Subgroups — C₂²×C₆₁×#470 #29563873

Order 15616.

2878 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂²⋊Q₈×C₆₁	C₂	1	2	yes	maximal
4	C₂×C₆₁×#470	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#106×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#103×C₆₁	C₂	1	2	yes	maximal, complemented
7	C₂³×Q₈×C₆₁	C₂²	1	2	yes	—
8	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
9	C₂²×Q₁₆×C₆₁	—	2	8	no	—
10	C₂²×Q₁₆×C₆₁	C₂²	1	12	yes	complemented
11	C₂²×M₁₆×C₆₁	—	2	8	no	—
12	C₂²×M₁₆×C₆₁	C₂²	1	12	yes	—
13	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₆₁	—	2	12	no	—
16	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	24	yes	—
17	C₂²×C₄⋊C₄×C₆₁	—	2	2	no	—
18	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
19	C₂²×C₂²⋊C₄×C₆₁	—	2	2	no	—
20	C₆₁×#470	C₂²	1	64	yes	direct product factor
21	C₂×#106×C₆₁	C₂²	1	12	yes	complemented
22	C₂×#103×C₆₁	C₂²	1	24	yes	complemented
23	C₂⁵×C₆₁	D₄	1	1	yes	characteristic
24	C₂²×Q₈×C₆₁	—	2	6	no	—
25	C₂²×Q₈×C₆₁	—	2	8	no	—
26	C₂²×Q₈×C₆₁	—	2	8	no	—
27	C₂²×Q₈×C₆₁	C₂³	1	12	yes	—
28	C₂³×C₄×C₆₁	—	2	4	no	—
29	C₂³×C₄×C₆₁	—	2	4	no	—
30	C₂³×C₄×C₆₁	—	2	2	no	—
31	C₂³×C₄×C₆₁	—	2	2	no	—
32	C₂³×C₄×C₆₁	D₄	1	6	yes	—
33	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₆₁	C₂³	1	6	yes	—
35	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
36	C₂×Q₁₆×C₆₁	—	2	48	no	—
37	C₂×Q₁₆×C₆₁	C₂³	1	16	yes	complemented
38	C₂×M₁₆×C₆₁	—	2	48	no	—
39	C₂×M₁₆×C₆₁	C₂³	1	16	yes	—
40	C₂²×C₈×C₆₁	—	2	4	no	—
41	C₂²×C₈×C₆₁	D₄	1	8	yes	—
42	C₂²×C₈×C₆₁	C₂³	1	6	yes	—
43	C₂²×C₈×C₆₁	C₂³	1	6	yes	—
44	C₂²⋊Q₈×C₆₁	—	2	48	no	—
45	C₂²⋊Q₈×C₆₁	C₂³	1	32	yes	—
46	C₂×C₄⋊C₄×C₆₁	—	2	24	no	—
47	C₂×C₄⋊C₄×C₆₁	—	2	6	no	—
48	C₂×C₄⋊C₄×C₆₁	C₂³	1	12	yes	—
49	C₂×C₂²⋊C₄×C₆₁	—	2	24	no	—
50	#106×C₆₁	C₂³	1	16	yes	complemented
51	#103×C₆₁	C₂³	1	32	yes	complemented
52	C₂⁴×C₆₁	—	2	4	no	—
53	C₂⁴×C₆₁	—	2	8	no	—
54	C₂⁴×C₆₁	C₂×D₄	1	6	yes	—
55	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₂×Q₈×C₆₁	—	2	12	no	—
57	C₂×Q₈×C₆₁	—	2	24	no	—
58	C₂×Q₈×C₆₁	—	2	24	no	—
59	C₂×Q₈×C₆₁	—	4	24	no	—
60	C₂×Q₈×C₆₁	C₂⁴	1	8	yes	—
61	C₂²×C₄×C₆₁	—	2	4	no	—
62	C₂²×C₄×C₆₁	—	2	24	no	—
63	C₂²×C₄×C₆₁	—	2	12	no	—
64	C₂²×C₄×C₆₁	—	2	12	no	—
65	C₂²×C₄×C₆₁	—	2	12	no	—
66	C₂²×C₄×C₆₁	—	2	12	no	—
67	C₂²×C₄×C₆₁	—	4	8	no	—
68	C₂²×C₄×C₆₁	—	4	8	no	—
69	C₂²×C₄×C₆₁	C₂×D₄	1	6	yes	—
70	C₂²×C₄×C₆₁	C₂×D₄	1	12	yes	—
71	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
72	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
