Subgroups — C₆₁×#10835 #29514268

Order 15616.

1014 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×#97×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₆₁×#336	C₂	1	4	yes	maximal
5	C₂³×C₈×C₆₁	C₂²	1	2	yes	—
6	C₂³×C₈×C₆₁	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₆₁	C₄	1	2	yes	—
8	C₂²×C₄²×C₆₁	C₂²	1	2	yes	—
9	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₈×C₆₁	—	2	2	no	—
11	C₂×C₄⋊C₈×C₆₁	—	2	2	no	—
12	C₂×C₄⋊C₈×C₆₁	C₄	1	4	yes	complemented
13	C₂×#97×C₆₁	—	2	2	no	—
14	C₂×#97×C₆₁	—	2	2	no	—
15	C₂×#97×C₆₁	C₄	1	4	yes	complemented
16	C₄×C₄⋊C₄×C₆₁	—	2	2	no	—
17	C₄×C₄⋊C₄×C₆₁	—	2	2	no	—
18	C₂×#95×C₆₁	C₄	1	2	yes	—
19	C₆₁×#336	—	2	8	no	—
20	C₂³×C₄×C₆₁	C₂×C₄	1	4	yes	—
21	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₆₁	—	2	2	no	—
25	C₂²×C₈×C₆₁	—	2	2	no	—
26	C₂²×C₈×C₆₁	—	2	2	no	—
27	C₂²×C₈×C₆₁	—	2	2	no	—
28	C₂²×C₈×C₆₁	—	2	4	no	—
29	C₂²×C₈×C₆₁	—	2	2	no	—
30	C₂²×C₈×C₆₁	—	2	2	no	—
31	C₂²×C₈×C₆₁	—	2	4	no	—
32	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₆₁	D₄	1	4	yes	—
35	C₂²×C₈×C₆₁	Q₈	1	4	yes	—
36	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
37	C₂×C₄⋊C₄×C₆₁	—	2	8	no	—
38	C₂×C₄²×C₆₁	—	2	4	no	—
39	C₂×C₄²×C₆₁	—	2	2	no	—
40	C₂×C₄²×C₆₁	—	2	2	no	—
41	C₂×C₄²×C₆₁	—	2	4	no	—
42	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₆₁	D₄	1	2	yes	—
45	C₂×C₄²×C₆₁	D₄	1	2	yes	—
46	C₂×C₄²×C₆₁	D₄	1	2	yes	—
47	C₂×C₄²×C₆₁	Q₈	1	2	yes	—
48	C₄⋊C₈×C₆₁	—	2	8	no	—
49	#97×C₆₁	—	2	8	no	—
50	#95×C₆₁	—	2	4	no	—
51	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
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	—	2	8	no	—
56	C₂²×C₄×C₆₁	—	2	4	no	—
57	C₂²×C₄×C₆₁	—	2	8	no	—
58	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
59	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
60	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₆₁	M₁₆	1	8	yes	—
65	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₆₁	#41	1	2	yes	—
74	C₂²×C₄×C₆₁	#41	1	2	yes	—
75	C₂×C₈×C₆₁	—	2	4	no	—
76	C₂×C₈×C₆₁	—	2	4	no	—
77	C₂×C₈×C₆₁	—	2	4	no	—
78	C₂×C₈×C₆₁	—	2	4	no	—
79	C₂×C₈×C₆₁	—	2	4	no	—
80	C₂×C₈×C₆₁	—	2	4	no	—
81	C₂×C₈×C₆₁	—	2	4	no	—
82	C₂×C₈×C₆₁	—	2	8	no	—
83	C₂×C₈×C₆₁	—	4	4	no	—
84	C₂×C₈×C₆₁	—	4	2	no	—
85	C₂×C₈×C₆₁	—	4	2	no	—
86	C₂×C₈×C₆₁	C₄⋊C₄	1	8	yes	complemented
87	C₄⋊C₄×C₆₁	—	2	16	no	—
88	C₄²×C₆₁	—	4	4	no	—
89	C₄²×C₆₁	—	4	2	no	—
90	C₄²×C₆₁	—	4	2	no	—
91	C₄²×C₆₁	C₂²⋊C₄	1	4	yes	—
92	C₄²×C₆₁	C₂²⋊C₄	1	4	yes	—
93	C₄²×C₆₁	C₄⋊C₄	1	4	yes	—
