Subgroups — C₆₁×#18718 #29522151

Order 15616.

1004 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3550	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3208	C₂	1	2	yes	maximal, complemented
3	C₆₁×#3151	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1623	C₂	1	4	yes	maximal, complemented
5	C₆₁×#1620	C₂	1	4	yes	maximal, complemented
6	C₆₁×#1543	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1485	C₂	1	2	yes	maximal, complemented
8	C₂×AGL(1,ℤ/8)×C₆₁	—	2	2	no	—
9	C₆₁×#563	—	2	2	no	—
10	C₆₁×#559	C₂²	1	2	yes	complemented
11	Q₈²×C₆₁	C₂²	1	2	yes	complemented
12	C₆₁×#556	—	2	2	no	—
13	D₄×Q₈×C₆₁	—	2	2	no	—
14	C₆₁×#542	—	2	4	no	—
15	C₆₁×#536	—	2	2	no	—
16	C₆₁×#533	C₂²	1	2	yes	—
17	C₆₁×#532	C₂²	1	1	yes	characteristic
18	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
19	C₆₁×#496	—	2	2	no	—
20	C₆₁×#493	C₂²	1	2	yes	—
21	C₆₁×#492	—	2	2	no	—
22	C₆₁×#492	C₂²	1	2	yes	—
23	C₆₁×#486	—	2	2	no	—
24	C₆₁×#460	C₂²	1	8	yes	—
25	C₆₁×#450	—	2	4	no	—
26	C₆₁×#431	C₂²	1	2	yes	—
27	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
28	C₆₁×#333	C₂²	1	2	yes	complemented
29	C₆₁×#331	C₂²	1	2	yes	complemented
30	C₆₁×Q₈⋊C₈	C₂²	1	8	yes	complemented
31	C₂×#41×C₆₁	D₄	1	2	yes	—
32	C₂²×Q₈×C₆₁	—	2	2	no	—
33	C₂²×D₄×C₆₁	—	2	1	no	—
34	C₂²×D₄×C₆₁	—	2	2	no	—
35	AGL(1,ℤ/8)×C₆₁	—	4	8	no	—
36	C₂×SD₁₆×C₆₁	—	2	4	no	—
37	C₂×SD₁₆×C₆₁	—	2	8	no	—
38	C₂×D₈×C₆₁	—	2	4	no	—
39	C₂×M₁₆×C₆₁	—	2	1	no	—
40	C₄⋊Q₈×C₆₁	—	2	4	no	—
41	C₄⋊Q₈×C₆₁	—	2	4	no	—
42	C₄⋊Q₈×C₆₁	D₄	1	2	yes	—
43	C₄⋊Q₈×C₆₁	D₄	1	2	yes	—
44	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
45	#127×C₆₁	—	2	2	no	—
46	#127×C₆₁	—	2	1	no	—
47	#127×C₆₁	C₂³	1	2	yes	—
48	#127×C₆₁	C₂³	1	1	yes	characteristic
49	#126×C₆₁	—	2	8	no	—
50	#125×C₆₁	—	2	2	no	—
51	#124×C₆₁	—	2	4	no	—
52	#124×C₆₁	—	2	4	no	—
53	#124×C₆₁	D₄	1	2	yes	—
54	#123×C₆₁	—	2	4	no	—
55	#123×C₆₁	—	2	4	no	—
56	C₂²⋊Q₈×C₆₁	—	2	4	no	—
57	C₂²⋊Q₈×C₆₁	—	2	8	no	—
58	C₂²⋊Q₈×C₆₁	—	2	2	no	—
59	C₂²⋊Q₈×C₆₁	D₄	1	4	yes	—
60	#121×C₆₁	—	2	2	no	—
61	C₄×Q₈×C₆₁	—	2	4	no	—
62	C₄×Q₈×C₆₁	—	2	2	no	—
63	C₄×Q₈×C₆₁	C₂³	1	4	yes	—
64	C₄×D₄×C₆₁	—	2	4	no	—
65	C₄×D₄×C₆₁	—	2	2	no	—
66	#117×C₆₁	—	2	2	no	—
67	#117×C₆₁	—	2	2	no	—
68	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
70	#102×C₆₁	—	2	8	no	—
71	C₂²⋊C₈×C₆₁	—	2	2	no	—
72	#97×C₆₁	—	2	1	no	—
73	C₄×C₈×C₆₁	—	2	2	no	—
74	C₄×C₈×C₆₁	C₂³	1	2	yes	—
75	C₂⁴×C₆₁	—	4	1	no	—
76	#41×C₆₁	—	4	4	no	—
77	C₂×Q₈×C₆₁	—	2	4	no	—
78	C₂×Q₈×C₆₁	—	2	2	no	—
79	C₂×Q₈×C₆₁	—	8	4	no	—
80	C₂×Q₈×C₆₁	C₂×D₄	1	4	yes	—
81	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
82	C₂×D₄×C₆₁	—	2	2	no	—
