Subgroups — C₆₁×#12991 #29516424

Order 15616.

626 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#466	C₂	1	1	yes	maximal, characteristic
2	C₂²×#107×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₆₁×#357	C₂	1	2	yes	maximal, complemented
4	C₂×C₆₁×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1380	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₆₁	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
8	C₂×#121×C₆₁	—	2	1	no	—
9	C₂²×C₄⋊C₄×C₆₁	—	2	1	no	—
10	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
11	C₆₁×#466	—	2	2	no	—
12	C₆₁×#466	C₄	1	4	yes	complemented
13	C₂×#107×C₆₁	—	2	1	no	—
14	C₂×#107×C₆₁	—	2	4	no	—
15	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#107×C₆₁	C₂²	1	2	yes	—
17	C₂×#102×C₆₁	—	2	1	no	—
18	C₆₁×#357	—	2	4	no	—
19	C₆₁×C₂²⋊C₁₆	—	2	2	no	—
20	C₂²×D₄×C₆₁	—	2	1	no	—
21	C₂²×D₄×C₆₁	—	4	1	no	—
22	C₂³×C₄×C₆₁	—	4	1	no	—
23	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
24	C₂×D₈×C₆₁	—	2	2	no	—
25	C₂×D₈×C₆₁	—	2	1	no	—
26	C₂×D₈×C₆₁	—	2	2	no	—
27	C₂×D₈×C₆₁	C₂×C₄	1	2	yes	complemented
28	C₂²×C₈×C₆₁	—	2	2	no	—
29	C₂²×C₈×C₆₁	—	2	2	no	—
30	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₁	D₄	1	2	yes	—
33	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
34	#121×C₆₁	—	2	4	no	—
35	#121×C₆₁	—	4	2	no	—
36	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
37	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
38	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₁	—	4	2	no	—
40	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
41	C₂×C₂²⋊C₄×C₆₁	—	4	1	no	—
42	C₂×C₁₆×C₆₁	—	2	2	no	—
43	C₂×C₁₆×C₆₁	—	2	2	no	—
44	C₂×C₁₆×C₆₁	—	2	2	no	—
45	#107×C₆₁	—	2	4	no	—
46	#107×C₆₁	—	2	2	no	—
47	#107×C₆₁	—	2	1	no	—
48	#107×C₆₁	C₂×C₄	1	2	yes	complemented
49	#107×C₆₁	D₄	1	4	yes	complemented
50	#102×C₆₁	—	2	4	no	—
51	C₂⁴×C₆₁	—	4	1	no	—
52	C₂⁴×C₆₁	D₈	1	1	yes	characteristic
53	C₂×D₄×C₆₁	—	2	2	no	—
54	C₂×D₄×C₆₁	—	4	4	no	—
55	C₂×D₄×C₆₁	—	4	1	no	—
56	C₂×D₄×C₆₁	—	4	1	no	—
57	C₂×D₄×C₆₁	—	4	1	no	—
58	C₂×D₄×C₆₁	—	4	1	no	—
59	C₂×D₄×C₆₁	—	4	1	no	—
60	C₂×D₄×C₆₁	—	8	2	no	—
61	C₂×D₄×C₆₁	—	8	2	no	—
62	C₂²×C₄×C₆₁	—	2	1	no	—
63	C₂²×C₄×C₆₁	—	2	1	no	—
64	C₂²×C₄×C₆₁	—	2	1	no	—
65	C₂²×C₄×C₆₁	—	2	1	no	—
66	C₂²×C₄×C₆₁	—	4	4	no	—
67	C₂²×C₄×C₆₁	—	4	1	no	—
68	C₂²×C₄×C₆₁	—	4	2	no	—
69	C₂²×C₄×C₆₁	—	8	2	no	—
70	C₂²×C₄×C₆₁	—	8	2	no	—
71	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₆₁	SD₁₆	1	2	yes	—
74	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
75	D₈×C₆₁	—	2	4	no	—
76	D₈×C₆₁	—	4	2	no	—
77	C₂×C₈×C₆₁	—	2	1	no	—
78	C₂×C₈×C₆₁	—	2	1	no	—
79	C₂×C₈×C₆₁	—	2	4	no	—
80	C₂×C₈×C₆₁	—	2	4	no	—
81	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₆₁	#41	1	2	yes	—
86	C₄⋊C₄×C₆₁	—	2	2	no	—
87	C₄⋊C₄×C₆₁	—	2	4	no	—
88	C₄⋊C₄×C₆₁	—	4	4	no	—
