Subgroups — #261×#585 #16809637

Order 3328.

11346 conjugacy classes of subgroups.

The subgroup lattice has 248 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	12	yes	maximal, complemented
2	H(5,𝔽₂)×#261	C₂	1	20	yes	maximal, complemented
3	#13131816	C₂	1	15	yes	maximal, complemented
4	C₂×#41×#261	C₂	1	15	yes	maximal, complemented
5	D₁₃×#585	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#585	C₄	1	1	yes	characteristic, complemented
7	#11719652	C₄	1	6	yes	complemented
8	D₁₃×2₋¹⁺⁴	C₂²	1	6	yes	—
9	#11719650	C₄	1	10	yes	complemented
10	D₁₃×H(5,𝔽₂)	C₂²	1	10	yes	—
11	#11719648	C₄	1	15	yes	complemented
12	C₂×#41×D₁₃	C₂²	1	15	yes	—
13	#11719631	C₂²	1	120	yes	complemented
14	#41×#261	C₂²	1	20	yes	complemented
15	#41×#261	C₂²	1	60	yes	complemented
16	#41×#261	C₂²	1	120	yes	complemented
17	#11719627	C₂²	1	30	yes	complemented
18	C₂×Q₈×#261	C₂²	1	30	yes	complemented
19	#11719623	C₂²	1	90	yes	complemented
20	C₂×D₄×#261	C₂²	1	90	yes	complemented
21	C₂×#88237	C₂²	1	45	yes	complemented
22	C₂²×C₄×#261	C₂²	1	15	yes	complemented
23	C₂³×#261	C₂³	1	30	yes	—
24	C₁₃×2₋¹⁺⁴	C₂×C₄	1	6	yes	complemented
25	C₁₃×H(5,𝔽₂)	C₂×C₄	1	10	yes	complemented
26	C₂×C₁₃×#41	C₂×C₄	1	15	yes	complemented
27	#88257	C₂×C₄	1	60	yes	complemented
28	#88256	C₂×C₄	1	60	yes	complemented
29	#41×D₁₃	C₂×C₄	1	60	yes	complemented
30	#41×D₁₃	C₂×C₄	1	60	yes	complemented
31	#41×D₁₃	C₂³	1	20	yes	—
32	#41×D₁₃	C₂³	1	60	yes	—
33	#88254	C₂×C₄	1	30	yes	complemented
34	C₂×#6131	C₂×C₄	1	15	yes	complemented
35	C₂×Q₈×D₁₃	C₂³	1	15	yes	—
36	#88251	C₂×C₄	1	90	yes	complemented
37	C₂×#6129	C₂×C₄	1	45	yes	complemented
38	C₂×D₄×D₁₃	C₂³	1	45	yes	—
39	C₂×#6127	C₂×C₄	1	45	yes	complemented
40	C₂²×C₄×D₁₃	C₂³	1	15	yes	—
41	C₂×C₂²⋊#261	C₂³	1	90	yes	—
42	Q₈×#261	C₂³	1	40	yes	—
43	Q₈×#261	C₂³	1	120	yes	—
44	D₄×#261	C₂³	1	120	yes	—
45	D₄×#261	C₂³	1	360	yes	—
46	#88237	C₂³	1	60	yes	—
47	#88237	C₂³	1	180	yes	—
48	C₂×C₄⋊#261	C₂³	1	90	yes	—
49	C₂×C₄×#261	C₂³	1	15	yes	—
50	C₂×C₄×#261	C₂³	1	45	yes	—
51	C₂×C₄×#261	C₂³	1	90	yes	—
52	C₄×#585	—	13	1	no	maximal, Sylow
53	C₂³×D₁₃	C₂⁴	1	15	yes	—
54	C₂²×#261	—	4	60	no	—
55	C₂²×#261	C₂⁴	1	30	yes	—
56	C₂²×#261	C₂⁴	1	90	yes	—
57	C₁₃×#41	C₂²×C₄	1	20	yes	complemented
58	C₁₃×#41	C₂²×C₄	1	60	yes	complemented
59	C₂×Q₈×C₁₃	C₂²×C₄	1	15	yes	complemented
60	C₂×D₄×C₁₃	C₂²×C₄	1	45	yes	complemented
61	C₂²×C₄×C₁₃	C₂²×C₄	1	15	yes	complemented
62	C₂×C₂²⋊D₁₃	C₂²×C₄	1	90	yes	complemented
63	C₂²×Dic₁₃	C₂²×C₄	1	15	yes	complemented