83	C₂×D₄×C₆₁	—	2	2	no	—
84	C₂×D₄×C₆₁	—	2	4	no	—
85	C₂×D₄×C₆₁	—	2	2	no	—
86	C₂×D₄×C₆₁	—	4	4	no	—
87	C₂×D₄×C₆₁	—	4	2	no	—
88	C₂×D₄×C₆₁	—	4	1	no	—
89	C₂×D₄×C₆₁	—	8	4	no	—
90	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₆₁	—	2	4	no	—
92	C₂²×C₄×C₆₁	—	2	2	no	—
93	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
95	SD₁₆×C₆₁	—	4	8	no	—
96	SD₁₆×C₆₁	—	8	8	no	—
97	D₈×C₆₁	—	4	8	no	—
98	M₁₆×C₆₁	—	2	4	no	—
99	C₂×C₈×C₆₁	—	2	2	no	—
100	C₂×C₈×C₆₁	—	2	4	no	—
101	C₄⋊C₄×C₆₁	—	2	8	no	—
102	C₄⋊C₄×C₆₁	—	2	2	no	—
103	C₄⋊C₄×C₆₁	—	2	2	no	—
104	C₄⋊C₄×C₆₁	—	2	2	no	—
105	C₄⋊C₄×C₆₁	—	2	4	no	—
106	C₄⋊C₄×C₆₁	—	2	2	no	—
107	C₄⋊C₄×C₆₁	—	2	4	no	—
108	C₄⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
109	C₂²⋊C₄×C₆₁	—	2	4	no	—
110	C₂²⋊C₄×C₆₁	—	2	8	no	—
111	C₂²⋊C₄×C₆₁	—	2	4	no	—
112	C₂²⋊C₄×C₆₁	—	2	4	no	—
113	C₄²×C₆₁	—	2	4	no	—
114	C₄²×C₆₁	—	2	4	no	—
115	C₄²×C₆₁	C₂×D₄	1	2	yes	—
116	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
118	C₂³×C₆₁	—	2	2	no	—
119	C₂³×C₆₁	—	4	2	no	—
120	C₂³×C₆₁	—	8	2	no	—
121	C₂³×C₆₁	—	8	4	no	—
122	C₂³×C₆₁	#120	1	1	yes	characteristic
123	Q₈×C₆₁	—	2	8	no	—
124	Q₈×C₆₁	—	4	2	no	—
125	Q₈×C₆₁	—	4	4	no	—
126	Q₈×C₆₁	—	8	4	no	—
127	D₄×C₆₁	—	4	2	no	—
128	D₄×C₆₁	—	4	4	no	—
129	D₄×C₆₁	—	4	4	no	—
130	D₄×C₆₁	—	4	8	no	—
131	D₄×C₆₁	—	8	4	no	—
132	D₄×C₆₁	—	8	2	no	—
133	D₄×C₆₁	—	8	2	no	—
134	C₂×C₄×C₆₁	—	2	4	no	—
135	C₂×C₄×C₆₁	—	2	2	no	—
136	C₂×C₄×C₆₁	—	2	4	no	—
137	C₂×C₄×C₆₁	—	2	2	no	—
138	C₂×C₄×C₆₁	—	2	4	no	—
139	C₂×C₄×C₆₁	—	4	4	no	—
140	C₂×C₄×C₆₁	—	8	4	no	—
141	C₂×C₄×C₆₁	#120	1	2	yes	—
142	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
143	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
145	C₈×C₆₁	—	2	4	no	—
146	C₈×C₆₁	—	4	4	no	—
147	#18718	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂²×C₆₁	—	2	2	no	—
149	C₂²×C₆₁	—	2	1	no	—
150	C₂²×C₆₁	—	4	4	no	—
151	C₂²×C₆₁	—	4	2	no	—
152	C₂²×C₆₁	—	8	4	no	—
153	C₂²×C₆₁	—	8	2	no	—
154	C₂²×C₆₁	—	16	4	no	—
155	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
156	C₄×C₆₁	—	2	2	no	—
157	C₄×C₆₁	—	2	1	no	—
158	C₄×C₆₁	—	4	4	no	—
159	C₄×C₆₁	—	4	2	no	—
160	C₄×C₆₁	—	4	4	no	—
161	C₄×C₆₁	#453	1	2	yes	—
162	C₄×C₆₁	#577	1	4	yes	—
163	#3550	C₂×C₆₁	1	1	yes	characteristic, complemented
164	#3208	C₂×C₆₁	1	2	yes	complemented
165	#3151	C₂×C₆₁	1	1	yes	characteristic
166	#1623	C₂×C₆₁	1	4	yes	complemented
167	#1620	C₂×C₆₁	1	4	yes	complemented
168	#1543	C₂×C₆₁	1	1	yes	characteristic, complemented
169	#1485	C₂×C₆₁	1	2	yes	complemented
170	C₂×C₆₁	—	4	1	no	—
171	C₂×C₆₁	—	8	2	no	—
172	C₂×C₆₁	—	16	2	no	—
173	C₂×C₆₁	#3010	1	2	yes	—
