Subgroups — C₆₁×#21203 #29524636

Order 15616.

2246 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#127×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#418	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2285	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1961	C₂	1	4	yes	maximal, complemented
6	C₆₁×#1957	C₂	1	4	yes	maximal, complemented
7	C₂³×D₄×C₆₁	—	2	2	no	—
8	C₂³×D₄×C₆₁	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
10	C₂×#127×C₆₁	—	2	2	no	—
11	C₂×#127×C₆₁	—	2	4	no	—
12	C₂×#127×C₆₁	C₄	1	8	yes	complemented
13	C₂×#127×C₆₁	C₂²	1	2	yes	—
14	C₂×#127×C₆₁	C₂²	1	2	yes	—
15	C₂×#117×C₆₁	C₂²	1	2	yes	—
16	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#418	—	2	16	no	—
18	C₂×#102×C₆₁	C₂²	1	16	yes	complemented
19	C₄×M₁₆×C₆₁	—	2	4	no	—
20	C₂×#97×C₆₁	C₂²	1	2	yes	complemented
21	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	complemented
22	C₆₁×#384	C₂²	1	2	yes	complemented
23	C₆₁×#382	C₂²	1	2	yes	complemented
24	C₆₁×#380	—	2	2	no	—
25	C₄×C₄⋊C₄×C₆₁	—	2	2	no	—
26	C₂⁵×C₆₁	—	2	2	no	—
27	C₂²×D₄×C₆₁	—	2	16	no	—
28	C₂²×D₄×C₆₁	—	2	8	no	—
29	C₂²×D₄×C₆₁	—	2	2	no	—
30	C₂²×D₄×C₆₁	—	2	4	no	—
31	C₂²×D₄×C₆₁	—	2	4	no	—
32	C₂²×D₄×C₆₁	—	2	4	no	—
33	C₂²×D₄×C₆₁	—	2	8	no	—
34	C₂²×D₄×C₆₁	—	4	16	no	—
35	C₂²×D₄×C₆₁	C₂×C₄	1	8	yes	—
36	C₂²×D₄×C₆₁	C₂³	1	4	yes	—
37	C₂³×C₄×C₆₁	D₄	1	2	yes	—
38	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₆₁	—	2	4	no	—
40	C₂×M₁₆×C₆₁	—	2	4	no	—
41	C₂×M₁₆×C₆₁	D₄	1	8	yes	complemented
42	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
43	#127×C₆₁	—	2	16	no	—
44	#127×C₆₁	—	2	4	no	—
45	#127×C₆₁	—	2	4	no	—
46	#127×C₆₁	C₂×C₄	1	8	yes	—
47	#127×C₆₁	C₂×C₄	1	8	yes	—
48	#117×C₆₁	—	2	8	no	—
49	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
50	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
51	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
52	C₂×C₄²×C₆₁	—	2	2	no	—
53	C₂×C₄²×C₆₁	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₆₁	D₄	1	2	yes	—
55	C₂×C₄²×C₆₁	D₄	1	4	yes	—
56	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
58	#102×C₆₁	—	2	32	no	—
59	#97×C₆₁	—	2	4	no	—
60	C₄×C₈×C₆₁	—	2	4	no	—
61	#95×C₆₁	—	2	4	no	—
62	C₂⁴×C₆₁	—	2	8	no	—
63	C₂⁴×C₆₁	—	2	4	no	—
64	C₂⁴×C₆₁	—	4	16	no	—
65	C₂⁴×C₆₁	—	4	4	no	—
66	C₂⁴×C₆₁	—	4	4	no	—
67	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
68	C₂×D₄×C₆₁	—	2	8	no	—
69	C₂×D₄×C₆₁	—	2	16	no	—
70	C₂×D₄×C₆₁	—	2	16	no	—
71	C₂×D₄×C₆₁	—	2	32	no	—
72	C₂×D₄×C₆₁	—	2	8	no	—
73	C₂×D₄×C₆₁	—	2	4	no	—
74	C₂×D₄×C₆₁	—	2	8	no	—
75	C₂×D₄×C₆₁	—	4	16	no	—
76	C₂×D₄×C₆₁	—	4	64	no	—
77	C₂×D₄×C₆₁	—	4	16	no	—
78	C₂×D₄×C₆₁	—	4	4	no	—
79	C₂×D₄×C₆₁	—	4	4	no	—
80	C₂×D₄×C₆₁	—	4	16	no	—
81	C₂×D₄×C₆₁	C₂²×C₄	1	8	yes	—
82	C₂²×C₄×C₆₁	—	2	8	no	—
83	C₂²×C₄×C₆₁	—	2	4	no	—
84	C₂²×C₄×C₆₁	C₂²⋊C₄	1	8	yes	—
85	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
91	C₂²×C₄×C₆₁	#41	1	4	yes	—
92	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
93	M₁₆×C₆₁	—	2	16	no	—
