Subgroups — C₂²×C₆₁×#469 #29563872

Order 15616.

3566 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#121×C₆₁	C₂	1	2	yes	maximal, complemented
4	C₂×C₆₁×#469	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#106×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#102×C₆₁	C₂	1	2	yes	maximal, complemented
7	C₂³×D₄×C₆₁	—	2	2	no	—
8	C₂³×D₄×C₆₁	C₂²	1	2	yes	—
9	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
10	C₂²×D₈×C₆₁	—	2	8	no	—
11	C₂²×D₈×C₆₁	C₂²	1	12	yes	complemented
12	C₂²×M₁₆×C₆₁	—	2	8	no	—
13	C₂²×M₁₆×C₆₁	C₂²	1	12	yes	complemented
14	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#121×C₆₁	—	2	12	no	—
17	C₂×#121×C₆₁	C₂²	1	24	yes	complemented
18	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
19	C₂²×C₂²⋊C₄×C₆₁	—	2	2	no	—
20	C₆₁×#469	C₂²	1	64	yes	direct product factor
21	C₂×#106×C₆₁	C₂²	1	12	yes	complemented
22	C₂×#102×C₆₁	C₂²	1	24	yes	complemented
23	C₂⁵×C₆₁	—	2	2	no	—
24	C₂⁵×C₆₁	D₄	1	1	yes	characteristic
25	C₂²×D₄×C₆₁	—	2	24	no	—
26	C₂²×D₄×C₆₁	—	2	8	no	—
27	C₂²×D₄×C₆₁	—	2	6	no	—
28	C₂²×D₄×C₆₁	—	2	8	no	—
29	C₂²×D₄×C₆₁	—	4	16	no	—
30	C₂²×D₄×C₆₁	C₂³	1	12	yes	—
31	C₂³×C₄×C₆₁	—	2	4	no	—
32	C₂³×C₄×C₆₁	—	2	4	no	—
33	C₂³×C₄×C₆₁	—	2	2	no	—
34	C₂³×C₄×C₆₁	D₄	1	6	yes	—
35	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₆₁	C₂³	1	6	yes	—
37	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
38	C₂×D₈×C₆₁	—	2	48	no	—
39	C₂×D₈×C₆₁	C₂³	1	16	yes	complemented
40	C₂×M₁₆×C₆₁	—	2	48	no	—
41	C₂×M₁₆×C₆₁	C₂³	1	16	yes	complemented
42	C₂²×C₈×C₆₁	—	2	4	no	—
43	C₂²×C₈×C₆₁	D₄	1	8	yes	complemented
44	C₂²×C₈×C₆₁	C₂³	1	6	yes	—
45	C₂²×C₈×C₆₁	C₂³	1	6	yes	—
46	#121×C₆₁	—	2	48	no	—
47	#121×C₆₁	C₂³	1	32	yes	complemented
48	C₂×C₄⋊C₄×C₆₁	—	2	6	no	—
49	C₂×C₄⋊C₄×C₆₁	C₂³	1	12	yes	—
50	C₂×C₂²⋊C₄×C₆₁	—	2	24	no	—
51	#106×C₆₁	C₂³	1	16	yes	complemented
52	#102×C₆₁	C₂³	1	32	yes	complemented
53	C₂⁴×C₆₁	—	2	4	no	—
54	C₂⁴×C₆₁	—	2	8	no	—
55	C₂⁴×C₆₁	—	2	12	no	—
56	C₂⁴×C₆₁	—	4	8	no	—
57	C₂⁴×C₆₁	—	4	8	no	—
58	C₂⁴×C₆₁	—	4	8	no	—
59	C₂⁴×C₆₁	C₂×D₄	1	6	yes	—
60	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
61	C₂×D₄×C₆₁	—	2	32	no	—
62	C₂×D₄×C₆₁	—	2	24	no	—
63	C₂×D₄×C₆₁	—	2	12	no	—
64	C₂×D₄×C₆₁	—	2	24	no	—
