Subgroups — C₆₁×#21268 #29524701

Order 15616.

838 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#433	C₂	1	2	yes	maximal, complemented
2	C₆₁×#2293	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1954	C₂	1	2	yes	maximal
4	C₆₁×#1939	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1924	C₂	1	4	yes	maximal, complemented
6	C₄×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1750	C₂	1	2	yes	maximal, complemented
8	C₄×#97×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×#117×C₆₁	C₂²	1	2	yes	—
11	C₆₁×#433	—	2	8	no	—
12	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₆₁	C₂²	1	4	yes	—
14	C₂×C₂²⋊C₈×C₆₁	C₂²	1	4	yes	—
15	C₂×C₂²⋊C₈×C₆₁	C₂²	1	4	yes	—
16	C₂×#97×C₆₁	C₂²	1	2	yes	—
17	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
21	C₆₁×#390	C₄	1	2	yes	—
22	C₆₁×#389	C₄	1	2	yes	—
23	C₆₁×#388	C₄	1	4	yes	—
24	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
25	C₄×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
26	C₄³×C₆₁	C₂²	1	1	yes	characteristic
27	C₆₁×#336	C₂²	1	2	yes	—
28	C₆₁×#336	C₂²	1	4	yes	—
29	C₂²×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
31	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
32	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
33	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
34	C₄×D₄×C₆₁	—	2	4	no	—
35	#117×C₆₁	—	2	4	no	—
36	#117×C₆₁	—	2	4	no	—
37	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
42	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
47	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₆₁	—	2	8	no	—
50	C₄⋊C₈×C₆₁	—	2	4	no	—
51	C₂²⋊C₈×C₆₁	—	2	8	no	—
52	C₂²⋊C₈×C₆₁	—	2	8	no	—
53	#97×C₆₁	—	2	4	no	—
54	#97×C₆₁	—	2	2	no	—
55	#97×C₆₁	D₄	1	4	yes	complemented
56	C₄×C₈×C₆₁	—	2	4	no	—
57	C₄×C₈×C₆₁	—	2	4	no	—
58	#95×C₆₁	C₂×C₄	1	4	yes	—
59	#95×C₆₁	C₂×C₄	1	2	yes	—
60	C₂⁴×C₆₁	C₂²×C₄	1	2	yes	—
61	C₂×D₄×C₆₁	—	2	2	no	—
62	C₂×D₄×C₆₁	—	2	4	no	—
63	C₂²×C₄×C₆₁	—	2	2	no	—
64	C₂²×C₄×C₆₁	—	2	4	no	—
65	C₂²×C₄×C₆₁	—	2	2	no	—
66	C₂²×C₄×C₆₁	—	2	4	no	—
67	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₆₁	—	2	2	no	—
75	C₂×C₈×C₆₁	—	2	2	no	—
76	C₂×C₈×C₆₁	—	2	2	no	—
77	C₂×C₈×C₆₁	—	2	2	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	C₂×D₄	1	4	yes	—
82	C₂×C₈×C₆₁	#41	1	4	yes	—
83	C₄⋊C₄×C₆₁	—	2	2	no	—
84	C₄⋊C₄×C₆₁	—	2	8	no	—
85	C₄⋊C₄×C₆₁	—	2	4	no	—
86	C₂²⋊C₄×C₆₁	—	2	4	no	—
87	C₂²⋊C₄×C₆₁	—	2	8	no	—
88	C₂²⋊C₄×C₆₁	—	4	4	no	—
89	C₄²×C₆₁	—	2	4	no	—
90	C₄²×C₆₁	—	2	2	no	—
91	C₄²×C₆₁	—	2	2	no	—
92	C₄²×C₆₁	—	2	2	no	—
93	C₄²×C₆₁	—	4	2	no	—
94	C₄²×C₆₁	C₂×D₄	1	2	yes	—
95	C₄²×C₆₁	#41	1	4	yes	—
96	C₄²×C₆₁	#41	1	2	yes	—
97	C₂³×C₆₁	—	2	2	no	—
98	C₂³×C₆₁	—	2	2	no	—
99	C₂³×C₆₁	—	2	2	no	—
100	C₂³×C₆₁	—	2	4	no	—
101	C₂³×C₆₁	—	2	4	no	—
102	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
103	D₄×C₆₁	—	4	4	no	—
104	C₂×C₄×C₆₁	—	2	2	no	—
105	C₂×C₄×C₆₁	—	2	2	no	—
106	C₂×C₄×C₆₁	—	2	2	no	—
107	C₂×C₄×C₆₁	—	2	2	no	—
108	C₂×C₄×C₆₁	—	2	4	no	—
109	C₂×C₄×C₆₁	—	2	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	4	no	—
114	C₂×C₄×C₆₁	—	2	2	no	—
115	C₂×C₄×C₆₁	—	2	4	no	—
116	C₂×C₄×C₆₁	—	4	4	no	—
117	C₂×C₄×C₆₁	—	4	4	no	—
118	C₂×C₄×C₆₁	#117	1	2	yes	—
119	C₂×C₄×C₆₁	#117	1	2	yes	—
120	C₂×C₄×C₆₁	C₄×D₄	1	4	yes	—
121	C₂×C₄×C₆₁	#131	1	4	yes	—
