Subgroups — C₂×2₋¹⁺⁴×#261 #16809636

Order 3328.

11210 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	2₋¹⁺⁴×#261	C₂	1	32	yes	maximal, direct product factor
2	C₂×#11719631	C₂	1	10	yes	maximal, complemented
3	C₂×#11719627	C₂	1	5	yes	maximal, complemented
4	C₂×#41×#261	C₂	1	10	yes	maximal, complemented
5	C₂²×Q₈×#261	C₂	1	5	yes	maximal, complemented
6	C₂×D₁₃×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×2₋¹⁺⁴	C₄	1	1	yes	characteristic, complemented
8	D₁₃×2₋¹⁺⁴	C₄	1	16	yes	complemented
9	D₁₃×2₋¹⁺⁴	C₂²	1	16	yes	—
10	C₂×#88257	C₄	1	10	yes	complemented
11	C₂×#41×D₁₃	C₂²	1	10	yes	—
12	C₂×#88254	C₄	1	5	yes	complemented
13	C₂²×Q₈×D₁₃	C₂²	1	5	yes	—
14	#11719631	C₂²	1	160	yes	complemented
15	#41×#261	C₂²	1	160	yes	complemented
16	#11719627	C₂²	1	80	yes	complemented
17	C₂×Q₈×#261	C₂²	1	10	yes	—
18	C₂×Q₈×#261	C₂²	1	20	yes	—
19	C₂×Q₈×#261	C₂²	1	10	yes	—
20	C₂×Q₈×#261	C₂²	1	80	yes	complemented
21	C₂×D₄×#261	C₂²	1	10	yes	—
22	C₂×D₄×#261	C₂²	1	30	yes	—
23	C₂×#88237	C₂²	1	30	yes	—
24	C₂²×C₄⋊#261	C₂²	1	15	yes	—
25	C₂²×C₄×#261	C₂²	1	15	yes	—
26	C₂³×#261	C₂³	1	5	yes	—
27	C₂³×#261	C₂³	1	15	yes	—
28	C₁₃×2₋¹⁺⁴	C₂×C₄	1	16	yes	complemented
29	C₂×C₁₃×#41	C₂×C₄	1	10	yes	complemented
30	C₂²×Q₈×C₁₃	C₂×C₄	1	5	yes	complemented
31	#88257	C₂×C₄	1	160	yes	complemented
32	#41×D₁₃	C₂×C₄	1	80	yes	complemented
33	#41×D₁₃	C₂³	1	80	yes	—
34	#88254	C₂×C₄	1	80	yes	complemented
35	C₂×#6131	C₂×C₄	1	10	yes	complemented
36	C₂×Q₈×D₁₃	C₂×C₄	1	20	yes	complemented
37	C₂×Q₈×D₁₃	C₂×C₄	1	40	yes	complemented
38	C₂×Q₈×D₁₃	C₂³	1	10	yes	—
39	C₂×Q₈×D₁₃	C₂³	1	40	yes	—
40	C₂×#6129	C₂×C₄	1	30	yes	complemented
41	C₂×D₄×D₁₃	C₂³	1	10	yes	—
42	C₂×#6127	C₂×C₄	1	30	yes	complemented
43	C₂²×C₄×D₁₃	C₂³	1	15	yes	—
44	C₂²×Dic₂₆	C₂×C₄	1	15	yes	complemented
45	C₂×C₂²⋊#261	C₂³	1	20	yes	—
46	Q₈×#261	C₂³	1	80	yes	—
47	Q₈×#261	C₂³	1	160	yes	—
48	Q₈×#261	C₂³	1	80	yes	—
49	D₄×#261	C₂³	1	80	yes	—
50	D₄×#261	C₂³	1	240	yes	—
51	#88237	C₂³	1	240	yes	—
52	C₂×C₄⋊#261	C₂³	1	30	yes	—
53	C₂×C₄⋊#261	C₂³	1	120	yes	—
54	C₂×C₄⋊#261	C₂³	1	10	yes	—
55	C₂×C₄×#261	C₂³	1	10	yes	—
56	C₂×C₄×#261	C₂³	1	30	yes	—
