Subgroups — C₆₁×#29124 #29532557

Order 15616.

854 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2864	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#127×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2100	C₂	1	2	yes	maximal, complemented
4	C₆₁×#2041	C₂	1	2	yes	maximal
5	C₆₁×#1934	C₂	1	4	yes	maximal, complemented
6	C₆₁×#1804	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1804	C₂	1	2	yes	maximal, complemented
8	C₄×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₆₁	—	2	2	no	—
11	C₂×C₄×D₄×C₆₁	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
13	C₂×#102×C₆₁	C₂²	1	4	yes	—
14	C₂×#102×C₆₁	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
16	C₆₁×#390	—	2	2	no	—
17	C₆₁×#390	C₂²	1	2	yes	—
18	C₆₁×#389	C₂²	1	2	yes	—
19	C₆₁×#389	C₂²	1	2	yes	—
20	C₆₁×#387	—	2	2	no	—
21	C₆₁×#387	—	2	2	no	—
22	C₆₁×#384	C₂²	1	1	yes	characteristic
23	C₆₁×#382	C₂²	1	1	yes	characteristic
24	C₆₁×#382	C₂²	1	1	yes	characteristic
25	C₄³×C₆₁	C₂²	1	1	yes	characteristic
26	C₆₁×#340	C₂²	1	4	yes	—
27	C₆₁×#340	C₂²	1	4	yes	—
28	C₆₁×#336	C₂²	1	2	yes	—
29	C₂²×D₄×C₆₁	—	2	2	no	—
30	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
31	C₂³×C₄×C₆₁	—	2	2	no	—
32	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
33	#127×C₆₁	—	2	4	no	—
34	C₄×D₄×C₆₁	—	2	4	no	—
35	C₄×D₄×C₆₁	—	2	4	no	—
36	C₄×D₄×C₆₁	—	4	8	no	—
37	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
43	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
44	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
45	C₂×C₄²×C₆₁	D₄	1	2	yes	—
46	C₂×C₄²×C₆₁	D₄	1	2	yes	—
47	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₆₁	—	2	4	no	—
51	#102×C₆₁	—	2	8	no	—
52	#102×C₆₁	—	4	4	no	—
53	C₄×C₈×C₆₁	—	4	2	no	—
54	#95×C₆₁	—	2	1	no	—
55	#95×C₆₁	—	2	1	no	—
56	#95×C₆₁	—	2	1	no	—
57	#95×C₆₁	—	2	1	no	—
58	#95×C₆₁	—	2	2	no	—
59	#95×C₆₁	—	2	2	no	—
60	C₂⁴×C₆₁	—	2	2	no	—
61	C₂×D₄×C₆₁	—	2	8	no	—
62	C₂×D₄×C₆₁	—	2	2	no	—
63	C₂×D₄×C₆₁	—	2	2	no	—
64	C₂×D₄×C₆₁	—	2	2	no	—
65	C₂×D₄×C₆₁	—	2	2	no	—
66	C₂×D₄×C₆₁	—	2	2	no	—
67	C₂×D₄×C₆₁	—	4	4	no	—
68	C₂×D₄×C₆₁	—	4	4	no	—
69	C₂×D₄×C₆₁	#41	1	4	yes	—
70	C₂²×C₄×C₆₁	—	2	2	no	—
71	C₂²×C₄×C₆₁	—	2	2	no	—
72	C₂²×C₄×C₆₁	—	2	2	no	—
73	C₂²×C₄×C₆₁	—	4	8	no	—
74	C₂²×C₄×C₆₁	—	4	4	no	—
75	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₆₁	—	2	4	no	—
81	C₂×C₈×C₆₁	—	2	4	no	—
82	C₂×C₈×C₆₁	—	2	4	no	—
83	C₄⋊C₄×C₆₁	—	2	4	no	—
84	C₄⋊C₄×C₆₁	—	2	2	no	—
85	C₄⋊C₄×C₆₁	—	2	2	no	—
86	C₄⋊C₄×C₆₁	—	2	2	no	—
87	C₄⋊C₄×C₆₁	—	2	2	no	—
88	C₄⋊C₄×C₆₁	—	4	4	no	—
89	C₄⋊C₄×C₆₁	—	4	4	no	—
90	C₄⋊C₄×C₆₁	—	4	4	no	—
91	C₂²⋊C₄×C₆₁	—	4	4	no	—
92	C₂²⋊C₄×C₆₁	—	4	8	no	—
93	C₄²×C₆₁	—	2	4	no	—
94	C₄²×C₆₁	—	2	4	no	—
95	C₄²×C₆₁	—	2	2	no	—
96	C₄²×C₆₁	—	2	1	no	—
97	C₄²×C₆₁	—	2	1	no	—
98	C₄²×C₆₁	#41	1	4	yes	—
99	C₂³×C₆₁	—	2	4	no	—
100	C₂³×C₆₁	—	4	2	no	—
101	C₂³×C₆₁	—	4	2	no	—
102	C₂³×C₆₁	—	4	4	no	—
103	C₂³×C₆₁	—	4	4	no	—
104	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
105	D₄×C₆₁	—	2	4	no	—
106	D₄×C₆₁	—	2	4	no	—
107	D₄×C₆₁	—	4	16	no	—
108	D₄×C₆₁	—	4	2	no	—
109	D₄×C₆₁	—	4	2	no	—
110	C₂×C₄×C₆₁	—	2	2	no	—
111	C₂×C₄×C₆₁	—	2	2	no	—
112	C₂×C₄×C₆₁	—	2	2	no	—