89	C₄⋊C₄×C₆₁	—	4	1	no	—
90	C₄⋊C₄×C₆₁	—	4	2	no	—
91	C₂²⋊C₄×C₆₁	—	4	4	no	—
92	C₁₆×C₆₁	—	4	2	no	—
93	C₂³×C₆₁	—	2	1	no	—
94	C₂³×C₆₁	—	2	1	no	—
95	C₂³×C₆₁	—	2	1	no	—
96	C₂³×C₆₁	—	2	1	no	—
97	C₂³×C₆₁	—	4	2	no	—
98	C₂³×C₆₁	—	8	1	no	—
99	C₂³×C₆₁	—	8	1	no	—
100	C₂³×C₆₁	—	8	1	no	—
101	C₂³×C₆₁	—	8	1	no	—
102	C₂³×C₆₁	—	8	1	no	—
103	C₂³×C₆₁	—	8	1	no	—
104	C₂³×C₆₁	#102	1	2	yes	—
105	C₂³×C₆₁	D₁₆	1	2	yes	—
106	C₂³×C₆₁	SD₃₂	1	2	yes	—
107	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
108	D₄×C₆₁	—	4	2	no	—
109	D₄×C₆₁	—	4	2	no	—
110	D₄×C₆₁	—	8	8	no	—
111	D₄×C₆₁	—	8	1	no	—
112	D₄×C₆₁	—	8	1	no	—
113	C₂×C₄×C₆₁	—	2	1	no	—
114	C₂×C₄×C₆₁	—	2	1	no	—
115	C₂×C₄×C₆₁	—	2	2	no	—
116	C₂×C₄×C₆₁	—	2	2	no	—
117	C₂×C₄×C₆₁	—	2	2	no	—
118	C₂×C₄×C₆₁	—	2	2	no	—
119	C₂×C₄×C₆₁	—	4	2	no	—
120	C₂×C₄×C₆₁	—	4	4	no	—
121	C₂×C₄×C₆₁	—	8	8	no	—
122	C₂×C₄×C₆₁	—	8	2	no	—
123	C₂×C₄×C₆₁	—	8	1	no	—
124	C₂×C₄×C₆₁	—	8	1	no	—
125	C₂×C₄×C₆₁	—	8	2	no	—
126	C₂×C₄×C₆₁	#102	1	2	yes	—
127	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
129	C₂×C₄×C₆₁	#123	1	1	yes	characteristic
130	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
133	C₈×C₆₁	—	2	2	no	—
134	C₈×C₆₁	C₄×D₄	1	2	yes	—
135	C₈×C₆₁	#121	1	2	yes	—
136	#12991	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂²×C₆₁	—	2	2	no	—
138	C₂²×C₆₁	—	2	2	no	—
139	C₂²×C₆₁	—	2	2	no	—
140	C₂²×C₆₁	—	2	2	no	—
141	C₂²×C₆₁	—	2	2	no	—
142	C₂²×C₆₁	—	2	2	no	—
143	C₂²×C₆₁	—	8	2	no	—
144	C₂²×C₆₁	—	8	2	no	—
145	C₂²×C₆₁	—	8	2	no	—
146	C₂²×C₆₁	—	16	1	no	—
147	C₂²×C₆₁	—	16	1	no	—
148	C₂²×C₆₁	—	16	1	no	—
149	C₂²×C₆₁	—	16	1	no	—
150	C₂²×C₆₁	#357	1	4	yes	complemented
151	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
152	C₂²×C₆₁	#447	1	1	yes	characteristic
153	C₂²×C₆₁	#480	1	1	yes	characteristic
154	C₂²×C₆₁	C₂×D₁₆	1	1	yes	characteristic
155	C₂²×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
156	C₂²×C₆₁	#509	1	1	yes	characteristic
157	C₂²×C₆₁	#510	1	1	yes	characteristic
158	C₄×C₆₁	—	2	2	no	—
159	C₄×C₆₁	—	8	2	no	—
160	C₄×C₆₁	—	8	4	no	—
161	C₄×C₆₁	#386	1	1	yes	characteristic
162	C₄×C₆₁	#417	1	1	yes	characteristic
163	C₄×C₆₁	#448	1	1	yes	characteristic
164	C₄×C₆₁	#481	1	1	yes	characteristic
165	C₂×#466	C₂×C₆₁	1	1	yes	characteristic
166	C₂²×#107	C₂×C₆₁	1	1	yes	characteristic, complemented
167	C₂×#357	C₂×C₆₁	1	2	yes	complemented
168	C₂×C₂²⋊C₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
169	#1380	C₂×C₆₁	1	2	yes	complemented
170	C₂×C₆₁	—	2	4	no	—
171	C₂×C₆₁	—	16	2	no	—
172	C₂×C₆₁	#1874	1	1	yes	characteristic
173	C₂×C₆₁	C₂×#357	1	1	yes	characteristic
174	C₂×C₆₁	#2139	1	1	yes	characteristic
175	C₂×C₆₁	#2184	1	1	yes	characteristic
176	C₂×C₆₁	#2185	1	1	yes	characteristic
177	C₂×C₆₁	#2232	1	1	yes	characteristic