64	#6131	C₂²×C₄	1	20	yes	complemented
65	#6131	C₂²×C₄	1	60	yes	complemented
66	Q₈×D₁₃	C₂²×C₄	1	60	yes	complemented
67	Q₈×D₁₃	C₂⁴	1	20	yes	—
68	#6129	C₂²×C₄	1	60	yes	complemented
69	#6129	C₂²×C₄	1	180	yes	complemented
70	D₄×D₁₃	C₂²×C₄	1	180	yes	complemented
71	D₄×D₁₃	C₂⁴	1	60	yes	—
72	#6127	C₂²×C₄	1	60	yes	complemented
73	#6127	C₂²×C₄	1	180	yes	complemented
74	C₂×D₅₂	C₂²×C₄	1	45	yes	complemented
75	C₂×C₄×D₁₃	C₂²×C₄	1	45	yes	complemented
76	C₂×C₄×D₁₃	C₂²×C₄	1	45	yes	complemented
77	C₂×C₄×D₁₃	C₂⁴	1	15	yes	—
78	C₂×C₄×D₁₃	C₂⁴	1	45	yes	—
79	C₂×Dic₂₆	C₂²×C₄	1	45	yes	complemented
80	C₂²⋊#261	C₂⁴	1	120	yes	—
81	C₄⋊#261	C₂⁴	1	120	yes	—
82	C₄×#261	C₂⁴	1	1	yes	characteristic
83	C₄×#261	C₂⁴	1	15	yes	—
84	C₄×#261	C₂⁴	1	30	yes	—
85	C₄×#261	C₂⁴	1	90	yes	—
86	C₂×#585	—	13	1	no	—
87	C₄×2₋¹⁺⁴	—	13	12	no	—
88	C₄×H(5,𝔽₂)	—	13	20	no	—
89	#3428	—	13	15	no	—
90	C₂×C₄×#41	—	13	15	no	—
91	C₂³×C₁₃	C₂³×C₄	1	15	yes	—
92	C₂²×D₁₃	—	4	30	no	—
93	C₂²×D₁₃	C₂³×C₄	1	45	yes	—
94	C₂²×D₁₃	C₂⁵	1	15	yes	—
95	C₂×#261	—	2	60	no	—
96	C₂×#261	—	4	90	no	—
97	C₂×#261	C₂⁵	1	30	yes	—
98	C₂×#261	C₂⁵	1	2	yes	—
99	Q₈×C₁₃	C₂³×C₄	1	20	yes	—
100	D₄×C₁₃	C₂³×C₄	1	60	yes	—
101	C₂×C₄×C₁₃	C₂³×C₄	1	45	yes	—
102	C₂×C₄×C₁₃	C₂³×C₄	1	15	yes	—
103	C₂²⋊D₁₃	C₂³×C₄	1	120	yes	—
104	C₂×Dic₁₃	C₂³×C₄	1	45	yes	—
105	C₂×Dic₁₃	C₂³×C₄	1	15	yes	—
106	D₅₂	C₂³×C₄	1	60	yes	—
107	C₄×D₁₃	C₂³×C₄	1	15	yes	—
108	C₄×D₁₃	C₂³×C₄	1	15	yes	—
109	C₄×D₁₃	C₂³×C₄	1	90	yes	—
110	C₄×D₁₃	C₂⁵	1	1	yes	characteristic
111	C₄×D₁₃	C₂⁵	1	15	yes	—
112	Dic₂₆	C₂³×C₄	1	60	yes	—
113	#585	—	13	6	no	—
114	#585	—	13	10	no	—
115	#585	—	13	15	no	—
116	#585	#261	1	1	yes	characteristic, direct product factor
117	C₂×2₋¹⁺⁴	—	13	6	no	—
118	C₂×H(5,𝔽₂)	—	13	10	no	—
119	C₂²×#41	—	13	15	no	—
120	#520	—	13	120	no	—
121	#519	—	13	30	no	—
122	#518	—	13	90	no	—
123	C₄×#41	—	13	20	no	—
124	C₄×#41	—	13	60	no	—
125	C₄×#41	—	13	120	no	—
126	C₂×C₄×Q₈	—	13	30	no	—
127	C₂×C₄×D₄	—	13	90	no	—
128	C₂×#117	—	13	45	no	—
129	C₂²×C₄²	—	13	15	no	—
130	C₂²×C₁₃	—	4	15	no	—
131	C₂²×C₁₃	C₂⁴×C₄	1	15	yes	—
132	C₂×D₁₃	—	2	30	no	—
133	C₂×D₁₃	—	4	45	no	—
134	C₂×D₁₃	C₂⁴×C₄	1	15	yes	—
135	C₂×D₁₃	C₂⁶	1	1	yes	characteristic
136	#261	—	2	30	no	—
137	#261	#585	1	4	yes	direct product factor
138	C₄×C₁₃	C₂⁴×C₄	1	15	yes	—
139	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
140	Dic₁₃	C₂⁴×C₄	1	15	yes	—
141	Dic₁₃	C₂⁴×C₄	1	1	yes	characteristic
142	2₋¹⁺⁴	—	13	60	no	—
143	2₋¹⁺⁴	—	13	30	no	—