57	C₂×C₄×#261	C₂³	1	120	yes	—
58	C₂×C₄×2₋¹⁺⁴	—	13	1	no	maximal, Sylow
59	C₂³×D₁₃	C₂⁴	1	5	yes	—
60	C₂²×#261	—	2	20	no	—
61	C₂²×#261	—	4	30	no	—
62	C₂²×#261	C₂⁴	1	5	yes	—
63	C₂²×#261	C₂⁴	1	20	yes	—
64	C₂²×#261	C₂⁴	1	10	yes	—
65	C₂²×#261	C₂⁴	1	60	yes	—
66	C₂²×#261	C₂⁴	1	1	yes	characteristic
67	C₁₃×#41	C₂²×C₄	1	80	yes	complemented
68	C₂×Q₈×C₁₃	C₂²×C₄	1	10	yes	—
69	C₂×Q₈×C₁₃	C₂²×C₄	1	40	yes	complemented
70	C₂×D₄×C₁₃	C₂²×C₄	1	10	yes	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	15	yes	—
72	C₂×C₂²⋊D₁₃	C₂²×C₄	1	20	yes	—
73	C₂²×Dic₁₃	C₂²×C₄	1	15	yes	—
74	#6131	C₂²×C₄	1	80	yes	complemented
75	Q₈×D₁₃	C₂²×C₄	1	40	yes	—
76	Q₈×D₁₃	C₂²×C₄	1	160	yes	complemented
77	Q₈×D₁₃	C₂⁴	1	40	yes	—
78	#6129	C₂²×C₄	1	240	yes	complemented
79	D₄×D₁₃	C₂²×C₄	1	40	yes	—
80	D₄×D₁₃	C₂⁴	1	40	yes	—
81	#6127	C₂²×C₄	1	240	yes	complemented
82	C₂×D₅₂	C₂²×C₄	1	10	yes	—
83	C₂×C₄×D₁₃	C₂²×C₄	1	30	yes	—
84	C₂×C₄×D₁₃	C₂²×C₄	1	60	yes	—
85	C₂×C₄×D₁₃	C₂⁴	1	10	yes	—
86	C₂×C₄×D₁₃	C₂⁴	1	60	yes	—
87	C₂×Dic₂₆	C₂²×C₄	1	30	yes	—
88	C₂×Dic₂₆	C₂²×C₄	1	120	yes	complemented
89	C₂²⋊#261	C₂⁴	1	80	yes	—
90	C₄⋊#261	C₂⁴	1	120	yes	—
91	C₄⋊#261	C₂⁴	1	40	yes	—
92	C₄×#261	C₂⁴	1	40	yes	—
93	C₄×#261	C₂⁴	1	120	yes	—
94	C₂²×2₋¹⁺⁴	—	13	1	no	—
95	C₄×2₋¹⁺⁴	—	13	32	no	—
96	C₂×#520	—	13	10	no	—
97	C₂×#519	—	13	5	no	—
98	C₂×C₄×#41	—	13	10	no	—
99	C₂²×C₄×Q₈	—	13	5	no	—
100	C₂³×C₁₃	C₂³×C₄	1	5	yes	—
101	C₂²×D₁₃	—	2	10	no	—
102	C₂²×D₁₃	—	2	10	no	—
103	C₂²×D₁₃	C₂³×C₄	1	5	yes	—
104	C₂²×D₁₃	C₂³×C₄	1	10	yes	—
105	C₂²×D₁₃	C₂⁵	1	10	yes	—
106	C₂²×D₁₃	C₂⁵	1	1	yes	characteristic
107	C₂×#261	—	2	10	no	—
108	C₂×#261	—	2	20	no	—
109	C₂×#261	—	2	40	no	—
110	C₂×#261	—	4	60	no	—
111	C₂×#261	2₋¹⁺⁴	1	4	yes	direct product factor
112	C₂×#261	C₂⁵	1	10	yes	—
113	C₂×#261	C₂⁵	1	20	yes	—
114	C₂×#261	C₂⁵	1	2	yes	—
115	Q₈×C₁₃	C₂³×C₄	1	40	yes	—
116	D₄×C₁₃	C₂³×C₄	1	40	yes	—
117	C₂×C₄×C₁₃	C₂³×C₄	1	10	yes	—
118	C₂×C₄×C₁₃	C₂³×C₄	1	60	yes	—
119	C₂²⋊D₁₃	C₂³×C₄	1	80	yes	—
120	C₂×Dic₁₃	C₂³×C₄	1	10	yes	—
121	C₂×Dic₁₃	C₂³×C₄	1	60	yes	—
122	D₅₂	C₂³×C₄	1	40	yes	—
123	C₄×D₁₃	C₂³×C₄	1	20	yes	—
124	C₄×D₁₃	C₂³×C₄	1	120	yes	—