174	C₂×C₆₁	C₂×#453	1	1	yes	characteristic
175	C₂×AGL(1,ℤ/8)	—	2	2	no	—
176	#563	—	2	2	no	—
177	#559	C₂²×C₆₁	1	2	yes	complemented
178	Q₈²	C₂²×C₆₁	1	2	yes	complemented
179	#556	—	2	2	no	—
180	D₄×Q₈	—	2	2	no	—
181	#542	—	2	4	no	—
182	#536	—	2	2	no	—
183	#533	C₂²×C₆₁	1	2	yes	—
184	#532	C₂²×C₆₁	1	1	yes	characteristic
185	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
186	#496	—	2	2	no	—
187	#493	C₂²×C₆₁	1	2	yes	—
188	#492	—	2	2	no	—
189	#492	C₂²×C₆₁	1	2	yes	—
190	#486	—	2	2	no	—
191	#460	C₂²×C₆₁	1	8	yes	—
192	#450	—	2	4	no	—
193	#431	C₂²×C₆₁	1	2	yes	—
194	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
195	#333	C₂²×C₆₁	1	2	yes	complemented
196	#331	C₂²×C₆₁	1	2	yes	complemented
197	Q₈⋊C₈	C₂²×C₆₁	1	8	yes	complemented
198	C₆₁	#18718	1	1	yes	characteristic, direct product factor, Sylow
199	C₂×#41	D₄×C₆₁	1	2	yes	—
200	C₂²×Q₈	—	2	2	no	—
201	C₂²×D₄	—	2	1	no	—
202	C₂²×D₄	—	2	2	no	—
203	AGL(1,ℤ/8)	—	4	8	no	—
204	C₂×SD₁₆	—	2	4	no	—
205	C₂×SD₁₆	—	2	8	no	—
206	C₂×D₈	—	2	4	no	—
207	C₂×M₁₆	—	2	1	no	—
208	C₄⋊Q₈	—	2	4	no	—
209	C₄⋊Q₈	—	2	4	no	—
210	C₄⋊Q₈	D₄×C₆₁	1	2	yes	—
211	C₄⋊Q₈	D₄×C₆₁	1	2	yes	—
212	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
213	#127	—	2	2	no	—
214	#127	—	2	1	no	—
215	#127	C₂³×C₆₁	1	2	yes	—
216	#127	C₂³×C₆₁	1	1	yes	characteristic
217	#126	—	2	8	no	—
218	#125	—	2	2	no	—
219	#124	—	2	4	no	—
220	#124	—	2	4	no	—
221	#124	D₄×C₆₁	1	2	yes	—
222	#123	—	2	4	no	—
223	#123	—	2	4	no	—
224	C₂²⋊Q₈	—	2	4	no	—
225	C₂²⋊Q₈	—	2	8	no	—
226	C₂²⋊Q₈	—	2	2	no	—
227	C₂²⋊Q₈	D₄×C₆₁	1	4	yes	—
228	#121	—	2	2	no	—
229	C₄×Q₈	—	2	4	no	—
230	C₄×Q₈	—	2	2	no	—
231	C₄×Q₈	C₂³×C₆₁	1	4	yes	—
232	C₄×D₄	—	2	4	no	—
233	C₄×D₄	—	2	2	no	—
234	#117	—	2	2	no	—
235	#117	—	2	2	no	—
236	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
237	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
238	#102	—	2	8	no	—
239	C₂²⋊C₈	—	2	2	no	—
240	#97	—	2	1	no	—
241	C₄×C₈	—	2	2	no	—
242	C₄×C₈	C₂³×C₆₁	1	2	yes	—
243	C₂⁴	—	4	1	no	—
244	#41	—	4	4	no	—
245	C₂×Q₈	—	2	4	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	8	4	no	—
248	C₂×Q₈	C₂×D₄×C₆₁	1	4	yes	—
249	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	4	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	4	4	no	—
255	C₂×D₄	—	4	2	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	8	4	no	—
258	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
259	C₂²×C₄	—	2	4	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
262	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
263	SD₁₆	—	4	8	no	—
264	SD₁₆	—	8	8	no	—
265	D₈	—	4	8	no	—
