Subgroups — C₆₁×#33717 #29537150

Order 15616.

918 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#509	C₂	1	4	yes	maximal, complemented
2	C₆₁×#3147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₆₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2250	C₂	1	4	yes	maximal, complemented
5	C₆₁×#2248	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2246	C₂	1	2	yes	maximal, complemented
7	C₄×M₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₆₁	—	2	1	no	—
9	C₂×#135×C₆₁	C₂²	1	2	yes	—
10	C₂²×D₈×C₆₁	—	2	1	no	—
11	C₂²×D₈×C₆₁	C₂²	1	2	yes	—
12	C₆₁×#532	—	2	1	no	—
13	C₂×#127×C₆₁	—	2	1	no	—
14	C₆₁×#509	—	2	16	no	—
15	C₂×SD₃₂×C₆₁	C₂²	1	8	yes	complemented
16	C₂×D₁₆×C₆₁	C₂²	1	8	yes	complemented
17	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
18	C₆₁×#495	—	2	1	no	—
19	C₆₁×#495	C₂²	1	2	yes	complemented
20	C₆₁×#494	C₂²	1	1	yes	characteristic, complemented
21	C₆₁×#493	—	2	1	no	—
22	C₆₁×#493	—	2	1	no	—
23	C₆₁×#493	C₂²	1	2	yes	—
24	C₆₁×#493	C₂²	1	2	yes	—
25	C₆₁×#493	C₂²	1	1	yes	characteristic, complemented
26	C₆₁×#492	—	2	1	no	—
27	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₆₁×#346	C₂²	1	2	yes	complemented
29	C₄×C₁₆×C₆₁	C₂²	1	2	yes	complemented
30	C₂×#41×C₆₁	—	2	2	no	—
31	C₂²×D₄×C₆₁	—	2	2	no	—
32	C₂²×D₄×C₆₁	—	4	2	no	—
33	#135×C₆₁	—	2	8	no	—
34	#135×C₆₁	—	4	4	no	—
35	C₂×Q₁₆×C₆₁	—	2	1	no	—
36	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
37	C₂×SD₁₆×C₆₁	—	2	2	no	—
38	C₂×SD₁₆×C₆₁	—	2	2	no	—
39	C₂×D₈×C₆₁	—	2	8	no	—
40	C₂×D₈×C₆₁	—	2	2	no	—
41	C₂×D₈×C₆₁	—	2	2	no	—
42	C₂×D₈×C₆₁	—	2	2	no	—
43	C₂×D₈×C₆₁	—	2	1	no	—
44	C₂×D₈×C₆₁	—	4	4	no	—
45	C₂×D₈×C₆₁	C₂³	1	4	yes	—
46	C₂×D₈×C₆₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₆₁	—	2	1	no	—
50	#127×C₆₁	—	2	2	no	—
51	#127×C₆₁	—	2	2	no	—
52	#127×C₆₁	—	2	1	no	—
53	#127×C₆₁	—	4	2	no	—
54	#124×C₆₁	—	2	2	no	—
55	#121×C₆₁	—	4	2	no	—
56	C₄×D₄×C₆₁	—	4	2	no	—
57	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
58	SD₃₂×C₆₁	—	2	16	no	—
59	D₁₆×C₆₁	—	2	16	no	—
60	M₃₂×C₆₁	D₄	1	8	yes	complemented
61	C₂×C₁₆×C₆₁	C₂³	1	4	yes	—
62	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
63	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
64	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
66	C₂⁴×C₆₁	—	4	2	no	—
67	#41×C₆₁	—	4	4	no	—
68	#41×C₆₁	—	4	4	no	—
69	C₂×Q₈×C₆₁	—	2	2	no	—
70	C₂×D₄×C₆₁	—	2	2	no	—
71	C₂×D₄×C₆₁	—	2	4	no	—
72	C₂×D₄×C₆₁	—	4	2	no	—
73	C₂×D₄×C₆₁	—	4	4	no	—
74	C₂×D₄×C₆₁	—	4	4	no	—
75	C₂×D₄×C₆₁	—	4	2	no	—
76	C₂×D₄×C₆₁	—	4	4	no	—
77	C₂×D₄×C₆₁	—	4	4	no	—
78	C₂×D₄×C₆₁	—	4	2	no	—
79	C₂×D₄×C₆₁	—	8	8	no	—
80	C₂²×C₄×C₆₁	—	4	2	no	—
81	C₂²×C₄×C₆₁	D₈	1	2	yes	—
82	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
