Subgroups — C₆₁×#19987 #29523420

Order 15616.

680 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3303	C₂	1	2	yes	maximal, complemented
2	C₆₁×#3294	C₂	1	2	yes	maximal, complemented
3	C₆₁×#1708	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1683	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1681	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1654	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1629	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1624	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×D₈×C₆₁	C₂	1	2	yes	maximal, complemented
10	C₆₁×#555	C₂²	1	1	yes	characteristic, complemented
11	C₆₁×#547	C₂²	1	2	yes	—
12	D₄²×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×#498	C₂²	1	1	yes	characteristic, complemented
14	C₆₁×#492	C₂²	1	1	yes	characteristic
15	C₆₁×#487	C₂²	1	2	yes	complemented
16	C₆₁×#486	C₂²	1	2	yes	—
17	C₆₁×#485	—	2	2	no	—
18	C₆₁×#480	—	2	2	no	—
19	C₆₁×#476	C₂²	1	2	yes	—
20	C₆₁×#474	C₂²	1	2	yes	—
21	C₆₁×#466	—	2	2	no	—
22	C₆₁×#465	—	2	2	no	—
23	C₆₁×#461	C₂²	1	2	yes	complemented
24	C₆₁×#459	C₂²	1	2	yes	—
25	C₆₁×#449	—	2	2	no	—
26	C₆₁×#447	—	2	2	no	—
27	C₄×D₈×C₆₁	C₂²	1	2	yes	—
28	C₈×D₄×C₆₁	C₂²	1	2	yes	—
29	C₈×D₄×C₆₁	C₂²	1	2	yes	complemented
30	C₆₁×#416	—	2	2	no	—
31	C₂×#102×C₆₁	—	2	2	no	—
32	C₆₁×#335	C₂²	1	2	yes	complemented
33	C₆₁×#333	C₂²	1	1	yes	characteristic, complemented
34	C₆₁×#332	C₂²	1	2	yes	complemented
35	C₆₁×#331	C₂²	1	1	yes	characteristic
36	C₆₁×D₄⋊C₈	C₂²	1	2	yes	—
37	C₆₁×D₄⋊C₈	C₂²	1	2	yes	complemented
38	C₈²×C₆₁	C₂²	1	1	yes	characteristic, complemented
39	C₂×#41×C₆₁	—	2	2	no	—
40	C₂²×D₄×C₆₁	—	2	2	no	—
41	C₂×SD₁₆×C₆₁	—	2	2	no	—
42	C₂×SD₁₆×C₆₁	—	4	1	no	—
43	C₂×D₈×C₆₁	—	2	2	no	—
44	C₂×D₈×C₆₁	D₄	1	2	yes	—
45	C₂²×C₈×C₆₁	—	2	2	no	—
46	C₂²×C₈×C₆₁	—	2	2	no	—
47	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
48	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
49	#127×C₆₁	C₂³	1	1	yes	characteristic
50	#124×C₆₁	—	4	1	no	—
51	#123×C₆₁	—	4	2	no	—
52	C₂²⋊Q₈×C₆₁	—	2	2	no	—
53	#121×C₆₁	—	2	2	no	—
54	#121×C₆₁	—	2	2	no	—
55	#121×C₆₁	—	2	2	no	—
56	#120×C₆₁	—	4	1	no	—
57	C₄×D₄×C₆₁	C₂³	1	2	yes	—
58	C₄×D₄×C₆₁	C₂³	1	2	yes	—
59	#117×C₆₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
61	#107×C₆₁	—	2	2	no	—
62	#107×C₆₁	D₄	1	2	yes	—
63	#106×C₆₁	—	2	2	no	—
64	#106×C₆₁	—	4	1	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	—	2	2	no	—
69	#103×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	D₄	1	2	yes	—
76	#102×C₆₁	D₄	1	2	yes	—
77	C₂²⋊C₈×C₆₁	—	2	2	no	—
78	C₂²⋊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	#41×C₆₁	—	4	4	no	—
83	#41×C₆₁	—	8	2	no	—
84	C₂×Q₈×C₆₁	—	2	2	no	—
85	C₂×Q₈×C₆₁	—	2	1	no	—
86	C₂×D₄×C₆₁	—	2	2	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	—	4	2	no	—
91	C₂×D₄×C₆₁	—	4	1	no	—
92	C₂×D₄×C₆₁	—	8	2	no	—
