Subgroups — C₆₁×#29886 #29533319

Order 15616.

620 conjugacy classes of subgroups.

The subgroup lattice has 300 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	C₆₁×#3079	C₂	1	2	yes	maximal
3	C₆₁×#2922	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#2089	C₂	1	4	yes	maximal, complemented
5	C₆₁×#2078	C₂	1	4	yes	maximal
6	C₆₁×#1997	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1837	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
9	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
10	C₆₁×#500	—	2	2	no	—
11	C₆₁×#500	—	2	2	no	—
12	C₆₁×#477	—	2	2	no	—
13	C₆₁×#476	—	2	2	no	—
14	C₆₁×#475	—	2	2	no	—
15	C₆₁×#474	—	2	2	no	—
16	C₆₁×#433	—	2	2	no	—
17	C₆₁×#431	—	2	2	no	—
18	C₆₁×#428	C₂²	1	4	yes	—
19	C₂×#107×C₆₁	C₂²	1	4	yes	—
20	C₆₁×#424	C₂²	1	2	yes	—
21	C₆₁×#423	C₂²	1	2	yes	—
22	C₆₁×#417	C₂²	1	4	yes	—
23	C₆₁×#408	C₂²	1	2	yes	—
24	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
26	C₆₁×#343	C₂²	1	8	yes	—
27	C₆₁×#341	C₂²	1	4	yes	—
28	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
29	C₂²×Q₈×C₆₁	—	2	2	no	—
30	C₂×M₁₆×C₆₁	C₂³	1	4	yes	—
31	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
32	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
33	C₄⋊Q₈×C₆₁	—	2	2	no	—
34	C₄⋊Q₈×C₆₁	—	2	2	no	—
35	C₄⋊Q₈×C₆₁	—	2	2	no	—
36	C₄⋊Q₈×C₆₁	—	2	2	no	—
37	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
38	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
39	C₄×D₄×C₆₁	—	2	1	no	—
40	C₄×D₄×C₆₁	—	2	1	no	—
41	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
42	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
43	#117×C₆₁	—	2	1	no	—
44	#117×C₆₁	C₂³	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
47	C₂×C₄²×C₆₁	—	2	1	no	—
48	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
49	#107×C₆₁	—	2	8	no	—
50	#107×C₆₁	—	2	4	no	—
51	#106×C₆₁	—	2	4	no	—
52	C₄⋊C₈×C₆₁	—	2	4	no	—
53	C₄⋊C₈×C₆₁	—	2	2	no	—
54	C₄⋊C₈×C₆₁	—	2	2	no	—
55	#103×C₆₁	—	2	4	no	—
56	#102×C₆₁	—	2	4	no	—
57	C₂²⋊C₈×C₆₁	—	2	4	no	—
58	#97×C₆₁	—	2	1	no	—
59	C₄×C₈×C₆₁	—	2	1	no	—
60	C₄×C₈×C₆₁	D₄	1	2	yes	—
61	C₄×C₈×C₆₁	D₄	1	2	yes	—
62	#41×C₆₁	—	2	4	no	—
63	C₂×Q₈×C₆₁	—	4	2	no	—
64	C₂×Q₈×C₆₁	—	4	2	no	—
65	C₂×Q₈×C₆₁	—	8	4	no	—
66	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
67	C₂×D₄×C₆₁	—	2	1	no	—
68	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₁	—	2	1	no	—
70	C₂²×C₄×C₆₁	—	2	1	no	—
71	C₂²×C₄×C₆₁	—	2	1	no	—
72	C₂²×C₄×C₆₁	—	2	4	no	—
73	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
74	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
75	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
76	M₁₆×C₆₁	—	2	8	no	—
77	M₁₆×C₆₁	—	2	4	no	—
78	C₂×C₈×C₆₁	—	2	4	no	—
79	C₂×C₈×C₆₁	—	2	2	no	—
80	C₂×C₈×C₆₁	—	2	4	no	—
81	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
82	C₄⋊C₄×C₆₁	—	2	1	no	—
83	C₄⋊C₄×C₆₁	—	2	1	no	—
84	C₄⋊C₄×C₆₁	—	2	4	no	—
85	C₄⋊C₄×C₆₁	—	2	4	no	—
86	C₄⋊C₄×C₆₁	—	4	2	no	—
87	C₄⋊C₄×C₆₁	—	4	2	no	—
88	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
89	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
90	C₂²⋊C₄×C₆₁	—	2	1	no	—
91	C₂²⋊C₄×C₆₁	—	2	1	no	—
92	C₂²⋊C₄×C₆₁	—	2	1	no	—
93	C₄²×C₆₁	—	2	1	no	—
94	C₄²×C₆₁	—	2	1	no	—
95	C₄²×C₆₁	—	2	1	no	—
