₈	—	212	8	no	—
211	C₂×C₈	—	212	2	no	—
212	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
213	C₂×C₈	#11719947	1	1	yes	characteristic
214	C₂×C₈	#11719947	1	1	yes	characteristic
215	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
216	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
217	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
218	C₄²	—	53	2	no	—
219	C₄²	—	106	1	no	—
220	C₁₆	—	106	2	no	—
221	C₁₆	C₂²×Dic₅₃	1	2	yes	—
222	C₁₆	C₂²×Dic₅₃	1	2	yes	—
223	C₂³	—	4	2	no	—
224	C₂³	#13133236	1	1	yes	characteristic
225	Q₈	—	2	4	no	—
226	Q₈	—	4	2	no	—
227	D₄	—	2	4	no	—
228	D₄	—	4	4	no	—
229	D₄	—	4	4	no	—
230	D₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	106	2	no	—
235	C₂×C₄	—	212	2	no	—
236	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
237	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
238	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
239	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
240	C₂×C₄	#13133236	1	1	yes	characteristic
241	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
242	C₈	—	53	4	no	—
243	C₈	—	212	4	no	—
244	C₈	D₄×Dic₅₃	1	1	yes	characteristic
245	C₈	D₄×Dic₅₃	1	1	yes	characteristic
246	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
247	C₈	C₂×#11719947	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	4	4	no	—
250	C₂²	—	8	4	no	—
251	C₂²	D₈×Dic₅₃	1	1	yes	characteristic
252	C₂²	D₈×Dic₅₃	1	1	yes	characteristic
253	C₂²	C₂×#13133236	1	1	yes	characteristic
254	C₄	—	4	4	no	—
255	C₄	—	212	2	no	—
256	C₄	D₈×Dic₅₃	1	1	yes	characteristic
257	C₄	D₈×Dic₅₃	1	1	yes	characteristic
258	C₄	C₂×D₄×Dic₅₃	1	1	yes	characteristic
259	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	8	4	no	—
263	C₂	#20276601	1	2	yes	direct product factor
264	C₂	C₂×D₈×Dic₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting, prime radical	C₂×C₅₃×#508	C₂
socle	C₂²×C₅₃	C₄×D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#508	D₅₃
hypocenter	C₅₃	C₂×#2179
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂×#16810740	C₂	semidirect
2. #20276601	C₂	direct (N × H)
3. C₂×#16810453	C₂	semidirect
4. C₂×#16810430	C₂	semidirect
5. C₂×#16810398	C₂	semidirect
10. #16810740	C₂²	semidirect
14. #16810453	C₂²	semidirect
15. #16810430	C₂²	semidirect
16. #16810398	C₂²	semidirect
18. C₅₃×#508	C₄	semidirect
20. C₂×Q₃₂×C₅₃	C₄	semidirect
21. C₂×SD₃₂×C₅₃	C₄	semidirect
22. C₂×D₁₆×C₅₃	C₄	semidirect
48. Q₃₂×C₅₃	C₂×C₄	semidirect
49. SD₃₂×C₅₃	C₂×C₄	semidirect
50. D₁₆×C₅₃	C₂×C₄	semidirect
137. C₂×C₅₃	#2179	semidirect
141. #508	Dic₅₃	semidirect
143. C₂×Q₃₂	Dic₅₃	semidirect
144. C₂×SD₃₂	Dic₅₃	semidirect
145. C₂×D₁₆	Dic₅₃	semidirect
156. C₅₃	C₂×#2179	semidirect
173. Q₃₂	C₂×Dic₅₃	semidirect
174. SD₃₂	C₂×Dic₅₃	semidirect
175. D₁₆	C₂×Dic₅₃	semidirect
263. C₂	#20276601	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×#16810740	2
#20276601	2
C₂×#16810453	2
C₂×#16810430	2
C₂×#16810398	2
C₂×C₅₃×#508	2
C₂×#2179	53

Subgroups — C2×#20276601 #27537498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276601 #27537498