122	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
125	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
126	C₈×C₆₁	—	2	4	no	—
127	C₈×C₆₁	—	2	4	no	—
128	C₈×C₆₁	—	4	4	no	—
129	#21268	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂²×C₆₁	—	4	2	no	—
131	C₂²×C₆₁	—	4	2	no	—
132	C₂²×C₆₁	—	4	4	no	—
133	C₂²×C₆₁	—	4	4	no	—
134	C₂²×C₆₁	—	4	2	no	—
135	C₂²×C₆₁	C₂×#117	1	1	yes	characteristic
136	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂²×C₆₁	C₄×#41	1	1	yes	characteristic
138	C₂²×C₆₁	C₂×#131	1	2	yes	—
139	C₂²×C₆₁	#568	1	2	yes	—
140	C₄×C₆₁	—	2	2	no	—
141	C₄×C₆₁	—	2	4	no	—
142	C₄×C₆₁	—	4	4	no	—
143	C₄×C₆₁	—	4	2	no	—
144	C₄×C₆₁	—	4	2	no	—
145	C₄×C₆₁	#433	1	4	yes	complemented
146	C₄×C₆₁	#529	1	2	yes	—
147	C₄×C₆₁	#532	1	2	yes	—
148	C₂×#433	C₂×C₆₁	1	2	yes	complemented
149	#2293	C₂×C₆₁	1	1	yes	characteristic
150	#1954	C₂×C₆₁	1	2	yes	—
151	#1939	C₂×C₆₁	1	2	yes	complemented
152	#1924	C₂×C₆₁	1	4	yes	complemented
153	C₄×C₄⋊C₈	C₂×C₆₁	1	1	yes	characteristic, complemented
154	#1750	C₂×C₆₁	1	2	yes	complemented
155	C₄×#97	C₂×C₆₁	1	1	yes	characteristic, complemented
156	C₂×C₆₁	—	8	2	no	—
157	C₂×C₆₁	#2293	1	1	yes	characteristic
158	C₂×C₆₁	C₂×#433	1	1	yes	characteristic
159	C₂×C₆₁	#2923	1	1	yes	characteristic
160	C₂×C₆₁	#2935	1	2	yes	—
161	C₂×C₆₁	#2969	1	2	yes	—
162	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
163	C₂×#117	C₂²×C₆₁	1	2	yes	—
164	#433	—	2	8	no	—
165	C₂×C₄⋊C₈	C₂²×C₆₁	1	4	yes	—
166	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
167	C₂×C₂²⋊C₈	C₂²×C₆₁	1	4	yes	—
168	C₂×C₂²⋊C₈	C₂²×C₆₁	1	4	yes	—
169	C₂×#97	C₂²×C₆₁	1	2	yes	—
170	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
171	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
172	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
173	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
174	#390	C₄×C₆₁	1	2	yes	—
175	#389	C₄×C₆₁	1	2	yes	—
176	#388	C₄×C₆₁	1	4	yes	—
177	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
178	C₄×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
179	C₄³	C₂²×C₆₁	1	1	yes	characteristic
180	#336	C₂²×C₆₁	1	2	yes	—
181	#336	C₂²×C₆₁	1	4	yes	—
182	C₆₁	#21268	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²×D₄	C₂×C₄×C₆₁	1	1	yes	characteristic
184	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
185	C₂²×C₈	C₂³×C₆₁	1	4	yes	—
186	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
187	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
188	C₄×D₄	—	2	4	no	—
189	#117	—	2	4	no	—
190	#117	—	2	4	no	—
191	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
192	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
193	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
194	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
195	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
196	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
197	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
198	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
199	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
200	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
201	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