96	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
100	C₂³×C₆₁	—	2	1	no	—
101	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
102	Q₈×C₆₁	—	2	2	no	—
103	Q₈×C₆₁	—	8	4	no	—
104	Q₈×C₆₁	—	8	4	no	—
105	D₄×C₆₁	—	2	2	no	—
106	D₄×C₆₁	—	4	2	no	—
107	C₂×C₄×C₆₁	—	2	2	no	—
108	C₂×C₄×C₆₁	—	2	2	no	—
109	C₂×C₄×C₆₁	—	2	1	no	—
110	C₂×C₄×C₆₁	—	2	1	no	—
111	C₂×C₄×C₆₁	—	2	1	no	—
112	C₂×C₄×C₆₁	—	4	2	no	—
113	C₂×C₄×C₆₁	—	4	2	no	—
114	C₂×C₄×C₆₁	—	4	2	no	—
115	C₂×C₄×C₆₁	—	4	4	no	—
116	C₂×C₄×C₆₁	—	8	4	no	—
117	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
118	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
119	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
120	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
121	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₁	2₋¹⁺⁴	1	4	yes	—
124	C₈×C₆₁	—	2	4	no	—
125	C₈×C₆₁	—	4	2	no	—
126	C₈×C₆₁	—	4	4	no	—
127	#29886	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂²×C₆₁	—	2	1	no	—
129	C₂²×C₆₁	—	2	1	no	—
130	C₂²×C₆₁	—	4	1	no	—
131	C₂²×C₆₁	—	4	1	no	—
132	C₂²×C₆₁	—	4	1	no	—
133	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
134	C₂²×C₆₁	#541	1	2	yes	—
135	C₄×C₆₁	—	2	1	no	—
136	C₄×C₆₁	—	2	1	no	—
137	C₄×C₆₁	—	4	1	no	—
138	C₄×C₆₁	—	4	1	no	—
139	C₄×C₆₁	—	4	1	no	—
140	C₄×C₆₁	—	8	4	no	—
141	C₄×C₆₁	#472	1	2	yes	—
142	C₄×C₆₁	#473	1	2	yes	—
143	C₄×C₆₁	#536	1	2	yes	—
144	C₄×C₆₁	#536	1	2	yes	—
145	C₂×#500	C₂×C₆₁	1	1	yes	characteristic, complemented
146	#3079	C₂×C₆₁	1	2	yes	—
147	#2922	C₂×C₆₁	1	1	yes	characteristic
148	#2089	C₂×C₆₁	1	4	yes	complemented
149	#2078	C₂×C₆₁	1	4	yes	—
150	#1997	C₂×C₆₁	1	2	yes	complemented
151	#1837	C₂×C₆₁	1	1	yes	characteristic, complemented
152	C₂×C₆₁	—	2	2	no	—
153	C₂×C₆₁	—	8	1	no	—
154	C₂×C₆₁	#2621	1	1	yes	characteristic
155	C₂×C₆₁	C₂×#472	1	1	yes	characteristic
156	C₂×C₆₁	C₂×#473	1	1	yes	characteristic
157	C₂×C₄⋊Q₈	C₂²×C₆₁	1	2	yes	—
158	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
159	#500	—	2	2	no	—
160	#500	—	2	2	no	—
161	#477	—	2	2	no	—
162	#476	—	2	2	no	—
163	#475	—	2	2	no	—
164	#474	—	2	2	no	—
165	#433	—	2	2	no	—
166	#431	—	2	2	no	—
167	#428	C₂²×C₆₁	1	4	yes	—
168	C₂×#107	C₂²×C₆₁	1	4	yes	—
169	#424	C₂²×C₆₁	1	2	yes	—
170	#423	C₂²×C₆₁	1	2	yes	—
171	#417	C₂²×C₆₁	1	4	yes	—
172	#408	C₂²×C₆₁	1	2	yes	—
173	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
174	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
175	#343	C₂²×C₆₁	1	8	yes	—
176	#341	C₂²×C₆₁	1	4	yes	—
177	C₆₁	#29886	1	1	yes	characteristic, direct product factor, Sylow
178	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
179	C₂²×Q₈	—	2	2	no	—
180	C₂×M₁₆	C₂³×C₆₁	1	4	yes	—
181	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
182	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
183	C₄⋊Q₈	—	2	2	no	—
184	C₄⋊Q₈	—	2	2	no	—
185	C₄⋊Q₈	—	2	2	no	—
186	C₄⋊Q₈	—	2	2	no	—
187	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
188	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
189	C₄×D₄	—	2	1	no	—
190	C₄×D₄	—	2	1	no	—
191	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
192	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
193	#117	—	2	1	no	—
194	#117	C₂³×C₆₁	1	1	yes	characteristic