94	C₂×C₈×C₆₁	—	2	4	no	—
95	C₂×C₈×C₆₁	—	2	4	no	—
96	C₂×C₈×C₆₁	C₂×D₄	1	8	yes	complemented
97	C₄⋊C₄×C₆₁	—	2	8	no	—
98	C₄⋊C₄×C₆₁	—	4	8	no	—
99	C₂²⋊C₄×C₆₁	—	4	4	no	—
100	C₄²×C₆₁	—	4	4	no	—
101	C₄²×C₆₁	C₂²⋊C₄	1	8	yes	—
102	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
103	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
104	C₄²×C₆₁	C₂×D₄	1	2	yes	—
105	C₄²×C₆₁	C₂×D₄	1	2	yes	—
106	C₂³×C₆₁	—	2	4	no	—
107	C₂³×C₆₁	—	2	8	no	—
108	C₂³×C₆₁	—	4	32	no	—
109	C₂³×C₆₁	—	4	4	no	—
110	C₂³×C₆₁	—	4	4	no	—
111	C₂³×C₆₁	—	4	16	no	—
112	C₂³×C₆₁	—	4	16	no	—
113	C₂³×C₆₁	—	8	16	no	—
114	C₂³×C₆₁	—	8	16	no	—
115	C₂³×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
116	C₂³×C₆₁	#124	1	2	yes	—
117	C₂³×C₆₁	#127	1	2	yes	—
118	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
119	D₄×C₆₁	—	2	16	no	—
120	D₄×C₆₁	—	4	32	no	—
121	D₄×C₆₁	—	4	8	no	—
122	D₄×C₆₁	—	4	16	no	—
123	D₄×C₆₁	—	4	32	no	—
124	C₂×C₄×C₆₁	—	2	4	no	—
125	C₂×C₄×C₆₁	—	2	16	no	—
126	C₂×C₄×C₆₁	—	4	4	no	—
127	C₂×C₄×C₆₁	—	4	8	no	—
128	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	4	yes	—
129	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
130	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	4	yes	—
131	C₂×C₄×C₆₁	C₄×D₄	1	8	yes	—
132	C₂×C₄×C₆₁	#121	1	8	yes	—
133	C₂×C₄×C₆₁	#124	1	2	yes	—
134	C₂×C₄×C₆₁	#127	1	2	yes	—
135	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	8	yes	—
136	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
138	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
140	C₈×C₆₁	—	2	8	no	—
141	#21203	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂²×C₆₁	—	2	4	no	—
143	C₂²×C₆₁	—	2	4	no	—
144	C₂²×C₆₁	—	2	4	no	—
145	C₂²×C₆₁	—	4	16	no	—
146	C₂²×C₆₁	—	4	8	no	—
147	C₂²×C₆₁	—	8	32	no	—
148	C₂²×C₆₁	—	8	8	no	—
149	C₂²×C₆₁	—	8	8	no	—
150	C₂²×C₆₁	#390	1	4	yes	—
151	C₂²×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₆₁	C₂×#124	1	1	yes	characteristic
153	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
154	C₂²×C₆₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
155	C₄×C₆₁	—	2	4	no	—
156	C₄×C₆₁	—	8	4	no	—
157	C₄×C₆₁	#390	1	4	yes	—
158	C₄×C₆₁	#418	1	8	yes	complemented
159	C₄×C₆₁	C₂×C₄×D₄	1	2	yes	—
160	C₄×C₆₁	C₂×#121	1	2	yes	—
161	C₂²×#127	C₂×C₆₁	1	1	yes	characteristic
162	C₂×#418	C₂×C₆₁	1	4	yes	complemented
163	C₂×C₄×M₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
164	#2285	C₂×C₆₁	1	1	yes	characteristic, complemented
165	#1961	C₂×C₆₁	1	4	yes	complemented
166	#1957	C₂×C₆₁	1	4	yes	complemented
167	C₂×C₆₁	—	2	4	no	—
168	C₂×C₆₁	—	8	8	no	—
169	C₂×C₆₁	C₂×#390	1	1	yes	characteristic
170	C₂×C₆₁	C₂×#418	1	2	yes	—
171	C₂×C₆₁	#3138	1	2	yes	—
172	C₂×C₆₁	#3151	1	2	yes	—
173	C₂³×D₄	—	2	2	no	—
174	C₂³×D₄	C₂²×C₆₁	1	2	yes	—
175	C₂²×M₁₆	C₂²×C₆₁	1	2	yes	—
176	C₂×#127	—	2	2	no	—
177	C₂×#127	—	2	4	no	—
178	C₂×#127	C₄×C₆₁	1	8	yes	complemented
179	C₂×#127	C₂²×C₆₁	1	2	yes	—
180	C₂×#127	C₂²×C₆₁	1	2	yes	—
181	C₂×#117	C₂²×C₆₁	1	2	yes	—
182	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