83	Q₁₆×C₆₁	—	2	4	no	—
84	Q₁₆×C₆₁	—	4	2	no	—
85	SD₁₆×C₆₁	—	4	4	no	—
86	SD₁₆×C₆₁	—	4	4	no	—
87	D₈×C₆₁	—	2	4	no	—
88	D₈×C₆₁	—	2	8	no	—
89	D₈×C₆₁	—	4	4	no	—
90	D₈×C₆₁	—	4	4	no	—
91	D₈×C₆₁	—	4	4	no	—
92	D₈×C₆₁	—	4	2	no	—
93	C₂×C₈×C₆₁	D₈	1	4	yes	—
94	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
96	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₆₁	—	4	2	no	—
100	C₂²⋊C₄×C₆₁	—	4	4	no	—
101	C₄²×C₆₁	D₈	1	2	yes	—
102	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
104	C₁₆×C₆₁	C₂×D₄	1	8	yes	complemented
105	C₂³×C₆₁	—	4	2	no	—
106	C₂³×C₆₁	—	4	4	no	—
107	C₂³×C₆₁	—	8	8	no	—
108	C₂³×C₆₁	—	8	4	no	—
109	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
110	Q₈×C₆₁	—	4	2	no	—
111	Q₈×C₆₁	—	4	2	no	—
112	D₄×C₆₁	—	4	4	no	—
113	D₄×C₆₁	—	4	2	no	—
114	D₄×C₆₁	—	4	4	no	—
115	D₄×C₆₁	—	4	2	no	—
116	D₄×C₆₁	—	8	8	no	—
117	D₄×C₆₁	—	8	4	no	—
118	D₄×C₆₁	—	8	4	no	—
119	D₄×C₆₁	—	8	4	no	—
120	D₄×C₆₁	—	8	8	no	—
121	C₂×C₄×C₆₁	—	2	4	no	—
122	C₂×C₄×C₆₁	—	4	2	no	—
123	C₂×C₄×C₆₁	—	8	4	no	—
124	C₂×C₄×C₆₁	#127	1	2	yes	—
125	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
127	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₆₁	#127	1	2	yes	—
131	C₈×C₆₁	C₂×D₈	1	2	yes	—
132	C₈×C₆₁	C₂×D₈	1	2	yes	—
133	C₈×C₆₁	C₂²×D₄	1	2	yes	—
134	#33717	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂²×C₆₁	—	2	2	no	—
136	C₂²×C₆₁	—	8	2	no	—
137	C₂²×C₆₁	—	8	4	no	—
138	C₂²×C₆₁	—	8	8	no	—
139	C₂²×C₆₁	—	8	4	no	—
140	C₂²×C₆₁	—	16	8	no	—
141	C₂²×C₆₁	#493	1	2	yes	—
142	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
143	C₄×C₆₁	—	2	2	no	—
144	C₄×C₆₁	—	8	2	no	—
145	C₄×C₆₁	#493	1	2	yes	—
146	C₄×C₆₁	#509	1	4	yes	complemented
147	C₄×C₆₁	C₂×#127	1	1	yes	characteristic
148	C₄×C₆₁	C₂²×D₈	1	1	yes	characteristic
149	C₂×#509	C₂×C₆₁	1	4	yes	complemented
150	#3147	C₂×C₆₁	1	1	yes	characteristic, complemented
151	C₂×#493	C₂×C₆₁	1	1	yes	characteristic, complemented
152	#2250	C₂×C₆₁	1	4	yes	complemented
153	#2248	C₂×C₆₁	1	2	yes	complemented
154	#2246	C₂×C₆₁	1	2	yes	complemented
155	C₄×M₃₂	C₂×C₆₁	1	1	yes	characteristic, complemented
156	C₂×C₆₁	—	2	2	no	—
157	C₂×C₆₁	—	16	2	no	—
158	C₂×C₆₁	—	16	4	no	—
159	C₂×C₆₁	C₂×#493	1	1	yes	characteristic
160	C₂×C₆₁	C₂×#509	1	2	yes	—
161	C₂×#135	—	2	1	no	—
162	C₂×#135	C₂²×C₆₁	1	2	yes	—
163	C₂²×D₈	—	2	1	no	—
164	C₂²×D₈	C₂²×C₆₁	1	2	yes	—
165	#532	—	2	1	no	—
166	C₂×#127	—	2	1	no	—
167	#509	—	2	16	no	—
168	C₂×SD₃₂	C₂²×C₆₁	1	8	yes	complemented
169	C₂×D₁₆	C₂²×C₆₁	1	8	yes	complemented
170	C₂×M₃₂	C₂²×C₆₁	1	2	yes	—
171	#495	—	2	1	no	—
172	#495	C₂²×C₆₁	1	2	yes	complemented
173	#494	C₂²×C₆₁	1	1	yes	characteristic, complemented
174	#493	—	2	1	no	—
175	#493	—	2	1	no	—
