Subgroups — #30626734

Order 15616.

11222 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	2₋¹⁺⁴×#6923	C₂	1	12	yes	maximal, complemented
2	H(5,𝔽₂)×#6923	C₂	1	20	yes	maximal, complemented
3	#21462799	C₂	1	15	yes	maximal, complemented
4	C₂×#16840831	C₂	1	15	yes	maximal, complemented
5	#585×D₆₁	C₂	1	1	yes	maximal, characteristic
6	#16840831	C₂²	1	60	yes	complemented
7	#16840831	C₂²	1	120	yes	complemented
8	#16840831	C₂²	1	20	yes	complemented
9	#16840830	C₂²	1	30	yes	complemented
10	#16840829	C₂²	1	90	yes	complemented
11	#41×#6923	C₂²	1	120	yes	complemented
12	C₂×Q₈×#6923	C₂²	1	30	yes	complemented
13	C₂×D₄×#6923	C₂²	1	90	yes	complemented
14	C₂×#14519856	C₂²	1	45	yes	complemented
15	C₂²×C₄⋊#6923	C₂²	1	15	yes	complemented
16	#16840599	C₄	1	6	yes	complemented
17	#16840598	C₄	1	15	yes	complemented
18	#16840597	C₄	1	10	yes	complemented
19	2₋¹⁺⁴×D₆₁	C₂²	1	6	yes	—
20	H(5,𝔽₂)×D₆₁	C₂²	1	10	yes	—
21	C₂×#41×D₆₁	C₂²	1	15	yes	—
22	C₆₁×#585	C₄	1	1	yes	characteristic, complemented
23	C₂³×#6923	C₂³	1	30	yes	—
24	Q₈×#6923	C₂³	1	120	yes	—
25	Q₈×#6923	C₂³	1	40	yes	—
26	D₄×#6923	C₂³	1	360	yes	—
27	D₄×#6923	C₂³	1	120	yes	—
28	#14519856	C₂³	1	60	yes	—
29	#14519856	C₂³	1	180	yes	—
30	C₂×C₄⋊#6923	C₂³	1	45	yes	—
31	C₂×C₄⋊#6923	C₂³	1	90	yes	—
32	C₂×C₄⋊#6923	C₂³	1	15	yes	—
33	C₂×C₂²⋊#6923	C₂³	1	90	yes	—
34	C₂×C₄×#6923	C₂³	1	90	yes	—
35	#14519825	C₂×C₄	1	30	yes	complemented
36	#14519824	C₂×C₄	1	60	yes	complemented
37	#14519823	C₂×C₄	1	60	yes	complemented
38	#14519822	C₂×C₄	1	90	yes	complemented
39	C₂×#11758226	C₂×C₄	1	15	yes	complemented
40	C₂×#11758222	C₂×C₄	1	45	yes	complemented
41	C₂×#11758224	C₂×C₄	1	45	yes	complemented
42	#41×D₆₁	C₂×C₄	1	60	yes	complemented
43	#41×D₆₁	C₂×C₄	1	60	yes	complemented
44	#41×D₆₁	C₂³	1	60	yes	—
45	#41×D₆₁	C₂³	1	20	yes	—
46	C₂×Q₈×D₆₁	C₂³	1	15	yes	—
47	C₂×D₄×D₆₁	C₂³	1	45	yes	—
48	C₂²×C₄×D₆₁	C₂³	1	15	yes	—
49	2₋¹⁺⁴×C₆₁	C₂×C₄	1	6	yes	complemented
50	H(5,𝔽₂)×C₆₁	C₂×C₄	1	10	yes	complemented
51	C₂×#41×C₆₁	C₂×C₄	1	15	yes	complemented
