Subgroups — C₆₁×#10840 #29514273

Order 15616.

894 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×#423	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2285	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#376	C₂	1	1	yes	maximal, characteristic
4	C₂×C₆₁×#339	C₂	1	4	yes	maximal
5	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
6	C₂×#117×C₆₁	C₂²	1	2	yes	—
7	C₂²×C₄²×C₆₁	C₂²	1	2	yes	—
8	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
9	C₆₁×#423	—	2	2	no	—
10	C₆₁×#423	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₈×C₆₁	—	2	2	no	—
12	C₆₁×#384	C₄	1	2	yes	complemented
13	C₆₁×#382	C₄	1	2	yes	complemented
14	C₆₁×#380	—	2	2	no	—
15	C₄×C₄⋊C₄×C₆₁	—	2	2	no	—
16	C₆₁×#376	—	2	2	no	—
17	C₆₁×#376	C₄	1	4	yes	complemented
18	C₂×#95×C₆₁	—	2	2	no	—
19	C₆₁×#339	—	2	8	no	—
20	C₂³×C₄×C₆₁	—	2	2	no	—
21	C₂³×C₄×C₆₁	D₄	1	2	yes	—
22	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₆₁	—	2	2	no	—
24	C₂×M₁₆×C₆₁	—	2	4	no	—
25	C₂×M₁₆×C₆₁	—	2	4	no	—
26	C₂×M₁₆×C₆₁	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₆₁	—	2	2	no	—
28	#117×C₆₁	—	2	4	no	—
29	#117×C₆₁	—	2	4	no	—
30	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
31	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
32	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
33	C₂×C₄²×C₆₁	—	2	2	no	—
34	C₂×C₄²×C₆₁	—	2	2	no	—
35	C₂×C₄²×C₆₁	—	2	2	no	—
36	C₂×C₄²×C₆₁	—	2	2	no	—
37	C₂×C₄²×C₆₁	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₁	D₄	1	4	yes	—
43	C₂×C₄²×C₆₁	D₄	1	2	yes	—
44	C₂×C₄²×C₆₁	D₄	1	2	yes	—
45	C₂×C₄²×C₆₁	Q₈	1	4	yes	—
46	C₂×C₄²×C₆₁	Q₈	1	2	yes	—
47	C₄⋊C₈×C₆₁	—	2	8	no	—
48	#95×C₆₁	—	2	8	no	—
49	#95×C₆₁	—	2	4	no	—
50	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	—	2	4	no	—
52	C₂²×C₄×C₆₁	—	2	4	no	—
53	C₂²×C₄×C₆₁	—	2	4	no	—
54	C₂²×C₄×C₆₁	—	2	4	no	—
55	C₂²×C₄×C₆₁	—	4	4	no	—
56	C₂²×C₄×C₆₁	—	4	8	no	—
57	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
60	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₆₁	#41	1	2	yes	—
70	C₂²×C₄×C₆₁	#41	1	2	yes	—
71	M₁₆×C₆₁	—	2	8	no	—
72	M₁₆×C₆₁	—	4	4	no	—
73	C₂×C₈×C₆₁	—	2	4	no	—
74	C₂×C₈×C₆₁	—	4	2	no	—
75	C₂×C₈×C₆₁	—	4	2	no	—
76	C₄⋊C₄×C₆₁	—	2	8	no	—
77	C₄⋊C₄×C₆₁	—	4	8	no	—
78	C₂²⋊C₄×C₆₁	—	4	4	no	—
79	C₄²×C₆₁	—	2	4	no	—
80	C₄²×C₆₁	—	2	4	no	—
81	C₄²×C₆₁	—	2	4	no	—
82	C₄²×C₆₁	—	4	2	no	—
83	C₄²×C₆₁	—	4	2	no	—
84	C₄²×C₆₁	C₄²	1	2	yes	—
85	C₄²×C₆₁	C₄²	1	2	yes	—
86	C₄²×C₆₁	C₂²⋊C₄	1	4	yes	—
87	C₄²×C₆₁	C₄⋊C₄	1	4	yes	—
88	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
89	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
90	C₄²×C₆₁	C₄⋊C₄	1	8	yes	complemented
91	C₂³×C₆₁	—	2	4	no	—
92	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂³×C₆₁	#121	1	2	yes	—
96	C₂³×C₆₁	C₂²⋊Q₈	1	2	yes	—
97	C₂×C₄×C₆₁	—	2	4	no	—
98	C₂×C₄×C₆₁	—	2	4	no	—
99	C₂×C₄×C₆₁	—	2	8	no	—
100	C₂×C₄×C₆₁	—	4	4	no	—
101	C₂×C₄×C₆₁	—	4	4	no	—
102	C₂×C₄×C₆₁	—	4	8	no	—
103	C₂×C₄×C₆₁	—	4	8	no	—
104	C₂×C₄×C₆₁	—	4	8	no	—
105	C₂×C₄×C₆₁	—	4	4	no	—
106	C₂×C₄×C₆₁	—	4	8	no	—
107	C₂×C₄×C₆₁	#95	1	4	yes	—
108	C₂×C₄×C₆₁	#95	1	4	yes	—
109	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₁	#117	1	2	yes	—
115	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
