Subgroups — C₂×#21460630 #30593910

Order 15616.

60570 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21460630	C₂	1	64	yes	maximal, direct product factor
2	C₂×#16840599	C₂	1	12	yes	maximal, complemented
3	C₂×#16840598	C₂	1	15	yes	maximal, complemented
4	C₂²×#14519823	C₂	1	15	yes	maximal, complemented
5	C₂×H(5,𝔽₂)×D₆₁	C₂	1	20	yes	maximal, complemented
6	C₂×C₆₁×#585	C₂	1	1	yes	maximal, characteristic, complemented
7	#16840599	C₂²	1	384	yes	complemented
8	#16840598	C₂²	1	480	yes	complemented
9	C₂×#14519825	C₂²	1	30	yes	complemented
10	C₂×#14519824	C₂²	1	20	yes	complemented
11	C₂×#14519823	C₂²	1	120	yes	complemented
12	C₂×#14519823	C₂²	1	60	yes	complemented
13	C₂×#14519823	C₂²	1	480	yes	complemented
14	C₂×#14519822	C₂²	1	90	yes	complemented
15	H(5,𝔽₂)×D₆₁	C₂²	1	640	yes	complemented
16	C₂²×#11758226	C₂²	1	30	yes	complemented
17	C₂²×#11758222	C₂²	1	45	yes	complemented
18	C₂×#41×D₆₁	C₂²	1	120	yes	complemented
19	C₂³×D₂₄₄	C₂²	1	15	yes	complemented
20	C₂²×D₄×D₆₁	C₂²	1	90	yes	complemented
21	C₆₁×#585	C₂²	1	32	yes	complemented
22	C₂×2₋¹⁺⁴×C₆₁	C₂²	1	6	yes	complemented
23	C₂×H(5,𝔽₂)×C₆₁	C₂²	1	10	yes	complemented
24	C₂²×#41×C₆₁	C₂²	1	15	yes	complemented
25	C₂⁴×D₆₁	C₂³	1	30	yes	complemented
26	#14519825	C₂³	1	480	yes	complemented
27	#14519824	C₂³	1	320	yes	complemented
28	#14519823	C₂³	1	1920	yes	complemented
29	#14519823	C₂³	1	960	yes	complemented
30	#14519822	C₂³	1	1440	yes	complemented
31	C₂×#11758226	C₂³	1	120	yes	complemented
32	C₂×#11758226	C₂³	1	480	yes	complemented
33	C₂×#11758222	C₂³	1	180	yes	complemented
34	C₂×#11758222	C₂³	1	60	yes	complemented
35	C₂×#11758222	C₂³	1	720	yes	complemented
36	C₂×#11758224	C₂³	1	120	yes	complemented
37	#41×D₆₁	C₂³	1	1920	yes	complemented
38	C₂²×Dic₁₂₂	C₂³	1	15	yes	complemented
39	C₂×Q₈×D₆₁	C₂³	1	40	yes	complemented
40	C₂²×D₂₄₄	C₂³	1	90	yes	complemented
41	C₂²×D₂₄₄	C₂³	1	45	yes	complemented
42	C₂²×D₂₄₄	C₂³	1	240	yes	complemented
43	C₂²×C₂²⋊D₆₁	C₂³	1	90	yes	complemented
44	C₂×D₄×D₆₁	C₂³	1	360	yes	complemented
45	C₂×D₄×D₆₁	C₂³	1	1440	yes	complemented