94	C₄²×C₆₁	C₄⋊C₄	1	4	yes	—
95	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
96	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂³×C₆₁	C₄×D₄	1	2	yes	—
99	C₂³×C₆₁	C₄×Q₈	1	2	yes	—
100	C₂³×C₆₁	C₂×M₁₆	1	4	yes	—
101	C₂³×C₆₁	#131	1	4	yes	—
102	C₂×C₄×C₆₁	—	2	4	no	—
103	C₂×C₄×C₆₁	—	2	4	no	—
104	C₂×C₄×C₆₁	—	2	4	no	—
105	C₂×C₄×C₆₁	—	2	4	no	—
106	C₂×C₄×C₆₁	—	2	4	no	—
107	C₂×C₄×C₆₁	—	2	4	no	—
108	C₂×C₄×C₆₁	—	2	8	no	—
109	C₂×C₄×C₆₁	—	2	16	no	—
110	C₂×C₄×C₆₁	—	4	4	no	—
111	C₂×C₄×C₆₁	—	4	4	no	—
112	C₂×C₄×C₆₁	—	4	8	no	—
113	C₂×C₄×C₆₁	#95	1	4	yes	—
114	C₂×C₄×C₆₁	#95	1	4	yes	—
115	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₆₁	#117	1	2	yes	—
121	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₆₁	#121	1	2	yes	—
124	C₂×C₄×C₆₁	#121	1	2	yes	—
125	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
128	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
129	C₂×C₄×C₆₁	#125	1	2	yes	—
130	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
131	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
132	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
133	C₈×C₆₁	—	2	8	no	—
134	C₈×C₆₁	—	4	4	no	—
135	#10835	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×C₆₁	C₂×#95	1	1	yes	characteristic
137	C₂²×C₆₁	C₄×C₄⋊C₄	1	2	yes	—
138	C₂²×C₆₁	#389	1	2	yes	—
139	C₂²×C₆₁	#390	1	1	yes	characteristic
140	C₂²×C₆₁	#391	1	1	yes	characteristic
141	C₂²×C₆₁	C₄×M₁₆	1	2	yes	—
142	C₂²×C₆₁	#405	1	2	yes	—
143	C₂²×C₆₁	#407	1	2	yes	—
144	C₂²×C₆₁	#408	1	2	yes	—
145	C₂²×C₆₁	#423	1	2	yes	—
146	C₂²×C₆₁	#424	1	2	yes	—
147	C₂²×C₆₁	#435	1	8	yes	—
148	C₂²×C₆₁	#446	1	8	yes	—
149	C₄×C₆₁	—	2	4	no	—
150	C₄×C₆₁	—	2	4	no	—
151	C₄×C₆₁	—	4	8	no	—
152	C₄×C₆₁	#380	1	2	yes	—
153	C₄×C₆₁	#382	1	1	yes	characteristic
154	C₄×C₆₁	#384	1	1	yes	characteristic
155	C₄×C₆₁	#389	1	2	yes	—
156	C₄×C₆₁	#390	1	1	yes	characteristic
157	C₄×C₆₁	#391	1	1	yes	characteristic
158	C₂²×C₄⋊C₈	C₂×C₆₁	1	1	yes	characteristic
159	C₂²×#97	C₂×C₆₁	1	1	yes	characteristic
160	C₂×C₄×C₄⋊C₄	C₂×C₆₁	1	1	yes	characteristic
161	C₂×#336	C₂×C₆₁	1	4	yes	—
162	C₂×C₆₁	C₄⋊#95	1	1	yes	characteristic
163	C₂×C₆₁	#1728	1	1	yes	characteristic
164	C₂×C₆₁	#1729	1	1	yes	characteristic
165	C₂×C₆₁	#1771	1	4	yes	—
166	C₂×C₆₁	#1940	1	4	yes	—
167	C₂×C₆₁	#1955	1	2	yes	—
168	C₂×C₆₁	#1985	1	2	yes	—
169	C₂³×C₈	C₂²×C₆₁	1	2	yes	—
170	C₂³×C₈	C₂²×C₆₁	1	2	yes	—
171	C₂²×C₄⋊C₄	C₄×C₆₁	1	2	yes	—
172	C₂²×C₄²	C₂²×C₆₁	1	2	yes	—