176	#493	C₂²×C₆₁	1	2	yes	—
177	#493	C₂²×C₆₁	1	2	yes	—
178	#493	C₂²×C₆₁	1	1	yes	characteristic, complemented
179	#492	—	2	1	no	—
180	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
181	#346	C₂²×C₆₁	1	2	yes	complemented
182	C₄×C₁₆	C₂²×C₆₁	1	2	yes	complemented
183	C₆₁	#33717	1	1	yes	characteristic, direct product factor, Sylow
184	C₂×#41	—	2	2	no	—
185	C₂²×D₄	—	2	2	no	—
186	C₂²×D₄	—	4	2	no	—
187	#135	—	2	8	no	—
188	#135	—	4	4	no	—
189	C₂×Q₁₆	—	2	1	no	—
190	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
191	C₂×SD₁₆	—	2	2	no	—
192	C₂×SD₁₆	—	2	2	no	—
193	C₂×D₈	—	2	8	no	—
194	C₂×D₈	—	2	2	no	—
195	C₂×D₈	—	2	2	no	—
196	C₂×D₈	—	2	2	no	—
197	C₂×D₈	—	2	1	no	—
198	C₂×D₈	—	4	4	no	—
199	C₂×D₈	C₂³×C₆₁	1	4	yes	—
200	C₂×D₈	C₂³×C₆₁	1	2	yes	—
201	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
203	C₄⋊Q₈	—	2	1	no	—
204	#127	—	2	2	no	—
205	#127	—	2	2	no	—
206	#127	—	2	1	no	—
207	#127	—	4	2	no	—
208	#124	—	2	2	no	—
209	#121	—	4	2	no	—
210	C₄×D₄	—	4	2	no	—
211	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
212	SD₃₂	—	2	16	no	—
213	D₁₆	—	2	16	no	—
214	M₃₂	D₄×C₆₁	1	8	yes	complemented
215	C₂×C₁₆	C₂³×C₆₁	1	4	yes	—
216	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
217	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
218	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
219	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
220	C₂⁴	—	4	2	no	—
221	#41	—	4	4	no	—
222	#41	—	4	4	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	2	4	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	4	no	—
228	C₂×D₄	—	4	4	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂×D₄	—	4	4	no	—
231	C₂×D₄	—	4	4	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	8	8	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	D₈×C₆₁	1	2	yes	—
236	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
237	Q₁₆	—	2	4	no	—
238	Q₁₆	—	4	2	no	—
239	SD₁₆	—	4	4	no	—
240	SD₁₆	—	4	4	no	—
241	D₈	—	2	4	no	—
242	D₈	—	2	8	no	—
243	D₈	—	4	4	no	—
244	D₈	—	4	4	no	—
245	D₈	—	4	4	no	—
246	D₈	—	4	2	no	—
247	C₂×C₈	D₈×C₆₁	1	4	yes	—
248	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
249	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
250	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
251	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
252	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
253	C₄⋊C₄	—	4	2	no	—
254	C₂²⋊C₄	—	4	4	no	—
255	C₄²	D₈×C₆₁	1	2	yes	—
256	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
257	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
258	C₁₆	C₂×D₄×C₆₁	1	8	yes	complemented
259	C₂³	—	4	2	no	—
260	C₂³	—	4	4	no	—
261	C₂³	—	8	8	no	—