65	C₂×D₄×C₆₁	—	4	96	no	—
66	C₂×D₄×C₆₁	—	4	24	no	—
67	C₂×D₄×C₆₁	C₂⁴	1	8	yes	—
68	C₂²×C₄×C₆₁	—	2	4	no	—
69	C₂²×C₄×C₆₁	—	2	24	no	—
70	C₂²×C₄×C₆₁	—	2	12	no	—
71	C₂²×C₄×C₆₁	—	2	12	no	—
72	C₂²×C₄×C₆₁	—	2	12	no	—
73	C₂²×C₄×C₆₁	—	4	8	no	—
74	C₂²×C₄×C₆₁	C₂×D₄	1	6	yes	—
75	C₂²×C₄×C₆₁	C₂×D₄	1	12	yes	—
76	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
77	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₆₁	#41	1	4	yes	—
79	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
80	C₂²×C₄×C₆₁	C₂⁴	1	3	yes	—
81	D₈×C₆₁	—	2	32	no	—
82	M₁₆×C₆₁	—	2	32	no	—
83	C₂×C₈×C₆₁	—	2	12	no	—
84	C₂×C₈×C₆₁	C₂×D₄	1	24	yes	complemented
85	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
86	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
87	C₄⋊C₄×C₆₁	—	2	12	no	—
88	C₄⋊C₄×C₆₁	C₂⁴	1	8	yes	—
89	C₂²⋊C₄×C₆₁	—	2	32	no	—
90	C₂³×C₆₁	—	2	12	no	—
91	C₂³×C₆₁	—	2	24	no	—
92	C₂³×C₆₁	—	2	8	no	—
93	C₂³×C₆₁	—	4	16	no	—
94	C₂³×C₆₁	—	4	24	no	—
95	C₂³×C₆₁	—	4	24	no	—
96	C₂³×C₆₁	—	4	24	no	—
97	C₂³×C₆₁	—	8	32	no	—
98	C₂³×C₆₁	#121	1	4	yes	—
99	C₂³×C₆₁	AGL(1,ℤ/8)	1	8	yes	—
100	C₂³×C₆₁	C₂²×D₄	1	4	yes	—
101	C₂³×C₆₁	C₂²×D₄	1	3	yes	—
102	D₄×C₆₁	—	2	8	no	—
103	D₄×C₆₁	—	2	8	no	—
104	D₄×C₆₁	—	4	64	no	—
105	D₄×C₆₁	—	4	24	no	—
106	C₂×C₄×C₆₁	—	2	12	no	—
107	C₂×C₄×C₆₁	—	2	24	no	—
108	C₂×C₄×C₆₁	—	2	4	no	—
109	C₂×C₄×C₆₁	—	2	4	no	—
110	C₂×C₄×C₆₁	—	2	8	no	—
111	C₂×C₄×C₆₁	—	4	24	no	—
112	C₂×C₄×C₆₁	#121	1	12	yes	—
113	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
114	C₂×C₄×C₆₁	C₂²×D₄	1	3	yes	—
115	C₂×C₄×C₆₁	C₂²×D₄	1	6	yes	—
116	C₂×C₄×C₆₁	C₂×#41	1	6	yes	—
117	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
118	C₈×C₆₁	—	2	4	no	—
119	C₈×C₆₁	C₂²×D₄	1	8	yes	complemented
120	C₂²×#469	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂²×C₆₁	—	2	4	no	—
122	C₂²×C₆₁	—	2	8	no	—
123	C₂²×C₆₁	—	4	24	no	—
124	C₂²×C₆₁	—	4	8	no	—
125	C₂²×C₆₁	—	4	8	no	—
126	C₂²×C₆₁	—	4	8	no	—
127	C₂²×C₆₁	—	8	48	no	—
128	C₂²×C₆₁	#469	1	16	yes	direct product factor
129	C₂²×C₆₁	C₂×#121	1	6	yes	—
130	C₂²×C₆₁	C₂×AGL(1,ℤ/8)	1	12	yes	—
131	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
132	C₄×C₆₁	—	2	4	no	—
133	C₄×C₆₁	—	4	8	no	—
134	C₄×C₆₁	C₂×#121	1	6	yes	—
