Subgroups — C₆₁×#9735 #29513168

Order 15616.

522 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#486	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1987	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#1982	C₂	1	1	yes	maximal, characteristic
4	C₈×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₆₁×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₆₁×#333	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₆₁×#331	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
9	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
10	C₆₁×#486	—	2	2	no	—
11	C₆₁×#486	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×#102×C₆₁	—	2	1	no	—
16	C₂×#102×C₆₁	—	2	1	no	—
17	C₂×#102×C₆₁	—	2	2	no	—
18	C₂×C₄×C₈×C₆₁	—	2	1	no	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
21	C₆₁×#340	—	2	2	no	—
22	C₆₁×#336	—	2	1	no	—
23	C₆₁×#334	—	2	2	no	—
24	C₆₁×#333	—	2	2	no	—
25	C₆₁×#331	—	2	2	no	—
26	C₂²×Q₈×C₆₁	—	2	1	no	—
27	C₂²×D₄×C₆₁	—	2	1	no	—
28	C₂²×D₄×C₆₁	—	2	1	no	—
29	C₂²×D₄×C₆₁	—	2	1	no	—
30	C₂²×C₈×C₆₁	—	2	1	no	—
31	C₂²×C₈×C₆₁	—	2	1	no	—
32	C₂²×C₈×C₆₁	—	2	1	no	—
33	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
35	C₄⋊Q₈×C₆₁	—	2	1	no	—
36	C₄⋊Q₈×C₆₁	—	2	1	no	—
37	C₄⋊Q₈×C₆₁	—	2	1	no	—
38	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
39	#127×C₆₁	—	2	1	no	—
40	#127×C₆₁	—	2	1	no	—
41	#127×C₆₁	—	2	1	no	—
42	#127×C₆₁	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
44	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
45	C₂×C₄⋊C₄×C₆₁	D₄	1	2	yes	—
46	C₂×C₄²×C₆₁	—	2	1	no	—
47	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₆₁	—	2	1	no	—
49	C₄⋊C₈×C₆₁	—	2	1	no	—
50	C₄⋊C₈×C₆₁	—	2	1	no	—
51	C₄⋊C₈×C₆₁	—	2	1	no	—
52	C₄⋊C₈×C₆₁	—	2	1	no	—
53	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
54	#102×C₆₁	—	2	4	no	—
55	#102×C₆₁	—	2	4	no	—
56	#102×C₆₁	—	4	4	no	—
57	C₄×C₈×C₆₁	—	4	2	no	—
58	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₆₁	D₄	1	2	yes	—
60	#95×C₆₁	—	2	1	no	—
61	C₂⁴×C₆₁	—	4	1	no	—
62	C₂×Q₈×C₆₁	—	2	2	no	—
63	C₂×Q₈×C₆₁	—	4	1	no	—
64	C₂×Q₈×C₆₁	—	8	2	no	—
65	C₂×D₄×C₆₁	—	2	2	no	—
66	C₂×D₄×C₆₁	—	2	2	no	—
67	C₂×D₄×C₆₁	—	2	2	no	—
68	C₂×D₄×C₆₁	—	4	1	no	—
69	C₂×D₄×C₆₁	—	4	1	no	—
70	C₂×D₄×C₆₁	—	4	1	no	—
71	C₂×D₄×C₆₁	—	4	2	no	—
72	C₂×D₄×C₆₁	—	4	1	no	—
73	C₂×D₄×C₆₁	—	4	1	no	—
74	C₂×D₄×C₆₁	—	4	2	no	—
75	C₂×D₄×C₆₁	—	4	1	no	—
76	C₂×D₄×C₆₁	—	4	1	no	—
77	C₂×D₄×C₆₁	—	8	2	no	—
78	C₂²×C₄×C₆₁	—	2	2	no	—
79	C₂²×C₄×C₆₁	—	4	1	no	—
80	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
83	C₂×C₈×C₆₁	—	2	1	no	—
84	C₂×C₈×C₆₁	—	2	1	no	—
85	C₂×C₈×C₆₁	—	2	1	no	—
86	C₂×C₈×C₆₁	—	2	1	no	—
87	C₂×C₈×C₆₁	—	2	2	no	—
88	C₂×C₈×C₆₁	—	4	1	no	—
89	C₂×C₈×C₆₁	—	4	1	no	—
90	C₂×C₈×C₆₁	—	4	1	no	—
91	C₂×C₈×C₆₁	—	4	1	no	—
92	C₂×C₈×C₆₁	—	4	2	no	—
93	C₂×C₈×C₆₁	—	4	2	no	—
94	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
95	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
96	C₄⋊C₄×C₆₁	—	2	2	no	—
97	C₄⋊C₄×C₆₁	—	2	2	no	—
98	C₄⋊C₄×C₆₁	—	2	2	no	—
99	C₄⋊C₄×C₆₁	—	4	1	no	—
100	C₄⋊C₄×C₆₁	—	4	1	no	—
101	C₄⋊C₄×C₆₁	SD₁₆	1	4	yes	—
102	C₄²×C₆₁	—	4	1	no	—
103	C₄²×C₆₁	—	4	1	no	—
104	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₆₁	#41	1	1	yes	characteristic
108	C₂³×C₆₁	—	4	2	no	—
109	C₂³×C₆₁	—	8	1	no	—
110	C₂³×C₆₁	—	8	1	no	—
111	C₂³×C₆₁	—	8	2	no	—
112	C₂³×C₆₁	—	8	2	no	—
113	C₂³×C₆₁	#121	1	1	yes	characteristic
114	Q₈×C₆₁	—	8	2	no	—
115	Q₈×C₆₁	—	8	2	no	—
116	D₄×C₆₁	—	4	2	no	—