144	2₋¹⁺⁴	C₂×#261	1	6	yes	complemented
145	H(5,𝔽₂)	—	13	60	no	—
146	H(5,𝔽₂)	—	13	90	no	—
147	H(5,𝔽₂)	C₂×#261	1	10	yes	complemented
148	C₂×#41	—	13	20	no	—
149	C₂×#41	—	13	60	no	—
150	C₂×#41	—	13	60	no	—
151	C₂×#41	—	13	60	no	—
152	C₂×#41	—	13	15	no	—
153	C₂×#41	—	13	45	no	—
154	C₂×#41	—	13	45	no	—
155	C₂×#41	C₂×#261	1	15	yes	complemented
156	C₂²×Q₈	—	13	15	no	—
157	C₂²×D₄	—	13	45	no	—
158	C₂³×C₄	—	13	15	no	—
159	C₂³×C₄	—	13	30	no	—
160	C₄×Q₈	—	13	40	no	—
161	C₄×Q₈	—	13	120	no	—
162	C₄×D₄	—	13	120	no	—
163	C₄×D₄	—	13	360	no	—
164	#117	—	13	60	no	—
165	#117	—	13	180	no	—
166	C₂×C₄⋊C₄	—	13	90	no	—
167	C₂×C₂²⋊C₄	—	13	90	no	—
168	C₂×C₄²	—	13	45	no	—
169	C₂×C₄²	—	13	90	no	—
170	C₂×C₄²	—	13	15	no	—
171	C₂×C₁₃	—	2	15	no	—
172	C₂×C₁₃	C₂⁵×C₄	1	1	yes	characteristic
173	D₁₃	—	2	15	no	—
174	D₁₃	C₂×#585	1	2	yes	—
175	C₂⁴	—	13	15	no	—
176	#41	—	13	20	no	—
177	#41	—	13	60	no	—
178	#41	—	13	60	no	—
179	#41	—	13	180	no	—
180	#41	—	13	60	no	—
181	#41	—	13	180	no	—
182	#41	C₂²×#261	1	20	yes	complemented
183	#41	C₂²×#261	1	60	yes	complemented
184	C₂×Q₈	—	13	20	no	—
185	C₂×Q₈	—	13	60	no	—
186	C₂×Q₈	—	13	45	no	—
187	C₂×Q₈	C₂²×#261	1	15	yes	complemented
188	C₂×D₄	—	13	90	no	—
189	C₂×D₄	—	13	60	no	—
190	C₂×D₄	—	13	180	no	—
191	C₂×D₄	—	13	45	no	—
192	C₂×D₄	C₂²×#261	1	45	yes	complemented
193	C₂²×C₄	—	13	15	no	—
194	C₂²×C₄	—	13	15	no	—
195	C₂²×C₄	—	13	45	no	—
196	C₂²×C₄	—	13	45	no	—
197	C₂²×C₄	—	13	45	no	—
198	C₂²×C₄	—	13	30	no	—
199	C₂²×C₄	—	13	90	no	—
200	C₂²×C₄	—	52	60	no	—
201	C₂²×C₄	C₂²×#261	1	15	yes	complemented
202	C₄⋊C₄	—	13	120	no	—
203	C₂²⋊C₄	—	13	120	no	—
204	C₄²	—	13	1	no	—
205	C₄²	—	13	15	no	—
206	C₄²	—	13	30	no	—
207	C₄²	—	13	90	no	—
208	C₁₃	C₄×#585	1	1	yes	characteristic, complemented, Sylow
209	C₂³	—	13	15	no	—
210	C₂³	—	13	45	no	—
211	C₂³	—	52	30	no	—
212	C₂³	C₂³×#261	1	15	yes	—
213	Q₈	—	13	60	no	—
214	Q₈	C₂³×#261	1	20	yes	—
215	D₄	—	13	120	no	—
216	D₄	—	13	60	no	—
217	D₄	C₂³×#261	1	60	yes	—
218	C₂×C₄	—	13	2	no	—
219	C₂×C₄	—	13	15	no	—
220	C₂×C₄	—	13	45	no	—
221	C₂×C₄	—	13	15	no	—
222	C₂×C₄	—	13	15	no	—
223	C₂×C₄	—	13	90	no	—
224	C₂×C₄	—	13	1	no	—
225	C₂×C₄	—	13	15	no	—
226	C₂×C₄	—	13	30	no	—
227	C₂×C₄	—	26	60	no	—
228	C₂×C₄	—	52	90	no	—
229	C₂×C₄	C₂³×#261	1	45	yes	—
230	C₂×C₄	C₂³×#261	1	15	yes	—
231	C₂²	—	4	15	no	—
232	C₂²	—	13	1	no	—
233	C₂²	—	13	15	no	—
234	C₂²	—	26	30	no	—
235	C₂²	—	52	45	no	—
236	C₂²	C₂⁴×#261	1	15	yes	—
237	C₄	—	13	4	no	—
238	C₄	—	13	1	no	—