113	C₂×C₄×C₆₁	—	2	2	no	—
114	C₂×C₄×C₆₁	—	2	2	no	—
115	C₂×C₄×C₆₁	—	2	2	no	—
116	C₂×C₄×C₆₁	—	4	2	no	—
117	C₂×C₄×C₆₁	—	4	2	no	—
118	C₂×C₄×C₆₁	—	4	2	no	—
119	C₂×C₄×C₆₁	—	4	2	no	—
120	C₂×C₄×C₆₁	—	4	2	no	—
121	C₂×C₄×C₆₁	—	4	4	no	—
122	C₂×C₄×C₆₁	—	4	2	no	—
123	C₂×C₄×C₆₁	—	4	2	no	—
124	C₂×C₄×C₆₁	—	8	8	no	—
125	C₂×C₄×C₆₁	#124	1	2	yes	—
126	C₂×C₄×C₆₁	#124	1	2	yes	—
127	C₂×C₄×C₆₁	#135	1	4	yes	—
128	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	4	yes	—
129	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
130	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
131	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
133	C₈×C₆₁	—	4	4	no	—
134	#29124	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂²×C₆₁	—	4	4	no	—
136	C₂²×C₆₁	—	4	8	no	—
137	C₂²×C₆₁	—	8	2	no	—
138	C₂²×C₆₁	—	8	2	no	—
139	C₂²×C₆₁	—	8	4	no	—
140	C₂²×C₆₁	#525	1	1	yes	characteristic
141	C₂²×C₆₁	C₂×#124	1	1	yes	characteristic
142	C₂²×C₆₁	#537	1	1	yes	characteristic
143	C₂²×C₆₁	C₂×#135	1	2	yes	—
144	C₂²×C₆₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
145	C₄×C₆₁	—	2	4	no	—
146	C₄×C₆₁	—	2	2	no	—
147	C₄×C₆₁	—	2	2	no	—
148	C₄×C₆₁	—	2	4	no	—
149	C₄×C₆₁	—	8	2	no	—
150	C₄×C₆₁	—	8	2	no	—
151	C₄×C₆₁	—	8	2	no	—
152	C₄×C₆₁	#552	1	2	yes	—
153	C₄×C₆₁	#553	1	2	yes	—
154	#2864	C₂×C₆₁	1	1	yes	characteristic, complemented
155	C₄×#127	C₂×C₆₁	1	1	yes	characteristic
156	#2100	C₂×C₆₁	1	2	yes	complemented
157	#2041	C₂×C₆₁	1	2	yes	—
158	#1934	C₂×C₆₁	1	4	yes	complemented
159	#1804	C₂×C₆₁	1	2	yes	complemented
160	#1804	C₂×C₆₁	1	2	yes	complemented
161	C₄×C₄⋊C₈	C₂×C₆₁	1	1	yes	characteristic, complemented
162	C₂×C₆₁	—	8	4	no	—
163	C₂×C₆₁	#2562	1	1	yes	characteristic
164	C₂×C₆₁	#3105	1	2	yes	—
165	C₂×C₆₁	#3131	1	1	yes	characteristic
166	C₂×C₆₁	#3138	1	1	yes	characteristic
167	C₂×C₆₁	#3247	1	2	yes	—
168	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
169	C₂×C₄×D₄	—	2	2	no	—
170	C₂×C₄×D₄	C₂²×C₆₁	1	2	yes	—
171	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
172	C₂×#102	C₂²×C₆₁	1	4	yes	—
173	C₂×#102	C₂²×C₆₁	1	4	yes	—
174	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
175	#390	—	2	2	no	—
176	#390	C₂²×C₆₁	1	2	yes	—
177	#389	C₂²×C₆₁	1	2	yes	—
178	#389	C₂²×C₆₁	1	2	yes	—
179	#387	—	2	2	no	—
180	#387	—	2	2	no	—
181	#384	C₂²×C₆₁	1	1	yes	characteristic
182	#382	C₂²×C₆₁	1	1	yes	characteristic
183	#382	C₂²×C₆₁	1	1	yes	characteristic
184	C₄³	C₂²×C₆₁	1	1	yes	characteristic
185	#340	C₂²×C₆₁	1	4	yes	—
186	#340	C₂²×C₆₁	1	4	yes	—
187	#336	C₂²×C₆₁	1	2	yes	—
188	C₆₁	#29124	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²×D₄	—	2	2	no	—
190	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
191	C₂³×C₄	—	2	2	no	—
192	C₂²×C₈	C₂³×C₆₁	1	4	yes	—
193	#127	—	2	4	no	—
194	C₄×D₄	—	2	4	no	—
195	C₄×D₄	—	2	4	no	—
196	C₄×D₄	—	4	8	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
201	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
202	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
203	C₂×C₂²⋊C₄	—	2	4	no	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₄²	D₄×C₆₁	1	2	yes	—
206	C₂×C₄²	D₄×C₆₁	1	2	yes	—
207	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