117	D₄×C₆₁	—	4	2	no	—
118	D₄×C₆₁	—	4	4	no	—
119	D₄×C₆₁	—	8	2	no	—
120	D₄×C₆₁	—	8	2	no	—
121	D₄×C₆₁	—	8	1	no	—
122	D₄×C₆₁	—	8	1	no	—
123	D₄×C₆₁	—	8	1	no	—
124	D₄×C₆₁	—	8	1	no	—
125	C₂×C₄×C₆₁	—	2	2	no	—
126	C₂×C₄×C₆₁	—	2	4	no	—
127	C₂×C₄×C₆₁	—	4	2	no	—
128	C₂×C₄×C₆₁	—	4	2	no	—
129	C₂×C₄×C₆₁	—	4	2	no	—
130	C₂×C₄×C₆₁	—	8	2	no	—
131	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
135	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
136	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
137	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
138	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
139	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
140	C₂×C₄×C₆₁	#135	1	2	yes	—
141	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
143	C₈×C₆₁	—	2	2	no	—
144	C₈×C₆₁	—	2	2	no	—
145	C₈×C₆₁	—	4	2	no	—
146	C₈×C₆₁	—	8	1	no	—
147	C₈×C₆₁	—	8	1	no	—
148	#9735	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂²×C₆₁	—	8	2	no	—
150	C₂²×C₆₁	—	8	2	no	—
151	C₂²×C₆₁	—	8	2	no	—
152	C₂²×C₆₁	—	16	2	no	—
153	C₂²×C₆₁	—	16	2	no	—
154	C₂²×C₆₁	#390	1	1	yes	characteristic
155	C₂²×C₆₁	#463	1	1	yes	characteristic
156	C₂²×C₆₁	#464	1	1	yes	characteristic
157	C₂²×C₆₁	#465	1	2	yes	—
158	C₂²×C₆₁	#472	1	2	yes	—
159	C₄×C₆₁	—	2	1	no	—
160	C₄×C₆₁	—	2	1	no	—
161	C₄×C₆₁	—	4	4	no	—
162	C₄×C₆₁	—	8	2	no	—
163	C₄×C₆₁	#416	1	1	yes	characteristic
164	C₄×C₆₁	#417	1	1	yes	characteristic
165	C₄×C₆₁	C₄×SD₁₆	1	2	yes	—
166	C₄×C₆₁	#463	1	1	yes	characteristic
167	C₄×C₆₁	#464	1	1	yes	characteristic
168	C₄×C₆₁	#492	1	1	yes	characteristic
169	C₄×C₆₁	#495	1	1	yes	characteristic
170	C₂×#486	C₂×C₆₁	1	1	yes	characteristic
171	#1987	C₂×C₆₁	1	1	yes	characteristic, complemented
172	#1982	C₂×C₆₁	1	1	yes	characteristic
173	C₈×C₄⋊C₄	C₂×C₆₁	1	1	yes	characteristic, complemented
174	C₂×#334	C₂×C₆₁	1	1	yes	characteristic, complemented
175	C₂×#333	C₂×C₆₁	1	1	yes	characteristic, complemented
176	C₂×#331	C₂×C₆₁	1	1	yes	characteristic
177	C₂×C₆₁	—	16	2	no	—
178	C₂×C₆₁	#1677	1	2	yes	—
179	C₂×C₆₁	#1678	1	2	yes	—
180	C₂×C₆₁	#1968	1	1	yes	characteristic
181	C₂×C₆₁	#1978	1	1	yes	characteristic
182	C₂×C₆₁	#1983	1	1	yes	characteristic
183	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
184	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
185	#486	—	2	2	no	—
186	#486	C₄×C₆₁	1	4	yes	complemented
187	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
188	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
189	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
190	C₂×#102	—	2	1	no	—
191	C₂×#102	—	2	1	no	—
192	C₂×#102	—	2	2	no	—
193	C₂×C₄×C₈	—	2	1	no	—
194	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
195	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
196	#340	—	2	2	no	—
197	#336	—	2	1	no	—
198	#334	—	2	2	no	—
199	#333	—	2	2	no	—
200	#331	—	2	2	no	—
201	C₆₁	#9735	1	1	yes	characteristic, direct product factor, Sylow
202	C₂²×Q₈	—	2	1	no	—
203	C₂²×D₄	—	2	1	no	—
204	C₂²×D₄	—	2	1	no	—
205	C₂²×D₄	—	2	1	no	—
206	C₂²×C₈	—	2	1	no	—
207	C₂²×C₈	—	2	1	no	—
208	C₂²×C₈	—	2	1	no	—
209	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
210	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	C₂×C₄×C₆₁	1	2	yes	—
215	#127	—	2	1	no	—
216	#127	—	2	1	no	—
217	#127	—	2	1	no	—
218	#127	C₂×C₄×C₆₁	1	2	yes	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	D₄×C₆₁	1	2	yes	—
222	C₂×C₄²	—	2	1	no	—
223	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	—	2	1	no	—
