Subgroups — C₆₁×#29876 #29533309

Order 15616.

818 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₆₁×#431	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2857	C₂	1	2	yes	maximal
4	C₆₁×#2086	C₂	1	4	yes	maximal, complemented
5	C₆₁×#2075	C₂	1	4	yes	maximal
6	C₆₁×#1990	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1834	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
9	C₂×C₂²⋊Q₈×C₆₁	—	2	2	no	—
10	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
11	C₆₁×#500	—	2	4	no	—
12	C₆₁×#431	—	2	4	no	—
13	C₂×#107×C₆₁	C₂²	1	4	yes	—
14	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₆₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#391	—	2	2	no	—
19	C₆₁×#389	—	2	2	no	—
20	C₆₁×#389	C₂²	1	4	yes	—
21	C₆₁×#384	C₂²	1	2	yes	—
22	C₆₁×#380	—	2	2	no	—
23	C₆₁×#380	C₂²	1	4	yes	—
24	C₆₁×#340	C₂²	1	4	yes	—
25	C₆₁×#340	C₂²	1	8	yes	—
26	C₂²×Q₈×C₆₁	—	2	2	no	—
27	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
31	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
32	C₄⋊Q₈×C₆₁	—	2	4	no	—
33	C₄⋊Q₈×C₆₁	—	2	4	no	—
34	C₂²⋊Q₈×C₆₁	—	4	8	no	—
35	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
39	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
41	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
42	C₂×C₄²×C₆₁	—	2	2	no	—
43	C₂×C₄²×C₆₁	—	2	2	no	—
44	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
48	#107×C₆₁	—	2	8	no	—
49	C₄⋊C₈×C₆₁	—	2	4	no	—
50	C₂²⋊C₈×C₆₁	—	2	4	no	—
51	C₄×C₈×C₆₁	—	2	2	no	—
52	C₄×C₈×C₆₁	D₄	1	4	yes	—
53	#95×C₆₁	—	2	4	no	—
54	#95×C₆₁	—	2	4	no	—
55	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₂×Q₈×C₆₁	—	4	4	no	—
57	C₂×Q₈×C₆₁	—	4	4	no	—
58	C₂×Q₈×C₆₁	—	4	4	no	—
59	C₂²×C₄×C₆₁	—	2	2	no	—
60	C₂²×C₄×C₆₁	—	2	2	no	—
61	C₂²×C₄×C₆₁	—	2	2	no	—
62	C₂²×C₄×C₆₁	—	2	4	no	—
63	C₂²×C₄×C₆₁	—	2	1	no	—
64	C₂²×C₄×C₆₁	—	2	1	no	—
65	C₂²×C₄×C₆₁	—	2	4	no	—
66	C₂²×C₄×C₆₁	—	2	4	no	—
67	C₂²×C₄×C₆₁	—	2	4	no	—
68	C₂²×C₄×C₆₁	D₈	1	4	yes	—
69	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
75	C₂×C₈×C₆₁	—	2	4	no	—
76	C₂×C₈×C₆₁	—	2	2	no	—
77	C₂×C₈×C₆₁	—	2	4	no	—
78	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
79	C₄⋊C₄×C₆₁	—	2	4	no	—
80	C₄⋊C₄×C₆₁	—	2	4	no	—
81	C₄⋊C₄×C₆₁	—	2	8	no	—
82	C₄⋊C₄×C₆₁	—	4	4	no	—
83	C₄⋊C₄×C₆₁	—	4	4	no	—
84	C₄⋊C₄×C₆₁	—	4	8	no	—
85	C₄⋊C₄×C₆₁	—	4	4	no	—
86	C₂²⋊C₄×C₆₁	—	4	8	no	—
87	C₄²×C₆₁	—	2	2	no	—
88	C₄²×C₆₁	—	2	2	no	—
89	C₄²×C₆₁	—	2	1	no	—
90	C₄²×C₆₁	—	2	1	no	—
91	C₄²×C₆₁	C₂×D₄	1	2	yes	—
92	C₄²×C₆₁	#41	1	2	yes	—
93	C₂³×C₆₁	—	2	1	no	—
94	C₂³×C₆₁	—	2	2	no	—
95	C₂³×C₆₁	—	2	1	no	—
96	C₂³×C₆₁	—	2	2	no	—
97	C₂³×C₆₁	C₂×D₈	1	2	yes	—
98	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
99	Q₈×C₆₁	—	8	8	no	—
100	C₂×C₄×C₆₁	—	2	1	no	—
101	C₂×C₄×C₆₁	—	2	1	no	—
102	C₂×C₄×C₆₁	—	2	8	no	—
103	C₂×C₄×C₆₁	—	2	1	no	—
104	C₂×C₄×C₆₁	—	2	1	no	—
105	C₂×C₄×C₆₁	—	2	2	no	—
106	C₂×C₄×C₆₁	—	2	2	no	—
107	C₂×C₄×C₆₁	—	4	2	no	—