118	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₆₁	#121	1	2	yes	—
120	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
122	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
123	C₂×C₄×C₆₁	#125	1	2	yes	—
124	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
125	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
126	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
127	C₈×C₆₁	—	4	4	no	—
128	#10840	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂²×C₆₁	—	2	4	no	—
130	C₂²×C₆₁	—	2	2	no	—
131	C₂²×C₆₁	—	2	2	no	—
132	C₂²×C₆₁	—	2	4	no	—
133	C₂²×C₆₁	C₂×#95	1	1	yes	characteristic
134	C₂²×C₆₁	#380	1	2	yes	—
135	C₂²×C₆₁	#389	1	2	yes	—
136	C₂²×C₆₁	#390	1	1	yes	characteristic
137	C₂²×C₆₁	#391	1	1	yes	characteristic
138	C₂²×C₆₁	#421	1	4	yes	—
139	C₄×C₆₁	—	2	4	no	—
140	C₄×C₆₁	—	2	4	no	—
141	C₄×C₆₁	—	4	8	no	—
142	C₄×C₆₁	—	8	4	no	—
143	C₄×C₆₁	C₄×C₄⋊C₄	1	2	yes	—
144	C₄×C₆₁	#382	1	1	yes	characteristic
145	C₄×C₆₁	#384	1	1	yes	characteristic
146	C₄×C₆₁	#389	1	2	yes	—
147	C₄×C₆₁	#390	1	1	yes	characteristic
148	C₄×C₆₁	#391	1	1	yes	characteristic
149	C₂×#423	C₂×C₆₁	1	1	yes	characteristic
150	#2285	C₂×C₆₁	1	1	yes	characteristic
151	C₂×#376	C₂×C₆₁	1	1	yes	characteristic
152	C₂×#339	C₂×C₆₁	1	4	yes	—
153	C₂×C₆₁	—	2	2	no	—
154	C₂×C₆₁	—	2	2	no	—
155	C₂×C₆₁	C₄⋊#95	1	1	yes	characteristic
156	C₂×C₆₁	#1733	1	2	yes	—
157	C₂×C₆₁	#1963	1	2	yes	—
158	C₂×C₆₁	#1995	1	2	yes	—
159	C₂²×M₁₆	C₂²×C₆₁	1	2	yes	—
160	C₂×#117	C₂²×C₆₁	1	2	yes	—
161	C₂²×C₄²	C₂²×C₆₁	1	2	yes	—
162	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
163	#423	—	2	2	no	—
164	#423	C₄×C₆₁	1	4	yes	complemented
165	C₂×C₄⋊C₈	—	2	2	no	—
166	#384	C₄×C₆₁	1	2	yes	complemented
167	#382	C₄×C₆₁	1	2	yes	complemented
168	#380	—	2	2	no	—
169	C₄×C₄⋊C₄	—	2	2	no	—
170	#376	—	2	2	no	—
171	#376	C₄×C₆₁	1	4	yes	complemented
172	C₂×#95	—	2	2	no	—
173	#339	—	2	8	no	—
174	C₆₁	#10840	1	1	yes	characteristic, direct product factor, Sylow
175	C₂³×C₄	—	2	2	no	—
176	C₂³×C₄	D₄×C₆₁	1	2	yes	—
177	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	4	no	—
180	C₂×M₁₆	—	2	4	no	—
181	C₂×M₁₆	C₂×C₄×C₆₁	1	4	yes	—
182	C₂²×C₈	—	2	2	no	—
183	#117	—	2	4	no	—
184	#117	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	4	no	—
186	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	C₂×C₄×C₆₁	1	4	yes	—
193	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
194	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
195	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
196	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
197	C₂×C₄²	D₄×C₆₁	1	4	yes	—
198	C₂×C₄²	D₄×C₆₁	1	2	yes	—
199	C₂×C₄²	D₄×C₆₁	1	2	yes	—
200	C₂×C₄²	Q₈×C₆₁	1	4	yes	—
201	C₂×C₄²	Q₈×C₆₁	1	2	yes	—
202	C₄⋊C₈	—	2	8	no	—
203	#95	—	2	8	no	—
204	#95	—	2	4	no	—
205	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	—	4	8	no	—
212	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
213	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
214	C₂²×C₄	C₂²⋊C₄×C₆₁	1	4	yes	—
215	C₂²×C₄	C₄⋊C₄×C₆₁	1	4	yes	—
216	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
217	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
218	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