93	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
94	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₆₁	—	2	2	no	—
96	C₂²×C₄×C₆₁	—	2	2	no	—
97	C₂²×C₄×C₆₁	—	4	2	no	—
98	SD₁₆×C₆₁	—	8	2	no	—
99	SD₁₆×C₆₁	—	8	2	no	—
100	D₈×C₆₁	—	2	4	no	—
101	D₈×C₆₁	—	8	2	no	—
102	C₂×C₈×C₆₁	—	2	1	no	—
103	C₂×C₈×C₆₁	—	2	2	no	—
104	C₂×C₈×C₆₁	—	2	2	no	—
105	C₂×C₈×C₆₁	—	8	2	no	—
106	C₂×C₈×C₆₁	—	8	2	no	—
107	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
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	2	no	—
113	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
114	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
115	C₂²⋊C₄×C₆₁	—	2	2	no	—
116	C₂²⋊C₄×C₆₁	—	2	2	no	—
117	C₂²⋊C₄×C₆₁	—	4	1	no	—
118	C₂²⋊C₄×C₆₁	—	4	2	no	—
119	C₂²⋊C₄×C₆₁	—	4	2	no	—
120	C₄²×C₆₁	—	4	1	no	—
121	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
122	C₂³×C₆₁	—	2	2	no	—
123	C₂³×C₆₁	—	2	2	no	—
124	C₂³×C₆₁	—	4	1	no	—
125	C₂³×C₆₁	—	8	4	no	—
126	C₂³×C₆₁	—	16	1	no	—
127	Q₈×C₆₁	—	4	2	no	—
128	Q₈×C₆₁	—	4	2	no	—
129	Q₈×C₆₁	—	8	1	no	—
130	D₄×C₆₁	—	2	2	no	—
131	D₄×C₆₁	—	2	2	no	—
132	D₄×C₆₁	—	2	2	no	—
133	D₄×C₆₁	—	4	2	no	—
134	D₄×C₆₁	—	4	2	no	—
135	D₄×C₆₁	—	8	1	no	—
136	D₄×C₆₁	—	8	4	no	—
137	D₄×C₆₁	—	8	4	no	—
138	D₄×C₆₁	—	16	1	no	—
139	D₄×C₆₁	#135	1	4	yes	—
140	C₂×C₄×C₆₁	—	2	2	no	—
141	C₂×C₄×C₆₁	—	2	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	2	no	—
146	C₂×C₄×C₆₁	—	4	1	no	—
147	C₂×C₄×C₆₁	—	4	2	no	—
148	C₂×C₄×C₆₁	—	8	4	no	—
149	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
150	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
151	C₈×C₆₁	—	2	2	no	—
152	C₈×C₆₁	—	2	2	no	—
153	C₈×C₆₁	—	2	2	no	—
154	C₈×C₆₁	—	4	2	no	—
155	C₈×C₆₁	—	4	2	no	—
156	#19987	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂²×C₆₁	—	2	4	no	—
158	C₂²×C₆₁	—	4	2	no	—
159	C₂²×C₆₁	—	4	2	no	—
160	C₂²×C₆₁	—	4	2	no	—
161	C₂²×C₆₁	—	4	2	no	—
162	C₂²×C₆₁	—	4	2	no	—
163	C₂²×C₆₁	—	8	2	no	—
164	C₂²×C₆₁	—	8	1	no	—
165	C₂²×C₆₁	—	16	4	no	—
166	C₂²×C₆₁	D₄²	1	1	yes	characteristic
167	C₄×C₆₁	—	2	1	no	—
168	C₄×C₆₁	—	4	2	no	—
169	C₄×C₆₁	—	4	2	no	—
170	C₄×C₆₁	—	8	2	no	—
171	C₄×C₆₁	—	8	1	no	—
172	C₄×C₆₁	C₂×#135	1	2	yes	—
173	C₄×C₆₁	#576	1	2	yes	—
174	#3303	C₂×C₆₁	1	2	yes	complemented
175	#3294	C₂×C₆₁	1	2	yes	complemented
176	#1708	C₂×C₆₁	1	1	yes	characteristic, complemented
177	#1683	C₂×C₆₁	1	2	yes	complemented
178	#1681	C₂×C₆₁	1	2	yes	complemented
179	#1654	C₂×C₆₁	1	2	yes	complemented
180	#1629	C₂×C₆₁	1	1	yes	characteristic, complemented
181	#1624	C₂×C₆₁	1	1	yes	characteristic, complemented
182	C₈×D₈	C₂×C₆₁	1	2	yes	complemented
183	C₂×C₆₁	—	4	4	no	—
184	C₂×C₆₁	—	8	2	no	—
185	C₂×C₆₁	—	16	1	no	—
186	C₂×C₆₁	#3280	1	2	yes	—
187	C₂×C₆₁	#3291	1	1	yes	characteristic
188	#555	C₂²×C₆₁	1	1	yes	characteristic, complemented
189	#547	C₂²×C₆₁	1	2	yes	—
190	D₄²	C₂²×C₆₁	1	1	yes	characteristic