178	C₂×C₆₁	#2233	1	1	yes	characteristic
179	C₂²×D₈	C₂²×C₆₁	1	1	yes	characteristic
180	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
181	C₂×#121	—	2	1	no	—
182	C₂²×C₄⋊C₄	—	2	1	no	—
183	C₂²×C₁₆	C₂²×C₆₁	1	2	yes	—
184	#466	—	2	2	no	—
185	#466	C₄×C₆₁	1	4	yes	complemented
186	C₂×#107	—	2	1	no	—
187	C₂×#107	—	2	4	no	—
188	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
189	C₂×#107	C₂²×C₆₁	1	2	yes	—
190	C₂×#102	—	2	1	no	—
191	#357	—	2	4	no	—
192	C₂²⋊C₁₆	—	2	2	no	—
193	C₆₁	#12991	1	1	yes	characteristic, direct product factor, Sylow
194	C₂²×D₄	—	2	1	no	—
195	C₂²×D₄	—	4	1	no	—
196	C₂³×C₄	—	4	1	no	—
197	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
198	C₂×D₈	—	2	2	no	—
199	C₂×D₈	—	2	1	no	—
200	C₂×D₈	—	2	2	no	—
201	C₂×D₈	C₂×C₄×C₆₁	1	2	yes	complemented
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	—	2	2	no	—
204	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
205	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
206	C₂²×C₈	D₄×C₆₁	1	2	yes	—
207	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
208	#121	—	2	4	no	—
209	#121	—	4	2	no	—
210	C₂×C₄⋊C₄	—	2	4	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	4	2	no	—
214	C₂×C₄⋊C₄	—	4	1	no	—
215	C₂×C₂²⋊C₄	—	4	1	no	—
216	C₂×C₁₆	—	2	2	no	—
217	C₂×C₁₆	—	2	2	no	—
218	C₂×C₁₆	—	2	2	no	—
219	#107	—	2	4	no	—
220	#107	—	2	2	no	—
221	#107	—	2	1	no	—
222	#107	C₂×C₄×C₆₁	1	2	yes	complemented
223	#107	D₄×C₆₁	1	4	yes	complemented
224	#102	—	2	4	no	—
225	C₂⁴	—	4	1	no	—
226	C₂⁴	D₈×C₆₁	1	1	yes	characteristic
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	4	no	—
229	C₂×D₄	—	4	1	no	—
230	C₂×D₄	—	4	1	no	—
231	C₂×D₄	—	4	1	no	—
232	C₂×D₄	—	4	1	no	—
233	C₂×D₄	—	4	1	no	—
234	C₂×D₄	—	8	2	no	—
235	C₂×D₄	—	8	2	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	4	4	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	—	8	2	no	—
244	C₂²×C₄	—	8	2	no	—
245	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
246	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
247	C₂²×C₄	SD₁₆×C₆₁	1	2	yes	—
248	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
249	D₈	—	2	4	no	—
250	D₈	—	4	2	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
256	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
257	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
258	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
259	C₂×C₈	#41×C₆₁	1	2	yes	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	4	4	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	4	no	—
266	C₁₆	—	4	2	no	—
267	C₂³	—	2	1	no	—
268	C₂³	—	2	1	no	—
269	C₂³	—	2	1	no	—
270	C₂³	—	2	1	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	8	1	no	—
273	C₂³	—	8	1	no	—
274	C₂³	—	8	1	no	—
275	C₂³	—	8	1	no	—
276	C₂³	—	8	1	no	—
277	C₂³	—	8	1	no	—