222	C₂²×C₄	C₂×Q₈×C₆₁	1	1	yes	characteristic
223	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
224	C₂²×C₄	#41×C₆₁	1	2	yes	—
225	C₂²×C₄	#41×C₆₁	1	2	yes	—
226	M₁₆	—	2	8	no	—
227	M₁₆	—	4	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₄⋊C₄	—	2	8	no	—
232	C₄⋊C₄	—	4	8	no	—
233	C₂²⋊C₄	—	4	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	4	2	no	—
238	C₄²	—	4	2	no	—
239	C₄²	C₄²×C₆₁	1	2	yes	—
240	C₄²	C₄²×C₆₁	1	2	yes	—
241	C₄²	C₂²⋊C₄×C₆₁	1	4	yes	—
242	C₄²	C₄⋊C₄×C₆₁	1	4	yes	—
243	C₄²	C₄⋊C₄×C₆₁	1	2	yes	—
244	C₄²	C₄⋊C₄×C₆₁	1	2	yes	—
245	C₄²	C₄⋊C₄×C₆₁	1	8	yes	complemented
246	C₂³	—	2	4	no	—
247	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
248	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
249	C₂³	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
250	C₂³	#121×C₆₁	1	2	yes	—
251	C₂³	C₂²⋊Q₈×C₆₁	1	2	yes	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	—	4	8	no	—
259	C₂×C₄	—	4	8	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	8	no	—
262	C₂×C₄	#95×C₆₁	1	4	yes	—
263	C₂×C₄	#95×C₆₁	1	4	yes	—
264	C₂×C₄	C₂×C₄²×C₆₁	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
266	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
267	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
268	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
269	C₂×C₄	#117×C₆₁	1	2	yes	—
270	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
271	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
272	C₂×C₄	C₄×Q₈×C₆₁	1	2	yes	—
273	C₂×C₄	C₄×Q₈×C₆₁	1	2	yes	—
274	C₂×C₄	#121×C₆₁	1	2	yes	—
275	C₂×C₄	C₂²⋊Q₈×C₆₁	1	2	yes	—
276	C₂×C₄	#124×C₆₁	1	1	yes	characteristic
277	C₂×C₄	#124×C₆₁	1	1	yes	characteristic
278	C₂×C₄	#125×C₆₁	1	2	yes	—
279	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
280	C₂×C₄	C₄⋊Q₈×C₆₁	1	2	yes	—
281	C₂×C₄	C₄⋊Q₈×C₆₁	1	1	yes	characteristic
282	C₈	—	4	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	4	no	—
287	C₂²	C₂×#95×C₆₁	1	1	yes	characteristic
288	C₂²	C₆₁×#380	1	2	yes	—
289	C₂²	C₆₁×#389	1	2	yes	—
290	C₂²	C₆₁×#390	1	1	yes	characteristic
291	C₂²	C₆₁×#391	1	1	yes	characteristic
292	C₂²	C₆₁×#421	1	4	yes	—
293	C₄	—	2	4	no	—
294	C₄	—	2	4	no	—
295	C₄	—	4	8	no	—
296	C₄	—	8	4	no	—
297	C₄	C₄×C₄⋊C₄×C₆₁	1	2	yes	—
298	C₄	C₆₁×#382	1	1	yes	characteristic
299	C₄	C₆₁×#384	1	1	yes	characteristic
300	C₄	C₆₁×#389	1	2	yes	—
301	C₄	C₆₁×#390	1	1	yes	characteristic
302	C₄	C₆₁×#391	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	2	2	no	—
305	C₂	C₆₁×C₄⋊#95	1	1	yes	characteristic
306	C₂	C₆₁×#1733	1	2	yes	—
307	C₂	C₆₁×#1963	1	2	yes	—
308	C₂	C₆₁×#1995	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
10. C₆₁×#423	C₄	semidirect
12. C₆₁×#384	C₄	semidirect
13. C₆₁×#382	C₄	semidirect
17. C₆₁×#376	C₄	semidirect
90. C₄²×C₆₁	C₄⋊C₄	semidirect
128. #10840	C₆₁	direct (N × H)
164. #423	C₄×C₆₁	semidirect
166. #384	C₄×C₆₁	semidirect
167. #382	C₄×C₆₁	semidirect
171. #376	C₄×C₆₁	semidirect
174. C₆₁	#10840	direct (N × H)
245. C₄²	C₄⋊C₄×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#423	2
C₆₁×#2285	2
C₂×C₆₁×#376	2
C₂×C₆₁×#339	2
#10840	61

Subgroups — C61×#10840 #29514273

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#10840 #29514273