209	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
210	C₄⋊C₈	—	2	4	no	—
211	#102	—	2	8	no	—
212	#102	—	4	4	no	—
213	C₄×C₈	—	4	2	no	—
214	#95	—	2	1	no	—
215	#95	—	2	1	no	—
216	#95	—	2	1	no	—
217	#95	—	2	1	no	—
218	#95	—	2	2	no	—
219	#95	—	2	2	no	—
220	C₂⁴	—	2	2	no	—
221	C₂×D₄	—	2	8	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	4	4	no	—
228	C₂×D₄	—	4	4	no	—
229	C₂×D₄	#41×C₆₁	1	4	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	4	8	no	—
234	C₂²×C₄	—	4	4	no	—
235	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
238	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
239	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	4	4	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₂²⋊C₄	—	4	4	no	—
252	C₂²⋊C₄	—	4	8	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	4	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	#41×C₆₁	1	4	yes	—
259	C₂³	—	2	4	no	—
260	C₂³	—	4	2	no	—
261	C₂³	—	4	4	no	—
262	C₂³	—	4	4	no	—
263	C₂³	—	4	2	no	—
264	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
265	D₄	—	2	4	no	—
266	D₄	—	2	4	no	—
267	D₄	—	4	16	no	—
268	D₄	—	4	2	no	—
269	D₄	—	4	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	8	8	no	—
285	C₂×C₄	#124×C₆₁	1	2	yes	—
286	C₂×C₄	#124×C₆₁	1	2	yes	—
287	C₂×C₄	#135×C₆₁	1	4	yes	—
288	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	4	yes	—
289	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
290	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
291	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
292	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
293	C₈	—	4	4	no	—
294	C₂²	—	4	4	no	—
295	C₂²	—	4	8	no	—
296	C₂²	—	8	2	no	—
297	C₂²	—	8	2	no	—
298	C₂²	—	8	4	no	—
299	C₂²	C₆₁×#525	1	1	yes	characteristic
300	C₂²	C₂×#124×C₆₁	1	1	yes	characteristic
301	C₂²	C₆₁×#537	1	1	yes	characteristic
302	C₂²	C₂×#135×C₆₁	1	2	yes	—
303	C₂²	C₂×AGL(1,ℤ/8)×C₆₁	1	2	yes	—
304	C₄	—	2	4	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	4	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	2	no	—
311	C₄	C₆₁×#552	1	2	yes	—
312	C₄	C₆₁×#553	1	2	yes	—
313	C₂	—	8	4	no	—
314	C₂	C₆₁×#2562	1	1	yes	characteristic
315	C₂	C₆₁×#3105	1	2	yes	—
316	C₂	C₆₁×#3131	1	1	yes	characteristic
317	C₂	C₆₁×#3138	1	1	yes	characteristic
318	C₂	C₆₁×#3247	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂×#127×C₆₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₄³×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#2864	C₂	semidirect
3. C₆₁×#2100	C₂	semidirect
5. C₆₁×#1934	C₂	semidirect
6. C₆₁×#1804	C₂	semidirect
7. C₆₁×#1804	C₂	semidirect
8. C₄×C₄⋊C₈×C₆₁	C₂	semidirect
134. #29124	C₆₁	direct (N × H)
154. #2864	C₂×C₆₁	semidirect
156. #2100	C₂×C₆₁	semidirect
158. #1934	C₂×C₆₁	semidirect
159. #1804	C₂×C₆₁	semidirect
160. #1804	C₂×C₆₁	semidirect
161. C₄×C₄⋊C₈	C₂×C₆₁	semidirect
188. C₆₁	#29124	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₁×#2864	2
C₄×#127×C₆₁	2
C₆₁×#2100	2
C₆₁×#2041	2
C₆₁×#1934	2
C₆₁×#1804	2
C₄×C₄⋊C₈×C₆₁	2
#29124	61

Subgroups — C61×#29124 #29532557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#29124 #29532557