Subgroups — C₆₁×#19996 #29523429

Order 15616.

650 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	C₆₁×#3361	C₂	1	2	yes	maximal, complemented
2	C₆₁×#3295	C₂	1	2	yes	maximal, complemented
3	C₆₁×#1709	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1678	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1677	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1632	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1620	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1619	C₂	1	1	yes	maximal, characteristic
9	C₈×SD₁₆×C₆₁	C₂	1	2	yes	maximal, complemented
10	C₆₁×#559	C₂²	1	1	yes	characteristic, complemented
11	C₆₁×#554	C₂²	1	2	yes	—
12	D₄²×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×#494	C₂²	1	1	yes	characteristic, complemented
14	C₆₁×#493	C₂²	1	1	yes	characteristic
15	C₆₁×#488	—	2	2	no	—
16	C₆₁×#486	C₂²	1	2	yes	—
17	C₆₁×#486	C₂²	1	2	yes	complemented
18	C₆₁×#481	—	2	2	no	—
19	C₆₁×#478	—	2	2	no	—
20	C₆₁×#477	C₂²	1	2	yes	complemented
21	C₆₁×#476	C₂²	1	2	yes	—
22	C₆₁×#465	—	2	2	no	—
23	C₆₁×#463	—	2	2	no	—
24	C₆₁×#460	C₂²	1	2	yes	—
25	C₆₁×#459	C₂²	1	2	yes	—
26	C₆₁×#450	—	2	2	no	—
27	C₈×Q₈×C₆₁	C₂²	1	2	yes	complemented
28	C₄×SD₁₆×C₆₁	C₂²	1	2	yes	—
29	C₄×D₈×C₆₁	—	2	2	no	—
30	C₈×D₄×C₆₁	C₂²	1	2	yes	—
31	C₂×#102×C₆₁	—	2	2	no	—
32	C₆₁×#334	C₂²	1	2	yes	complemented
33	C₆₁×#333	C₂²	1	2	yes	complemented
34	C₆₁×#331	C₂²	1	1	yes	characteristic
35	C₆₁×#331	C₂²	1	1	yes	characteristic, complemented
36	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	complemented
37	C₆₁×D₄⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₆₁	C₂²	1	1	yes	characteristic, complemented
39	C₂²×D₄×C₆₁	—	2	2	no	—
40	C₂×Q₁₆×C₆₁	—	4	1	no	—
41	C₂×SD₁₆×C₆₁	—	2	2	no	—
42	C₂×SD₁₆×C₆₁	D₄	1	2	yes	—
43	C₂×D₈×C₆₁	—	2	2	no	—
44	C₂×D₈×C₆₁	—	4	1	no	—
45	C₂²×C₈×C₆₁	—	2	2	no	—
46	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
47	#127×C₆₁	C₂³	1	1	yes	characteristic
48	#127×C₆₁	C₂³	1	1	yes	characteristic
49	#125×C₆₁	—	2	2	no	—
50	#124×C₆₁	—	2	2	no	—
51	#123×C₆₁	—	4	1	no	—
52	C₂²⋊Q₈×C₆₁	—	2	2	no	—
53	C₂²⋊Q₈×C₆₁	—	4	2	no	—
54	#121×C₆₁	—	2	2	no	—
55	#120×C₆₁	—	4	1	no	—
56	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
57	C₄×D₄×C₆₁	—	2	2	no	—
58	C₄×D₄×C₆₁	—	2	2	no	—
59	C₄×D₄×C₆₁	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
61	#107×C₆₁	—	2	2	no	—
62	#106×C₆₁	—	2	2	no	—
63	#106×C₆₁	D₄	1	2	yes	—
64	C₄⋊C₈×C₆₁	—	2	2	no	—
65	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
66	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
67	#103×C₆₁	—	2	2	no	—
68	#103×C₆₁	D₄	1	2	yes	—
69	#102×C₆₁	—	2	2	no	—
70	#102×C₆₁	—	2	2	no	—
71	#102×C₆₁	—	2	2	no	—
72	#102×C₆₁	—	2	2	no	—
73	#102×C₆₁	—	2	2	no	—
74	#102×C₆₁	—	2	2	no	—
75	#102×C₆₁	—	2	2	no	—
76	#102×C₆₁	D₄	1	2	yes	—
77	C₂²⋊C₈×C₆₁	—	2	2	no	—
78	C₄×C₈×C₆₁	—	2	2	no	—
79	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₆₁	C₂³	1	2	yes	—