239	C₄	—	13	15	no	—
240	C₄	—	26	30	no	—
241	C₄	C₂⁴×#261	1	15	yes	—
242	C₄	C₂⁴×#261	1	1	yes	characteristic
243	C₂	—	2	15	no	—
244	C₂	—	13	2	no	—
245	C₂	—	26	15	no	—
246	C₂	C₂⁵×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂⁴×#261
commutator, socle	C₂×C₁₃	C₂⁵×C₄
Frattini	C₂	C₂⁵×#261
Fitting	C₁₃×#585	C₄
prime radical	D₁₃×#585	C₂
cosocle	C₂×D₁₃	C₂⁶
hypercenter	#585	#261
hypocenter	C₁₃	C₄×#585
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ #585

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴×#261	C₂	semidirect
2. H(5,𝔽₂)×#261	C₂	semidirect
3. #13131816	C₂	semidirect
4. C₂×#41×#261	C₂	semidirect
6. C₁₃×#585	C₄	semidirect
7. #11719652	C₄	semidirect
9. #11719650	C₄	semidirect
11. #11719648	C₄	semidirect
13. #11719631	C₂²	semidirect
14. #41×#261	C₂²	semidirect
15. #41×#261	C₂²	semidirect
16. #41×#261	C₂²	semidirect
17. #11719627	C₂²	semidirect
18. C₂×Q₈×#261	C₂²	semidirect
19. #11719623	C₂²	semidirect
20. C₂×D₄×#261	C₂²	semidirect
21. C₂×#88237	C₂²	semidirect
22. C₂²×C₄×#261	C₂²	semidirect
24. C₁₃×2₋¹⁺⁴	C₂×C₄	semidirect
25. C₁₃×H(5,𝔽₂)	C₂×C₄	semidirect
26. C₂×C₁₃×#41	C₂×C₄	semidirect
27. #88257	C₂×C₄	semidirect
28. #88256	C₂×C₄	semidirect
29. #41×D₁₃	C₂×C₄	semidirect
30. #41×D₁₃	C₂×C₄	semidirect
33. #88254	C₂×C₄	semidirect
34. C₂×#6131	C₂×C₄	semidirect
36. #88251	C₂×C₄	semidirect
37. C₂×#6129	C₂×C₄	semidirect
39. C₂×#6127	C₂×C₄	semidirect
57. C₁₃×#41	C₂²×C₄	semidirect
58. C₁₃×#41	C₂²×C₄	semidirect
59. C₂×Q₈×C₁₃	C₂²×C₄	semidirect
60. C₂×D₄×C₁₃	C₂²×C₄	semidirect
61. C₂²×C₄×C₁₃	C₂²×C₄	semidirect
62. C₂×C₂²⋊D₁₃	C₂²×C₄	semidirect
63. C₂²×Dic₁₃	C₂²×C₄	semidirect
64. #6131	C₂²×C₄	semidirect
65. #6131	C₂²×C₄	semidirect
66. Q₈×D₁₃	C₂²×C₄	semidirect
68. #6129	C₂²×C₄	semidirect
69. #6129	C₂²×C₄	semidirect
70. D₄×D₁₃	C₂²×C₄	semidirect
72. #6127	C₂²×C₄	semidirect
73. #6127	C₂²×C₄	semidirect
74. C₂×D₅₂	C₂²×C₄	semidirect
75. C₂×C₄×D₁₃	C₂²×C₄	semidirect
76. C₂×C₄×D₁₃	C₂²×C₄	semidirect
79. C₂×Dic₂₆	C₂²×C₄	semidirect
116. #585	#261	direct (N × H)
137. #261	#585	direct (N × H)
144. 2₋¹⁺⁴	C₂×#261	semidirect
147. H(5,𝔽₂)	C₂×#261	semidirect
155. C₂×#41	C₂×#261	semidirect
182. #41	C₂²×#261	semidirect
183. #41	C₂²×#261	semidirect
187. C₂×Q₈	C₂²×#261	semidirect
192. C₂×D₄	C₂²×#261	semidirect
201. C₂²×C₄	C₂²×#261	semidirect
208. C₁₃	C₄×#585	semidirect

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
H(5,𝔽₂)×#261	2
#13131816	2
C₂×#41×#261	2
D₁₃×#585	2
C₄×#585	13