Subgroups — #30593877

Order 15616.

10796 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21460630	C₂	1	1	yes	maximal, characteristic, complemented
2	#21460606	C₂	1	15	yes	maximal, complemented
3	#21460598	C₂	1	15	yes	maximal, complemented
4	#21460590	C₂	1	15	yes	maximal, complemented
5	C₂×#16840570	C₂	1	15	yes	maximal, complemented
6	#21460531	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×2₋¹⁺⁶	C₂	1	1	yes	maximal, characteristic, complemented
8	#16840599	—	2	6	no	—
9	#16840598	C₂²	1	15	yes	complemented
10	C₂×#14519823	C₂²	1	15	yes	complemented
11	H(5,𝔽₂)×D₆₁	—	2	10	no	—
12	#16840577	C₂²	1	60	yes	complemented
13	#16840573	C₂²	1	20	yes	complemented
14	#16840570	C₂²	1	60	yes	complemented
15	#16840570	C₂²	1	120	yes	complemented
16	#16840565	C₂²	1	60	yes	complemented
17	#16840560	C₂²	1	90	yes	complemented
18	C₂×#14519808	C₂²	1	45	yes	complemented
19	C₂×#14519802	C₂²	1	45	yes	complemented
20	C₂×#14519797	C₂²	1	45	yes	complemented
21	C₂²×#11758201	C₂²	1	15	yes	complemented
22	#16840529	C₂²	1	15	yes	complemented
23	C₂×#14519784	C₂²	1	15	yes	complemented
24	C₆₁×#585	—	2	10	no	—
25	C₆₁×#585	C₂²	1	1	yes	characteristic
26	C₆₁×#585	C₂²	1	15	yes	complemented
27	C₂×2₋¹⁺⁴×C₆₁	—	2	6	no	—
28	C₂×2₋¹⁺⁴×C₆₁	C₂²	1	15	yes	complemented
29	#14519825	—	2	15	no	—
30	#14519824	C₂³	1	20	yes	—
31	#14519823	—	2	60	no	—
32	#14519823	C₂³	1	60	yes	—
33	#14519822	—	2	45	no	—
34	C₂×#11758226	—	2	15	no	—
35	C₂×#11758222	C₂³	1	45	yes	—
36	#41×D₆₁	—	2	60	no	—
37	C₂²×D₂₄₄	C₂³	1	15	yes	—
38	C₂×D₄×D₆₁	—	2	45	no	—
39	#14519808	C₂³	1	180	yes	complemented
40	#14519807	C₂³	1	60	yes	complemented
41	#14519803	C₂³	1	60	yes	complemented
42	#14519802	C₂³	1	180	yes	complemented
43	#14519797	C₂³	1	60	yes	complemented
44	#14519797	C₂³	1	180	yes	complemented
45	C₂×#11758202	C₂³	1	45	yes	complemented
46	C₂×#11758200	C₂³	1	15	yes	complemented
47	C₂×#11758201	C₂³	1	45	yes	complemented
48	C₂×#11758201	C₂³	1	90	yes	complemented
49	C₂×#11758199	C₂³	1	45	yes	complemented
50	#14519784	C₂³	1	20	yes	complemented
51	#14519784	C₂³	1	60	yes	complemented