46	C₂²×C₄×D₆₁	C₂³	1	90	yes	complemented
47	2₋¹⁺⁴×C₆₁	C₂³	1	96	yes	complemented
48	H(5,𝔽₂)×C₆₁	C₂³	1	160	yes	complemented
49	C₂×#41×C₆₁	C₂³	1	20	yes	complemented
50	C₂×#41×C₆₁	C₂³	1	60	yes	complemented
51	C₂×#41×C₆₁	C₂³	1	240	yes	complemented
52	C₂²×Q₈×C₆₁	C₂³	1	15	yes	complemented
53	C₂²×D₄×C₆₁	C₂³	1	45	yes	complemented
54	C₂³×C₄×C₆₁	C₂³	1	15	yes	complemented
55	C₂⁴×C₆₁	C₂⁴	1	15	yes	—
56	C₂³×D₆₁	—	8	60	no	—
57	C₂³×D₆₁	C₂⁴	1	90	yes	—
58	C₂³×D₆₁	C₂⁴	1	240	yes	complemented
59	#41×C₆₁	C₂⁴	1	480	yes	complemented
60	#41×C₆₁	C₂⁴	1	160	yes	complemented
61	C₂×Q₈×C₆₁	C₂⁴	1	20	yes	—
62	C₂×Q₈×C₆₁	C₂⁴	1	120	yes	complemented
63	C₂×D₄×C₆₁	C₂⁴	1	60	yes	—
64	C₂×D₄×C₆₁	C₂⁴	1	360	yes	complemented
65	C₂²×C₄×C₆₁	C₂⁴	1	45	yes	—
66	C₂²×C₄×C₆₁	C₂⁴	1	15	yes	—
67	C₂²×C₄×C₆₁	C₂⁴	1	120	yes	complemented
68	C₂×C₂²⋊D₆₁	C₂⁴	1	120	yes	—
69	C₂×C₂²⋊D₆₁	C₂⁴	1	720	yes	complemented
70	C₂²×Dic₆₁	C₂⁴	1	30	yes	—
71	#11758226	C₂⁴	1	960	yes	complemented
72	Q₈×D₆₁	C₂⁴	1	320	yes	complemented
73	#11758224	C₂⁴	1	960	yes	complemented
74	D₄×D₆₁	C₂⁴	1	2880	yes	complemented
75	#11758222	C₂⁴	1	1440	yes	complemented
76	#11758222	C₂⁴	1	480	yes	complemented
77	C₂×D₂₄₄	C₂⁴	1	90	yes	—
78	C₂×D₂₄₄	C₂⁴	1	10	yes	—
79	C₂×D₂₄₄	C₂⁴	1	720	yes	complemented
80	C₂×D₂₄₄	C₂⁴	1	360	yes	complemented
81	C₂×C₄×D₆₁	C₂⁴	1	120	yes	—
82	C₂×C₄×D₆₁	C₂⁴	1	720	yes	complemented
83	C₂×Dic₁₂₂	C₂⁴	1	30	yes	—
84	C₂×Dic₁₂₂	C₂⁴	1	6	yes	—
85	C₂×Dic₁₂₂	C₂⁴	1	120	yes	complemented
86	C₂³×C₆₁	—	4	30	no	—
87	C₂³×C₆₁	C₂⁵	1	15	yes	—
88	C₂³×C₆₁	C₂⁵	1	60	yes	—
89	C₂²×D₆₁	—	4	180	no	—
90	C₂²×D₆₁	—	8	240	no	—
91	C₂²×D₆₁	C₂⁵	1	20	yes	—
92	C₂²×D₆₁	C₂⁵	1	360	yes	—
93	Q₈×C₆₁	C₂⁵	1	80	yes	—
94	D₄×C₆₁	C₂⁵	1	240	yes	—
95	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₂⁵	1	15	yes	—
97	C₂×C₄×C₆₁	C₂⁵	1	60	yes	—
98	C₂×C₄×C₆₁	C₂⁵	1	180	yes	—
99	C₂²⋊D₆₁	C₂⁵	1	480	yes	—
100	C₂×Dic₆₁	C₂⁵	1	12	yes	—
101	C₂×Dic₆₁	C₂⁵	1	120	yes	—
102	D₂₄₄	C₂⁵	1	40	yes	—
103	D₂₄₄	C₂⁵	1	360	yes	—
104	C₄×D₆₁	C₂⁵	1	480	yes	—
105	Dic₁₂₂	C₂⁵	1	24	yes	—
106	Dic₁₂₂	C₂⁵	1	120	yes	—
107	C₂×H(7,𝔽₂)	—	61	1	no	maximal, Sylow