81	C₂⁴×C₆₁	—	4	1	no	—
82	C₂×Q₈×C₆₁	—	2	2	no	—
83	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
84	C₂×D₄×C₆₁	—	2	2	no	—
85	C₂×D₄×C₆₁	—	2	2	no	—
86	C₂×D₄×C₆₁	—	2	1	no	—
87	C₂×D₄×C₆₁	—	2	1	no	—
88	C₂×D₄×C₆₁	—	4	2	no	—
89	C₂×D₄×C₆₁	—	4	4	no	—
90	C₂×D₄×C₆₁	—	8	2	no	—
91	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₆₁	—	2	2	no	—
93	C₂²×C₄×C₆₁	—	4	1	no	—
94	C₂²×C₄×C₆₁	—	4	2	no	—
95	Q₁₆×C₆₁	—	8	2	no	—
96	SD₁₆×C₆₁	—	2	4	no	—
97	SD₁₆×C₆₁	—	8	2	no	—
98	D₈×C₆₁	—	4	4	no	—
99	D₈×C₆₁	—	8	2	no	—
100	C₂×C₈×C₆₁	—	2	1	no	—
101	C₂×C₈×C₆₁	—	2	2	no	—
102	C₂×C₈×C₆₁	—	2	2	no	—
103	C₂×C₈×C₆₁	—	8	2	no	—
104	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
106	C₄⋊C₄×C₆₁	—	2	2	no	—
107	C₄⋊C₄×C₆₁	—	2	2	no	—
108	C₄⋊C₄×C₆₁	—	2	2	no	—
109	C₄⋊C₄×C₆₁	—	4	2	no	—
110	C₄⋊C₄×C₆₁	—	4	2	no	—
111	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
112	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
113	C₂²⋊C₄×C₆₁	—	2	2	no	—
114	C₂²⋊C₄×C₆₁	—	4	2	no	—
115	C₂²⋊C₄×C₆₁	—	4	1	no	—
116	C₂²⋊C₄×C₆₁	—	4	1	no	—
117	C₂²⋊C₄×C₆₁	—	4	2	no	—
118	C₄²×C₆₁	—	2	2	no	—
119	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
120	C₂³×C₆₁	—	2	2	no	—
121	C₂³×C₆₁	—	4	1	no	—
122	C₂³×C₆₁	—	4	1	no	—
123	C₂³×C₆₁	—	8	4	no	—
124	C₂³×C₆₁	—	16	1	no	—
125	Q₈×C₆₁	—	2	2	no	—
126	Q₈×C₆₁	—	2	2	no	—
127	Q₈×C₆₁	—	8	2	no	—
128	D₄×C₆₁	—	2	2	no	—
129	D₄×C₆₁	—	4	2	no	—
130	D₄×C₆₁	—	4	2	no	—
131	D₄×C₆₁	—	4	2	no	—
132	D₄×C₆₁	—	4	2	no	—
133	D₄×C₆₁	—	8	1	no	—
134	D₄×C₆₁	—	8	1	no	—
135	D₄×C₆₁	—	8	4	no	—
136	D₄×C₆₁	#135	1	4	yes	—
137	C₂×C₄×C₆₁	—	2	2	no	—
138	C₂×C₄×C₆₁	—	2	2	no	—
139	C₂×C₄×C₆₁	—	2	2	no	—
140	C₂×C₄×C₆₁	—	4	2	no	—
141	C₂×C₄×C₆₁	—	4	2	no	—
142	C₂×C₄×C₆₁	—	4	2	no	—
143	C₂×C₄×C₆₁	—	16	1	no	—
144	C₂×C₄×C₆₁	—	16	2	no	—
145	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
146	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
147	C₈×C₆₁	—	2	2	no	—
148	C₈×C₆₁	—	2	2	no	—
149	C₈×C₆₁	—	2	2	no	—
150	C₈×C₆₁	—	4	2	no	—
151	C₈×C₆₁	—	4	2	no	—
152	#19996	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂²×C₆₁	—	2	4	no	—
154	C₂²×C₆₁	—	4	2	no	—
155	C₂²×C₆₁	—	4	2	no	—
156	C₂²×C₆₁	—	8	2	no	—
157	C₂²×C₆₁	—	8	2	no	—
158	C₂²×C₆₁	—	8	1	no	—
159	C₂²×C₆₁	—	8	1	no	—
160	C₂²×C₆₁	—	16	4	no	—
161	C₂²×C₆₁	D₄²	1	1	yes	characteristic
162	C₄×C₆₁	—	2	1	no	—
163	C₄×C₆₁	—	4	2	no	—
164	C₄×C₆₁	—	4	2	no	—
165	C₄×C₆₁	—	4	2	no	—
166	C₄×C₆₁	—	8	2	no	—
167	C₄×C₆₁	C₂×#135	1	2	yes	—
168	C₄×C₆₁	#577	1	2	yes	—
169	#3361	C₂×C₆₁	1	2	yes	complemented
170	#3295	C₂×C₆₁	1	2	yes	complemented
171	#1709	C₂×C₆₁	1	1	yes	characteristic, complemented
172	#1678	C₂×C₆₁	1	2	yes	complemented
173	#1677	C₂×C₆₁	1	2	yes	complemented
174	#1632	C₂×C₆₁	1	2	yes	complemented
175	#1620	C₂×C₆₁	1	1	yes	characteristic, complemented
176	#1619	C₂×C₆₁	1	1	yes	characteristic
177	C₈×SD₁₆	C₂×C₆₁	1	2	yes	complemented
178	C₂×C₆₁	—	4	4	no	—