125	C₄×D₁₃	C₂⁵	1	20	yes	—
126	Dic₂₆	C₂³×C₄	1	120	yes	—
127	C₂×2₋¹⁺⁴	—	13	16	no	—
128	C₂×2₋¹⁺⁴	—	13	16	no	—
129	C₂×2₋¹⁺⁴	—	13	10	no	—
130	C₂×2₋¹⁺⁴	—	13	5	no	—
131	C₂×2₋¹⁺⁴	#261	1	1	yes	characteristic, direct product factor
132	C₂²×#41	—	13	10	no	—
133	C₂³×Q₈	—	13	5	no	—
134	#520	—	13	160	no	—
135	#519	—	13	80	no	—
136	C₄×#41	—	13	160	no	—
137	C₂×C₄×Q₈	—	13	10	no	—
138	C₂×C₄×Q₈	—	13	20	no	—
139	C₂×C₄×Q₈	—	13	80	no	—
140	C₂×C₄×Q₈	—	13	10	no	—
141	C₂×C₄×D₄	—	13	10	no	—
142	C₂×C₄×D₄	—	13	30	no	—
143	C₂×#117	—	13	30	no	—
144	C₂²×C₄⋊C₄	—	13	15	no	—
145	C₂²×C₄²	—	13	15	no	—
146	C₂²×C₁₃	—	2	10	no	—
147	C₂²×C₁₃	C₂⁴×C₄	1	10	yes	—
148	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
149	C₂×D₁₃	—	2	10	no	—
150	C₂×D₁₃	—	2	10	no	—
151	C₂×D₁₃	—	2	10	no	—
152	C₂×D₁₃	—	2	20	no	—
153	C₂×D₁₃	C₂⁴×C₄	1	10	yes	—
154	C₂×D₁₃	C₂⁴×C₄	1	1	yes	characteristic
155	C₂×D₁₃	C₂×2₋¹⁺⁴	1	2	yes	—
156	C₂×D₁₃	#585	1	2	yes	—
157	C₂×D₁₃	C₂⁶	1	1	yes	characteristic
158	#261	—	2	10	no	—
159	#261	—	2	20	no	—
160	#261	C₂×2₋¹⁺⁴	1	4	yes	direct product factor
161	C₄×C₁₃	C₂⁴×C₄	1	20	yes	—
162	Dic₁₃	C₂⁴×C₄	1	20	yes	—
163	2₋¹⁺⁴	—	13	160	no	—
164	2₋¹⁺⁴	—	13	80	no	—
165	2₋¹⁺⁴	C₂×#261	1	16	yes	direct product factor
166	C₂×#41	—	13	80	no	—
167	C₂×#41	—	13	80	no	—
168	C₂×#41	—	13	10	no	—
169	C₂×#41	—	13	30	no	—
170	C₂×#41	—	13	30	no	—
171	C₂×#41	C₂×#261	1	10	yes	complemented
172	C₂²×Q₈	—	13	10	no	—
173	C₂²×Q₈	—	13	20	no	—
174	C₂²×Q₈	—	13	40	no	—
175	C₂²×Q₈	—	13	40	no	—
176	C₂²×Q₈	—	13	15	no	—
177	C₂²×Q₈	C₂×#261	1	5	yes	complemented
178	C₂²×D₄	—	13	10	no	—
179	C₂³×C₄	—	13	15	no	—
180	C₂³×C₄	—	13	5	no	—
181	C₂³×C₄	—	13	15	no	—
182	C₄×Q₈	—	13	80	no	—
183	C₄×Q₈	—	13	160	no	—
184	C₄×Q₈	—	13	80	no	—
185	C₄×D₄	—	13	80	no	—
186	C₄×D₄	—	13	240	no	—
187	#117	—	13	240	no	—
188	C₂×C₄⋊C₄	—	13	30	no	—
189	C₂×C₄⋊C₄	—	13	120	no	—
190	C₂×C₄⋊C₄	—	13	10	no	—
191	C₂×C₂²⋊C₄	—	13	20	no	—
192	C₂×C₄²	—	13	30	no	—
193	C₂×C₄²	—	13	10	no	—
194	C₂×C₄²	—	13	120	no	—
195	C₂×C₁₃	—	2	10	no	—
196	C₂×C₁₃	C₄×2₋¹⁺⁴	1	2	yes	complemented
197	C₂×C₁₃	C₂⁵×C₄	1	1	yes	characteristic
198	D₁₃	—	2	10	no	—