191	#498	C₂²×C₆₁	1	1	yes	characteristic, complemented
192	#492	C₂²×C₆₁	1	1	yes	characteristic
193	#487	C₂²×C₆₁	1	2	yes	complemented
194	#486	C₂²×C₆₁	1	2	yes	—
195	#485	—	2	2	no	—
196	#480	—	2	2	no	—
197	#476	C₂²×C₆₁	1	2	yes	—
198	#474	C₂²×C₆₁	1	2	yes	—
199	#466	—	2	2	no	—
200	#465	—	2	2	no	—
201	#461	C₂²×C₆₁	1	2	yes	complemented
202	#459	C₂²×C₆₁	1	2	yes	—
203	#449	—	2	2	no	—
204	#447	—	2	2	no	—
205	C₄×D₈	C₂²×C₆₁	1	2	yes	—
206	C₈×D₄	C₂²×C₆₁	1	2	yes	—
207	C₈×D₄	C₂²×C₆₁	1	2	yes	complemented
208	#416	—	2	2	no	—
209	C₂×#102	—	2	2	no	—
210	#335	C₂²×C₆₁	1	2	yes	complemented
211	#333	C₂²×C₆₁	1	1	yes	characteristic, complemented
212	#332	C₂²×C₆₁	1	2	yes	complemented
213	#331	C₂²×C₆₁	1	1	yes	characteristic
214	D₄⋊C₈	C₂²×C₆₁	1	2	yes	—
215	D₄⋊C₈	C₂²×C₆₁	1	2	yes	complemented
216	C₈²	C₂²×C₆₁	1	1	yes	characteristic, complemented
217	C₆₁	#19987	1	1	yes	characteristic, direct product factor, Sylow
218	C₂×#41	—	2	2	no	—
219	C₂²×D₄	—	2	2	no	—
220	C₂×SD₁₆	—	2	2	no	—
221	C₂×SD₁₆	—	4	1	no	—
222	C₂×D₈	—	2	2	no	—
223	C₂×D₈	D₄×C₆₁	1	2	yes	—
224	C₂²×C₈	—	2	2	no	—
225	C₂²×C₈	—	2	2	no	—
226	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
227	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
228	#127	C₂³×C₆₁	1	1	yes	characteristic
229	#124	—	4	1	no	—
230	#123	—	4	2	no	—
231	C₂²⋊Q₈	—	2	2	no	—
232	#121	—	2	2	no	—
233	#121	—	2	2	no	—
234	#121	—	2	2	no	—
235	#120	—	4	1	no	—
236	C₄×D₄	C₂³×C₆₁	1	2	yes	—
237	C₄×D₄	C₂³×C₆₁	1	2	yes	—
238	#117	—	2	2	no	—
239	C₂×C₄⋊C₄	—	2	2	no	—
240	#107	—	2	2	no	—
241	#107	D₄×C₆₁	1	2	yes	—
242	#106	—	2	2	no	—
243	#106	—	4	1	no	—
244	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
245	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
246	#103	—	2	2	no	—
247	#103	—	2	2	no	—
248	#103	—	2	2	no	—
249	#102	—	2	2	no	—
250	#102	—	2	2	no	—
251	#102	—	2	2	no	—
252	#102	—	2	2	no	—
253	#102	—	2	2	no	—
254	#102	D₄×C₆₁	1	2	yes	—
255	#102	D₄×C₆₁	1	2	yes	—
256	C₂²⋊C₈	—	2	2	no	—
257	C₂²⋊C₈	—	2	2	no	—
258	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
259	C₄×C₈	C₂³×C₆₁	1	2	yes	—
260	C₂⁴	—	4	1	no	—
261	#41	—	4	4	no	—
262	#41	—	8	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	2	1	no	—
267	C₂×D₄	—	4	2	no	—
268	C₂×D₄	—	4	4	no	—
269	C₂×D₄	—	4	2	no	—
270	C₂×D₄	—	4	1	no	—
271	C₂×D₄	—	8	2	no	—
272	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
273	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	4	2	no	—
277	SD₁₆	—	8	2	no	—
278	SD₁₆	—	8	2	no	—
279	D₈	—	2	4	no	—
280	D₈	—	8	2	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	8	2	no	—
285	C₂×C₈	—	8	2	no	—
286	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
287	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	4	2	no	—
292	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
293	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