108	C₂×C₄×C₆₁	—	4	2	no	—
109	C₂×C₄×C₆₁	—	4	4	no	—
110	C₂×C₄×C₆₁	—	4	4	no	—
111	C₂×C₄×C₆₁	—	4	4	no	—
112	C₂×C₄×C₆₁	—	4	4	no	—
113	C₂×C₄×C₆₁	—	4	4	no	—
114	C₂×C₄×C₆₁	—	4	4	no	—
115	C₂×C₄×C₆₁	—	4	4	no	—
116	C₂×C₄×C₆₁	#121	1	2	yes	—
117	C₂×C₄×C₆₁	#121	1	2	yes	—
118	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₆₁	#135	1	2	yes	—
121	C₂×C₄×C₆₁	#137	1	4	yes	—
122	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₆₁	2₋¹⁺⁴	1	4	yes	—
125	C₈×C₆₁	—	2	4	no	—
126	C₈×C₆₁	—	4	2	no	—
127	#29876	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂²×C₆₁	—	2	4	no	—
129	C₂²×C₆₁	—	2	2	no	—
130	C₂²×C₆₁	—	4	1	no	—
131	C₂²×C₆₁	—	4	2	no	—
132	C₂²×C₆₁	—	4	1	no	—
133	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
134	C₂²×C₆₁	#536	1	2	yes	—
135	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
136	C₂²×C₆₁	C₂×#135	1	1	yes	characteristic
137	C₂²×C₆₁	C₂×#137	1	2	yes	—
138	C₄×C₆₁	—	2	2	no	—
139	C₄×C₆₁	—	4	1	no	—
140	C₄×C₆₁	—	4	1	no	—
141	C₄×C₆₁	—	8	4	no	—
142	C₄×C₆₁	—	8	4	no	—
143	C₄×C₆₁	#466	1	4	yes	—
144	C₄×C₆₁	#470	1	4	yes	—
145	C₄×C₆₁	#536	1	2	yes	—
146	C₄×C₆₁	#541	1	2	yes	—
147	C₂×#500	C₂×C₆₁	1	1	yes	characteristic, complemented
148	C₂×#431	C₂×C₆₁	1	1	yes	characteristic
149	#2857	C₂×C₆₁	1	2	yes	—
150	#2086	C₂×C₆₁	1	4	yes	complemented
151	#2075	C₂×C₆₁	1	4	yes	—
152	#1990	C₂×C₆₁	1	2	yes	complemented
153	#1834	C₂×C₆₁	1	1	yes	characteristic, complemented
154	C₂×C₆₁	—	4	2	no	—
155	C₂×C₆₁	#2621	1	1	yes	characteristic
156	C₂×C₆₁	C₂×#466	1	1	yes	characteristic
157	C₂×C₆₁	C₂×#470	1	1	yes	characteristic
158	C₂×C₆₁	#3226	1	2	yes	—
159	C₂×C₆₁	#3231	1	2	yes	—
160	C₂×C₄⋊Q₈	C₂²×C₆₁	1	2	yes	—
161	C₂×C₂²⋊Q₈	—	2	2	no	—
162	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
163	#500	—	2	4	no	—
164	#431	—	2	4	no	—
165	C₂×#107	C₂²×C₆₁	1	4	yes	—
166	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
167	C₂×C₂²⋊C₈	C₂²×C₆₁	1	2	yes	—
168	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
169	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
170	#391	—	2	2	no	—
171	#389	—	2	2	no	—
172	#389	C₂²×C₆₁	1	4	yes	—
173	#384	C₂²×C₆₁	1	2	yes	—
174	#380	—	2	2	no	—
175	#380	C₂²×C₆₁	1	4	yes	—
176	#340	C₂²×C₆₁	1	4	yes	—
177	#340	C₂²×C₆₁	1	8	yes	—
178	C₆₁	#29876	1	1	yes	characteristic, direct product factor, Sylow
179	C₂²×Q₈	—	2	2	no	—
180	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
181	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
182	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
183	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
184	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
185	C₄⋊Q₈	—	2	4	no	—
186	C₄⋊Q₈	—	2	4	no	—
187	C₂²⋊Q₈	—	4	8	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	4	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
193	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
194	C₂×C₂²⋊C₄	—	2	4	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