52	C₂×#11758194	C₂³	1	45	yes	complemented
53	C₂²×#92418	C₂³	1	15	yes	complemented
54	2₋¹⁺⁴×C₆₁	—	2	60	no	—
55	2₋¹⁺⁴×C₆₁	—	2	15	no	—
56	2₋¹⁺⁴×C₆₁	D₄	1	6	yes	—
57	2₋¹⁺⁴×C₆₁	C₂³	1	60	yes	complemented
58	H(5,𝔽₂)×C₆₁	—	2	45	no	—
59	H(5,𝔽₂)×C₆₁	D₄	1	10	yes	—
60	H(5,𝔽₂)×C₆₁	C₂³	1	20	yes	complemented
61	C₂×#41×C₆₁	—	2	60	no	—
62	C₂×#41×C₆₁	—	2	45	no	—
63	C₂×#41×C₆₁	C₂³	1	15	yes	—
64	C₂×#41×C₆₁	C₂³	1	45	yes	complemented
65	C₂²×Q₈×C₆₁	—	2	15	no	—
66	C₂²×Q₈×C₆₁	C₂³	1	15	yes	complemented
67	C₂³×D₆₁	—	2	15	no	—
68	#41×C₆₁	—	2	20	no	—
69	#41×C₆₁	—	2	180	no	—
70	#41×C₆₁	—	2	90	no	—
71	#41×C₆₁	C₂×D₄	1	60	yes	—
72	#41×C₆₁	C₂⁴	1	60	yes	—
73	#41×C₆₁	C₂⁴	1	20	yes	—
74	C₂×Q₈×C₆₁	—	2	60	no	—
75	C₂×Q₈×C₆₁	—	2	45	no	—
76	C₂×Q₈×C₆₁	C₂×D₄	1	15	yes	—
77	C₂×Q₈×C₆₁	C₂⁴	1	45	yes	—
78	C₂×D₄×C₆₁	—	2	45	no	—
79	C₂×D₄×C₆₁	—	2	60	no	—
80	C₂×D₄×C₆₁	C₂×D₄	1	45	yes	—
81	C₂×D₄×C₆₁	C₂⁴	1	15	yes	—
82	C₂²×C₄×C₆₁	—	2	15	no	—
83	C₂²×C₄×C₆₁	—	2	45	no	—
84	C₂²×C₄×C₆₁	C₂⁴	1	15	yes	—
85	C₂×C₂²⋊D₆₁	—	2	45	no	—
86	#11758226	—	2	60	no	—
87	Q₈×D₆₁	—	2	20	no	—
88	#11758224	—	2	60	no	—
89	D₄×D₆₁	—	2	180	no	—
90	#11758222	—	2	90	no	—
91	#11758222	C₂⁴	1	60	yes	—
92	C₂×D₂₄₄	—	2	45	no	—
93	C₂×D₂₄₄	C₂⁴	1	45	yes	—
94	C₂×C₄×D₆₁	—	2	45	no	—
95	C₂×Dic₁₂₂	C₂⁴	1	15	yes	—
96	#11758202	C₂⁴	1	60	yes	—
97	#11758201	C₂⁴	1	90	yes	—
98	#11758201	C₂⁴	1	10	yes	—
99	#11758200	C₂⁴	1	6	yes	—
100	#11758200	C₂⁴	1	30	yes	—
101	#11758199	C₂⁴	1	60	yes	—
102	#11758194	C₂⁴	1	60	yes	—
103	C₂×#92418	C₂⁴	1	15	yes	—
104	C₂×#92418	C₂⁴	1	45	yes	—
105	C₂³×C₆₁	—	2	15	no	—
106	C₂³×C₆₁	C₂²×D₄	1	15	yes	—
107	C₂²×D₆₁	—	2	45	no	—
108	C₂²×D₆₁	—	16	15	no	—
109	Q₈×C₆₁	—	2	45	no	—
110	Q₈×C₆₁	C₂²×D₄	1	20	yes	—
111	Q₈×C₆₁	C₂⁵	1	10	yes	—
112	D₄×C₆₁	—	2	60	no	—
113	D₄×C₆₁	—	2	15	no	—
114	D₄×C₆₁	C₂²×D₄	1	60	yes	—
115	D₄×C₆₁	C₂⁵	1	6	yes	—
116	C₂×C₄×C₆₁	—	2	45	no	—
117	C₂×C₄×C₆₁	—	2	60	no	—
118	C₂×C₄×C₆₁	C₂²×D₄	1	45	yes	—
119	C₂×C₄×C₆₁	C₂⁵	1	15	yes	—
120	C₂²⋊D₆₁	—	2	60	no	—
121	C₂×Dic₆₁	—	2	15	no	—
122	D₂₄₄	—	2	45	no	—
123	D₂₄₄	C₂⁵	1	10	yes	—
124	C₄×D₆₁	—	2	60	no	—
125	Dic₁₂₂	—	2	15	no	—