266	M₁₆	—	2	4	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	4	no	—
269	C₄⋊C₄	—	2	8	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	4	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
277	C₂²⋊C₄	—	2	4	no	—
278	C₂²⋊C₄	—	2	8	no	—
279	C₂²⋊C₄	—	2	4	no	—
280	C₂²⋊C₄	—	2	4	no	—
281	C₄²	—	2	4	no	—
282	C₄²	—	2	4	no	—
283	C₄²	C₂×D₄×C₆₁	1	2	yes	—
284	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
285	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
286	C₂³	—	2	2	no	—
287	C₂³	—	4	2	no	—
288	C₂³	—	8	2	no	—
289	C₂³	—	8	4	no	—
290	C₂³	#120×C₆₁	1	1	yes	characteristic
291	Q₈	—	2	8	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	4	4	no	—
294	Q₈	—	8	4	no	—
295	D₄	—	4	2	no	—
296	D₄	—	4	4	no	—
297	D₄	—	4	4	no	—
298	D₄	—	4	8	no	—
299	D₄	—	8	4	no	—
300	D₄	—	8	2	no	—
301	D₄	—	8	2	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	8	4	no	—
309	C₂×C₄	#120×C₆₁	1	2	yes	—
310	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
311	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
312	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
313	C₈	—	2	4	no	—
314	C₈	—	4	4	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	8	4	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	16	4	no	—
322	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	4	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	4	no	—
328	C₄	C₆₁×#453	1	2	yes	—
329	C₄	C₆₁×#577	1	4	yes	—
330	C₂	—	4	1	no	—
331	C₂	—	8	2	no	—
332	C₂	—	16	2	no	—
333	C₂	C₆₁×#3010	1	2	yes	—
334	C₂	C₂×C₆₁×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
prime radical	C₆₁×#3151	C₂
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×#3550	C₂	semidirect
2. C₆₁×#3208	C₂	semidirect
4. C₆₁×#1623	C₂	semidirect
5. C₆₁×#1620	C₂	semidirect
6. C₆₁×#1543	C₂	semidirect
7. C₆₁×#1485	C₂	semidirect
10. C₆₁×#559	C₂²	semidirect
11. Q₈²×C₆₁	C₂²	semidirect
28. C₆₁×#333	C₂²	semidirect
29. C₆₁×#331	C₂²	semidirect
30. C₆₁×Q₈⋊C₈	C₂²	semidirect
147. #18718	C₆₁	direct (N × H)
163. #3550	C₂×C₆₁	semidirect
164. #3208	C₂×C₆₁	semidirect
166. #1623	C₂×C₆₁	semidirect
167. #1620	C₂×C₆₁	semidirect
168. #1543	C₂×C₆₁	semidirect
169. #1485	C₂×C₆₁	semidirect
177. #559	C₂²×C₆₁	semidirect
178. Q₈²	C₂²×C₆₁	semidirect
195. #333	C₂²×C₆₁	semidirect
196. #331	C₂²×C₆₁	semidirect
197. Q₈⋊C₈	C₂²×C₆₁	semidirect
198. C₆₁	#18718	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×#3550	2
C₆₁×#3208	2
C₆₁×#3151	2
C₆₁×#1623	2
C₆₁×#1620	2
C₆₁×#1543	2
C₆₁×#1485	2
#18718	61

Subgroups — C61×#18718 #29522151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#18718 #29522151