Subgroups — C₆₁×#33948 #29537381

Order 15616.

462 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3383	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3380	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2253	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2229	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2228	C₂	1	2	yes	maximal
7	C₆₁×#2227	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#2224	C₂	1	1	yes	maximal, characteristic
9	Q₈×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×SD₃₂×C₆₁	C₂	1	2	yes	maximal
11	C₄×SD₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
12	C₆₁×#557	—	2	1	no	—
13	C₆₁×#554	—	2	1	no	—
14	C₂×SD₃₂×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#500	C₂²	1	1	yes	characteristic
16	C₆₁×#500	C₂²	1	1	yes	characteristic
17	C₆₁×#499	C₂²	1	2	yes	—
18	C₆₁×#499	C₂²	1	2	yes	—
19	C₆₁×#479	—	2	2	no	—
20	C₆₁×#478	—	2	2	no	—
21	C₆₁×#476	—	2	1	no	—
22	C₆₁×#475	—	2	1	no	—
23	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
25	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
26	C₄×D₈×C₆₁	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₆₁	C₂²	1	2	yes	—
28	C₆₁×#367	C₂²	1	1	yes	characteristic
29	C₆₁×#367	C₂²	1	2	yes	—
30	C₆₁×#367	C₂²	1	2	yes	—
31	C₆₁×#366	C₂²	1	2	yes	—
32	C₆₁×#366	C₂²	1	2	yes	—
33	C₆₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
35	C₆₁×#358	C₂²	1	2	yes	—
36	C₆₁×#358	C₂²	1	1	yes	characteristic
37	C₆₁×#357	C₂²	1	2	yes	—
38	C₆₁×#357	C₂²	1	1	yes	characteristic
39	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
40	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆×C₆₁	C₂³	1	1	yes	characteristic
42	C₂×D₈×C₆₁	C₂³	1	1	yes	characteristic
43	C₄⋊Q₈×C₆₁	—	2	1	no	—
44	C₄⋊Q₈×C₆₁	—	2	1	no	—
45	C₄⋊Q₈×C₆₁	—	4	1	no	—
46	#125×C₆₁	—	2	2	no	—
47	#125×C₆₁	—	2	2	no	—
48	#125×C₆₁	—	4	2	no	—
49	C₂²⋊Q₈×C₆₁	—	4	1	no	—
50	C₂²⋊Q₈×C₆₁	—	4	2	no	—
51	C₄×Q₈×C₆₁	—	2	1	no	—
52	C₄×Q₈×C₆₁	—	2	2	no	—
53	C₄×Q₈×C₆₁	—	4	1	no	—
54	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
55	C₄×D₄×C₆₁	—	2	1	no	—
56	C₄×D₄×C₆₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
58	SD₃₂×C₆₁	Q₈	1	4	yes	complemented
59	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
60	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
61	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
62	#107×C₆₁	—	2	1	no	—
63	#107×C₆₁	—	2	1	no	—
64	#107×C₆₁	C₂³	1	1	yes	characteristic
65	#107×C₆₁	C₂³	1	2	yes	—
66	#107×C₆₁	C₂³	1	1	yes	characteristic
67	#107×C₆₁	C₂³	1	2	yes	—
68	#106×C₆₁	—	2	2	no	—
69	#106×C₆₁	—	2	2	no	—
70	C₄⋊C₈×C₆₁	D₄	1	1	yes	characteristic