108	C₂²×C₆₁	—	2	30	no	—
109	C₂²×C₆₁	—	4	60	no	—
110	C₂²×C₆₁	C₂⁶	1	30	yes	—
111	C₂²×C₆₁	C₂⁶	1	1	yes	characteristic
112	C₂×D₆₁	—	2	40	no	—
113	C₂×D₆₁	—	4	360	no	—
114	C₂×D₆₁	C₂⁶	1	40	yes	—
115	C₄×C₆₁	C₂⁶	1	2	yes	—
116	C₄×C₆₁	C₂⁶	1	30	yes	—
117	Dic₆₁	C₂⁶	1	24	yes	—
118	H(7,𝔽₂)	—	61	64	no	—
119	C₂×#585	—	61	12	no	—
120	C₂×#585	—	61	15	no	—
121	C₂×#585	D₆₁	1	1	yes	characteristic, complemented
122	C₂²×H(5,𝔽₂)	—	61	20	no	—
123	C₂²×H(5,𝔽₂)	—	61	15	no	—
124	C₂×C₆₁	—	2	30	no	—
125	C₂×C₆₁	H(7,𝔽₂)	1	2	yes	complemented
126	C₂×C₆₁	C₂⁷	1	1	yes	characteristic
127	D₆₁	—	2	40	no	—
128	#585	—	61	384	no	—
129	#585	—	61	480	no	—
130	#585	C₂×D₆₁	1	32	yes	complemented
131	C₂×2₋¹⁺⁴	—	61	20	no	—
132	C₂×2₋¹⁺⁴	—	61	30	no	—
133	C₂×2₋¹⁺⁴	C₂×D₆₁	1	6	yes	complemented
134	C₂×H(5,𝔽₂)	—	61	640	no	—
135	C₂×H(5,𝔽₂)	—	61	120	no	—
136	C₂×H(5,𝔽₂)	—	61	60	no	—
137	C₂×H(5,𝔽₂)	—	61	480	no	—
138	C₂×H(5,𝔽₂)	—	61	90	no	—
139	C₂×H(5,𝔽₂)	C₂×D₆₁	1	10	yes	complemented
140	C₂²×#41	—	61	120	no	—
141	C₂²×#41	—	61	30	no	—
142	C₂²×#41	—	61	45	no	—
143	C₂²×#41	C₂×D₆₁	1	15	yes	complemented
144	C₂³×D₄	—	61	90	no	—
145	C₂³×D₄	—	61	15	no	—
146	C₆₁	C₂×H(7,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
147	C₂⁵	—	61	30	no	—
148	2₋¹⁺⁴	—	61	320	no	—
149	2₋¹⁺⁴	—	61	480	no	—
150	2₋¹⁺⁴	C₂²×D₆₁	1	96	yes	complemented
151	H(5,𝔽₂)	—	61	1920	no	—
152	H(5,𝔽₂)	—	61	960	no	—
153	H(5,𝔽₂)	—	61	1440	no	—
154	H(5,𝔽₂)	C₂²×D₆₁	1	160	yes	complemented
155	C₂×#41	—	61	1920	no	—
156	C₂×#41	—	61	120	no	—
157	C₂×#41	—	61	480	no	—
158	C₂×#41	—	61	120	no	—
159	C₂×#41	—	61	180	no	—
160	C₂×#41	—	61	60	no	—
161	C₂×#41	—	61	720	no	—
162	C₂×#41	C₂²×D₆₁	1	60	yes	complemented
163	C₂×#41	C₂²×D₆₁	1	20	yes	complemented
164	C₂×#41	C₂²×D₆₁	1	240	yes	complemented
165	C₂²×Q₈	—	61	40	no	—
166	C₂²×Q₈	—	61	15	no	—
167	C₂²×Q₈	C₂²×D₆₁	1	15	yes	complemented
168	C₂²×D₄	—	61	90	no	—
169	C₂²×D₄	—	61	360	no	—
170	C₂²×D₄	—	61	1440	no	—
171	C₂²×D₄	—	61	90	no	—
172	C₂²×D₄	—	61	45	no	—
173	C₂²×D₄	—	61	240	no	—
174	C₂²×D₄	C₂²×D₆₁	1	45	yes	complemented
175	C₂³×C₄	—	61	90	no	—