179	C₂×C₆₁	—	16	1	no	—
180	C₂×C₆₁	—	16	1	no	—
181	C₂×C₆₁	#3282	1	2	yes	—
182	C₂×C₆₁	#3292	1	1	yes	characteristic
183	#559	C₂²×C₆₁	1	1	yes	characteristic, complemented
184	#554	C₂²×C₆₁	1	2	yes	—
185	D₄²	C₂²×C₆₁	1	1	yes	characteristic
186	#494	C₂²×C₆₁	1	1	yes	characteristic, complemented
187	#493	C₂²×C₆₁	1	1	yes	characteristic
188	#488	—	2	2	no	—
189	#486	C₂²×C₆₁	1	2	yes	—
190	#486	C₂²×C₆₁	1	2	yes	complemented
191	#481	—	2	2	no	—
192	#478	—	2	2	no	—
193	#477	C₂²×C₆₁	1	2	yes	complemented
194	#476	C₂²×C₆₁	1	2	yes	—
195	#465	—	2	2	no	—
196	#463	—	2	2	no	—
197	#460	C₂²×C₆₁	1	2	yes	—
198	#459	C₂²×C₆₁	1	2	yes	—
199	#450	—	2	2	no	—
200	C₈×Q₈	C₂²×C₆₁	1	2	yes	complemented
201	C₄×SD₁₆	C₂²×C₆₁	1	2	yes	—
202	C₄×D₈	—	2	2	no	—
203	C₈×D₄	C₂²×C₆₁	1	2	yes	—
204	C₂×#102	—	2	2	no	—
205	#334	C₂²×C₆₁	1	2	yes	complemented
206	#333	C₂²×C₆₁	1	2	yes	complemented
207	#331	C₂²×C₆₁	1	1	yes	characteristic
208	#331	C₂²×C₆₁	1	1	yes	characteristic, complemented
209	Q₈⋊C₈	C₂²×C₆₁	1	2	yes	complemented
210	D₄⋊C₈	C₂²×C₆₁	1	2	yes	—
211	C₈²	C₂²×C₆₁	1	1	yes	characteristic, complemented
212	C₆₁	#19996	1	1	yes	characteristic, direct product factor, Sylow
213	C₂²×D₄	—	2	2	no	—
214	C₂×Q₁₆	—	4	1	no	—
215	C₂×SD₁₆	—	2	2	no	—
216	C₂×SD₁₆	D₄×C₆₁	1	2	yes	—
217	C₂×D₈	—	2	2	no	—
218	C₂×D₈	—	4	1	no	—
219	C₂²×C₈	—	2	2	no	—
220	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
221	#127	C₂³×C₆₁	1	1	yes	characteristic
222	#127	C₂³×C₆₁	1	1	yes	characteristic
223	#125	—	2	2	no	—
224	#124	—	2	2	no	—
225	#123	—	4	1	no	—
226	C₂²⋊Q₈	—	2	2	no	—
227	C₂²⋊Q₈	—	4	2	no	—
228	#121	—	2	2	no	—
229	#120	—	4	1	no	—
230	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
231	C₄×D₄	—	2	2	no	—
232	C₄×D₄	—	2	2	no	—
233	C₄×D₄	C₂³×C₆₁	1	2	yes	—
234	C₂×C₄⋊C₄	—	2	2	no	—
235	#107	—	2	2	no	—
236	#106	—	2	2	no	—
237	#106	D₄×C₆₁	1	2	yes	—
238	C₄⋊C₈	—	2	2	no	—
239	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
240	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
241	#103	—	2	2	no	—
242	#103	D₄×C₆₁	1	2	yes	—
243	#102	—	2	2	no	—
244	#102	—	2	2	no	—
245	#102	—	2	2	no	—
246	#102	—	2	2	no	—
247	#102	—	2	2	no	—
248	#102	—	2	2	no	—
249	#102	—	2	2	no	—
250	#102	D₄×C₆₁	1	2	yes	—
251	C₂²⋊C₈	—	2	2	no	—
252	C₄×C₈	—	2	2	no	—
253	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
254	C₄×C₈	C₂³×C₆₁	1	2	yes	—
255	C₂⁴	—	4	1	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	2	1	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	—	4	4	no	—
264	C₂×D₄	—	8	2	no	—
265	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	4	1	no	—
268	C₂²×C₄	—	4	2	no	—
269	Q₁₆	—	8	2	no	—
270	SD₁₆	—	2	4	no	—
271	SD₁₆	—	8	2	no	—
272	D₈	—	4	4	no	—
273	D₈	—	8	2	no	—
274	C₂×C₈	—	2	1	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	8	2	no	—
278	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
279	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