199	D₁₃	C₄×2₋¹⁺⁴	1	2	yes	complemented
200	D₁₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
201	D₁₃	C₂×#585	1	1	yes	characteristic
202	C₂⁴	—	13	5	no	—
203	#41	—	13	80	no	—
204	#41	—	13	240	no	—
205	#41	—	13	240	no	—
206	#41	C₂²×#261	1	80	yes	complemented
207	C₂×Q₈	—	13	40	no	—
208	C₂×Q₈	—	13	40	no	—
209	C₂×Q₈	—	13	160	no	—
210	C₂×Q₈	—	13	30	no	—
211	C₂×Q₈	—	13	120	no	—
212	C₂×Q₈	C₂²×#261	1	10	yes	—
213	C₂×Q₈	C₂²×#261	1	40	yes	complemented
214	C₂×D₄	—	13	20	no	—
215	C₂×D₄	—	13	40	no	—
216	C₂×D₄	—	13	40	no	—
217	C₂×D₄	—	13	10	no	—
218	C₂×D₄	C₂²×#261	1	10	yes	—
219	C₂²×C₄	—	13	1	no	—
220	C₂²×C₄	—	13	15	no	—
221	C₂²×C₄	—	13	10	no	—
222	C₂²×C₄	—	13	60	no	—
223	C₂²×C₄	—	13	60	no	—
224	C₂²×C₄	—	13	30	no	—
225	C₂²×C₄	—	13	5	no	—
226	C₂²×C₄	—	13	20	no	—
227	C₂²×C₄	—	13	10	no	—
228	C₂²×C₄	—	13	60	no	—
229	C₂²×C₄	—	26	20	no	—
230	C₂²×C₄	—	52	30	no	—
231	C₂²×C₄	C₂²×#261	1	15	yes	—
232	C₄⋊C₄	—	13	120	no	—
233	C₄⋊C₄	—	13	40	no	—
234	C₂²⋊C₄	—	13	80	no	—
235	C₄²	—	13	40	no	—
236	C₄²	—	13	120	no	—
237	C₁₃	C₂×C₄×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
238	C₂³	—	13	1	no	—
239	C₂³	—	13	10	no	—
240	C₂³	—	13	10	no	—
241	C₂³	—	13	5	no	—
242	C₂³	—	26	10	no	—
243	C₂³	—	26	10	no	—
244	C₂³	C₂³×#261	1	5	yes	—
245	Q₈	—	13	120	no	—
246	Q₈	C₂³×#261	1	40	yes	—
247	D₄	—	13	80	no	—
248	D₄	—	13	40	no	—
249	D₄	C₂³×#261	1	40	yes	—
250	C₂×C₄	—	13	2	no	—
251	C₂×C₄	—	13	4	no	—
252	C₂×C₄	—	13	60	no	—
253	C₂×C₄	—	13	10	no	—
254	C₂×C₄	—	13	20	no	—
255	C₂×C₄	—	13	20	no	—
256	C₂×C₄	—	13	120	no	—
257	C₂×C₄	—	13	10	no	—
258	C₂×C₄	—	13	20	no	—
259	C₂×C₄	—	26	40	no	—
260	C₂×C₄	—	26	10	no	—
261	C₂×C₄	—	26	20	no	—
262	C₂×C₄	—	52	60	no	—
263	C₂×C₄	C₂³×#261	1	60	yes	—
264	C₂×C₄	C₂³×#261	1	10	yes	—
265	C₂²	—	2	10	no	—
266	C₂²	—	13	2	no	—
267	C₂²	—	13	2	no	—
268	C₂²	—	13	1	no	—
269	C₂²	—	13	1	no	—
270	C₂²	—	13	10	no	—
271	C₂²	—	26	10	no	—
272	C₂²	—	26	10	no	—
273	C₂²	—	26	20	no	—
274	C₂²	—	26	10	no	—
275	C₂²	C₂⁴×#261	1	10	yes	—
276	C₂²	C₂⁴×#261	1	1	yes	characteristic
277	C₄	—	13	4	no	—
278	C₄	—	13	20	no	—
279	C₄	—	26	10	no	—
280	C₄	—	26	20	no	—
281	C₄	C₂⁴×#261	1	20	yes	—