126	Dic₁₂₂	C₂⁵	1	6	yes	—
127	#92418	C₂⁵	1	1	yes	characteristic
128	#92418	C₂⁵	1	15	yes	—
129	#63092	—	61	1	no	maximal, Sylow
130	C₂²×C₆₁	—	2	6	no	—
131	C₂²×C₆₁	—	4	15	no	—
132	C₂²×C₆₁	—	8	15	no	—
133	C₂²×C₆₁	C₂³×D₄	1	15	yes	—
134	C₂×D₆₁	—	2	10	no	—
135	C₂×D₆₁	—	8	45	no	—
136	C₄×C₆₁	—	2	10	no	—
137	C₄×C₆₁	C₂³×D₄	1	15	yes	—
138	C₄×C₆₁	C₂⁶	1	1	yes	characteristic
139	Dic₆₁	—	2	6	no	—
140	2₋¹⁺⁶	D₆₁	1	1	yes	characteristic, complemented
141	H(7,𝔽₂)	—	61	1	no	—
142	#3586	—	61	15	no	—
143	#3585	—	61	15	no	—
144	#3584	—	61	15	no	—
145	C₂×#577	—	61	15	no	—
146	#3573	—	61	1	no	—
147	C₂×C₆₁	—	2	15	no	—
148	C₂×C₆₁	—	4	6	no	—
149	C₂×C₆₁	C₂⁴×D₄	1	1	yes	characteristic
150	D₆₁	—	4	10	no	—
151	#585	—	2	10	no	—
152	#585	—	61	15	no	—
153	#585	—	122	6	no	—
154	#585	C₂×D₆₁	1	1	yes	characteristic
155	#585	C₂×D₆₁	1	15	yes	complemented
156	C₂×2₋¹⁺⁴	—	2	6	no	—
157	C₂×2₋¹⁺⁴	C₂×D₆₁	1	15	yes	complemented
158	C₂×H(5,𝔽₂)	—	61	15	no	—
159	C₂×H(5,𝔽₂)	—	122	10	no	—
160	#578	—	61	60	no	—
161	#577	—	61	60	no	—
162	#577	—	61	120	no	—
163	#577	—	61	20	no	—
164	#576	—	61	60	no	—
165	#575	—	61	90	no	—
166	C₂×#137	—	61	45	no	—
167	C₂×AGL(1,ℤ/8)	—	61	45	no	—
168	C₂×#135	—	61	45	no	—
169	C₂²×SD₁₆	—	61	15	no	—
170	#568	—	61	15	no	—
171	C₂×#131	—	61	15	no	—
172	C₆₁	#63092	1	1	yes	characteristic, complemented, Sylow
173	2₋¹⁺⁴	—	2	60	no	—
174	2₋¹⁺⁴	—	2	15	no	—
175	2₋¹⁺⁴	—	61	20	no	—
176	2₋¹⁺⁴	—	122	15	no	—
177	2₋¹⁺⁴	C₂²⋊D₆₁	1	6	yes	—
178	2₋¹⁺⁴	C₂²×D₆₁	1	60	yes	complemented
179	H(5,𝔽₂)	—	2	45	no	—
180	H(5,𝔽₂)	—	61	60	no	—
181	H(5,𝔽₂)	—	122	60	no	—
182	H(5,𝔽₂)	—	122	45	no	—
183	H(5,𝔽₂)	C₂²⋊D₆₁	1	10	yes	—
184	H(5,𝔽₂)	C₂²×D₆₁	1	20	yes	complemented
185	C₂×#41	—	2	60	no	—
186	C₂×#41	—	2	45	no	—
187	C₂×#41	—	61	45	no	—
188	C₂×#41	—	122	60	no	—
189	C₂×#41	—	122	15	no	—
190	C₂×#41	C₂²×D₆₁	1	15	yes	—
191	C₂×#41	C₂²×D₆₁	1	45	yes	complemented
192	C₂²×Q₈	—	2	15	no	—
193	C₂²×Q₈	C₂²×D₆₁	1	15	yes	complemented
194	C₂²×D₄	—	61	15	no	—
195	C₂²×D₄	—	122	45	no	—
196	#137	—	61	60	no	—
197	#137	—	61	180	no	—
198	AGL(1,ℤ/8)	—	61	180	no	—
199	AGL(1,ℤ/8)	—	61	60	no	—