73	C₂²×C₄×C₆₁	#41	1	4	yes	—
74	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
75	C₂²×C₄×C₆₁	C₂⁴	1	3	yes	—
76	Q₁₆×C₆₁	—	2	32	no	—
77	M₁₆×C₆₁	—	2	32	no	—
78	C₂×C₈×C₆₁	—	2	12	no	—
79	C₂×C₈×C₆₁	C₂×D₄	1	24	yes	—
80	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
81	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
82	C₄⋊C₄×C₆₁	—	2	32	no	—
83	C₄⋊C₄×C₆₁	—	2	12	no	—
84	C₄⋊C₄×C₆₁	C₂⁴	1	8	yes	—
85	C₂²⋊C₄×C₆₁	—	2	32	no	—
86	C₂³×C₆₁	—	2	12	no	—
87	C₂³×C₆₁	—	2	24	no	—
88	C₂³×C₆₁	—	4	16	no	—
89	C₂³×C₆₁	#121	1	4	yes	—
90	C₂³×C₆₁	#137	1	8	yes	—
91	C₂³×C₆₁	C₂²×D₄	1	4	yes	—
92	C₂³×C₆₁	C₂²×D₄	1	3	yes	—
93	Q₈×C₆₁	—	2	8	no	—
94	Q₈×C₆₁	—	2	8	no	—
95	Q₈×C₆₁	—	4	24	no	—
96	C₂×C₄×C₆₁	—	2	12	no	—
97	C₂×C₄×C₆₁	—	2	24	no	—
98	C₂×C₄×C₆₁	—	2	4	no	—
99	C₂×C₄×C₆₁	—	2	4	no	—
100	C₂×C₄×C₆₁	—	2	8	no	—
101	C₂×C₄×C₆₁	—	2	8	no	—
102	C₂×C₄×C₆₁	—	4	24	no	—
103	C₂×C₄×C₆₁	—	4	24	no	—
104	C₂×C₄×C₆₁	#121	1	12	yes	—
105	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
106	C₂×C₄×C₆₁	C₂²×D₄	1	3	yes	—
107	C₂×C₄×C₆₁	C₂²×D₄	1	6	yes	—
108	C₂×C₄×C₆₁	C₂×#41	1	6	yes	—
109	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
110	C₈×C₆₁	—	2	4	no	—
111	C₈×C₆₁	C₂²×D₄	1	8	yes	—
112	C₂²×#470	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₆₁	—	2	4	no	—
114	C₂²×C₆₁	—	2	8	no	—
115	C₂²×C₆₁	—	4	24	no	—
116	C₂²×C₆₁	#470	1	16	yes	direct product factor
117	C₂²×C₆₁	C₂×#121	1	6	yes	—
118	C₂²×C₆₁	C₂×#137	1	12	yes	—
119	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
120	C₄×C₆₁	—	2	4	no	—
121	C₄×C₆₁	—	4	8	no	—
122	C₄×C₆₁	—	4	8	no	—
123	C₄×C₆₁	C₂×#121	1	6	yes	—
124	C₄×C₆₁	C₂³×D₄	1	1	yes	characteristic
125	C₄×C₆₁	C₂²×#41	1	1	yes	characteristic
126	C₂³×Q₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
127	C₂³×M₁₆	C₂×C₆₁	1	1	yes	characteristic
128	C₂²×C₂²⋊Q₈	C₂×C₆₁	1	2	yes	—
129	C₂×#470	C₂×C₆₁	1	24	yes	direct product factor
130	C₂²×#106	C₂×C₆₁	1	1	yes	characteristic, complemented
131	C₂²×#103	C₂×C₆₁	1	2	yes	complemented
132	C₂×C₆₁	—	4	4	no	—
133	C₂×C₆₁	C₂×#470	1	12	yes	direct product factor
134	C₂×C₆₁	C₂²×#121	1	1	yes	characteristic
135	C₂×C₆₁	C₂²×#137	1	2	yes	—
136	C₂³×Q₈	C₂²×C₆₁	1	2	yes	—
137	C₂⁴×C₄	C₂²×C₆₁	1	1	yes	characteristic
138	C₂²×Q₁₆	—	2	8	no	—
139	C₂²×Q₁₆	C₂²×C₆₁	1	12	yes	complemented
140	C₂²×M₁₆	—	2	8	no	—
141	C₂²×M₁₆	C₂²×C₆₁	1	12	yes	—
142	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
143	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
144	C₂×C₂²⋊Q₈	—	2	12	no	—