135	C₄×C₆₁	C₂³×D₄	1	1	yes	characteristic
136	C₄×C₆₁	C₂²×#41	1	1	yes	characteristic
137	C₂³×D₈	C₂×C₆₁	1	1	yes	characteristic, complemented
138	C₂³×M₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
139	C₂²×#121	C₂×C₆₁	1	2	yes	complemented
140	C₂×#469	C₂×C₆₁	1	24	yes	direct product factor
141	C₂²×#106	C₂×C₆₁	1	1	yes	characteristic, complemented
142	C₂²×#102	C₂×C₆₁	1	2	yes	complemented
143	C₂×C₆₁	—	4	4	no	—
144	C₂×C₆₁	—	8	8	no	—
145	C₂×C₆₁	C₂×#469	1	12	yes	direct product factor
146	C₂×C₆₁	C₂²×#121	1	1	yes	characteristic
147	C₂×C₆₁	C₂²×AGL(1,ℤ/8)	1	2	yes	—
148	C₂³×D₄	—	2	2	no	—
149	C₂³×D₄	C₂²×C₆₁	1	2	yes	—
150	C₂⁴×C₄	C₂²×C₆₁	1	1	yes	characteristic
151	C₂²×D₈	—	2	8	no	—
152	C₂²×D₈	C₂²×C₆₁	1	12	yes	complemented
153	C₂²×M₁₆	—	2	8	no	—
154	C₂²×M₁₆	C₂²×C₆₁	1	12	yes	complemented
155	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
156	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
157	C₂×#121	—	2	12	no	—
158	C₂×#121	C₂²×C₆₁	1	24	yes	complemented
159	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
160	C₂²×C₂²⋊C₄	—	2	2	no	—
161	#469	C₂²×C₆₁	1	64	yes	direct product factor
162	C₂×#106	C₂²×C₆₁	1	12	yes	complemented
163	C₂×#102	C₂²×C₆₁	1	24	yes	complemented
164	C₆₁	C₂²×#469	1	1	yes	characteristic, direct product factor, Sylow
165	C₂⁵	—	2	2	no	—
166	C₂⁵	D₄×C₆₁	1	1	yes	characteristic
167	C₂²×D₄	—	2	24	no	—
168	C₂²×D₄	—	2	8	no	—
169	C₂²×D₄	—	2	6	no	—
170	C₂²×D₄	—	2	8	no	—
171	C₂²×D₄	—	4	16	no	—
172	C₂²×D₄	C₂³×C₆₁	1	12	yes	—
173	C₂³×C₄	—	2	4	no	—
174	C₂³×C₄	—	2	4	no	—
175	C₂³×C₄	—	2	2	no	—
176	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
177	C₂³×C₄	D₄×C₆₁	1	6	yes	—
178	C₂³×C₄	C₂³×C₆₁	1	6	yes	—
179	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
180	C₂×D₈	—	2	48	no	—
181	C₂×D₈	C₂³×C₆₁	1	16	yes	complemented
182	C₂×M₁₆	—	2	48	no	—
183	C₂×M₁₆	C₂³×C₆₁	1	16	yes	complemented
184	C₂²×C₈	—	2	4	no	—
185	C₂²×C₈	D₄×C₆₁	1	8	yes	complemented
186	C₂²×C₈	C₂³×C₆₁	1	6	yes	—
187	C₂²×C₈	C₂³×C₆₁	1	6	yes	—
188	#121	—	2	48	no	—
189	#121	C₂³×C₆₁	1	32	yes	complemented
190	C₂×C₄⋊C₄	—	2	6	no	—
191	C₂×C₄⋊C₄	C₂³×C₆₁	1	12	yes	—
192	C₂×C₂²⋊C₄	—	2	24	no	—
193	#106	C₂³×C₆₁	1	16	yes	complemented