173	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
174	C₂×C₄⋊C₈	—	2	2	no	—
175	C₂×C₄⋊C₈	—	2	2	no	—
176	C₂×C₄⋊C₈	C₄×C₆₁	1	4	yes	complemented
177	C₂×#97	—	2	2	no	—
178	C₂×#97	—	2	2	no	—
179	C₂×#97	C₄×C₆₁	1	4	yes	complemented
180	C₄×C₄⋊C₄	—	2	2	no	—
181	C₄×C₄⋊C₄	—	2	2	no	—
182	C₂×#95	C₄×C₆₁	1	2	yes	—
183	#336	—	2	8	no	—
184	C₆₁	#10835	1	1	yes	characteristic, direct product factor, Sylow
185	C₂³×C₄	C₂×C₄×C₆₁	1	4	yes	—
186	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
187	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
188	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	2	4	no	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	2	4	no	—
197	C₂²×C₈	C₂×C₄×C₆₁	1	4	yes	—
198	C₂²×C₈	C₂×C₄×C₆₁	1	4	yes	—
199	C₂²×C₈	D₄×C₆₁	1	4	yes	—
200	C₂²×C₈	Q₈×C₆₁	1	4	yes	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄⋊C₄	—	2	8	no	—
203	C₂×C₄²	—	2	4	no	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	2	4	no	—
207	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
208	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
209	C₂×C₄²	D₄×C₆₁	1	2	yes	—
210	C₂×C₄²	D₄×C₆₁	1	2	yes	—
211	C₂×C₄²	D₄×C₆₁	1	2	yes	—
212	C₂×C₄²	Q₈×C₆₁	1	2	yes	—
213	C₄⋊C₈	—	2	8	no	—
214	#97	—	2	8	no	—
215	#95	—	2	4	no	—
216	C₂⁴	C₂²×C₄×C₆₁	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	C₄²×C₆₁	1	2	yes	—
224	C₂²×C₄	C₄²×C₆₁	1	2	yes	—
225	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
226	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
227	C₂²×C₄	C₄⋊C₄×C₆₁	1	2	yes	—
228	C₂²×C₄	C₄⋊C₄×C₆₁	1	2	yes	—
229	C₂²×C₄	M₁₆×C₆₁	1	8	yes	—
230	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
231	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
236	C₂²×C₄	C₂×Q₈×C₆₁	1	1	yes	characteristic
237	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
238	C₂²×C₄	#41×C₆₁	1	2	yes	—
239	C₂²×C₄	#41×C₆₁	1	2	yes	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	8	no	—
248	C₂×C₈	—	4	4	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	C₄⋊C₄×C₆₁	1	8	yes	complemented
252	C₄⋊C₄	—	2	16	no	—
253	C₄²	—	4	4	no	—
254	C₄²	—	4	2	no	—
255	C₄²	—	4	2	no	—
256	C₄²	C₂²⋊C₄×C₆₁	1	4	yes	—
257	C₄²	C₂²⋊C₄×C₆₁	1	4	yes	—
258	C₄²	C₄⋊C₄×C₆₁	1	4	yes	—
259	C₄²	C₄⋊C₄×C₆₁	1	4	yes	—
260	C₂³	C₂×C₄²×C₆₁	1	1	yes	characteristic
261	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
262	C₂³	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
263	C₂³	C₄×D₄×C₆₁	1	2	yes	—
264	C₂³	C₄×Q₈×C₆₁	1	2	yes	—
