Subgroups — C₆₁×#29868 #29533301

Order 15616.

846 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#476	C₂	1	2	yes	maximal
2	C₆₁×#2868	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#2293	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#2084	C₂	1	4	yes	maximal, complemented
5	C₆₁×#2071	C₂	1	4	yes	maximal
6	C₆₁×#1987	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1831	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊Q₈×C₆₁	—	2	2	no	—
9	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
11	C₂×#117×C₆₁	—	2	1	no	—
12	C₆₁×#476	—	2	8	no	—
13	C₂×#106×C₆₁	C₂²	1	4	yes	—
14	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
15	C₂×#102×C₆₁	C₂²	1	4	yes	—
16	C₆₁×#390	C₂²	1	2	yes	—
17	C₆₁×#389	C₂²	1	2	yes	—
18	C₆₁×#388	—	2	2	no	—
19	C₆₁×#384	—	2	1	no	—
20	C₆₁×#384	—	2	1	no	—
21	C₆₁×#384	C₂²	1	4	yes	—
22	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊C₄×C₆₁	—	2	1	no	—
24	C₄³×C₆₁	C₂²	1	1	yes	characteristic
25	C₆₁×#340	C₂²	1	8	yes	—
26	C₆₁×#336	C₂²	1	4	yes	—
27	C₂²×Q₈×C₆₁	—	2	2	no	—
28	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
29	C₂³×C₄×C₆₁	—	2	1	no	—
30	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
31	C₄⋊Q₈×C₆₁	—	2	4	no	—
32	C₄⋊Q₈×C₆₁	—	2	4	no	—
33	C₄⋊Q₈×C₆₁	—	4	8	no	—
34	C₄×D₄×C₆₁	—	2	2	no	—
35	C₄×D₄×C₆₁	D₄	1	4	yes	—
36	#117×C₆₁	—	4	4	no	—
37	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
38	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
39	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
45	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
46	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
47	C₂×C₄²×C₆₁	—	2	1	no	—
48	C₂×C₄²×C₆₁	D₄	1	2	yes	—
49	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
52	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
53	#106×C₆₁	—	2	8	no	—
54	C₄⋊C₈×C₆₁	—	2	4	no	—
55	#102×C₆₁	—	2	8	no	—
56	#95×C₆₁	—	2	2	no	—
57	#95×C₆₁	—	2	1	no	—
58	C₂⁴×C₆₁	—	2	1	no	—
59	C₂×Q₈×C₆₁	—	4	8	no	—
60	C₂×Q₈×C₆₁	—	4	4	no	—
61	C₂×D₄×C₆₁	—	2	1	no	—
62	C₂×D₄×C₆₁	—	2	4	no	—
63	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₆₁	—	2	1	no	—
65	C₂²×C₄×C₆₁	—	2	4	no	—
66	C₂²×C₄×C₆₁	—	2	2	no	—
67	C₂²×C₄×C₆₁	—	2	4	no	—
68	C₂²×C₄×C₆₁	—	4	2	no	—
69	C₂²×C₄×C₆₁	—	4	2	no	—
70	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
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	8	no	—
77	C₄⋊C₄×C₆₁	—	2	1	no	—
78	C₄⋊C₄×C₆₁	—	2	4	no	—
79	C₄⋊C₄×C₆₁	—	2	4	no	—
80	C₄⋊C₄×C₆₁	—	2	4	no	—
81	C₄⋊C₄×C₆₁	—	4	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	C₂×D₄	1	2	yes	—
86	C₂²⋊C₄×C₆₁	—	2	4	no	—
87	C₂²⋊C₄×C₆₁	—	4	4	no	—
88	C₂²⋊C₄×C₆₁	—	4	4	no	—
89	C₄²×C₆₁	—	2	4	no	—
90	C₄²×C₆₁	—	2	2	no	—
91	C₄²×C₆₁	—	2	4	no	—
92	C₄²×C₆₁	—	4	2	no	—
93	C₄²×C₆₁	SD₁₆	1	4	yes	—
94	C₄²×C₆₁	C₂×D₄	1	2	yes	—
95	C₄²×C₆₁	#41	1	2	yes	—
96	C₂³×C₆₁	—	2	2	no	—
97	C₂³×C₆₁	—	4	2	no	—
98	C₂³×C₆₁	—	4	1	no	—
99	C₂³×C₆₁	—	4	2	no	—
100	C₂³×C₆₁	—	4	1	no	—
101	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
102	Q₈×C₆₁	—	8	8	no	—
103	D₄×C₆₁	—	2	4	no	—
104	D₄×C₆₁	—	4	2	no	—
105	C₂×C₄×C₆₁	—	2	2	no	—
106	C₂×C₄×C₆₁	—	2	2	no	—
107	C₂×C₄×C₆₁	—	2	2	no	—
108	C₂×C₄×C₆₁	—	2	2	no	—
