Subgroups — C₆₁×#23811 #29527244

Order 15616.

1038 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#440	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×SD₁₆×C₆₁	C₂	1	2	yes	maximal, complemented
3	C₆₁×#2328	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#2327	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2097	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2092	C₂	1	2	yes	maximal, complemented
7	C₆₁×#2014	C₂	1	2	yes	maximal
8	C₆₁×#2013	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1746	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×SD₁₆×C₆₁	C₂²	1	2	yes	—
11	C₂²×SD₁₆×C₆₁	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₆₁	C₂²	1	2	yes	—
16	C₆₁×#440	—	2	8	no	—
17	C₄×SD₁₆×C₆₁	—	2	8	no	—
18	C₂×#107×C₆₁	C₂²	1	2	yes	—
19	C₂×#106×C₆₁	C₂²	1	2	yes	—
20	C₂×#103×C₆₁	C₂²	1	2	yes	—
21	C₂×#103×C₆₁	C₂²	1	2	yes	—
22	C₂×#102×C₆₁	C₂²	1	2	yes	—
23	C₂×#102×C₆₁	C₂²	1	2	yes	—
24	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
26	C₆₁×#391	C₂²	1	2	yes	—
27	C₆₁×#390	C₂²	1	2	yes	—
28	C₆₁×#387	—	2	4	no	—
29	C₆₁×#386	—	2	4	no	—
30	C₆₁×#382	C₂²	1	1	yes	characteristic
31	C₆₁×#382	C₂²	1	1	yes	characteristic
32	C₆₁×#376	C₂²	1	1	yes	characteristic
33	C₆₁×#336	C₂²	1	2	yes	—
34	C₆₁×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
36	C₂²×D₄×C₆₁	—	2	2	no	—
37	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
38	C₂³×C₄×C₆₁	—	2	2	no	—
39	C₂×SD₁₆×C₆₁	—	2	4	no	—
40	C₂×SD₁₆×C₆₁	—	2	4	no	—
41	C₂×SD₁₆×C₆₁	—	2	4	no	—
42	C₂×SD₁₆×C₆₁	—	2	4	no	—
43	C₂×SD₁₆×C₆₁	—	2	4	no	—
44	C₂×SD₁₆×C₆₁	—	2	4	no	—
45	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
46	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₆₁	—	2	4	no	—
48	#127×C₆₁	—	2	4	no	—
49	C₄×Q₈×C₆₁	—	2	4	no	—
50	C₄×Q₈×C₆₁	—	2	4	no	—
51	C₄×D₄×C₆₁	—	2	4	no	—
52	C₄×D₄×C₆₁	—	2	4	no	—
53	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
54	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
55	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
57	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
58	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
59	C₂×C₄²×C₆₁	—	2	2	no	—
60	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
63	#107×C₆₁	—	2	4	no	—
64	#106×C₆₁	—	2	4	no	—
65	#103×C₆₁	—	2	4	no	—
66	#103×C₆₁	—	2	4	no	—
67	#102×C₆₁	—	2	4	no	—
68	#102×C₆₁	—	2	4	no	—
69	#97×C₆₁	D₄	1	4	yes	complemented
70	C₄×C₈×C₆₁	D₄	1	4	yes	complemented
71	#95×C₆₁	—	2	2	no	—
72	#95×C₆₁	—	2	2	no	—
73	#95×C₆₁	—	2	4	no	—
74	#95×C₆₁	D₄	1	4	yes	—
75	C₂⁴×C₆₁	—	2	2	no	—
76	C₂×Q₈×C₆₁	—	2	2	no	—
77	C₂×Q₈×C₆₁	—	2	2	no	—
78	C₂×Q₈×C₆₁	—	2	2	no	—
79	C₂×Q₈×C₆₁	—	2	2	no	—
80	C₂×Q₈×C₆₁	—	2	2	no	—
81	C₂×Q₈×C₆₁	#41	1	4	yes	—
82	C₂×D₄×C₆₁	—	2	8	no	—
83	C₂×D₄×C₆₁	—	2	2	no	—
84	C₂×D₄×C₆₁	—	2	2	no	—