282	C₂	—	2	10	no	—
283	C₂	—	13	1	no	—
284	C₂	—	13	1	no	—
285	C₂	—	13	2	no	—
286	C₂	—	26	10	no	—
287	C₂	2₋¹⁺⁴×#261	1	2	yes	direct product factor
288	C₂	C₂⁵×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂⁴×#261
commutator	C₂×C₁₃	C₂⁵×C₄
Frattini	C₂	C₂⁵×#261
Fitting	C₂×C₁₃×2₋¹⁺⁴	C₄
socle	C₂²×C₁₃	C₂⁴×C₄
prime radical	C₂×D₁₃×2₋¹⁺⁴	C₂
cosocle	C₂×D₁₃	C₂⁶
hypercenter	C₂×2₋¹⁺⁴	#261
hypocenter	C₁₃	C₂×C₄×2₋¹⁺⁴
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴×#261	C₂	direct (N × H)
2. C₂×#11719631	C₂	semidirect
3. C₂×#11719627	C₂	semidirect
4. C₂×#41×#261	C₂	semidirect
5. C₂²×Q₈×#261	C₂	semidirect
7. C₂×C₁₃×2₋¹⁺⁴	C₄	semidirect
8. D₁₃×2₋¹⁺⁴	C₄	semidirect
10. C₂×#88257	C₄	semidirect
12. C₂×#88254	C₄	semidirect
14. #11719631	C₂²	semidirect
15. #41×#261	C₂²	semidirect
16. #11719627	C₂²	semidirect
20. C₂×Q₈×#261	C₂²	semidirect
28. C₁₃×2₋¹⁺⁴	C₂×C₄	semidirect
29. C₂×C₁₃×#41	C₂×C₄	semidirect
30. C₂²×Q₈×C₁₃	C₂×C₄	semidirect
31. #88257	C₂×C₄	semidirect
32. #41×D₁₃	C₂×C₄	semidirect
34. #88254	C₂×C₄	semidirect
35. C₂×#6131	C₂×C₄	semidirect
36. C₂×Q₈×D₁₃	C₂×C₄	semidirect
37. C₂×Q₈×D₁₃	C₂×C₄	semidirect
40. C₂×#6129	C₂×C₄	semidirect
42. C₂×#6127	C₂×C₄	semidirect
44. C₂²×Dic₂₆	C₂×C₄	semidirect
67. C₁₃×#41	C₂²×C₄	semidirect
69. C₂×Q₈×C₁₃	C₂²×C₄	semidirect
74. #6131	C₂²×C₄	semidirect
76. Q₈×D₁₃	C₂²×C₄	semidirect
78. #6129	C₂²×C₄	semidirect
81. #6127	C₂²×C₄	semidirect
88. C₂×Dic₂₆	C₂²×C₄	semidirect
111. C₂×#261	2₋¹⁺⁴	direct (N × H)
131. C₂×2₋¹⁺⁴	#261	direct (N × H)
160. #261	C₂×2₋¹⁺⁴	direct (N × H)
165. 2₋¹⁺⁴	C₂×#261	direct (N × H)
171. C₂×#41	C₂×#261	semidirect
177. C₂²×Q₈	C₂×#261	semidirect
196. C₂×C₁₃	C₄×2₋¹⁺⁴	semidirect
199. D₁₃	C₄×2₋¹⁺⁴	semidirect
206. #41	C₂²×#261	semidirect
213. C₂×Q₈	C₂²×#261	semidirect
237. C₁₃	C₂×C₄×2₋¹⁺⁴	semidirect
287. C₂	2₋¹⁺⁴×#261	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
2₋¹⁺⁴×#261	2
C₂×#11719631	2
C₂×#11719627	2
C₂×#41×#261	2
C₂²×Q₈×#261	2
C₂×D₁₃×2₋¹⁺⁴	2
C₂×C₄×2₋¹⁺⁴	13

Subgroups — C2×2−1+4×#261 #16809636

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×2₋¹⁺⁴×#261 #16809636