52	C₂³×D₆₁	C₂⁴	1	15	yes	—
53	C₂²×#6923	—	8	30	no	—
54	C₂²×#6923	C₂⁴	1	30	yes	—
55	C₂²×#6923	C₂⁴	1	90	yes	—
56	#41×C₆₁	C₂²×C₄	1	20	yes	complemented
57	#41×C₆₁	C₂²×C₄	1	60	yes	complemented
58	C₂×Q₈×C₆₁	C₂²×C₄	1	15	yes	complemented
59	C₂×D₄×C₆₁	C₂²×C₄	1	45	yes	complemented
60	C₂²×C₄×C₆₁	C₂²×C₄	1	15	yes	complemented
61	C₂×C₂²⋊D₆₁	C₂²×C₄	1	90	yes	complemented
62	C₂²×Dic₆₁	C₂²×C₄	1	15	yes	complemented
63	#11758226	C₂²×C₄	1	20	yes	complemented
64	#11758226	C₂²×C₄	1	60	yes	complemented
65	Q₈×D₆₁	C₂²×C₄	1	60	yes	complemented
66	Q₈×D₆₁	C₂⁴	1	20	yes	—
67	#11758224	C₂²×C₄	1	60	yes	complemented
68	#11758224	C₂²×C₄	1	180	yes	complemented
69	D₄×D₆₁	C₂²×C₄	1	180	yes	complemented
70	D₄×D₆₁	C₂⁴	1	60	yes	—
71	#11758222	C₂²×C₄	1	60	yes	complemented
72	#11758222	C₂²×C₄	1	180	yes	complemented
73	C₂×D₂₄₄	C₂²×C₄	1	45	yes	complemented
74	C₂×C₄×D₆₁	C₂²×C₄	1	45	yes	complemented
75	C₂×C₄×D₆₁	C₂²×C₄	1	45	yes	complemented
76	C₂×C₄×D₆₁	C₂⁴	1	15	yes	—
77	C₂×C₄×D₆₁	C₂⁴	1	45	yes	—
78	C₂×Dic₁₂₂	C₂²×C₄	1	45	yes	complemented
79	C₂²⋊#6923	C₂⁴	1	120	yes	—
80	C₄⋊#6923	C₂⁴	1	10	yes	—
81	C₄⋊#6923	C₂⁴	1	6	yes	—
82	C₄⋊#6923	C₂⁴	1	90	yes	—
83	C₄⋊#6923	C₂⁴	1	30	yes	—
84	C₄×#6923	C₂⁴	1	120	yes	—
85	C₂³×C₆₁	C₂³×C₄	1	15	yes	—
86	C₂²×D₆₁	—	4	15	no	—
87	C₂²×D₆₁	—	4	15	no	—
88	C₂²×D₆₁	C₂³×C₄	1	45	yes	—
89	C₂²×D₆₁	C₂⁵	1	15	yes	—
90	C₂×#6923	—	4	90	no	—
91	C₂×#6923	—	4	30	no	—
92	C₂×#6923	C₂⁵	1	12	yes	—
93	C₂×#6923	C₂⁵	1	20	yes	—
94	Q₈×C₆₁	C₂³×C₄	1	20	yes	—
95	D₄×C₆₁	C₂³×C₄	1	60	yes	—
96	C₂×C₄×C₆₁	C₂³×C₄	1	45	yes	—
97	C₂×C₄×C₆₁	C₂³×C₄	1	15	yes	—
98	C₂²⋊D₆₁	C₂³×C₄	1	120	yes	—
99	C₂×Dic₆₁	C₂³×C₄	1	45	yes	—
100	C₂×Dic₆₁	C₂³×C₄	1	15	yes	—
101	D₂₄₄	C₂³×C₄	1	60	yes	—
102	C₄×D₆₁	C₂³×C₄	1	15	yes	—
103	C₄×D₆₁	C₂³×C₄	1	15	yes	—
104	C₄×D₆₁	C₂³×C₄	1	90	yes	—
105	C₄×D₆₁	C₂⁵	1	15	yes	—
106	C₄×D₆₁	C₂⁵	1	1	yes	characteristic
107	Dic₁₂₂	C₂³×C₄	1	60	yes	—
108	#62654	—	61	1	no	maximal, Sylow
109	C₂²×C₆₁	—	4	15	no	—
110	C₂²×C₆₁	C₂⁴×C₄	1	15	yes	—
111	C₂×D₆₁	—	2	15	no	—
112	C₂×D₆₁	—	2	15	no	—