85	C₂×D₄×C₆₁	—	2	2	no	—
86	C₂×D₄×C₆₁	—	2	2	no	—
87	C₂×D₄×C₆₁	—	2	2	no	—
88	C₂×D₄×C₆₁	—	8	4	no	—
89	C₂×D₄×C₆₁	#41	1	4	yes	—
90	C₂²×C₄×C₆₁	—	2	2	no	—
91	C₂²×C₄×C₆₁	—	2	2	no	—
92	C₂²×C₄×C₆₁	—	2	2	no	—
93	C₂²×C₄×C₆₁	—	4	4	no	—
94	C₂²×C₄×C₆₁	—	8	4	no	—
95	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
99	SD₁₆×C₆₁	—	2	16	no	—
100	SD₁₆×C₆₁	—	2	16	no	—
101	C₂×C₈×C₆₁	—	2	2	no	—
102	C₂×C₈×C₆₁	—	2	2	no	—
103	C₂×C₈×C₆₁	—	2	2	no	—
104	C₂×C₈×C₆₁	—	2	2	no	—
105	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
106	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
107	C₄⋊C₄×C₆₁	—	2	8	no	—
108	C₄⋊C₄×C₆₁	—	2	2	no	—
109	C₄⋊C₄×C₆₁	—	2	2	no	—
110	C₄⋊C₄×C₆₁	—	2	2	no	—
111	C₄⋊C₄×C₆₁	—	2	2	no	—
112	C₄⋊C₄×C₆₁	—	4	4	no	—
113	C₂²⋊C₄×C₆₁	—	4	4	no	—
114	C₂²⋊C₄×C₆₁	—	8	4	no	—
115	C₄²×C₆₁	—	2	2	no	—
116	C₄²×C₆₁	—	2	2	no	—
117	C₄²×C₆₁	—	4	4	no	—
118	C₄²×C₆₁	C₂×D₄	1	2	yes	—
119	C₄²×C₆₁	C₂×D₄	1	2	yes	—
120	C₂³×C₆₁	—	2	4	no	—
121	C₂³×C₆₁	—	4	2	no	—
122	C₂³×C₆₁	—	4	2	no	—
123	C₂³×C₆₁	—	4	4	no	—
124	C₂³×C₆₁	—	4	4	no	—
125	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
126	Q₈×C₆₁	—	2	4	no	—
127	Q₈×C₆₁	—	2	4	no	—
128	Q₈×C₆₁	—	4	2	no	—
129	Q₈×C₆₁	—	4	2	no	—
130	D₄×C₆₁	—	2	4	no	—
131	D₄×C₆₁	—	2	4	no	—
132	D₄×C₆₁	—	4	2	no	—
133	D₄×C₆₁	—	4	2	no	—
134	D₄×C₆₁	—	8	8	no	—
135	C₂×C₄×C₆₁	—	2	4	no	—
136	C₂×C₄×C₆₁	—	2	2	no	—
137	C₂×C₄×C₆₁	—	2	2	no	—
138	C₂×C₄×C₆₁	—	2	4	no	—
139	C₂×C₄×C₆₁	—	2	4	no	—
140	C₂×C₄×C₆₁	—	2	4	no	—
141	C₂×C₄×C₆₁	—	4	2	no	—
142	C₂×C₄×C₆₁	—	4	2	no	—
143	C₂×C₄×C₆₁	—	4	2	no	—
144	C₂×C₄×C₆₁	—	4	2	no	—
145	C₂×C₄×C₆₁	—	4	4	no	—
146	C₂×C₄×C₆₁	—	4	4	no	—
147	C₂×C₄×C₆₁	—	8	8	no	—
148	C₂×C₄×C₆₁	#127	1	4	yes	—
149	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
152	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
153	C₈×C₆₁	—	2	4	no	—
154	C₈×C₆₁	—	2	4	no	—
155	#23811	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂²×C₆₁	—	4	8	no	—
157	C₂²×C₆₁	—	4	4	no	—
158	C₂²×C₆₁	—	8	4	no	—
159	C₂²×C₆₁	—	8	2	no	—
160	C₂²×C₆₁	—	8	2	no	—
161	C₂²×C₆₁	#525	1	1	yes	characteristic
162	C₂²×C₆₁	#525	1	1	yes	characteristic
163	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
164	C₂²×C₆₁	#577	1	4	yes	—
165	C₄×C₆₁	—	2	4	no	—
166	C₄×C₆₁	—	4	2	no	—
167	C₄×C₆₁	—	4	2	no	—
168	C₄×C₆₁	—	4	4	no	—
169	C₄×C₆₁	—	8	2	no	—
170	C₄×C₆₁	—	8	2	no	—
171	C₄×C₆₁	#525	1	2	yes	—
172	C₄×C₆₁	#525	1	2	yes	—
173	C₂×#440	C₂×C₆₁	1	2	yes	complemented
174	C₂×C₄×SD₁₆	C₂×C₆₁	1	2	yes	complemented
175	#2328	C₂×C₆₁	1	1	yes	characteristic
176	#2327	C₂×C₆₁	1	1	yes	characteristic, complemented
177	#2097	C₂×C₆₁	1	2	yes	complemented
178	#2092	C₂×C₆₁	1	2	yes	complemented
179	#2014	C₂×C₆₁	1	2	yes	—
180	#2013	C₂×C₆₁	1	2	yes	complemented
181	#1746	C₂×C₆₁	1	1	yes	characteristic, complemented