200	#135	—	61	60	no	—
201	#135	—	61	180	no	—
202	C₂×Q₁₆	—	61	45	no	—
203	C₂×SD₁₆	—	61	45	no	—
204	C₂×SD₁₆	—	61	90	no	—
205	C₂×SD₁₆	—	61	15	no	—
206	C₂×D₈	—	61	45	no	—
207	#131	—	61	20	no	—
208	#131	—	61	60	no	—
209	C₂×M₁₆	—	61	45	no	—
210	C₂²×C₈	—	61	15	no	—
211	C₂⁴	—	122	15	no	—
212	#41	—	2	20	no	—
213	#41	—	2	180	no	—
214	#41	—	2	90	no	—
215	#41	—	61	60	no	—
216	#41	—	122	60	no	—
217	#41	—	122	60	no	—
218	#41	—	122	90	no	—
219	#41	C₂×C₂²⋊D₆₁	1	60	yes	—
220	#41	C₂³×D₆₁	1	60	yes	—
221	#41	C₂³×D₆₁	1	20	yes	—
222	C₂×Q₈	—	2	60	no	—
223	C₂×Q₈	—	2	45	no	—
224	C₂×Q₈	—	61	15	no	—
225	C₂×Q₈	—	122	20	no	—
226	C₂×Q₈	C₂×C₂²⋊D₆₁	1	15	yes	—
227	C₂×Q₈	C₂³×D₆₁	1	45	yes	—
228	C₂×D₄	—	2	45	no	—
229	C₂×D₄	—	2	60	no	—
230	C₂×D₄	—	61	45	no	—
231	C₂×D₄	—	122	45	no	—
232	C₂×D₄	—	122	180	no	—
233	C₂×D₄	—	122	45	no	—
234	C₂×D₄	C₂×C₂²⋊D₆₁	1	45	yes	—
235	C₂×D₄	C₂³×D₆₁	1	15	yes	—
236	C₂²×C₄	—	2	15	no	—
237	C₂²×C₄	—	2	45	no	—
238	C₂²×C₄	—	122	45	no	—
239	C₂²×C₄	C₂³×D₆₁	1	15	yes	—
240	Q₁₆	—	61	60	no	—
241	SD₁₆	—	61	10	no	—
242	SD₁₆	—	61	90	no	—
243	SD₁₆	—	61	6	no	—
244	SD₁₆	—	61	30	no	—
245	D₈	—	61	60	no	—
246	M₁₆	—	61	60	no	—
247	C₂×C₈	—	61	15	no	—
248	C₂×C₈	—	61	45	no	—
249	C₂³	—	2	15	no	—
250	C₂³	—	122	45	no	—
251	C₂³	—	976	15	no	—
252	C₂³	C₂²×C₂²⋊D₆₁	1	15	yes	—
253	Q₈	—	2	45	no	—
254	Q₈	—	61	6	no	—
255	Q₈	—	122	15	no	—
256	Q₈	C₂²×C₂²⋊D₆₁	1	20	yes	—
257	Q₈	C₂⁴×D₆₁	1	10	yes	—
258	D₄	—	2	60	no	—
259	D₄	—	2	15	no	—
260	D₄	—	61	10	no	—
261	D₄	—	122	60	no	—
262	D₄	—	122	45	no	—
263	D₄	C₂²×C₂²⋊D₆₁	1	60	yes	—
264	D₄	C₂⁴×D₆₁	1	6	yes	—
265	C₂×C₄	—	2	45	no	—
266	C₂×C₄	—	2	60	no	—
267	C₂×C₄	—	122	15	no	—
268	C₂×C₄	—	122	60	no	—
269	C₂×C₄	C₂²×C₂²⋊D₆₁	1	45	yes	—
270	C₂×C₄	C₂⁴×D₆₁	1	15	yes	—
271	C₈	—	61	1	no	—
272	C₈	—	61	15	no	—
273	C₂²	—	2	6	no	—
274	C₂²	—	4	15	no	—
275	C₂²	—	8	15	no	—
276	C₂²	—	122	10	no	—
277	C₂²	—	488	45	no	—
278	C₂²	C₂³×C₂²⋊D₆₁	1	15	yes	—
279	C₄	—	2	10	no	—
280	C₄	—	122	6	no	—
281	C₄	C₂³×C₂²⋊D₆₁	1	15	yes	—
282	C₄	C₂⁵×D₆₁	1	1	yes	characteristic
283	C₂	—	2	15	no	—
284	C₂	—	4	6	no	—
285	C₂	—	244	10	no	—