113	C₂×D₆₁	—	4	45	no	—
114	C₂×D₆₁	C₂⁴×C₄	1	15	yes	—
115	C₂×D₆₁	C₂⁶	1	1	yes	characteristic
116	#6923	—	2	20	no	—
117	#6923	—	2	12	no	—
118	C₄×C₆₁	C₂⁴×C₄	1	15	yes	—
119	C₄×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
120	Dic₆₁	C₂⁴×C₄	1	15	yes	—
121	Dic₆₁	C₂⁴×C₄	1	1	yes	characteristic
122	C₂×#585	—	61	1	no	—
123	C₄×2₋¹⁺⁴	—	61	12	no	—
124	C₄×H(5,𝔽₂)	—	61	20	no	—
125	#3428	—	61	15	no	—
126	C₂×#520	—	61	15	no	—
127	C₂×C₆₁	—	2	15	no	—
128	C₂×C₆₁	C₂⁵×C₄	1	1	yes	characteristic
129	D₆₁	—	2	15	no	—
130	D₆₁	H(7,𝔽₂)	1	1	yes	characteristic
131	D₆₁	2₋¹⁺⁶	1	1	yes	characteristic
132	#585	—	61	6	no	—
133	#585	—	61	10	no	—
134	#585	—	61	15	no	—
135	#585	#6923	1	1	yes	characteristic, complemented
136	C₂×2₋¹⁺⁴	—	61	6	no	—
137	C₂×H(5,𝔽₂)	—	61	10	no	—
138	C₂²×#41	—	61	15	no	—
139	#520	—	61	60	no	—
140	#520	—	61	120	no	—
141	#520	—	61	20	no	—
142	#519	—	61	30	no	—
143	#518	—	61	90	no	—
144	C₄×#41	—	61	120	no	—
145	C₂×C₄×Q₈	—	61	30	no	—
146	C₂×C₄×D₄	—	61	90	no	—
147	C₂×#117	—	61	45	no	—
148	C₂²×C₄⋊C₄	—	61	15	no	—
149	C₆₁	#62654	1	1	yes	characteristic, complemented, Sylow
150	2₋¹⁺⁴	—	61	60	no	—
151	2₋¹⁺⁴	—	61	30	no	—
152	2₋¹⁺⁴	C₂×#6923	1	6	yes	complemented
153	H(5,𝔽₂)	—	61	60	no	—
154	H(5,𝔽₂)	—	61	90	no	—
155	H(5,𝔽₂)	C₂×#6923	1	10	yes	complemented
156	C₂×#41	—	61	60	no	—
157	C₂×#41	—	61	20	no	—
158	C₂×#41	—	61	60	no	—
159	C₂×#41	—	61	60	no	—
160	C₂×#41	—	61	15	no	—
161	C₂×#41	—	61	45	no	—
162	C₂×#41	—	61	45	no	—
163	C₂×#41	C₂×#6923	1	15	yes	complemented
164	C₂²×Q₈	—	61	15	no	—
165	C₂²×D₄	—	61	45	no	—
166	C₂³×C₄	—	61	15	no	—
167	C₂³×C₄	—	61	30	no	—
168	C₄×Q₈	—	61	120	no	—
169	C₄×Q₈	—	61	40	no	—
170	C₄×D₄	—	61	360	no	—
171	C₄×D₄	—	61	120	no	—
172	#117	—	61	60	no	—
173	#117	—	61	180	no	—
174	C₂×C₄⋊C₄	—	61	45	no	—
175	C₂×C₄⋊C₄	—	61	90	no	—
176	C₂×C₄⋊C₄	—	61	15	no	—
177	C₂×C₂²⋊C₄	—	61	90	no	—
178	C₂×C₄²	—	61	90	no	—
179	C₂⁴	—	61	15	no	—
180	#41	—	61	20	no	—
181	#41	—	61	60	no	—
182	#41	—	61	60	no	—
183	#41	—	61	180	no	—
184	#41	—	61	60	no	—