109	C₂×C₄×C₆₁	—	2	2	no	—
110	C₂×C₄×C₆₁	—	2	2	no	—
111	C₂×C₄×C₆₁	—	4	1	no	—
112	C₂×C₄×C₆₁	—	4	1	no	—
113	C₂×C₄×C₆₁	—	4	8	no	—
114	C₂×C₄×C₆₁	—	4	4	no	—
115	C₂×C₄×C₆₁	—	4	2	no	—
116	C₂×C₄×C₆₁	—	4	2	no	—
117	C₂×C₄×C₆₁	—	4	2	no	—
118	C₂×C₄×C₆₁	—	4	8	no	—
119	C₂×C₄×C₆₁	#121	1	2	yes	—
120	C₂×C₄×C₆₁	#121	1	2	yes	—
121	C₂×C₄×C₆₁	C₂×SD₁₆	1	4	yes	—
122	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
123	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	4	yes	—
124	C₂×C₄×C₆₁	#137	1	2	yes	—
125	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₆₁	2₋¹⁺⁴	1	4	yes	—
128	C₈×C₆₁	—	4	4	no	—
129	#29868	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂²×C₆₁	—	4	2	no	—
131	C₂²×C₆₁	—	4	2	no	—
132	C₂²×C₆₁	—	4	2	no	—
133	C₂²×C₆₁	—	8	2	no	—
134	C₂²×C₆₁	—	8	2	no	—
135	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
136	C₂²×C₆₁	#536	1	2	yes	—
137	C₂²×C₆₁	C₂²×SD₁₆	1	1	yes	characteristic
138	C₂²×C₆₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
139	C₂²×C₆₁	C₂×#137	1	1	yes	characteristic
140	C₄×C₆₁	—	2	4	no	—
141	C₄×C₆₁	—	2	4	no	—
142	C₄×C₆₁	—	2	2	no	—
143	C₄×C₆₁	—	4	2	no	—
144	C₄×C₆₁	—	8	2	no	—
145	C₄×C₆₁	—	8	4	no	—
146	C₄×C₆₁	#461	1	4	yes	—
147	C₄×C₆₁	#536	1	2	yes	—
148	C₄×C₆₁	#541	1	2	yes	—
149	C₂×#476	C₂×C₆₁	1	2	yes	—
150	#2868	C₂×C₆₁	1	1	yes	characteristic, complemented
151	#2293	C₂×C₆₁	1	1	yes	characteristic
152	#2084	C₂×C₆₁	1	4	yes	complemented
153	#2071	C₂×C₆₁	1	4	yes	—
154	#1987	C₂×C₆₁	1	2	yes	complemented
155	#1831	C₂×C₆₁	1	1	yes	characteristic, complemented
156	C₂×C₆₁	—	8	2	no	—
157	C₂×C₆₁	#2621	1	1	yes	characteristic
158	C₂×C₆₁	C₂×#461	1	1	yes	characteristic
159	C₂×C₆₁	#3038	1	1	yes	characteristic
160	C₂×C₆₁	#3227	1	2	yes	—
161	C₂×C₆₁	#3238	1	2	yes	—
162	C₂×C₄⋊Q₈	—	2	2	no	—
163	C₂×C₄⋊Q₈	C₂²×C₆₁	1	2	yes	—
164	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
165	C₂×#117	—	2	1	no	—
166	#476	—	2	8	no	—
167	C₂×#106	C₂²×C₆₁	1	4	yes	—
168	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
169	C₂×#102	C₂²×C₆₁	1	4	yes	—
170	#390	C₂²×C₆₁	1	2	yes	—
171	#389	C₂²×C₆₁	1	2	yes	—
172	#388	—	2	2	no	—
173	#384	—	2	1	no	—
174	#384	—	2	1	no	—
175	#384	C₂²×C₆₁	1	4	yes	—
176	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
177	C₄×C₂²⋊C₄	—	2	1	no	—
178	C₄³	C₂²×C₆₁	1	1	yes	characteristic
179	#340	C₂²×C₆₁	1	8	yes	—
180	#336	C₂²×C₆₁	1	4	yes	—
181	C₆₁	#29868	1	1	yes	characteristic, direct product factor, Sylow
182	C₂²×Q₈	—	2	2	no	—
183	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
184	C₂³×C₄	—	2	1	no	—
185	C₂²×C₈	C₂³×C₆₁	1	4	yes	—
186	C₄⋊Q₈	—	2	4	no	—
187	C₄⋊Q₈	—	2	4	no	—
188	C₄⋊Q₈	—	4	8	no	—
189	C₄×D₄	—	2	2	no	—
190	C₄×D₄	D₄×C₆₁	1	4	yes	—
191	#117	—	4	4	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	—	2	4	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
197	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
198	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
199	C₂×C₂²⋊C₄	—	2	1	no	—
200	C₂×C₂²⋊C₄	—	2	2	no	—
201	C₂×C₂²⋊C₄	—	2	2	no	—
202	C₂×C₄²	—	2	1	no	—
203	C₂×C₄²	D₄×C₆₁	1	2	yes	—
204	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
205	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