278	C₂³	#102×C₆₁	1	2	yes	—
279	C₂³	D₁₆×C₆₁	1	2	yes	—
280	C₂³	SD₃₂×C₆₁	1	2	yes	—
281	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	D₄	—	8	8	no	—
285	D₄	—	8	1	no	—
286	D₄	—	8	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	8	8	no	—
296	C₂×C₄	—	8	2	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	8	2	no	—
300	C₂×C₄	#102×C₆₁	1	2	yes	—
301	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
302	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
303	C₂×C₄	#123×C₆₁	1	1	yes	characteristic
304	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
305	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
306	C₂×C₄	#137×C₆₁	1	1	yes	characteristic
307	C₈	—	2	2	no	—
308	C₈	C₄×D₄×C₆₁	1	2	yes	—
309	C₈	#121×C₆₁	1	2	yes	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	16	1	no	—
320	C₂²	—	16	1	no	—
321	C₂²	—	16	1	no	—
322	C₂²	—	16	1	no	—
323	C₂²	C₆₁×#357	1	4	yes	complemented
324	C₂²	C₂×#102×C₆₁	1	1	yes	characteristic
325	C₂²	C₆₁×#447	1	1	yes	characteristic
326	C₂²	C₆₁×#480	1	1	yes	characteristic
327	C₂²	C₂×D₁₆×C₆₁	1	1	yes	characteristic
328	C₂²	C₂×SD₃₂×C₆₁	1	1	yes	characteristic
329	C₂²	C₆₁×#509	1	1	yes	characteristic
330	C₂²	C₆₁×#510	1	1	yes	characteristic
331	C₄	—	2	2	no	—
332	C₄	—	8	2	no	—
333	C₄	—	8	4	no	—
334	C₄	C₆₁×#386	1	1	yes	characteristic
335	C₄	C₆₁×#417	1	1	yes	characteristic
336	C₄	C₆₁×#448	1	1	yes	characteristic
337	C₄	C₆₁×#481	1	1	yes	characteristic
338	C₂	—	2	4	no	—
339	C₂	—	16	2	no	—
340	C₂	C₆₁×#1874	1	1	yes	characteristic
341	C₂	C₂×C₆₁×#357	1	1	yes	characteristic
342	C₂	C₆₁×#2139	1	1	yes	characteristic
343	C₂	C₆₁×#2184	1	1	yes	characteristic
344	C₂	C₆₁×#2185	1	1	yes	characteristic
345	C₂	C₆₁×#2232	1	1	yes	characteristic
346	C₂	C₆₁×#2233	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
prime radical	C₂×C₆₁×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ C₂³×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#107×C₆₁	C₂	semidirect
3. C₂×C₆₁×#357	C₂	semidirect
4. C₂×C₆₁×C₂²⋊C₁₆	C₂	semidirect
5. C₆₁×#1380	C₂	semidirect
12. C₆₁×#466	C₄	semidirect
27. C₂×D₈×C₆₁	C₂×C₄	semidirect
48. #107×C₆₁	C₂×C₄	semidirect
49. #107×C₆₁	D₄	semidirect
136. #12991	C₆₁	direct (N × H)
150. C₂²×C₆₁	#357	semidirect
166. C₂²×#107	C₂×C₆₁	semidirect
167. C₂×#357	C₂×C₆₁	semidirect
168. C₂×C₂²⋊C₁₆	C₂×C₆₁	semidirect
169. #1380	C₂×C₆₁	semidirect
185. #466	C₄×C₆₁	semidirect
193. C₆₁	#12991	direct (N × H)
201. C₂×D₈	C₂×C₄×C₆₁	semidirect
222. #107	C₂×C₄×C₆₁	semidirect
223. #107	D₄×C₆₁	semidirect
323. C₂²	C₆₁×#357	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#466	2
C₂²×#107×C₆₁	2
C₂×C₆₁×#357	2
C₂×C₆₁×C₂²⋊C₁₆	2
C₆₁×#1380	2
#12991	61

Subgroups — C61×#12991 #29516424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#12991 #29516424