Subgroups — C₆₁×#18696 #29522129

Order 15616.

946 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3548	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3253	C₂	1	2	yes	maximal, complemented
3	C₆₁×#3131	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1635	C₂	1	4	yes	maximal, complemented
5	C₆₁×#1625	C₂	1	4	yes	maximal, complemented
6	C₆₁×#1561	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1492	C₂	1	2	yes	maximal, complemented
8	C₆₁×#562	—	2	2	no	—
9	C₆₁×#554	—	2	2	no	—
10	C₆₁×#550	—	2	2	no	—
11	C₆₁×#547	C₂²	1	4	yes	—
12	C₆₁×#540	—	2	2	no	—
13	C₆₁×#537	—	2	4	no	—
14	C₆₁×#533	C₂²	1	1	yes	characteristic
15	C₆₁×#532	C₂²	1	1	yes	characteristic
16	C₆₁×#532	C₂²	1	2	yes	—
17	C₆₁×#488	—	2	2	no	—
18	C₆₁×#487	C₂²	1	2	yes	—
19	C₆₁×#486	C₂²	1	2	yes	—
20	C₆₁×#476	—	2	2	no	—
21	C₆₁×#476	C₂²	1	4	yes	—
22	C₆₁×#469	—	2	4	no	—
23	C₆₁×#459	—	2	2	no	—
24	C₆₁×#459	C₂²	1	4	yes	—
25	C₆₁×#423	C₂²	1	2	yes	—
26	C₆₁×#416	—	2	2	no	—
27	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₆₁×#332	C₂²	1	4	yes	—
29	C₆₁×D₄⋊C₈	C₂²	1	8	yes	—
30	C₂×#41×C₆₁	—	2	4	no	—
31	C₂×#41×C₆₁	—	2	2	no	—
32	C₂×#41×C₆₁	—	2	1	no	—
33	C₂×D₈×C₆₁	—	2	4	no	—
34	C₂×M₁₆×C₆₁	—	2	2	no	—
35	C₂²×C₈×C₆₁	—	2	1	no	—
36	C₄⋊Q₈×C₆₁	D₄	1	2	yes	—
37	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
38	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
39	#127×C₆₁	—	2	2	no	—
40	#127×C₆₁	D₄	1	2	yes	—
41	#127×C₆₁	C₂³	1	1	yes	characteristic
42	#125×C₆₁	—	2	2	no	—
43	#125×C₆₁	D₄	1	2	yes	—
44	#124×C₆₁	—	2	2	no	—
45	#124×C₆₁	—	2	1	no	—
46	#123×C₆₁	—	2	8	no	—
47	C₂²⋊Q₈×C₆₁	—	2	8	no	—
48	C₂²⋊Q₈×C₆₁	—	2	2	no	—
49	C₂²⋊Q₈×C₆₁	—	2	4	no	—
50	#121×C₆₁	—	2	8	no	—
51	#121×C₆₁	—	2	8	no	—
52	#121×C₆₁	—	2	4	no	—
53	#121×C₆₁	—	2	2	no	—
54	#121×C₆₁	—	2	2	no	—
55	#121×C₆₁	D₄	1	4	yes	—
56	C₄×Q₈×C₆₁	—	2	2	no	—
57	C₄×D₄×C₆₁	—	2	4	no	—
58	C₄×D₄×C₆₁	—	2	2	no	—
59	C₄×D₄×C₆₁	—	2	2	no	—
60	C₄×D₄×C₆₁	C₂³	1	4	yes	—
61	#117×C₆₁	D₄	1	2	yes	—
62	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
63	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
64	#106×C₆₁	—	2	4	no	—
65	C₄⋊C₈×C₆₁	—	2	2	no	—
66	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
67	#103×C₆₁	—	2	4	no	—
68	#102×C₆₁	—	2	4	no	—
69	#102×C₆₁	—	2	4	no	—
70	#102×C₆₁	—	2	4	no	—
71	C₄×C₈×C₆₁	—	2	1	no	—
72	C₄×C₈×C₆₁	C₂³	1	2	yes	—
73	#41×C₆₁	—	4	4	no	—
74	#41×C₆₁	—	8	2	no	—
75	#41×C₆₁	—	8	2	no	—
76	#41×C₆₁	—	8	4	no	—
77	#41×C₆₁	—	8	4	no	—