183	#418	—	2	16	no	—
184	C₂×#102	C₂²×C₆₁	1	16	yes	complemented
185	C₄×M₁₆	—	2	4	no	—
186	C₂×#97	C₂²×C₆₁	1	2	yes	complemented
187	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	complemented
188	#384	C₂²×C₆₁	1	2	yes	complemented
189	#382	C₂²×C₆₁	1	2	yes	complemented
190	#380	—	2	2	no	—
191	C₄×C₄⋊C₄	—	2	2	no	—
192	C₆₁	#21203	1	1	yes	characteristic, direct product factor, Sylow
193	C₂⁵	—	2	2	no	—
194	C₂²×D₄	—	2	16	no	—
195	C₂²×D₄	—	2	8	no	—
196	C₂²×D₄	—	2	2	no	—
197	C₂²×D₄	—	2	4	no	—
198	C₂²×D₄	—	2	4	no	—
199	C₂²×D₄	—	2	4	no	—
200	C₂²×D₄	—	2	8	no	—
201	C₂²×D₄	—	4	16	no	—
202	C₂²×D₄	C₂×C₄×C₆₁	1	8	yes	—
203	C₂²×D₄	C₂³×C₆₁	1	4	yes	—
204	C₂³×C₄	D₄×C₆₁	1	2	yes	—
205	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
206	C₂×M₁₆	—	2	4	no	—
207	C₂×M₁₆	—	2	4	no	—
208	C₂×M₁₆	D₄×C₆₁	1	8	yes	complemented
209	C₂²×C₈	C₂³×C₆₁	1	4	yes	—
210	#127	—	2	16	no	—
211	#127	—	2	4	no	—
212	#127	—	2	4	no	—
213	#127	C₂×C₄×C₆₁	1	8	yes	—
214	#127	C₂×C₄×C₆₁	1	8	yes	—
215	#117	—	2	8	no	—
216	C₂×C₄⋊C₄	—	2	4	no	—
217	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
218	C₂×C₂²⋊C₄	—	2	2	no	—
219	C₂×C₄²	—	2	2	no	—
220	C₂×C₄²	C₂×C₄×C₆₁	1	4	yes	—
221	C₂×C₄²	D₄×C₆₁	1	2	yes	—
222	C₂×C₄²	D₄×C₆₁	1	4	yes	—
223	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
224	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
225	#102	—	2	32	no	—
226	#97	—	2	4	no	—
227	C₄×C₈	—	2	4	no	—
228	#95	—	2	4	no	—
229	C₂⁴	—	2	8	no	—
230	C₂⁴	—	2	4	no	—
231	C₂⁴	—	4	16	no	—
232	C₂⁴	—	4	4	no	—
233	C₂⁴	—	4	4	no	—
234	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
235	C₂×D₄	—	2	8	no	—
236	C₂×D₄	—	2	16	no	—
237	C₂×D₄	—	2	16	no	—
238	C₂×D₄	—	2	32	no	—
239	C₂×D₄	—	2	8	no	—
240	C₂×D₄	—	2	4	no	—
241	C₂×D₄	—	2	8	no	—
242	C₂×D₄	—	4	16	no	—
243	C₂×D₄	—	4	64	no	—
244	C₂×D₄	—	4	16	no	—
245	C₂×D₄	—	4	4	no	—
246	C₂×D₄	—	4	4	no	—
247	C₂×D₄	—	4	16	no	—
248	C₂×D₄	C₂²×C₄×C₆₁	1	8	yes	—
249	C₂²×C₄	—	2	8	no	—
250	C₂²×C₄	—	2	4	no	—
251	C₂²×C₄	C₂²⋊C₄×C₆₁	1	8	yes	—
252	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
254	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
256	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
257	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
258	C₂²×C₄	#41×C₆₁	1	4	yes	—
259	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
260	M₁₆	—	2	16	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	C₂×D₄×C₆₁	1	8	yes	complemented
264	C₄⋊C₄	—	2	8	no	—
265	C₄⋊C₄	—	4	8	no	—
266	C₂²⋊C₄	—	4	4	no	—
267	C₄²	—	4	4	no	—
268	C₄²	C₂²⋊C₄×C₆₁	1	8	yes	—
269	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
270	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
271	C₄²	C₂×D₄×C₆₁	1	2	yes	—
272	C₄²	C₂×D₄×C₆₁	1	2	yes	—
273	C₂³	—	2	4	no	—
274	C₂³	—	2	8	no	—
275	C₂³	—	4	32	no	—
276	C₂³	—	4	4	no	—
277	C₂³	—	4	16	no	—
278	C₂³	—	4	16	no	—
279	C₂³	—	4	4	no	—
280	C₂³	—	8	16	no	—
281	C₂³	—	8	16	no	—
282	C₂³	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