185	#41	—	61	180	no	—
186	#41	C₂²×#6923	1	20	yes	complemented
187	#41	C₂²×#6923	1	60	yes	complemented
188	C₂×Q₈	—	61	20	no	—
189	C₂×Q₈	—	61	60	no	—
190	C₂×Q₈	—	61	45	no	—
191	C₂×Q₈	C₂²×#6923	1	15	yes	complemented
192	C₂×D₄	—	61	90	no	—
193	C₂×D₄	—	61	60	no	—
194	C₂×D₄	—	61	180	no	—
195	C₂×D₄	—	61	45	no	—
196	C₂×D₄	C₂²×#6923	1	45	yes	complemented
197	C₂²×C₄	—	61	15	no	—
198	C₂²×C₄	—	61	45	no	—
199	C₂²×C₄	—	61	15	no	—
200	C₂²×C₄	—	61	45	no	—
201	C₂²×C₄	—	61	45	no	—
202	C₂²×C₄	—	61	30	no	—
203	C₂²×C₄	—	61	90	no	—
204	C₂²×C₄	—	488	30	no	—
205	C₂²×C₄	C₂²×#6923	1	15	yes	complemented
206	C₄⋊C₄	—	61	10	no	—
207	C₄⋊C₄	—	61	6	no	—
208	C₄⋊C₄	—	61	90	no	—
209	C₄⋊C₄	—	61	30	no	—
210	C₂²⋊C₄	—	61	120	no	—
211	C₄²	—	61	120	no	—
212	C₂³	—	61	15	no	—
213	C₂³	—	61	45	no	—
214	C₂³	—	244	15	no	—
215	C₂³	—	244	15	no	—
216	C₂³	C₂³×#6923	1	15	yes	—
217	Q₈	—	61	60	no	—
218	Q₈	C₂³×#6923	1	20	yes	—
219	D₄	—	61	120	no	—
220	D₄	—	61	60	no	—
221	D₄	C₂³×#6923	1	60	yes	—
222	C₂×C₄	—	61	15	no	—
223	C₂×C₄	—	61	45	no	—
224	C₂×C₄	—	61	15	no	—
225	C₂×C₄	—	61	15	no	—
226	C₂×C₄	—	61	90	no	—
227	C₂×C₄	—	61	15	no	—
228	C₂×C₄	—	61	1	no	—
229	C₂×C₄	—	61	12	no	—
230	C₂×C₄	—	61	20	no	—
231	C₂×C₄	—	244	30	no	—
232	C₂×C₄	—	244	90	no	—
233	C₂×C₄	C₂³×#6923	1	15	yes	—
234	C₂×C₄	C₂³×#6923	1	45	yes	—
235	C₂²	—	4	15	no	—
236	C₂²	—	61	1	no	—
237	C₂²	—	61	15	no	—
238	C₂²	—	122	15	no	—
239	C₂²	—	122	15	no	—
240	C₂²	—	244	45	no	—
241	C₂²	C₂⁴×#6923	1	15	yes	—
242	C₄	—	61	1	no	—
243	C₄	—	61	15	no	—
244	C₄	—	122	20	no	—
245	C₄	—	122	12	no	—
246	C₄	C₂⁴×#6923	1	1	yes	characteristic
247	C₄	C₂⁴×#6923	1	15	yes	—
248	C₂	—	2	15	no	—
249	C₂	—	61	1	no	—
250	C₂	—	61	1	no	—
251	C₂	—	122	15	no	—
252	C₂	C₂⁵×#6923	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁵×#6923
commutator, socle	C₂×C₆₁	C₂⁵×C₄
Fitting	C₆₁×#585	C₄
prime radical	#585×D₆₁	C₂
cosocle	C₂×D₆₁	C₂⁶
hypercenter	#585	#6923
hypocenter	C₆₁	#62654
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ #585