78	C₂×Q₈×C₆₁	—	2	2	no	—
79	C₂×Q₈×C₆₁	—	2	2	no	—
80	C₂×Q₈×C₆₁	—	2	1	no	—
81	C₂×D₄×C₆₁	—	2	8	no	—
82	C₂×D₄×C₆₁	—	2	4	no	—
83	C₂×D₄×C₆₁	—	2	2	no	—
84	C₂×D₄×C₆₁	—	2	2	no	—
85	C₂×D₄×C₆₁	—	2	1	no	—
86	C₂×D₄×C₆₁	—	2	2	no	—
87	C₂×D₄×C₆₁	—	4	1	no	—
88	C₂×D₄×C₆₁	C₂×D₄	1	4	yes	—
89	C₂²×C₄×C₆₁	—	2	8	no	—
90	C₂²×C₄×C₆₁	—	2	4	no	—
91	C₂²×C₄×C₆₁	—	4	1	no	—
92	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
94	D₈×C₆₁	—	8	4	no	—
95	M₁₆×C₆₁	—	8	2	no	—
96	C₂×C₈×C₆₁	—	2	2	no	—
97	C₂×C₈×C₆₁	—	2	4	no	—
98	C₂×C₈×C₆₁	—	4	2	no	—
99	C₄⋊C₄×C₆₁	—	2	8	no	—
100	C₄⋊C₄×C₆₁	—	2	2	no	—
101	C₄⋊C₄×C₆₁	—	2	4	no	—
102	C₄⋊C₄×C₆₁	—	2	2	no	—
103	C₄⋊C₄×C₆₁	—	2	2	no	—
104	C₄⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
105	C₄⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
106	C₂²⋊C₄×C₆₁	—	2	8	no	—
107	C₂²⋊C₄×C₆₁	—	2	2	no	—
108	C₂²⋊C₄×C₆₁	—	2	4	no	—
109	C₂²⋊C₄×C₆₁	—	2	4	no	—
110	C₂²⋊C₄×C₆₁	—	4	2	no	—
111	C₄²×C₆₁	—	2	2	no	—
112	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₆₁	C₂×D₄	1	2	yes	—
114	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
115	C₂³×C₆₁	—	2	4	no	—
116	C₂³×C₆₁	—	4	1	no	—
117	C₂³×C₆₁	#120	1	1	yes	characteristic
118	Q₈×C₆₁	—	4	2	no	—
119	Q₈×C₆₁	—	4	2	no	—
120	Q₈×C₆₁	—	4	2	no	—
121	Q₈×C₆₁	—	8	2	no	—
122	D₄×C₆₁	—	2	4	no	—
123	D₄×C₆₁	—	2	4	no	—
124	D₄×C₆₁	—	4	2	no	—
125	D₄×C₆₁	—	4	2	no	—
126	D₄×C₆₁	—	4	2	no	—
127	D₄×C₆₁	—	4	4	no	—
128	D₄×C₆₁	—	8	2	no	—
129	D₄×C₆₁	—	8	4	no	—
130	D₄×C₆₁	—	8	8	no	—
131	D₄×C₆₁	—	8	2	no	—
132	C₂×C₄×C₆₁	—	2	2	no	—
133	C₂×C₄×C₆₁	—	2	2	no	—
134	C₂×C₄×C₆₁	—	2	2	no	—
135	C₂×C₄×C₆₁	—	2	4	no	—
136	C₂×C₄×C₆₁	—	2	4	no	—
137	C₂×C₄×C₆₁	—	4	4	no	—
138	C₂×C₄×C₆₁	—	4	4	no	—
139	C₂×C₄×C₆₁	—	4	1	no	—
140	C₂×C₄×C₆₁	—	8	8	no	—
141	C₂×C₄×C₆₁	—	8	2	no	—
142	C₂×C₄×C₆₁	#120	1	2	yes	—
143	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
144	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
146	C₈×C₆₁	—	4	2	no	—
147	C₈×C₆₁	—	4	4	no	—
148	#18696	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂²×C₆₁	—	2	2	no	—
150	C₂²×C₆₁	—	2	1	no	—
151	C₂²×C₆₁	—	4	4	no	—
152	C₂²×C₆₁	—	4	4	no	—
153	C₂²×C₆₁	—	4	4	no	—
154	C₂²×C₆₁	—	8	2	no	—
155	C₂²×C₆₁	—	8	1	no	—
156	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
157	C₄×C₆₁	—	2	2	no	—
158	C₄×C₆₁	—	2	1	no	—
159	C₄×C₆₁	—	4	4	no	—
160	C₄×C₆₁	—	4	4	no	—
161	C₄×C₆₁	—	8	1	no	—
162	C₄×C₆₁	#454	1	2	yes	—
163	C₄×C₆₁	#576	1	2	yes	—
164	C₄×C₆₁	#577	1	2	yes	—
165	#3548	C₂×C₆₁	1	1	yes	characteristic, complemented