182	C₂×C₆₁	—	8	4	no	—
183	C₂×C₆₁	#2397	1	1	yes	characteristic
184	C₂×C₆₁	#3123	1	2	yes	—
185	C₂×C₆₁	#3126	1	2	yes	—
186	C₂×C₆₁	#3154	1	2	yes	—
187	C₂²×SD₁₆	C₂²×C₆₁	1	2	yes	—
188	C₂²×SD₁₆	C₂²×C₆₁	1	2	yes	—
189	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
190	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
191	C₂×C₄×Q₈	C₂²×C₆₁	1	2	yes	—
192	C₂×C₄×D₄	C₂²×C₆₁	1	2	yes	—
193	#440	—	2	8	no	—
194	C₄×SD₁₆	—	2	8	no	—
195	C₂×#107	C₂²×C₆₁	1	2	yes	—
196	C₂×#106	C₂²×C₆₁	1	2	yes	—
197	C₂×#103	C₂²×C₆₁	1	2	yes	—
198	C₂×#103	C₂²×C₆₁	1	2	yes	—
199	C₂×#102	C₂²×C₆₁	1	2	yes	—
200	C₂×#102	C₂²×C₆₁	1	2	yes	—
201	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
202	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
203	#391	C₂²×C₆₁	1	2	yes	—
204	#390	C₂²×C₆₁	1	2	yes	—
205	#387	—	2	4	no	—
206	#386	—	2	4	no	—
207	#382	C₂²×C₆₁	1	1	yes	characteristic
208	#382	C₂²×C₆₁	1	1	yes	characteristic
209	#376	C₂²×C₆₁	1	1	yes	characteristic
210	#336	C₂²×C₆₁	1	2	yes	—
211	#336	C₂²×C₆₁	1	2	yes	—
212	C₆₁	#23811	1	1	yes	characteristic, direct product factor, Sylow
213	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
214	C₂²×D₄	—	2	2	no	—
215	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
216	C₂³×C₄	—	2	2	no	—
217	C₂×SD₁₆	—	2	4	no	—
218	C₂×SD₁₆	—	2	4	no	—
219	C₂×SD₁₆	—	2	4	no	—
220	C₂×SD₁₆	—	2	4	no	—
221	C₂×SD₁₆	—	2	4	no	—
222	C₂×SD₁₆	—	2	4	no	—
223	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
224	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
225	C₄⋊Q₈	—	2	4	no	—
226	#127	—	2	4	no	—
227	C₄×Q₈	—	2	4	no	—
228	C₄×Q₈	—	2	4	no	—
229	C₄×D₄	—	2	4	no	—
230	C₄×D₄	—	2	4	no	—
231	C₂×C₄⋊C₄	—	2	2	no	—
232	C₂×C₄⋊C₄	—	2	2	no	—
233	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
234	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
235	C₂×C₂²⋊C₄	—	2	2	no	—
236	C₂×C₂²⋊C₄	—	2	4	no	—
237	C₂×C₄²	—	2	2	no	—
238	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
239	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
240	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
241	#107	—	2	4	no	—
242	#106	—	2	4	no	—
243	#103	—	2	4	no	—
244	#103	—	2	4	no	—
245	#102	—	2	4	no	—
246	#102	—	2	4	no	—
247	#97	D₄×C₆₁	1	4	yes	complemented
248	C₄×C₈	D₄×C₆₁	1	4	yes	complemented
249	#95	—	2	2	no	—
250	#95	—	2	2	no	—
251	#95	—	2	4	no	—
252	#95	D₄×C₆₁	1	4	yes	—
253	C₂⁴	—	2	2	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	2	2	no	—
259	C₂×Q₈	#41×C₆₁	1	4	yes	—
260	C₂×D₄	—	2	8	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	8	4	no	—
267	C₂×D₄	#41×C₆₁	1	4	yes	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	4	4	no	—
272	C₂²×C₄	—	8	4	no	—
273	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
274	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