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
197	C₂×C₄²	—	2	1	no	—
198	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
199	#107	—	2	8	no	—
200	#107	—	2	4	no	—
201	#106	—	2	4	no	—
202	C₄⋊C₈	—	2	4	no	—
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	—	2	2	no	—
205	#103	—	2	4	no	—
206	#102	—	2	4	no	—
207	C₂²⋊C₈	—	2	4	no	—
208	#97	—	2	1	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	D₄×C₆₁	1	2	yes	—
211	C₄×C₈	D₄×C₆₁	1	2	yes	—
212	#41	—	2	4	no	—
213	C₂×Q₈	—	4	2	no	—
214	C₂×Q₈	—	4	2	no	—
215	C₂×Q₈	—	8	4	no	—
216	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
217	C₂×D₄	—	2	1	no	—
218	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
224	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
225	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
226	M₁₆	—	2	8	no	—
227	M₁₆	—	2	4	no	—
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	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
239	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
240	C₂²⋊C₄	—	2	1	no	—
241	C₂²⋊C₄	—	2	1	no	—
242	C₂²⋊C₄	—	2	1	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	1	no	—
246	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
247	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
248	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
249	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
250	C₂³	—	2	1	no	—
251	C₂³	C₂×#41×C₆₁	1	1	yes	characteristic
252	Q₈	—	2	2	no	—
253	Q₈	—	8	4	no	—
254	Q₈	—	8	4	no	—
255	D₄	—	2	2	no	—
256	D₄	—	4	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	8	4	no	—
267	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
268	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
269	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
270	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
271	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
272	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
273	C₂×C₄	2₋¹⁺⁴×C₆₁	1	4	yes	—
274	C₈	—	2	4	no	—
275	C₈	—	4	2	no	—
276	C₈	—	4	4	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	C₂×#121×C₆₁	1	1	yes	characteristic
283	C₂²	C₆₁×#541	1	2	yes	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	4	1	no	—
287	C₄	—	4	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	8	4	no	—
290	C₄	C₆₁×#472	1	2	yes	—
291	C₄	C₆₁×#473	1	2	yes	—
292	C₄	C₆₁×#536	1	2	yes	—
293	C₄	C₆₁×#536	1	2	yes	—
294	C₂	—	2	2	no	—
295	C₂	—	8	1	no	—
296	C₂	C₆₁×#2621	1	1	yes	characteristic
297	C₂	C₂×C₆₁×#472	1	1	yes	characteristic
298	C₂	C₂×C₆₁×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂×#41×C₆₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×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×#2089	C₂	semidirect
6. C₆₁×#1997	C₂	semidirect
7. C₆₁×#1837	C₂	semidirect
127. #29886	C₆₁	direct (N × H)
145. C₂×#500	C₂×C₆₁	semidirect
148. #2089	C₂×C₆₁	semidirect
150. #1997	C₂×C₆₁	semidirect
151. #1837	C₂×C₆₁	semidirect
177. C₆₁	#29886	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
C₆₁×#3079	2
C₆₁×#2922	2
C₆₁×#2089	2
C₆₁×#2078	2
C₆₁×#1997	2
C₆₁×#1837	2
#29886	61

Subgroups — C61×#29886 #29533319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#29886 #29533319