71	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
72	#103×C₆₁	—	2	2	no	—
73	#103×C₆₁	—	2	1	no	—
74	#102×C₆₁	—	2	2	no	—
75	#102×C₆₁	—	2	1	no	—
76	C₄×C₈×C₆₁	C₂³	1	2	yes	—
77	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
78	C₂×Q₈×C₆₁	—	2	1	no	—
79	C₂×Q₈×C₆₁	—	4	1	no	—
80	C₂×Q₈×C₆₁	—	4	1	no	—
81	C₂×Q₈×C₆₁	D₈	1	1	yes	characteristic
82	C₂×D₄×C₆₁	—	2	1	no	—
83	C₂²×C₄×C₆₁	—	4	1	no	—
84	C₂²×C₄×C₆₁	—	4	2	no	—
85	Q₁₆×C₆₁	—	2	1	no	—
86	Q₁₆×C₆₁	C₂×Q₈	1	2	yes	—
87	D₈×C₆₁	—	2	1	no	—
88	D₈×C₆₁	C₂×Q₈	1	2	yes	—
89	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₆₁	—	2	1	no	—
93	C₄⋊C₄×C₆₁	—	2	2	no	—
94	C₄⋊C₄×C₆₁	—	2	1	no	—
95	C₄⋊C₄×C₆₁	—	2	2	no	—
96	C₄⋊C₄×C₆₁	—	4	2	no	—
97	C₄⋊C₄×C₆₁	—	4	1	no	—
98	C₄⋊C₄×C₆₁	—	4	1	no	—
99	C₄⋊C₄×C₆₁	—	4	2	no	—
100	C₄⋊C₄×C₆₁	—	4	2	no	—
101	C₄⋊C₄×C₆₁	—	4	2	no	—
102	C₄⋊C₄×C₆₁	—	4	2	no	—
103	C₄⋊C₄×C₆₁	—	4	1	no	—
104	C₄⋊C₄×C₆₁	D₈	1	1	yes	characteristic
105	C₄⋊C₄×C₆₁	D₈	1	2	yes	—
106	C₂²⋊C₄×C₆₁	—	4	1	no	—
107	C₂²⋊C₄×C₆₁	—	4	2	no	—
108	C₄²×C₆₁	—	4	2	no	—
109	C₄²×C₆₁	—	4	1	no	—
110	C₄²×C₆₁	C₂×D₄	1	2	yes	—
111	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₆₁	—	2	2	no	—
113	C₁₆×C₆₁	—	2	1	no	—
114	C₁₆×C₆₁	C₂×Q₈	1	2	yes	—
115	C₂³×C₆₁	—	4	1	no	—
116	Q₈×C₆₁	—	2	2	no	—
117	Q₈×C₆₁	—	2	2	no	—
118	Q₈×C₆₁	—	4	1	no	—
119	Q₈×C₆₁	—	8	2	no	—
120	Q₈×C₆₁	—	16	1	no	—
121	D₄×C₆₁	—	2	2	no	—
122	D₄×C₆₁	—	4	1	no	—
123	C₂×C₄×C₆₁	—	4	1	no	—
124	C₂×C₄×C₆₁	—	4	1	no	—
125	C₂×C₄×C₆₁	—	4	2	no	—
126	C₂×C₄×C₆₁	—	4	1	no	—
127	C₂×C₄×C₆₁	—	4	2	no	—
128	C₂×C₄×C₆₁	—	8	2	no	—
129	C₂×C₄×C₆₁	—	16	2	no	—
130	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
135	C₈×C₆₁	—	2	2	no	—
136	C₈×C₆₁	—	2	1	no	—
137	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
138	C₈×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
139	#33948	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂²×C₆₁	—	4	2	no	—
141	C₂²×C₆₁	—	8	1	no	—
142	C₂²×C₆₁	—	8	1	no	—
143	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
144	C₄×C₆₁	—	2	2	no	—
145	C₄×C₆₁	—	2	2	no	—
146	C₄×C₆₁	—	2	1	no	—
147	C₄×C₆₁	—	4	2	no	—
148	C₄×C₆₁	—	8	1	no	—
149	C₄×C₆₁	—	8	2	no	—
150	C₄×C₆₁	—	8	1	no	—
151	C₄×C₆₁	—	8	2	no	—
152	C₄×C₆₁	#508	1	2	yes	—
153	C₄×C₆₁	D₄×Q₈	1	1	yes	characteristic
154	C₄×C₆₁	#578	1	1	yes	characteristic
155	#3383	C₂×C₆₁	1	1	yes	characteristic, complemented
156	#3380	C₂×C₆₁	1	1	yes	characteristic
157	#2254	C₂×C₆₁	1	1	yes	characteristic, complemented
158	#2253	C₂×C₆₁	1	2	yes	complemented
159	#2229	C₂×C₆₁	1	2	yes	complemented
160	#2228	C₂×C₆₁	1	2	yes	—
161	#2227	C₂×C₆₁	1	1	yes	characteristic, complemented
162	#2224	C₂×C₆₁	1	1	yes	characteristic