176	C₂³×C₄	C₂²×D₆₁	1	15	yes	complemented
177	C₂⁴	—	61	240	no	—
178	C₂⁴	—	61	90	no	—
179	C₂⁴	—	488	60	no	—
180	C₂⁴	C₂³×D₆₁	1	15	yes	—
181	#41	—	61	960	no	—
182	#41	—	61	960	no	—
183	#41	—	61	1440	no	—
184	#41	—	61	480	no	—
185	#41	C₂³×D₆₁	1	480	yes	complemented
186	#41	C₂³×D₆₁	1	160	yes	complemented
187	C₂×Q₈	—	61	320	no	—
188	C₂×Q₈	—	61	30	no	—
189	C₂×Q₈	—	61	6	no	—
190	C₂×Q₈	—	61	120	no	—
191	C₂×Q₈	C₂³×D₆₁	1	20	yes	—
192	C₂×Q₈	C₂³×D₆₁	1	120	yes	complemented
193	C₂×D₄	—	61	120	no	—
194	C₂×D₄	—	61	720	no	—
195	C₂×D₄	—	61	2880	no	—
196	C₂×D₄	—	61	90	no	—
197	C₂×D₄	—	61	10	no	—
198	C₂×D₄	—	61	720	no	—
199	C₂×D₄	—	61	360	no	—
200	C₂×D₄	C₂³×D₆₁	1	60	yes	—
201	C₂×D₄	C₂³×D₆₁	1	360	yes	complemented
202	C₂²×C₄	—	61	30	no	—
203	C₂²×C₄	—	61	120	no	—
204	C₂²×C₄	—	61	720	no	—
205	C₂²×C₄	C₂³×D₆₁	1	45	yes	—
206	C₂²×C₄	C₂³×D₆₁	1	15	yes	—
207	C₂²×C₄	C₂³×D₆₁	1	120	yes	complemented
208	C₂³	—	4	30	no	—
209	C₂³	—	61	20	no	—
210	C₂³	—	61	360	no	—
211	C₂³	—	488	240	no	—
212	C₂³	—	244	180	no	—
213	C₂³	C₂⁴×D₆₁	1	15	yes	—
214	C₂³	C₂⁴×D₆₁	1	60	yes	—
215	Q₈	—	61	120	no	—
216	Q₈	—	61	24	no	—
217	Q₈	C₂⁴×D₆₁	1	80	yes	—
218	D₄	—	61	480	no	—
219	D₄	—	61	360	no	—
220	D₄	—	61	40	no	—
221	D₄	C₂⁴×D₆₁	1	240	yes	—
222	C₂×C₄	—	61	120	no	—
223	C₂×C₄	—	61	12	no	—
224	C₂×C₄	—	61	480	no	—
225	C₂×C₄	C₂⁴×D₆₁	1	180	yes	—
226	C₂×C₄	C₂⁴×D₆₁	1	60	yes	—
227	C₂×C₄	C₂⁴×D₆₁	1	15	yes	—
228	C₂×C₄	C₂⁴×D₆₁	1	1	yes	characteristic
229	C₂²	—	2	30	no	—
230	C₂²	—	4	60	no	—
231	C₂²	—	61	40	no	—
232	C₂²	—	122	40	no	—
233	C₂²	—	244	360	no	—
234	C₂²	C₂⁵×D₆₁	1	30	yes	—
235	C₂²	C₂⁵×D₆₁	1	1	yes	characteristic
236	C₄	—	61	24	no	—
237	C₄	C₂⁵×D₆₁	1	30	yes	—
238	C₄	C₂⁵×D₆₁	1	2	yes	—
239	C₂	—	2	30	no	—
240	C₂	—	122	40	no	—
241	C₂	#21460630	1	2	yes	direct product factor
242	C₂	C₂⁶×D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂⁵×D₆₁
commutator, cosocle	C₂×C₆₁	C₂⁷
Frattini	C₂	C₂⁶×D₆₁
Fitting	C₂×C₆₁×#585	C₂
socle	C₂²×C₆₁	C₂⁶
hypercenter	C₂×#585	D₆₁
hypocenter	C₆₁	C₂×H(7,𝔽₂)
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×#585