145	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	24	yes	—
146	C₂²×C₄⋊C₄	—	2	2	no	—
147	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
148	C₂²×C₂²⋊C₄	—	2	2	no	—
149	#470	C₂²×C₆₁	1	64	yes	direct product factor
150	C₂×#106	C₂²×C₆₁	1	12	yes	complemented
151	C₂×#103	C₂²×C₆₁	1	24	yes	complemented
152	C₆₁	C₂²×#470	1	1	yes	characteristic, direct product factor, Sylow
153	C₂⁵	D₄×C₆₁	1	1	yes	characteristic
154	C₂²×Q₈	—	2	6	no	—
155	C₂²×Q₈	—	2	8	no	—
156	C₂²×Q₈	—	2	8	no	—
157	C₂²×Q₈	C₂³×C₆₁	1	12	yes	—
158	C₂³×C₄	—	2	4	no	—
159	C₂³×C₄	—	2	4	no	—
160	C₂³×C₄	—	2	2	no	—
161	C₂³×C₄	—	2	2	no	—
162	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
163	C₂³×C₄	D₄×C₆₁	1	6	yes	—
164	C₂³×C₄	C₂³×C₆₁	1	6	yes	—
165	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
166	C₂×Q₁₆	—	2	48	no	—
167	C₂×Q₁₆	C₂³×C₆₁	1	16	yes	complemented
168	C₂×M₁₆	—	2	48	no	—
169	C₂×M₁₆	C₂³×C₆₁	1	16	yes	—
170	C₂²×C₈	—	2	4	no	—
171	C₂²×C₈	D₄×C₆₁	1	8	yes	—
172	C₂²×C₈	C₂³×C₆₁	1	6	yes	—
173	C₂²×C₈	C₂³×C₆₁	1	6	yes	—
174	C₂²⋊Q₈	—	2	48	no	—
175	C₂²⋊Q₈	C₂³×C₆₁	1	32	yes	—
176	C₂×C₄⋊C₄	—	2	24	no	—
177	C₂×C₄⋊C₄	—	2	6	no	—
178	C₂×C₄⋊C₄	C₂³×C₆₁	1	12	yes	—
179	C₂×C₂²⋊C₄	—	2	24	no	—
180	#106	C₂³×C₆₁	1	16	yes	complemented
181	#103	C₂³×C₆₁	1	32	yes	complemented
182	C₂⁴	—	2	4	no	—
183	C₂⁴	—	2	8	no	—
184	C₂⁴	C₂×D₄×C₆₁	1	6	yes	—
185	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
186	C₂×Q₈	—	2	12	no	—
187	C₂×Q₈	—	2	24	no	—
188	C₂×Q₈	—	2	24	no	—
189	C₂×Q₈	—	4	24	no	—
190	C₂×Q₈	C₂⁴×C₆₁	1	8	yes	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	24	no	—
193	C₂²×C₄	—	2	12	no	—
194	C₂²×C₄	—	2	12	no	—
195	C₂²×C₄	—	2	12	no	—
196	C₂²×C₄	—	2	12	no	—
197	C₂²×C₄	—	4	8	no	—
198	C₂²×C₄	—	4	8	no	—
199	C₂²×C₄	C₂×D₄×C₆₁	1	6	yes	—
200	C₂²×C₄	C₂×D₄×C₆₁	1	12	yes	—
201	C₂²×C₄	C₂×D₄×C₆₁	1	3	yes	—
202	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
203	C₂²×C₄	#41×C₆₁	1	4	yes	—
204	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
205	C₂²×C₄	C₂⁴×C₆₁	1	3	yes	—
206	Q₁₆	—	2	32	no	—
207	M₁₆	—	2	32	no	—
208	C₂×C₈	—	2	12	no	—
209	C₂×C₈	C₂×D₄×C₆₁	1	24	yes	—
210	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
211	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
212	C₄⋊C₄	—	2	32	no	—
213	C₄⋊C₄	—	2	12	no	—
214	C₄⋊C₄	C₂⁴×C₆₁	1	8	yes	—
215	C₂²⋊C₄	—	2	32	no	—
216	C₂³	—	2	24	no	—
217	C₂³	—	2	12	no	—
218	C₂³	—	4	16	no	—
219	C₂³	#121×C₆₁	1	4	yes	—
220	C₂³	#137×C₆₁	1	8	yes	—
221	C₂³	C₂²×D₄×C₆₁	1	4	yes	—