206	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
207	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
208	#106	—	2	8	no	—
209	C₄⋊C₈	—	2	4	no	—
210	#102	—	2	8	no	—
211	#95	—	2	2	no	—
212	#95	—	2	1	no	—
213	C₂⁴	—	2	1	no	—
214	C₂×Q₈	—	4	8	no	—
215	C₂×Q₈	—	4	4	no	—
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	—	2	4	no	—
218	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
227	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
228	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
229	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	8	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	8	no	—
237	C₄⋊C₄	—	4	4	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	—	4	8	no	—
240	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₂²⋊C₄	—	4	4	no	—
243	C₂²⋊C₄	—	4	4	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	4	no	—
247	C₄²	—	4	2	no	—
248	C₄²	SD₁₆×C₆₁	1	4	yes	—
249	C₄²	C₂×D₄×C₆₁	1	2	yes	—
250	C₄²	#41×C₆₁	1	2	yes	—
251	C₂³	—	2	2	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	4	1	no	—
254	C₂³	—	4	2	no	—
255	C₂³	—	4	1	no	—
256	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
257	Q₈	—	8	8	no	—
258	D₄	—	2	4	no	—
259	D₄	—	4	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	8	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	8	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	#121×C₆₁	1	2	yes	—
275	C₂×C₄	#121×C₆₁	1	2	yes	—
276	C₂×C₄	C₂×SD₁₆×C₆₁	1	4	yes	—
277	C₂×C₄	C₂×SD₁₆×C₆₁	1	2	yes	—
278	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	4	yes	—
279	C₂×C₄	#137×C₆₁	1	2	yes	—
280	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
281	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
282	C₂×C₄	2₋¹⁺⁴×C₆₁	1	4	yes	—
283	C₈	—	4	4	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	8	2	no	—
288	C₂²	—	8	2	no	—
289	C₂²	C₂×#121×C₆₁	1	1	yes	characteristic
290	C₂²	C₆₁×#536	1	2	yes	—
291	C₂²	C₂²×SD₁₆×C₆₁	1	1	yes	characteristic
292	C₂²	C₂×AGL(1,ℤ/8)×C₆₁	1	2	yes	—
293	C₂²	C₂×#137×C₆₁	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	2	4	no	—
296	C₄	—	2	4	no	—
297	C₄	—	4	2	no	—
298	C₄	—	8	4	no	—
299	C₄	—	8	2	no	—
300	C₄	C₆₁×#461	1	4	yes	—
301	C₄	C₆₁×#536	1	2	yes	—
302	C₄	C₆₁×#541	1	2	yes	—
303	C₂	—	8	2	no	—
304	C₂	C₆₁×#2621	1	1	yes	characteristic
305	C₂	C₂×C₆₁×#461	1	1	yes	characteristic
306	C₂	C₆₁×#3038	1	1	yes	characteristic
307	C₂	C₆₁×#3227	1	2	yes	—
308	C₂	C₆₁×#3238	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂²×D₄×C₆₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₄³×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#2868	C₂	semidirect
4. C₆₁×#2084	C₂	semidirect
6. C₆₁×#1987	C₂	semidirect
7. C₆₁×#1831	C₂	semidirect
129. #29868	C₆₁	direct (N × H)
150. #2868	C₂×C₆₁	semidirect
152. #2084	C₂×C₆₁	semidirect
154. #1987	C₂×C₆₁	semidirect
155. #1831	C₂×C₆₁	semidirect
181. C₆₁	#29868	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×#476	2
C₆₁×#2868	2
C₆₁×#2293	2
C₆₁×#2084	2
C₆₁×#2071	2
C₆₁×#1987	2
C₆₁×#1831	2
#29868	61

Subgroups — C61×#29868 #29533301

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#29868 #29533301