202	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
203	C₄⋊C₈	—	2	8	no	—
204	C₄⋊C₈	—	2	4	no	—
205	C₂²⋊C₈	—	2	8	no	—
206	C₂²⋊C₈	—	2	8	no	—
207	#97	—	2	4	no	—
208	#97	—	2	2	no	—
209	#97	D₄×C₆₁	1	4	yes	complemented
210	C₄×C₈	—	2	4	no	—
211	C₄×C₈	—	2	4	no	—
212	#95	C₂×C₄×C₆₁	1	4	yes	—
213	#95	C₂×C₄×C₆₁	1	2	yes	—
214	C₂⁴	C₂²×C₄×C₆₁	1	2	yes	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	2	4	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
222	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
223	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
224	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
225	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
226	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
227	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
236	C₂×C₈	#41×C₆₁	1	4	yes	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	8	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₂²⋊C₄	—	2	8	no	—
242	C₂²⋊C₄	—	4	4	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	2	no	—
248	C₄²	C₂×D₄×C₆₁	1	2	yes	—
249	C₄²	#41×C₆₁	1	4	yes	—
250	C₄²	#41×C₆₁	1	2	yes	—
251	C₂³	—	2	4	no	—
252	C₂³	—	2	4	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	C₂³×C₄×C₆₁	1	1	yes	characteristic
257	D₄	—	4	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	#117×C₆₁	1	2	yes	—
273	C₂×C₄	#117×C₆₁	1	2	yes	—
274	C₂×C₄	C₄×D₄×C₆₁	1	4	yes	—
275	C₂×C₄	#131×C₆₁	1	4	yes	—
276	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
277	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
278	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
279	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
280	C₈	—	2	4	no	—
281	C₈	—	2	4	no	—
282	C₈	—	4	4	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	4	no	—
287	C₂²	—	4	4	no	—
288	C₂²	C₂×#117×C₆₁	1	1	yes	characteristic
289	C₂²	C₂×C₄×D₄×C₆₁	1	1	yes	characteristic
290	C₂²	C₄×#41×C₆₁	1	1	yes	characteristic
291	C₂²	C₂×#131×C₆₁	1	2	yes	—
292	C₂²	C₆₁×#568	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	2	4	no	—
295	C₄	—	4	2	no	—
296	C₄	—	4	4	no	—
297	C₄	—	4	2	no	—
298	C₄	C₆₁×#433	1	4	yes	complemented
299	C₄	C₆₁×#529	1	2	yes	—
300	C₄	C₆₁×#532	1	2	yes	—
301	C₂	—	8	2	no	—
302	C₂	C₆₁×#2293	1	1	yes	characteristic
303	C₂	C₂×C₆₁×#433	1	1	yes	characteristic
304	C₂	C₆₁×#2923	1	1	yes	characteristic
305	C₂	C₆₁×#2935	1	2	yes	—
306	C₂	C₆₁×#2969	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×#433	C₂	semidirect
4. C₆₁×#1939	C₂	semidirect
5. C₆₁×#1924	C₂	semidirect
6. C₄×C₄⋊C₈×C₆₁	C₂	semidirect
7. C₆₁×#1750	C₂	semidirect
8. C₄×#97×C₆₁	C₂	semidirect
55. #97×C₆₁	D₄	semidirect
129. #21268	C₆₁	direct (N × H)
145. C₄×C₆₁	#433	semidirect
148. C₂×#433	C₂×C₆₁	semidirect
151. #1939	C₂×C₆₁	semidirect
152. #1924	C₂×C₆₁	semidirect
153. C₄×C₄⋊C₈	C₂×C₆₁	semidirect
154. #1750	C₂×C₆₁	semidirect
155. C₄×#97	C₂×C₆₁	semidirect
182. C₆₁	#21268	direct (N × H)
209. #97	D₄×C₆₁	semidirect
298. C₄	C₆₁×#433	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#433	2
C₆₁×#2293	2
C₆₁×#1954	2
C₆₁×#1939	2
C₆₁×#1924	2
C₄×C₄⋊C₈×C₆₁	2
C₆₁×#1750	2
C₄×#97×C₆₁	2
#21268	61

Subgroups — C61×#21268 #29524701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#21268 #29524701