163	Q₈×C₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
164	C₄×SD₃₂	C₂×C₆₁	1	2	yes	—
165	C₄×SD₃₂	C₂×C₆₁	1	1	yes	characteristic
166	C₂×C₆₁	—	8	2	no	—
167	C₂×C₆₁	Q₈×D₈	1	1	yes	characteristic
168	C₂×C₆₁	C₂×#508	1	1	yes	characteristic
169	C₂×C₆₁	#3416	1	1	yes	characteristic
170	#557	—	2	1	no	—
171	#554	—	2	1	no	—
172	C₂×SD₃₂	C₂²×C₆₁	1	1	yes	characteristic
173	#500	C₂²×C₆₁	1	1	yes	characteristic
174	#500	C₂²×C₆₁	1	1	yes	characteristic
175	#499	C₂²×C₆₁	1	2	yes	—
176	#499	C₂²×C₆₁	1	2	yes	—
177	#479	—	2	2	no	—
178	#478	—	2	2	no	—
179	#476	—	2	1	no	—
180	#475	—	2	1	no	—
181	C₈×Q₈	C₂²×C₆₁	1	1	yes	characteristic
182	C₄×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
183	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
184	C₄×D₈	C₂²×C₆₁	1	1	yes	characteristic
185	C₄×D₈	C₂²×C₆₁	1	2	yes	—
186	#367	C₂²×C₆₁	1	1	yes	characteristic
187	#367	C₂²×C₆₁	1	2	yes	—
188	#367	C₂²×C₆₁	1	2	yes	—
189	#366	C₂²×C₆₁	1	2	yes	—
190	#366	C₂²×C₆₁	1	2	yes	—
191	C₄⋊C₁₆	C₂²×C₆₁	1	1	yes	characteristic
192	C₄⋊C₁₆	C₂²×C₆₁	1	2	yes	—
193	#358	C₂²×C₆₁	1	2	yes	—
194	#358	C₂²×C₆₁	1	1	yes	characteristic
195	#357	C₂²×C₆₁	1	2	yes	—
196	#357	C₂²×C₆₁	1	1	yes	characteristic
197	C₄×C₁₆	C₂²×C₆₁	1	2	yes	—
198	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
199	C₆₁	#33948	1	1	yes	characteristic, direct product factor, Sylow
200	C₂×Q₁₆	C₂³×C₆₁	1	1	yes	characteristic
201	C₂×D₈	C₂³×C₆₁	1	1	yes	characteristic
202	C₄⋊Q₈	—	2	1	no	—
203	C₄⋊Q₈	—	2	1	no	—
204	C₄⋊Q₈	—	4	1	no	—
205	#125	—	2	2	no	—
206	#125	—	2	2	no	—
207	#125	—	4	2	no	—
208	C₂²⋊Q₈	—	4	1	no	—
209	C₂²⋊Q₈	—	4	2	no	—
210	C₄×Q₈	—	2	1	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	4	1	no	—
213	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
214	C₄×D₄	—	2	1	no	—
215	C₄×D₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	4	1	no	—
217	SD₃₂	Q₈×C₆₁	1	4	yes	complemented
218	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
219	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
220	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
221	#107	—	2	1	no	—
222	#107	—	2	1	no	—
223	#107	C₂³×C₆₁	1	1	yes	characteristic
224	#107	C₂³×C₆₁	1	2	yes	—
225	#107	C₂³×C₆₁	1	1	yes	characteristic
226	#107	C₂³×C₆₁	1	2	yes	—
227	#106	—	2	2	no	—
228	#106	—	2	2	no	—
229	C₄⋊C₈	D₄×C₆₁	1	1	yes	characteristic
230	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
231	#103	—	2	2	no	—
232	#103	—	2	1	no	—
233	#102	—	2	2	no	—
234	#102	—	2	1	no	—
235	C₄×C₈	C₂³×C₆₁	1	2	yes	—
236	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
237	C₂×Q₈	—	2	1	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	D₈×C₆₁	1	1	yes	characteristic