194	#102	C₂³×C₆₁	1	32	yes	complemented
195	C₂⁴	—	2	4	no	—
196	C₂⁴	—	2	8	no	—
197	C₂⁴	—	2	12	no	—
198	C₂⁴	—	4	8	no	—
199	C₂⁴	—	4	8	no	—
200	C₂⁴	—	4	8	no	—
201	C₂⁴	C₂×D₄×C₆₁	1	6	yes	—
202	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
203	C₂×D₄	—	2	32	no	—
204	C₂×D₄	—	2	24	no	—
205	C₂×D₄	—	2	12	no	—
206	C₂×D₄	—	2	24	no	—
207	C₂×D₄	—	4	96	no	—
208	C₂×D₄	—	4	24	no	—
209	C₂×D₄	C₂⁴×C₆₁	1	8	yes	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	24	no	—
212	C₂²×C₄	—	2	12	no	—
213	C₂²×C₄	—	2	12	no	—
214	C₂²×C₄	—	2	12	no	—
215	C₂²×C₄	—	4	8	no	—
216	C₂²×C₄	C₂×D₄×C₆₁	1	6	yes	—
217	C₂²×C₄	C₂×D₄×C₆₁	1	12	yes	—
218	C₂²×C₄	C₂×D₄×C₆₁	1	3	yes	—
219	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
220	C₂²×C₄	#41×C₆₁	1	4	yes	—
221	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
222	C₂²×C₄	C₂⁴×C₆₁	1	3	yes	—
223	D₈	—	2	32	no	—
224	M₁₆	—	2	32	no	—
225	C₂×C₈	—	2	12	no	—
226	C₂×C₈	C₂×D₄×C₆₁	1	24	yes	complemented
227	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
228	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
229	C₄⋊C₄	—	2	12	no	—
230	C₄⋊C₄	C₂⁴×C₆₁	1	8	yes	—
231	C₂²⋊C₄	—	2	32	no	—
232	C₂³	—	2	24	no	—
233	C₂³	—	2	12	no	—
234	C₂³	—	2	8	no	—
235	C₂³	—	4	16	no	—
236	C₂³	—	4	24	no	—
237	C₂³	—	4	24	no	—
238	C₂³	—	4	24	no	—
239	C₂³	—	8	32	no	—
240	C₂³	#121×C₆₁	1	4	yes	—
241	C₂³	AGL(1,ℤ/8)×C₆₁	1	8	yes	—
242	C₂³	C₂²×D₄×C₆₁	1	4	yes	—
243	C₂³	C₂²×D₄×C₆₁	1	3	yes	—
244	D₄	—	2	8	no	—
245	D₄	—	2	8	no	—
246	D₄	—	4	64	no	—
247	D₄	—	4	24	no	—
248	C₂×C₄	—	2	12	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	24	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	4	24	no	—
254	C₂×C₄	#121×C₆₁	1	12	yes	—
255	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
256	C₂×C₄	C₂²×D₄×C₆₁	1	6	yes	—
257	C₂×C₄	C₂²×D₄×C₆₁	1	3	yes	—
258	C₂×C₄	C₂×#41×C₆₁	1	6	yes	—
259	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
260	C₈	—	2	4	no	—
261	C₈	C₂²×D₄×C₆₁	1	8	yes	complemented
262	C₂²	—	2	8	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	4	24	no	—
265	C₂²	—	4	8	no	—
266	C₂²	—	4	8	no	—
267	C₂²	—	4	8	no	—
268	C₂²	—	8	48	no	—
269	C₂²	C₆₁×#469	1	16	yes	direct product factor
270	C₂²	C₂×#121×C₆₁	1	6	yes	—