286	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
287	C₂²⋊C₄	—	2	2	no	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₂²⋊C₄	—	4	1	no	—
290	C₂²⋊C₄	—	4	1	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₄²	—	2	2	no	—
293	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
294	C₂³	—	2	2	no	—
295	C₂³	—	4	1	no	—
296	C₂³	—	4	1	no	—
297	C₂³	—	8	4	no	—
298	C₂³	—	16	1	no	—
299	Q₈	—	2	2	no	—
300	Q₈	—	2	2	no	—
301	Q₈	—	8	2	no	—
302	D₄	—	2	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	4	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	8	1	no	—
308	D₄	—	8	1	no	—
309	D₄	—	8	4	no	—
310	D₄	#135×C₆₁	1	4	yes	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	16	1	no	—
318	C₂×C₄	—	16	2	no	—
319	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
320	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	—	2	2	no	—
323	C₈	—	2	2	no	—
324	C₈	—	4	2	no	—
325	C₈	—	4	2	no	—
326	C₂²	—	2	4	no	—
327	C₂²	—	4	2	no	—
328	C₂²	—	4	2	no	—
329	C₂²	—	8	1	no	—
330	C₂²	—	8	1	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	16	4	no	—
334	C₂²	D₄²×C₆₁	1	1	yes	characteristic
335	C₄	—	2	1	no	—
336	C₄	—	4	2	no	—
337	C₄	—	4	2	no	—
338	C₄	—	4	2	no	—
339	C₄	—	8	2	no	—
340	C₄	C₂×#135×C₆₁	1	2	yes	—
341	C₄	C₆₁×#577	1	2	yes	—
342	C₂	—	4	4	no	—
343	C₂	—	16	1	no	—
344	C₂	—	16	1	no	—
345	C₂	C₆₁×#3282	1	2	yes	—
346	C₂	C₆₁×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
prime radical	C₆₁×#1619	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3361	C₂	semidirect
2. C₆₁×#3295	C₂	semidirect
3. C₆₁×#1709	C₂	semidirect
4. C₆₁×#1678	C₂	semidirect
5. C₆₁×#1677	C₂	semidirect
6. C₆₁×#1632	C₂	semidirect
7. C₆₁×#1620	C₂	semidirect
9. C₈×SD₁₆×C₆₁	C₂	semidirect
10. C₆₁×#559	C₂²	semidirect
13. C₆₁×#494	C₂²	semidirect
17. C₆₁×#486	C₂²	semidirect
20. C₆₁×#477	C₂²	semidirect
27. C₈×Q₈×C₆₁	C₂²	semidirect
32. C₆₁×#334	C₂²	semidirect
33. C₆₁×#333	C₂²	semidirect
35. C₆₁×#331	C₂²	semidirect
36. C₆₁×Q₈⋊C₈	C₂²	semidirect
38. C₈²×C₆₁	C₂²	semidirect
152. #19996	C₆₁	direct (N × H)
169. #3361	C₂×C₆₁	semidirect
170. #3295	C₂×C₆₁	semidirect
171. #1709	C₂×C₆₁	semidirect
172. #1678	C₂×C₆₁	semidirect
173. #1677	C₂×C₆₁	semidirect
174. #1632	C₂×C₆₁	semidirect
175. #1620	C₂×C₆₁	semidirect
177. C₈×SD₁₆	C₂×C₆₁	semidirect
183. #559	C₂²×C₆₁	semidirect
186. #494	C₂²×C₆₁	semidirect
190. #486	C₂²×C₆₁	semidirect
193. #477	C₂²×C₆₁	semidirect
200. C₈×Q₈	C₂²×C₆₁	semidirect
205. #334	C₂²×C₆₁	semidirect
206. #333	C₂²×C₆₁	semidirect
208. #331	C₂²×C₆₁	semidirect
209. Q₈⋊C₈	C₂²×C₆₁	semidirect
211. C₈²	C₂²×C₆₁	semidirect
212. C₆₁	#19996	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×#3361	2
C₆₁×#3295	2
C₆₁×#1709	2
C₆₁×#1678	2
C₆₁×#1677	2
C₆₁×#1632	2
C₆₁×#1620	2
C₆₁×#1619	2
C₈×SD₁₆×C₆₁	2
#19996	61

Subgroups — C61×#19996 #29523429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19996 #29523429