265	C₂³	C₂×M₁₆×C₆₁	1	4	yes	—
266	C₂³	#131×C₆₁	1	4	yes	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	16	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	8	no	—
278	C₂×C₄	#95×C₆₁	1	4	yes	—
279	C₂×C₄	#95×C₆₁	1	4	yes	—
280	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
281	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
282	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
283	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
285	C₂×C₄	#117×C₆₁	1	2	yes	—
286	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
287	C₂×C₄	C₄×Q₈×C₆₁	1	2	yes	—
288	C₂×C₄	#121×C₆₁	1	2	yes	—
289	C₂×C₄	#121×C₆₁	1	2	yes	—
290	C₂×C₄	C₂²⋊Q₈×C₆₁	1	2	yes	—
291	C₂×C₄	C₂²⋊Q₈×C₆₁	1	2	yes	—
292	C₂×C₄	#124×C₆₁	1	1	yes	characteristic
293	C₂×C₄	#124×C₆₁	1	1	yes	characteristic
294	C₂×C₄	#125×C₆₁	1	2	yes	—
295	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
296	C₂×C₄	C₄⋊Q₈×C₆₁	1	2	yes	—
297	C₂×C₄	C₄⋊Q₈×C₆₁	1	1	yes	characteristic
298	C₈	—	2	8	no	—
299	C₈	—	4	4	no	—
300	C₂²	C₂×#95×C₆₁	1	1	yes	characteristic
301	C₂²	C₄×C₄⋊C₄×C₆₁	1	2	yes	—
302	C₂²	C₆₁×#389	1	2	yes	—
303	C₂²	C₆₁×#390	1	1	yes	characteristic
304	C₂²	C₆₁×#391	1	1	yes	characteristic
305	C₂²	C₄×M₁₆×C₆₁	1	2	yes	—
306	C₂²	C₆₁×#405	1	2	yes	—
307	C₂²	C₆₁×#407	1	2	yes	—
308	C₂²	C₆₁×#408	1	2	yes	—
309	C₂²	C₆₁×#423	1	2	yes	—
310	C₂²	C₆₁×#424	1	2	yes	—
311	C₂²	C₆₁×#435	1	8	yes	—
312	C₂²	C₆₁×#446	1	8	yes	—
313	C₄	—	2	4	no	—
314	C₄	—	2	4	no	—
315	C₄	—	4	8	no	—
316	C₄	C₆₁×#380	1	2	yes	—
317	C₄	C₆₁×#382	1	1	yes	characteristic
318	C₄	C₆₁×#384	1	1	yes	characteristic
319	C₄	C₆₁×#389	1	2	yes	—
320	C₄	C₆₁×#390	1	1	yes	characteristic
321	C₄	C₆₁×#391	1	1	yes	characteristic
322	C₂	C₆₁×C₄⋊#95	1	1	yes	characteristic
323	C₂	C₆₁×#1728	1	1	yes	characteristic
324	C₂	C₆₁×#1729	1	1	yes	characteristic
325	C₂	C₆₁×#1771	1	4	yes	—
326	C₂	C₆₁×#1940	1	4	yes	—
327	C₂	C₆₁×#1955	1	2	yes	—
328	C₂	C₆₁×#1985	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
12. C₂×C₄⋊C₈×C₆₁	C₄	semidirect
15. C₂×#97×C₆₁	C₄	semidirect
86. C₂×C₈×C₆₁	C₄⋊C₄	semidirect
135. #10835	C₆₁	direct (N × H)
176. C₂×C₄⋊C₈	C₄×C₆₁	semidirect
179. C₂×#97	C₄×C₆₁	semidirect
184. C₆₁	#10835	direct (N × H)
251. C₂×C₈	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
C₂²×C₄⋊C₈×C₆₁	2
C₂²×#97×C₆₁	2
C₂×C₄×C₄⋊C₄×C₆₁	2
C₂×C₆₁×#336	2
#10835	61

Subgroups — C61×#10835 #29514268

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#10835 #29514268