283	C₂³	#124×C₆₁	1	2	yes	—
284	C₂³	#127×C₆₁	1	2	yes	—
285	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
286	D₄	—	2	16	no	—
287	D₄	—	4	32	no	—
288	D₄	—	4	16	no	—
289	D₄	—	4	32	no	—
290	D₄	—	4	8	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	16	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	8	no	—
295	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	4	yes	—
296	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
297	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	4	yes	—
298	C₂×C₄	C₄×D₄×C₆₁	1	8	yes	—
299	C₂×C₄	#121×C₆₁	1	8	yes	—
300	C₂×C₄	#124×C₆₁	1	2	yes	—
301	C₂×C₄	#127×C₆₁	1	2	yes	—
302	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	8	yes	—
303	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
305	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
306	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
307	C₈	—	2	8	no	—
308	C₂²	—	2	4	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	4	8	no	—
312	C₂²	—	4	16	no	—
313	C₂²	—	8	32	no	—
314	C₂²	—	8	8	no	—
315	C₂²	—	8	8	no	—
316	C₂²	C₆₁×#390	1	4	yes	—
317	C₂²	C₂²×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
318	C₂²	C₂×#124×C₆₁	1	1	yes	characteristic
319	C₂²	C₂×#127×C₆₁	1	1	yes	characteristic
320	C₂²	C₂×AGL(1,ℤ/8)×C₆₁	1	4	yes	—
321	C₄	—	2	4	no	—
322	C₄	—	8	4	no	—
323	C₄	C₆₁×#390	1	4	yes	—
324	C₄	C₆₁×#418	1	8	yes	complemented
325	C₄	C₂×C₄×D₄×C₆₁	1	2	yes	—
326	C₄	C₂×#121×C₆₁	1	2	yes	—
327	C₂	—	2	4	no	—
328	C₂	—	8	8	no	—
329	C₂	C₂×C₆₁×#390	1	1	yes	characteristic
330	C₂	C₂×C₆₁×#418	1	2	yes	—
331	C₂	C₆₁×#3138	1	2	yes	—
332	C₂	C₆₁×#3151	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂²×#127×C₆₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂²×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₆₁×#418	C₂	semidirect
3. C₂×C₄×M₁₆×C₆₁	C₂	semidirect
4. C₆₁×#2285	C₂	semidirect
5. C₆₁×#1961	C₂	semidirect
6. C₆₁×#1957	C₂	semidirect
12. C₂×#127×C₆₁	C₄	semidirect
18. C₂×#102×C₆₁	C₂²	semidirect
20. C₂×#97×C₆₁	C₂²	semidirect
21. C₂×C₄×C₈×C₆₁	C₂²	semidirect
22. C₆₁×#384	C₂²	semidirect
23. C₆₁×#382	C₂²	semidirect
41. C₂×M₁₆×C₆₁	D₄	semidirect
96. C₂×C₈×C₆₁	C₂×D₄	semidirect
141. #21203	C₆₁	direct (N × H)
158. C₄×C₆₁	#418	semidirect
162. C₂×#418	C₂×C₆₁	semidirect
163. C₂×C₄×M₁₆	C₂×C₆₁	semidirect
164. #2285	C₂×C₆₁	semidirect
165. #1961	C₂×C₆₁	semidirect
166. #1957	C₂×C₆₁	semidirect
178. C₂×#127	C₄×C₆₁	semidirect
184. C₂×#102	C₂²×C₆₁	semidirect
186. C₂×#97	C₂²×C₆₁	semidirect
187. C₂×C₄×C₈	C₂²×C₆₁	semidirect
188. #384	C₂²×C₆₁	semidirect
189. #382	C₂²×C₆₁	semidirect
192. C₆₁	#21203	direct (N × H)
208. C₂×M₁₆	D₄×C₆₁	semidirect
263. C₂×C₈	C₂×D₄×C₆₁	semidirect
324. C₄	C₆₁×#418	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#127×C₆₁	2
C₂×C₆₁×#418	2
C₂×C₄×M₁₆×C₆₁	2
C₆₁×#2285	2
C₆₁×#1961	2
C₆₁×#1957	2
#21203	61

Subgroups — C61×#21203 #29524636

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#21203 #29524636