294	C₂²⋊C₄	—	2	2	no	—
295	C₂²⋊C₄	—	2	2	no	—
296	C₂²⋊C₄	—	4	1	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	4	2	no	—
299	C₄²	—	4	1	no	—
300	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
301	C₂³	—	2	2	no	—
302	C₂³	—	2	2	no	—
303	C₂³	—	4	1	no	—
304	C₂³	—	8	4	no	—
305	C₂³	—	16	1	no	—
306	Q₈	—	4	2	no	—
307	Q₈	—	4	2	no	—
308	Q₈	—	8	1	no	—
309	D₄	—	2	2	no	—
310	D₄	—	2	2	no	—
311	D₄	—	2	2	no	—
312	D₄	—	4	2	no	—
313	D₄	—	4	2	no	—
314	D₄	—	8	1	no	—
315	D₄	—	8	4	no	—
316	D₄	—	8	4	no	—
317	D₄	—	16	1	no	—
318	D₄	#135×C₆₁	1	4	yes	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	8	4	no	—
328	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
329	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
330	C₈	—	2	2	no	—
331	C₈	—	2	2	no	—
332	C₈	—	2	2	no	—
333	C₈	—	4	2	no	—
334	C₈	—	4	2	no	—
335	C₂²	—	2	4	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	2	no	—
340	C₂²	—	4	2	no	—
341	C₂²	—	8	1	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	16	4	no	—
344	C₂²	D₄²×C₆₁	1	1	yes	characteristic
345	C₄	—	2	1	no	—
346	C₄	—	4	2	no	—
347	C₄	—	4	2	no	—
348	C₄	—	8	1	no	—
349	C₄	—	8	2	no	—
350	C₄	C₂×#135×C₆₁	1	2	yes	—
351	C₄	C₆₁×#576	1	2	yes	—
352	C₂	—	4	4	no	—
353	C₂	—	8	2	no	—
354	C₂	—	16	1	no	—
355	C₂	C₆₁×#3280	1	2	yes	—
356	C₂	C₆₁×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
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×#3303	C₂	semidirect
2. C₆₁×#3294	C₂	semidirect
3. C₆₁×#1708	C₂	semidirect
4. C₆₁×#1683	C₂	semidirect
5. C₆₁×#1681	C₂	semidirect
6. C₆₁×#1654	C₂	semidirect
7. C₆₁×#1629	C₂	semidirect
8. C₆₁×#1624	C₂	semidirect
9. C₈×D₈×C₆₁	C₂	semidirect
10. C₆₁×#555	C₂²	semidirect
13. C₆₁×#498	C₂²	semidirect
15. C₆₁×#487	C₂²	semidirect
23. C₆₁×#461	C₂²	semidirect
29. C₈×D₄×C₆₁	C₂²	semidirect
32. C₆₁×#335	C₂²	semidirect
33. C₆₁×#333	C₂²	semidirect
34. C₆₁×#332	C₂²	semidirect
37. C₆₁×D₄⋊C₈	C₂²	semidirect
38. C₈²×C₆₁	C₂²	semidirect
156. #19987	C₆₁	direct (N × H)
174. #3303	C₂×C₆₁	semidirect
175. #3294	C₂×C₆₁	semidirect
176. #1708	C₂×C₆₁	semidirect
177. #1683	C₂×C₆₁	semidirect
178. #1681	C₂×C₆₁	semidirect
179. #1654	C₂×C₆₁	semidirect
180. #1629	C₂×C₆₁	semidirect
181. #1624	C₂×C₆₁	semidirect
182. C₈×D₈	C₂×C₆₁	semidirect
188. #555	C₂²×C₆₁	semidirect
191. #498	C₂²×C₆₁	semidirect
193. #487	C₂²×C₆₁	semidirect
201. #461	C₂²×C₆₁	semidirect
207. C₈×D₄	C₂²×C₆₁	semidirect
210. #335	C₂²×C₆₁	semidirect
211. #333	C₂²×C₆₁	semidirect
212. #332	C₂²×C₆₁	semidirect
215. D₄⋊C₈	C₂²×C₆₁	semidirect
216. C₈²	C₂²×C₆₁	semidirect
217. C₆₁	#19987	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×#3303	2
C₆₁×#3294	2
C₆₁×#1708	2
C₆₁×#1683	2
C₆₁×#1681	2
C₆₁×#1654	2
C₆₁×#1629	2
C₆₁×#1624	2
C₈×D₈×C₆₁	2
#19987	61

Subgroups — C61×#19987 #29523420

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19987 #29523420