198	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
199	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
200	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
201	#107	—	2	8	no	—
202	C₄⋊C₈	—	2	4	no	—
203	C₂²⋊C₈	—	2	4	no	—
204	C₄×C₈	—	2	2	no	—
205	C₄×C₈	D₄×C₆₁	1	4	yes	—
206	#95	—	2	4	no	—
207	#95	—	2	4	no	—
208	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
209	C₂×Q₈	—	4	4	no	—
210	C₂×Q₈	—	4	4	no	—
211	C₂×Q₈	—	4	4	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	D₈×C₆₁	1	4	yes	—
222	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
227	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₄⋊C₄	—	4	8	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₂²⋊C₄	—	4	8	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	2	1	no	—
244	C₄²	C₂×D₄×C₆₁	1	2	yes	—
245	C₄²	#41×C₆₁	1	2	yes	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	1	no	—
250	C₂³	C₂×D₈×C₆₁	1	2	yes	—
251	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
252	Q₈	—	8	8	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	#121×C₆₁	1	2	yes	—
270	C₂×C₄	#121×C₆₁	1	2	yes	—
271	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
272	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
273	C₂×C₄	#135×C₆₁	1	2	yes	—
274	C₂×C₄	#137×C₆₁	1	4	yes	—
275	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
276	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
277	C₂×C₄	2₋¹⁺⁴×C₆₁	1	4	yes	—
278	C₈	—	2	4	no	—
279	C₈	—	4	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	2	no	—
285	C₂²	C₂×#121×C₆₁	1	1	yes	characteristic
286	C₂²	C₆₁×#536	1	2	yes	—
287	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
288	C₂²	C₂×#135×C₆₁	1	1	yes	characteristic
289	C₂²	C₂×#137×C₆₁	1	2	yes	—
290	C₄	—	2	2	no	—
291	C₄	—	4	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	8	4	no	—
294	C₄	—	8	4	no	—
295	C₄	C₆₁×#466	1	4	yes	—
296	C₄	C₆₁×#470	1	4	yes	—
297	C₄	C₆₁×#536	1	2	yes	—
298	C₄	C₆₁×#541	1	2	yes	—
299	C₂	—	4	2	no	—
300	C₂	C₆₁×#2621	1	1	yes	characteristic
301	C₂	C₂×C₆₁×#466	1	1	yes	characteristic
302	C₂	C₂×C₆₁×#470	1	1	yes	characteristic
303	C₂	C₆₁×#3226	1	2	yes	—
304	C₂	C₆₁×#3231	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂⁴×C₆₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂²×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×#500	C₂	semidirect
4. C₆₁×#2086	C₂	semidirect
6. C₆₁×#1990	C₂	semidirect
7. C₆₁×#1834	C₂	semidirect
127. #29876	C₆₁	direct (N × H)
147. C₂×#500	C₂×C₆₁	semidirect
150. #2086	C₂×C₆₁	semidirect
152. #1990	C₂×C₆₁	semidirect
153. #1834	C₂×C₆₁	semidirect
178. C₆₁	#29876	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#500	2
C₂×C₆₁×#431	2
C₆₁×#2857	2
C₆₁×#2086	2
C₆₁×#2075	2
C₆₁×#1990	2
C₆₁×#1834	2
#29876	61

Subgroups — C61×#29876 #29533309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#29876 #29533309