166	#3253	C₂×C₆₁	1	2	yes	complemented
167	#3131	C₂×C₆₁	1	1	yes	characteristic, complemented
168	#1635	C₂×C₆₁	1	4	yes	complemented
169	#1625	C₂×C₆₁	1	4	yes	complemented
170	#1561	C₂×C₆₁	1	1	yes	characteristic, complemented
171	#1492	C₂×C₆₁	1	2	yes	complemented
172	C₂×C₆₁	—	4	1	no	—
173	C₂×C₆₁	—	8	4	no	—
174	C₂×C₆₁	—	16	1	no	—
175	C₂×C₆₁	#3012	1	2	yes	—
176	C₂×C₆₁	C₂×#454	1	1	yes	characteristic
177	#562	—	2	2	no	—
178	#554	—	2	2	no	—
179	#550	—	2	2	no	—
180	#547	C₂²×C₆₁	1	4	yes	—
181	#540	—	2	2	no	—
182	#537	—	2	4	no	—
183	#533	C₂²×C₆₁	1	1	yes	characteristic
184	#532	C₂²×C₆₁	1	1	yes	characteristic
185	#532	C₂²×C₆₁	1	2	yes	—
186	#488	—	2	2	no	—
187	#487	C₂²×C₆₁	1	2	yes	—
188	#486	C₂²×C₆₁	1	2	yes	—
189	#476	—	2	2	no	—
190	#476	C₂²×C₆₁	1	4	yes	—
191	#469	—	2	4	no	—
192	#459	—	2	2	no	—
193	#459	C₂²×C₆₁	1	4	yes	—
194	#423	C₂²×C₆₁	1	2	yes	—
195	#416	—	2	2	no	—
196	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
197	#332	C₂²×C₆₁	1	4	yes	—
198	D₄⋊C₈	C₂²×C₆₁	1	8	yes	—
199	C₆₁	#18696	1	1	yes	characteristic, direct product factor, Sylow
200	C₂×#41	—	2	4	no	—
201	C₂×#41	—	2	2	no	—
202	C₂×#41	—	2	1	no	—
203	C₂×D₈	—	2	4	no	—
204	C₂×M₁₆	—	2	2	no	—
205	C₂²×C₈	—	2	1	no	—
206	C₄⋊Q₈	D₄×C₆₁	1	2	yes	—
207	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
208	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
209	#127	—	2	2	no	—
210	#127	D₄×C₆₁	1	2	yes	—
211	#127	C₂³×C₆₁	1	1	yes	characteristic
212	#125	—	2	2	no	—
213	#125	D₄×C₆₁	1	2	yes	—
214	#124	—	2	2	no	—
215	#124	—	2	1	no	—
216	#123	—	2	8	no	—
217	C₂²⋊Q₈	—	2	8	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	C₂²⋊Q₈	—	2	4	no	—
220	#121	—	2	8	no	—
221	#121	—	2	8	no	—
222	#121	—	2	4	no	—
223	#121	—	2	2	no	—
224	#121	—	2	2	no	—
225	#121	D₄×C₆₁	1	4	yes	—
226	C₄×Q₈	—	2	2	no	—
227	C₄×D₄	—	2	4	no	—
228	C₄×D₄	—	2	2	no	—
229	C₄×D₄	—	2	2	no	—
230	C₄×D₄	C₂³×C₆₁	1	4	yes	—
231	#117	D₄×C₆₁	1	2	yes	—
232	C₂×C₄⋊C₄	—	2	4	no	—
233	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
234	#106	—	2	4	no	—
235	C₄⋊C₈	—	2	2	no	—
236	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
237	#103	—	2	4	no	—
238	#102	—	2	4	no	—
239	#102	—	2	4	no	—
240	#102	—	2	4	no	—
241	C₄×C₈	—	2	1	no	—
242	C₄×C₈	C₂³×C₆₁	1	2	yes	—
243	#41	—	4	4	no	—
244	#41	—	8	2	no	—
245	#41	—	8	2	no	—
246	#41	—	8	4	no	—
247	#41	—	8	4	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	2	1	no	—
251	C₂×D₄	—	2	8	no	—