241	C₂×D₄	—	2	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	2	no	—
244	Q₁₆	—	2	1	no	—
245	Q₁₆	C₂×Q₈×C₆₁	1	2	yes	—
246	D₈	—	2	1	no	—
247	D₈	C₂×Q₈×C₆₁	1	2	yes	—
248	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
249	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
250	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	D₈×C₆₁	1	1	yes	characteristic
264	C₄⋊C₄	D₈×C₆₁	1	2	yes	—
265	C₂²⋊C₄	—	4	1	no	—
266	C₂²⋊C₄	—	4	2	no	—
267	C₄²	—	4	2	no	—
268	C₄²	—	4	1	no	—
269	C₄²	C₂×D₄×C₆₁	1	2	yes	—
270	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
271	C₁₆	—	2	2	no	—
272	C₁₆	—	2	1	no	—
273	C₁₆	C₂×Q₈×C₆₁	1	2	yes	—
274	C₂³	—	4	1	no	—
275	Q₈	—	2	2	no	—
276	Q₈	—	2	2	no	—
277	Q₈	—	4	1	no	—
278	Q₈	—	8	2	no	—
279	Q₈	—	16	1	no	—
280	D₄	—	2	2	no	—
281	D₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	8	2	no	—
288	C₂×C₄	—	16	2	no	—
289	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
290	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
291	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
292	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	—	2	1	no	—
296	C₈	C₂²×Q₈×C₆₁	1	1	yes	characteristic
297	C₈	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
298	C₂²	—	4	2	no	—
299	C₂²	—	8	1	no	—
300	C₂²	—	8	1	no	—
301	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	1	no	—
305	C₄	—	4	2	no	—
306	C₄	—	8	1	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	1	no	—
309	C₄	—	8	2	no	—
310	C₄	C₆₁×#508	1	2	yes	—
311	C₄	D₄×Q₈×C₆₁	1	1	yes	characteristic
312	C₄	C₆₁×#578	1	1	yes	characteristic
313	C₂	—	8	2	no	—
314	C₂	Q₈×D₈×C₆₁	1	1	yes	characteristic
315	C₂	C₂×C₆₁×#508	1	1	yes	characteristic
316	C₂	C₆₁×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
prime radical	C₂×D₈×C₆₁	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄×Q₈×C₆₁ ◁ C₈×Q₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3383	C₂	semidirect
3. C₆₁×#2254	C₂	semidirect
4. C₆₁×#2253	C₂	semidirect
5. C₆₁×#2229	C₂	semidirect
7. C₆₁×#2227	C₂	semidirect
9. Q₈×C₁₆×C₆₁	C₂	semidirect
58. SD₃₂×C₆₁	Q₈	semidirect
139. #33948	C₆₁	direct (N × H)
155. #3383	C₂×C₆₁	semidirect
157. #2254	C₂×C₆₁	semidirect
158. #2253	C₂×C₆₁	semidirect
159. #2229	C₂×C₆₁	semidirect
161. #2227	C₂×C₆₁	semidirect
163. Q₈×C₁₆	C₂×C₆₁	semidirect
199. C₆₁	#33948	direct (N × H)
217. SD₃₂	Q₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3383	2
C₆₁×#3380	2
C₆₁×#2254	2
C₆₁×#2253	2
C₆₁×#2229	2
C₆₁×#2228	2
C₆₁×#2227	2
C₆₁×#2224	2
Q₈×C₁₆×C₆₁	2
C₄×SD₃₂×C₆₁	2
#33948	61

Subgroups — C61×#33948 #29537381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33948 #29537381