275	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
276	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
277	SD₁₆	—	2	16	no	—
278	SD₁₆	—	2	16	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
284	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
285	C₄⋊C₄	—	2	8	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	4	4	no	—
291	C₂²⋊C₄	—	4	4	no	—
292	C₂²⋊C₄	—	8	4	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	4	4	no	—
296	C₄²	C₂×D₄×C₆₁	1	2	yes	—
297	C₄²	C₂×D₄×C₆₁	1	2	yes	—
298	C₂³	—	2	4	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	4	no	—
301	C₂³	—	4	4	no	—
302	C₂³	—	4	2	no	—
303	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
304	Q₈	—	2	4	no	—
305	Q₈	—	2	4	no	—
306	Q₈	—	4	2	no	—
307	Q₈	—	4	2	no	—
308	D₄	—	2	4	no	—
309	D₄	—	2	4	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	8	8	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	—	4	4	no	—
325	C₂×C₄	—	8	8	no	—
326	C₂×C₄	#127×C₆₁	1	4	yes	—
327	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
328	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
329	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
330	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
331	C₈	—	2	4	no	—
332	C₈	—	2	4	no	—
333	C₂²	—	4	8	no	—
334	C₂²	—	4	4	no	—
335	C₂²	—	8	4	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	C₆₁×#525	1	1	yes	characteristic
339	C₂²	C₆₁×#525	1	1	yes	characteristic
340	C₂²	C₂×#127×C₆₁	1	1	yes	characteristic
341	C₂²	C₆₁×#577	1	4	yes	—
342	C₄	—	2	4	no	—
343	C₄	—	4	2	no	—
344	C₄	—	4	2	no	—
345	C₄	—	4	4	no	—
346	C₄	—	8	2	no	—
347	C₄	—	8	2	no	—
348	C₄	C₆₁×#525	1	2	yes	—
349	C₄	C₆₁×#525	1	2	yes	—
350	C₂	—	8	4	no	—
351	C₂	C₆₁×#2397	1	1	yes	characteristic
352	C₂	C₆₁×#3123	1	2	yes	—
353	C₂	C₆₁×#3126	1	2	yes	—
354	C₂	C₆₁×#3154	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 ◁ C₆₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×#440	C₂	semidirect
2. C₂×C₄×SD₁₆×C₆₁	C₂	semidirect
4. C₆₁×#2327	C₂	semidirect
5. C₆₁×#2097	C₂	semidirect
6. C₆₁×#2092	C₂	semidirect
8. C₆₁×#2013	C₂	semidirect
9. C₆₁×#1746	C₂	semidirect
69. #97×C₆₁	D₄	semidirect
70. C₄×C₈×C₆₁	D₄	semidirect
155. #23811	C₆₁	direct (N × H)
173. C₂×#440	C₂×C₆₁	semidirect
174. C₂×C₄×SD₁₆	C₂×C₆₁	semidirect
176. #2327	C₂×C₆₁	semidirect
177. #2097	C₂×C₆₁	semidirect
178. #2092	C₂×C₆₁	semidirect
180. #2013	C₂×C₆₁	semidirect
181. #1746	C₂×C₆₁	semidirect
212. C₆₁	#23811	direct (N × H)
247. #97	D₄×C₆₁	semidirect
248. C₄×C₈	D₄×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#440	2
C₂×C₄×SD₁₆×C₆₁	2
C₆₁×#2328	2
C₆₁×#2327	2
C₆₁×#2097	2
C₆₁×#2092	2
C₆₁×#2014	2
C₆₁×#2013	2
C₆₁×#1746	2
#23811	61

Subgroups — C61×#23811 #29527244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#23811 #29527244