286	C₂	C₂⁴×C₂²⋊D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂⁴×C₂²⋊D₆₁
commutator, cosocle	C₄×C₆₁	C₂⁶
Frattini	C₄	C₂⁵×D₆₁
Fitting	C₆₁×2₋¹⁺⁶	C₂
socle	C₂×C₆₁	C₂⁴×D₄
hypercenter	2₋¹⁺⁶	D₆₁
hypocenter	C₆₁	#63092
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ #585 ◁ 2₋¹⁺⁶

Complements

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

N	H ≅ G/N	type
1. #21460630	C₂	semidirect
2. #21460606	C₂	semidirect
3. #21460598	C₂	semidirect
4. #21460590	C₂	semidirect
5. C₂×#16840570	C₂	semidirect
6. #21460531	C₂	semidirect
7. C₆₁×2₋¹⁺⁶	C₂	semidirect
9. #16840598	C₂²	semidirect
10. C₂×#14519823	C₂²	semidirect
12. #16840577	C₂²	semidirect
13. #16840573	C₂²	semidirect
14. #16840570	C₂²	semidirect
15. #16840570	C₂²	semidirect
16. #16840565	C₂²	semidirect
17. #16840560	C₂²	semidirect
18. C₂×#14519808	C₂²	semidirect
19. C₂×#14519802	C₂²	semidirect
20. C₂×#14519797	C₂²	semidirect
21. C₂²×#11758201	C₂²	semidirect
22. #16840529	C₂²	semidirect
23. C₂×#14519784	C₂²	semidirect
26. C₆₁×#585	C₂²	semidirect
28. C₂×2₋¹⁺⁴×C₆₁	C₂²	semidirect
39. #14519808	C₂³	semidirect
40. #14519807	C₂³	semidirect
41. #14519803	C₂³	semidirect
42. #14519802	C₂³	semidirect
43. #14519797	C₂³	semidirect
44. #14519797	C₂³	semidirect
45. C₂×#11758202	C₂³	semidirect
46. C₂×#11758200	C₂³	semidirect
47. C₂×#11758201	C₂³	semidirect
48. C₂×#11758201	C₂³	semidirect
49. C₂×#11758199	C₂³	semidirect
50. #14519784	C₂³	semidirect
51. #14519784	C₂³	semidirect
52. C₂×#11758194	C₂³	semidirect
53. C₂²×#92418	C₂³	semidirect
57. 2₋¹⁺⁴×C₆₁	C₂³	semidirect
60. H(5,𝔽₂)×C₆₁	C₂³	semidirect
64. C₂×#41×C₆₁	C₂³	semidirect
66. C₂²×Q₈×C₆₁	C₂³	semidirect
140. 2₋¹⁺⁶	D₆₁	semidirect
155. #585	C₂×D₆₁	semidirect
157. C₂×2₋¹⁺⁴	C₂×D₆₁	semidirect
172. C₆₁	#63092	semidirect
178. 2₋¹⁺⁴	C₂²×D₆₁	semidirect
184. H(5,𝔽₂)	C₂²×D₆₁	semidirect
191. C₂×#41	C₂²×D₆₁	semidirect
193. C₂²×Q₈	C₂²×D₆₁	semidirect

Maximal subgroups

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

subgroup	index
#21460630	2
#21460606	2
#21460598	2
#21460590	2
C₂×#16840570	2
#21460531	2
C₆₁×2₋¹⁺⁶	2
#63092	61