Complements

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

N	H ≅ G/N	type
1. #21460630	C₂	direct (N × H)
2. C₂×#16840599	C₂	semidirect
3. C₂×#16840598	C₂	semidirect
4. C₂²×#14519823	C₂	semidirect
5. C₂×H(5,𝔽₂)×D₆₁	C₂	semidirect
6. C₂×C₆₁×#585	C₂	semidirect
7. #16840599	C₂²	semidirect
8. #16840598	C₂²	semidirect
9. C₂×#14519825	C₂²	semidirect
10. C₂×#14519824	C₂²	semidirect
11. C₂×#14519823	C₂²	semidirect
12. C₂×#14519823	C₂²	semidirect
13. C₂×#14519823	C₂²	semidirect
14. C₂×#14519822	C₂²	semidirect
15. H(5,𝔽₂)×D₆₁	C₂²	semidirect
16. C₂²×#11758226	C₂²	semidirect
17. C₂²×#11758222	C₂²	semidirect
18. C₂×#41×D₆₁	C₂²	semidirect
19. C₂³×D₂₄₄	C₂²	semidirect
20. C₂²×D₄×D₆₁	C₂²	semidirect
21. C₆₁×#585	C₂²	semidirect
22. C₂×2₋¹⁺⁴×C₆₁	C₂²	semidirect
23. C₂×H(5,𝔽₂)×C₆₁	C₂²	semidirect
24. C₂²×#41×C₆₁	C₂²	semidirect
25. C₂⁴×D₆₁	C₂³	semidirect
26. #14519825	C₂³	semidirect
27. #14519824	C₂³	semidirect
28. #14519823	C₂³	semidirect
29. #14519823	C₂³	semidirect
30. #14519822	C₂³	semidirect
31. C₂×#11758226	C₂³	semidirect
32. C₂×#11758226	C₂³	semidirect
33. C₂×#11758222	C₂³	semidirect
34. C₂×#11758222	C₂³	semidirect
35. C₂×#11758222	C₂³	semidirect
36. C₂×#11758224	C₂³	semidirect
37. #41×D₆₁	C₂³	semidirect
38. C₂²×Dic₁₂₂	C₂³	semidirect
39. C₂×Q₈×D₆₁	C₂³	semidirect
40. C₂²×D₂₄₄	C₂³	semidirect
41. C₂²×D₂₄₄	C₂³	semidirect
42. C₂²×D₂₄₄	C₂³	semidirect
43. C₂²×C₂²⋊D₆₁	C₂³	semidirect
44. C₂×D₄×D₆₁	C₂³	semidirect
45. C₂×D₄×D₆₁	C₂³	semidirect
46. C₂²×C₄×D₆₁	C₂³	semidirect
47. 2₋¹⁺⁴×C₆₁	C₂³	semidirect
48. H(5,𝔽₂)×C₆₁	C₂³	semidirect
49. C₂×#41×C₆₁	C₂³	semidirect
50. C₂×#41×C₆₁	C₂³	semidirect
51. C₂×#41×C₆₁	C₂³	semidirect
52. C₂²×Q₈×C₆₁	C₂³	semidirect
53. C₂²×D₄×C₆₁	C₂³	semidirect
54. C₂³×C₄×C₆₁	C₂³	semidirect
58. C₂³×D₆₁	C₂⁴	semidirect
59. #41×C₆₁	C₂⁴	semidirect
60. #41×C₆₁	C₂⁴	semidirect
62. C₂×Q₈×C₆₁	C₂⁴	semidirect
64. C₂×D₄×C₆₁	C₂⁴	semidirect
67. C₂²×C₄×C₆₁	C₂⁴	semidirect
69. C₂×C₂²⋊D₆₁	C₂⁴	semidirect
71. #11758226	C₂⁴	semidirect
72. Q₈×D₆₁	C₂⁴	semidirect
73. #11758224	C₂⁴	semidirect
74. D₄×D₆₁	C₂⁴	semidirect
75. #11758222	C₂⁴	semidirect
76. #11758222	C₂⁴	semidirect
79. C₂×D₂₄₄	C₂⁴	semidirect
80. C₂×D₂₄₄	C₂⁴	semidirect
82. C₂×C₄×D₆₁	C₂⁴	semidirect
85. C₂×Dic₁₂₂	C₂⁴	semidirect
121. C₂×#585	D₆₁	semidirect
125. C₂×C₆₁	H(7,𝔽₂)	semidirect
130. #585	C₂×D₆₁	semidirect
133. C₂×2₋¹⁺⁴	C₂×D₆₁	semidirect
139. C₂×H(5,𝔽₂)	C₂×D₆₁	semidirect
143. C₂²×#41	C₂×D₆₁	semidirect
146. C₆₁	C₂×H(7,𝔽₂)	semidirect
150. 2₋¹⁺⁴	C₂²×D₆₁	semidirect
154. H(5,𝔽₂)	C₂²×D₆₁	semidirect
162. C₂×#41	C₂²×D₆₁	semidirect
163. C₂×#41	C₂²×D₆₁	semidirect
164. C₂×#41	C₂²×D₆₁	semidirect
167. C₂²×Q₈	C₂²×D₆₁	semidirect
174. C₂²×D₄	C₂²×D₆₁	semidirect
176. C₂³×C₄	C₂²×D₆₁	semidirect
185. #41	C₂³×D₆₁	semidirect
186. #41	C₂³×D₆₁	semidirect
192. C₂×Q₈	C₂³×D₆₁	semidirect
201. C₂×D₄	C₂³×D₆₁	semidirect
207. C₂²×C₄	C₂³×D₆₁	semidirect
241. C₂	#21460630	direct (N × H)

Maximal subgroups

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

subgroup	index
#21460630	2
C₂×#16840599	2
C₂×#16840598	2
C₂²×#14519823	2
C₂×H(5,𝔽₂)×D₆₁	2
C₂×C₆₁×#585	2
C₂×H(7,𝔽₂)	61

Subgroups — C2×#21460630 #30593910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21460630 #30593910