262	C₂³	—	8	4	no	—
263	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
264	Q₈	—	4	2	no	—
265	Q₈	—	4	2	no	—
266	D₄	—	4	4	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	4	no	—
269	D₄	—	4	2	no	—
270	D₄	—	8	8	no	—
271	D₄	—	8	4	no	—
272	D₄	—	8	4	no	—
273	D₄	—	8	4	no	—
274	D₄	—	8	8	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	8	4	no	—
278	C₂×C₄	#127×C₆₁	1	2	yes	—
279	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
280	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
281	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
282	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
283	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
284	C₈	#127×C₆₁	1	2	yes	—
285	C₈	C₂×D₈×C₆₁	1	2	yes	—
286	C₈	C₂×D₈×C₆₁	1	2	yes	—
287	C₈	C₂²×D₄×C₆₁	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	8	2	no	—
290	C₂²	—	8	4	no	—
291	C₂²	—	8	8	no	—
292	C₂²	—	8	4	no	—
293	C₂²	—	16	8	no	—
294	C₂²	C₆₁×#493	1	2	yes	—
295	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	8	2	no	—
298	C₄	C₆₁×#493	1	2	yes	—
299	C₄	C₆₁×#509	1	4	yes	complemented
300	C₄	C₂×#127×C₆₁	1	1	yes	characteristic
301	C₄	C₂²×D₈×C₆₁	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	16	4	no	—
304	C₂	—	16	2	no	—
305	C₂	C₂×C₆₁×#493	1	1	yes	characteristic
306	C₂	C₂×C₆₁×#509	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×#509	C₂	semidirect
2. C₆₁×#3147	C₂	semidirect
3. C₂×C₆₁×#493	C₂	semidirect
4. C₆₁×#2250	C₂	semidirect
5. C₆₁×#2248	C₂	semidirect
6. C₆₁×#2246	C₂	semidirect
7. C₄×M₃₂×C₆₁	C₂	semidirect
15. C₂×SD₃₂×C₆₁	C₂²	semidirect
16. C₂×D₁₆×C₆₁	C₂²	semidirect
19. C₆₁×#495	C₂²	semidirect
20. C₆₁×#494	C₂²	semidirect
25. C₆₁×#493	C₂²	semidirect
28. C₆₁×#346	C₂²	semidirect
29. C₄×C₁₆×C₆₁	C₂²	semidirect
60. M₃₂×C₆₁	D₄	semidirect
104. C₁₆×C₆₁	C₂×D₄	semidirect
134. #33717	C₆₁	direct (N × H)
146. C₄×C₆₁	#509	semidirect
149. C₂×#509	C₂×C₆₁	semidirect
150. #3147	C₂×C₆₁	semidirect
151. C₂×#493	C₂×C₆₁	semidirect
152. #2250	C₂×C₆₁	semidirect
153. #2248	C₂×C₆₁	semidirect
154. #2246	C₂×C₆₁	semidirect
155. C₄×M₃₂	C₂×C₆₁	semidirect
168. C₂×SD₃₂	C₂²×C₆₁	semidirect
169. C₂×D₁₆	C₂²×C₆₁	semidirect
172. #495	C₂²×C₆₁	semidirect
173. #494	C₂²×C₆₁	semidirect
178. #493	C₂²×C₆₁	semidirect
181. #346	C₂²×C₆₁	semidirect
182. C₄×C₁₆	C₂²×C₆₁	semidirect
183. C₆₁	#33717	direct (N × H)
214. M₃₂	D₄×C₆₁	semidirect
258. C₁₆	C₂×D₄×C₆₁	semidirect
299. C₄	C₆₁×#509	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#509	2
C₆₁×#3147	2
C₂×C₆₁×#493	2
C₆₁×#2250	2
C₆₁×#2248	2
C₆₁×#2246	2
C₄×M₃₂×C₆₁	2
#33717	61

Subgroups — C61×#33717 #29537150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33717 #29537150