229	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
230	#102	—	2	4	no	—
231	#102	—	2	4	no	—
232	#102	—	4	4	no	—
233	C₄×C₈	—	4	2	no	—
234	C₄×C₈	C₂×C₄×C₆₁	1	2	yes	—
235	C₄×C₈	D₄×C₆₁	1	2	yes	—
236	#95	—	2	1	no	—
237	C₂⁴	—	4	1	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	8	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	8	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
258	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	2	no	—
269	C₂×C₈	—	4	2	no	—
270	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
271	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	SD₁₆×C₆₁	1	4	yes	—
278	C₄²	—	4	1	no	—
279	C₄²	—	4	1	no	—
280	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
283	C₄²	#41×C₆₁	1	1	yes	characteristic
284	C₂³	—	4	2	no	—
285	C₂³	—	8	1	no	—
286	C₂³	—	8	1	no	—
287	C₂³	—	8	2	no	—
288	C₂³	—	8	2	no	—
289	C₂³	#121×C₆₁	1	1	yes	characteristic
290	Q₈	—	8	2	no	—
291	Q₈	—	8	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	4	no	—
295	D₄	—	8	2	no	—
296	D₄	—	8	2	no	—
297	D₄	—	8	1	no	—
298	D₄	—	8	1	no	—
299	D₄	—	8	1	no	—
300	D₄	—	8	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
308	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
309	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
310	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
311	C₂×C₄	#124×C₆₁	1	1	yes	characteristic
312	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
313	C₂×C₄	C₂×SD₁₆×C₆₁	1	2	yes	—
314	C₂×C₄	#135×C₆₁	1	1	yes	characteristic
315	C₂×C₄	#135×C₆₁	1	1	yes	characteristic
316	C₂×C₄	#135×C₆₁	1	2	yes	—
317	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
318	C₂×C₄	#137×C₆₁	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	2	2	no	—
321	C₈	—	4	2	no	—
322	C₈	—	8	1	no	—
323	C₈	—	8	1	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	16	2	no	—
328	C₂²	—	16	2	no	—
329	C₂²	C₆₁×#390	1	1	yes	characteristic
330	C₂²	C₆₁×#463	1	1	yes	characteristic
331	C₂²	C₆₁×#464	1	1	yes	characteristic
332	C₂²	C₆₁×#465	1	2	yes	—
333	C₂²	C₆₁×#472	1	2	yes	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	4	4	no	—
337	C₄	—	8	2	no	—
338	C₄	C₆₁×#416	1	1	yes	characteristic
339	C₄	C₆₁×#417	1	1	yes	characteristic
340	C₄	C₄×SD₁₆×C₆₁	1	2	yes	—
341	C₄	C₆₁×#463	1	1	yes	characteristic
342	C₄	C₆₁×#464	1	1	yes	characteristic
343	C₄	C₆₁×#492	1	1	yes	characteristic
344	C₄	C₆₁×#495	1	1	yes	characteristic
345	C₂	—	16	2	no	—
346	C₂	C₆₁×#1677	1	2	yes	—
347	C₂	C₆₁×#1678	1	2	yes	—
348	C₂	C₆₁×#1968	1	1	yes	characteristic
349	C₂	C₆₁×#1978	1	1	yes	characteristic
350	C₂	C₆₁×#1983	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	#121
commutator	C₄²	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
prime radical	C₂×#127×C₆₁	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#1987	C₂	semidirect
4. C₈×C₄⋊C₄×C₆₁	C₂	semidirect
5. C₂×C₆₁×#334	C₂	semidirect
6. C₂×C₆₁×#333	C₂	semidirect
11. C₆₁×#486	C₄	semidirect
148. #9735	C₆₁	direct (N × H)
171. #1987	C₂×C₆₁	semidirect
173. C₈×C₄⋊C₄	C₂×C₆₁	semidirect
174. C₂×#334	C₂×C₆₁	semidirect
175. C₂×#333	C₂×C₆₁	semidirect
186. #486	C₄×C₆₁	semidirect
201. C₆₁	#9735	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×#486	2
C₆₁×#1987	2
C₆₁×#1982	2
C₈×C₄⋊C₄×C₆₁	2
C₂×C₆₁×#334	2
C₂×C₆₁×#333	2
C₂×C₆₁×#331	2
#9735	61

Subgroups — C61×#9735 #29513168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#9735 #29513168