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴×#6923	C₂	semidirect
2. H(5,𝔽₂)×#6923	C₂	semidirect
3. #21462799	C₂	semidirect
4. C₂×#16840831	C₂	semidirect
6. #16840831	C₂²	semidirect
7. #16840831	C₂²	semidirect
8. #16840831	C₂²	semidirect
9. #16840830	C₂²	semidirect
10. #16840829	C₂²	semidirect
11. #41×#6923	C₂²	semidirect
12. C₂×Q₈×#6923	C₂²	semidirect
13. C₂×D₄×#6923	C₂²	semidirect
14. C₂×#14519856	C₂²	semidirect
15. C₂²×C₄⋊#6923	C₂²	semidirect
16. #16840599	C₄	semidirect
17. #16840598	C₄	semidirect
18. #16840597	C₄	semidirect
22. C₆₁×#585	C₄	semidirect
35. #14519825	C₂×C₄	semidirect
36. #14519824	C₂×C₄	semidirect
37. #14519823	C₂×C₄	semidirect
38. #14519822	C₂×C₄	semidirect
39. C₂×#11758226	C₂×C₄	semidirect
40. C₂×#11758222	C₂×C₄	semidirect
41. C₂×#11758224	C₂×C₄	semidirect
42. #41×D₆₁	C₂×C₄	semidirect
43. #41×D₆₁	C₂×C₄	semidirect
49. 2₋¹⁺⁴×C₆₁	C₂×C₄	semidirect
50. H(5,𝔽₂)×C₆₁	C₂×C₄	semidirect
51. C₂×#41×C₆₁	C₂×C₄	semidirect
56. #41×C₆₁	C₂²×C₄	semidirect
57. #41×C₆₁	C₂²×C₄	semidirect
58. C₂×Q₈×C₆₁	C₂²×C₄	semidirect
59. C₂×D₄×C₆₁	C₂²×C₄	semidirect
60. C₂²×C₄×C₆₁	C₂²×C₄	semidirect
61. C₂×C₂²⋊D₆₁	C₂²×C₄	semidirect
62. C₂²×Dic₆₁	C₂²×C₄	semidirect
63. #11758226	C₂²×C₄	semidirect
64. #11758226	C₂²×C₄	semidirect
65. Q₈×D₆₁	C₂²×C₄	semidirect
67. #11758224	C₂²×C₄	semidirect
68. #11758224	C₂²×C₄	semidirect
69. D₄×D₆₁	C₂²×C₄	semidirect
71. #11758222	C₂²×C₄	semidirect
72. #11758222	C₂²×C₄	semidirect
73. C₂×D₂₄₄	C₂²×C₄	semidirect
74. C₂×C₄×D₆₁	C₂²×C₄	semidirect
75. C₂×C₄×D₆₁	C₂²×C₄	semidirect
78. C₂×Dic₁₂₂	C₂²×C₄	semidirect
135. #585	#6923	semidirect
149. C₆₁	#62654	semidirect
152. 2₋¹⁺⁴	C₂×#6923	semidirect
155. H(5,𝔽₂)	C₂×#6923	semidirect
163. C₂×#41	C₂×#6923	semidirect
186. #41	C₂²×#6923	semidirect
187. #41	C₂²×#6923	semidirect
191. C₂×Q₈	C₂²×#6923	semidirect
196. C₂×D₄	C₂²×#6923	semidirect
205. C₂²×C₄	C₂²×#6923	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×#6923	2
H(5,𝔽₂)×#6923	2
#21462799	2
C₂×#16840831	2
#585×D₆₁	2
#62654	61