222	C₂³	C₂²×D₄×C₆₁	1	3	yes	—
223	Q₈	—	2	8	no	—
224	Q₈	—	2	8	no	—
225	Q₈	—	4	24	no	—
226	C₂×C₄	—	2	12	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	24	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	4	24	no	—
233	C₂×C₄	—	4	24	no	—
234	C₂×C₄	#121×C₆₁	1	12	yes	—
235	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
236	C₂×C₄	C₂²×D₄×C₆₁	1	6	yes	—
237	C₂×C₄	C₂²×D₄×C₆₁	1	3	yes	—
238	C₂×C₄	C₂×#41×C₆₁	1	6	yes	—
239	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
240	C₈	—	2	4	no	—
241	C₈	C₂²×D₄×C₆₁	1	8	yes	—
242	C₂²	—	2	8	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	4	24	no	—
245	C₂²	C₆₁×#470	1	16	yes	direct product factor
246	C₂²	C₂×#121×C₆₁	1	6	yes	—
247	C₂²	C₂×#137×C₆₁	1	12	yes	—
248	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
249	C₄	—	2	4	no	—
250	C₄	—	4	8	no	—
251	C₄	—	4	8	no	—
252	C₄	C₂×#121×C₆₁	1	6	yes	—
253	C₄	C₂³×D₄×C₆₁	1	1	yes	characteristic
254	C₄	C₂²×#41×C₆₁	1	1	yes	characteristic
255	C₂	—	4	4	no	—
256	C₂	C₂×C₆₁×#470	1	12	yes	direct product factor
257	C₂	C₂²×#121×C₆₁	1	1	yes	characteristic
258	C₂	C₂²×#137×C₆₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
prime radical	C₂⁵×C₆₁	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₁ ◁ C₂⁴×C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₁₆×C₆₁	C₂	semidirect
4. C₂×C₆₁×#470	C₂	direct (N × H)
5. C₂²×#106×C₆₁	C₂	semidirect
6. C₂²×#103×C₆₁	C₂	semidirect
10. C₂²×Q₁₆×C₆₁	C₂²	semidirect
20. C₆₁×#470	C₂²	direct (N × H)
21. C₂×#106×C₆₁	C₂²	semidirect
22. C₂×#103×C₆₁	C₂²	semidirect
37. C₂×Q₁₆×C₆₁	C₂³	semidirect
50. #106×C₆₁	C₂³	semidirect
51. #103×C₆₁	C₂³	semidirect
112. C₂²×#470	C₆₁	direct (N × H)
116. C₂²×C₆₁	#470	direct (N × H)
126. C₂³×Q₁₆	C₂×C₆₁	semidirect
129. C₂×#470	C₂×C₆₁	direct (N × H)
130. C₂²×#106	C₂×C₆₁	semidirect
131. C₂²×#103	C₂×C₆₁	semidirect
133. C₂×C₆₁	C₂×#470	direct (N × H)
139. C₂²×Q₁₆	C₂²×C₆₁	semidirect
149. #470	C₂²×C₆₁	direct (N × H)
150. C₂×#106	C₂²×C₆₁	semidirect
151. C₂×#103	C₂²×C₆₁	semidirect
152. C₆₁	C₂²×#470	direct (N × H)
167. C₂×Q₁₆	C₂³×C₆₁	semidirect
180. #106	C₂³×C₆₁	semidirect
181. #103	C₂³×C₆₁	semidirect
245. C₂²	C₆₁×#470	direct (N × H)
256. C₂	C₂×C₆₁×#470	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₁₆×C₆₁	2
C₂³×M₁₆×C₆₁	2
C₂²×C₂²⋊Q₈×C₆₁	2
C₂×C₆₁×#470	2
C₂²×#106×C₆₁	2
C₂²×#103×C₆₁	2
C₂²×#470	61

Subgroups — C22×C61×#470 #29563873

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₆₁×#470 #29563873