271	C₂²	C₂×AGL(1,ℤ/8)×C₆₁	1	12	yes	—
272	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	4	8	no	—
275	C₄	C₂×#121×C₆₁	1	6	yes	—
276	C₄	C₂³×D₄×C₆₁	1	1	yes	characteristic
277	C₄	C₂²×#41×C₆₁	1	1	yes	characteristic
278	C₂	—	4	4	no	—
279	C₂	—	8	8	no	—
280	C₂	C₂×C₆₁×#469	1	12	yes	direct product factor
281	C₂	C₂²×#121×C₆₁	1	1	yes	characteristic
282	C₂	C₂²×AGL(1,ℤ/8)×C₆₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
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×D₈×C₆₁	C₂	semidirect
2. C₂³×M₁₆×C₆₁	C₂	semidirect
3. C₂²×#121×C₆₁	C₂	semidirect
4. C₂×C₆₁×#469	C₂	direct (N × H)
5. C₂²×#106×C₆₁	C₂	semidirect
6. C₂²×#102×C₆₁	C₂	semidirect
11. C₂²×D₈×C₆₁	C₂²	semidirect
13. C₂²×M₁₆×C₆₁	C₂²	semidirect
17. C₂×#121×C₆₁	C₂²	semidirect
20. C₆₁×#469	C₂²	direct (N × H)
21. C₂×#106×C₆₁	C₂²	semidirect
22. C₂×#102×C₆₁	C₂²	semidirect
39. C₂×D₈×C₆₁	C₂³	semidirect
41. C₂×M₁₆×C₆₁	C₂³	semidirect
43. C₂²×C₈×C₆₁	D₄	semidirect
47. #121×C₆₁	C₂³	semidirect
51. #106×C₆₁	C₂³	semidirect
52. #102×C₆₁	C₂³	semidirect
84. C₂×C₈×C₆₁	C₂×D₄	semidirect
119. C₈×C₆₁	C₂²×D₄	semidirect
120. C₂²×#469	C₆₁	direct (N × H)
128. C₂²×C₆₁	#469	direct (N × H)
137. C₂³×D₈	C₂×C₆₁	semidirect
138. C₂³×M₁₆	C₂×C₆₁	semidirect
139. C₂²×#121	C₂×C₆₁	semidirect
140. C₂×#469	C₂×C₆₁	direct (N × H)
141. C₂²×#106	C₂×C₆₁	semidirect
142. C₂²×#102	C₂×C₆₁	semidirect
145. C₂×C₆₁	C₂×#469	direct (N × H)
152. C₂²×D₈	C₂²×C₆₁	semidirect
154. C₂²×M₁₆	C₂²×C₆₁	semidirect
158. C₂×#121	C₂²×C₆₁	semidirect
161. #469	C₂²×C₆₁	direct (N × H)
162. C₂×#106	C₂²×C₆₁	semidirect
163. C₂×#102	C₂²×C₆₁	semidirect
164. C₆₁	C₂²×#469	direct (N × H)
181. C₂×D₈	C₂³×C₆₁	semidirect
183. C₂×M₁₆	C₂³×C₆₁	semidirect
185. C₂²×C₈	D₄×C₆₁	semidirect
189. #121	C₂³×C₆₁	semidirect
193. #106	C₂³×C₆₁	semidirect
194. #102	C₂³×C₆₁	semidirect
226. C₂×C₈	C₂×D₄×C₆₁	semidirect
261. C₈	C₂²×D₄×C₆₁	semidirect
269. C₂²	C₆₁×#469	direct (N × H)
280. C₂	C₂×C₆₁×#469	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×D₈×C₆₁	2
C₂³×M₁₆×C₆₁	2
C₂²×#121×C₆₁	2
C₂×C₆₁×#469	2
C₂²×#106×C₆₁	2
C₂²×#102×C₆₁	2
C₂²×#469	61

Subgroups — C22×C61×#469 #29563872

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₆₁×#469 #29563872