252	C₂×D₄	—	2	4	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	C₂×D₄×C₆₁	1	4	yes	—
259	C₂²×C₄	—	2	8	no	—
260	C₂²×C₄	—	2	4	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
264	D₈	—	8	4	no	—
265	M₁₆	—	8	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	4	no	—
268	C₂×C₈	—	4	2	no	—
269	C₄⋊C₄	—	2	8	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
275	C₄⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
276	C₂²⋊C₄	—	2	8	no	—
277	C₂²⋊C₄	—	2	2	no	—
278	C₂²⋊C₄	—	2	4	no	—
279	C₂²⋊C₄	—	2	4	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
283	C₄²	C₂×D₄×C₆₁	1	2	yes	—
284	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
285	C₂³	—	2	4	no	—
286	C₂³	—	4	1	no	—
287	C₂³	#120×C₆₁	1	1	yes	characteristic
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	8	2	no	—
292	D₄	—	2	4	no	—
293	D₄	—	2	4	no	—
294	D₄	—	4	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	4	no	—
298	D₄	—	8	2	no	—
299	D₄	—	8	4	no	—
300	D₄	—	8	8	no	—
301	D₄	—	8	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	8	8	no	—
311	C₂×C₄	—	8	2	no	—
312	C₂×C₄	#120×C₆₁	1	2	yes	—
313	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
315	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
316	C₈	—	4	2	no	—
317	C₈	—	4	4	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	4	4	no	—
321	C₂²	—	4	4	no	—
322	C₂²	—	4	4	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	1	no	—
325	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	4	no	—
329	C₄	—	4	4	no	—
330	C₄	—	8	1	no	—
331	C₄	C₆₁×#454	1	2	yes	—
332	C₄	C₆₁×#576	1	2	yes	—
333	C₄	C₆₁×#577	1	2	yes	—
334	C₂	—	4	1	no	—
335	C₂	—	8	4	no	—
336	C₂	—	16	1	no	—
337	C₂	C₆₁×#3012	1	2	yes	—
338	C₂	C₂×C₆₁×#454	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3548	C₂	semidirect
2. C₆₁×#3253	C₂	semidirect
3. C₆₁×#3131	C₂	semidirect
4. C₆₁×#1635	C₂	semidirect
5. C₆₁×#1625	C₂	semidirect
6. C₆₁×#1561	C₂	semidirect
7. C₆₁×#1492	C₂	semidirect
148. #18696	C₆₁	direct (N × H)
165. #3548	C₂×C₆₁	semidirect
166. #3253	C₂×C₆₁	semidirect
167. #3131	C₂×C₆₁	semidirect
168. #1635	C₂×C₆₁	semidirect
169. #1625	C₂×C₆₁	semidirect
170. #1561	C₂×C₆₁	semidirect
171. #1492	C₂×C₆₁	semidirect
199. C₆₁	#18696	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×#3548	2
C₆₁×#3253	2
C₆₁×#3131	2
C₆₁×#1635	2
C₆₁×#1625	2
C₆₁×#1561	2
C₆₁×#1492	2
#18696	61

Subgroups — C61×#18696 #29522129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#18696 #29522129