Subgroups — C₆₁×#33682 #29537115

Order 15616.

580 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3176	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#2948	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#2234	C₂	1	2	yes	maximal, complemented
4	C₆₁×#2228	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2206	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2184	C₂	1	2	yes	maximal, complemented
7	C₆₁×#2177	C₂	1	2	yes	maximal, complemented
8	C₄×D₁₆×C₆₁	C₂	1	2	yes	maximal, complemented
9	C₆₁×#2164	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₆₁×#540	—	2	1	no	—
12	C₆₁×#518	—	2	1	no	—
13	C₂×D₁₆×C₆₁	C₂²	1	2	yes	—
14	C₂×D₁₆×C₆₁	C₂²	1	2	yes	—
15	C₆₁×#499	C₂²	1	1	yes	characteristic, complemented
16	C₆₁×#493	C₂²	1	1	yes	characteristic, complemented
17	C₆₁×#490	—	2	1	no	—
18	C₆₁×#469	—	2	1	no	—
19	C₆₁×#466	C₂²	1	2	yes	—
20	C₆₁×#442	—	2	1	no	—
21	C₆₁×#442	—	2	1	no	—
22	C₆₁×#442	—	2	1	no	—
23	C₄×D₈×C₆₁	C₂²	1	2	yes	—
24	C₄×D₈×C₆₁	C₂²	1	2	yes	—
25	C₄×D₈×C₆₁	C₂²	1	2	yes	complemented
26	C₆₁×#427	C₂²	1	1	yes	characteristic
27	C₆₁×#418	—	2	1	no	—
28	C₆₁×#405	C₂²	1	1	yes	characteristic
29	C₆₁×#367	C₂²	1	2	yes	complemented
30	C₆₁×#366	C₂²	1	2	yes	complemented
31	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	complemented
32	C₆₁×#357	C₂²	1	2	yes	—
33	C₆₁×#357	C₂²	1	2	yes	—
34	C₆₁×#357	C₂²	1	2	yes	complemented
35	C₆₁×#357	C₂²	1	2	yes	complemented
36	C₆₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
37	C₆₁×#346	C₂²	1	1	yes	characteristic, complemented
38	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic, complemented
39	C₂²×D₄×C₆₁	—	2	1	no	—
40	C₂×D₈×C₆₁	—	2	4	no	—
41	C₂×D₈×C₆₁	—	2	1	no	—
42	C₂×D₈×C₆₁	—	2	1	no	—
43	C₂×D₈×C₆₁	C₂³	1	2	yes	—
44	C₂×D₈×C₆₁	C₂³	1	2	yes	—
45	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
47	#127×C₆₁	—	2	1	no	—
48	#125×C₆₁	—	2	1	no	—
49	#123×C₆₁	—	4	2	no	—
50	#121×C₆₁	—	2	2	no	—
51	#121×C₆₁	—	4	2	no	—
52	C₄×D₄×C₆₁	—	2	2	no	—
53	C₄×D₄×C₆₁	—	2	2	no	—
54	C₄×D₄×C₆₁	—	2	2	no	—
55	C₄×D₄×C₆₁	—	4	2	no	—
56	#117×C₆₁	—	2	1	no	—
57	#117×C₆₁	D₄	1	1	yes	characteristic
58	C₂×C₂²⋊C₄×C₆₁	—	4	2	no	—
59	D₁₆×C₆₁	—	2	4	no	—
60	D₁₆×C₆₁	—	2	4	no	—
61	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
62	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
63	#107×C₆₁	C₂³	1	2	yes	—
64	#107×C₆₁	C₂³	1	2	yes	—
65	#106×C₆₁	—	2	1	no	—
66	#106×C₆₁	—	2	1	no	—
67	#102×C₆₁	—	2	2	no	—
68	#102×C₆₁	—	2	2	no	—
69	#102×C₆₁	—	2	2	no	—
70	#102×C₆₁	—	2	2	no	—
71	#97×C₆₁	D₄	1	1	yes	characteristic
72	#97×C₆₁	D₄	1	1	yes	characteristic
73	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
75	C₂⁴×C₆₁	—	4	1	no	—
76	C₂×D₄×C₆₁	—	2	4	no	—
77	C₂×D₄×C₆₁	—	2	2	no	—
78	C₂×D₄×C₆₁	—	2	2	no	—
79	C₂×D₄×C₆₁	—	4	2	no	—
80	C₂×D₄×C₆₁	—	4	2	no	—
81	C₂×D₄×C₆₁	—	4	1	no	—
82	C₂×D₄×C₆₁	—	4	2	no	—
83	C₂²×C₄×C₆₁	—	4	2	no	—
84	C₂²×C₄×C₆₁	—	4	2	no	—
85	C₂²×C₄×C₆₁	—	4	2	no	—
86	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
87	D₈×C₆₁	—	2	4	no	—
88	D₈×C₆₁	—	2	2	no	—
89	D₈×C₆₁	—	2	2	no	—
90	D₈×C₆₁	—	2	2	no	—
91	D₈×C₆₁	—	4	2	no	—
92	D₈×C₆₁	#41	1	4	yes	—
93	M₁₆×C₆₁	—	4	1	no	—
94	C₂×C₈×C₆₁	—	2	2	no	—
95	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₆₁	—	2	2	no	—
100	C₄⋊C₄×C₆₁	—	2	2	no	—
101	C₄⋊C₄×C₆₁	—	4	2	no	—
102	C₄⋊C₄×C₆₁	D₈	1	2	yes	—
103	C₂²⋊C₄×C₆₁	—	4	1	no	—
104	C₂²⋊C₄×C₆₁	—	4	2	no	—
105	C₂²⋊C₄×C₆₁	—	4	2	no	—
106	C₂²⋊C₄×C₆₁	—	4	2	no	—
107	C₂²⋊C₄×C₆₁	—	4	2	no	—
108	C₂²⋊C₄×C₆₁	D₈	1	2	yes	—
109	C₄²×C₆₁	—	4	1	no	—
110	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₆₁	—	2	2	no	—
113	C₁₆×C₆₁	—	2	2	no	—
114	C₂³×C₆₁	—	4	4	no	—
115	C₂³×C₆₁	—	4	2	no	—
116	C₂³×C₆₁	—	4	2	no	—
117	C₂³×C₆₁	—	16	2	no	—
118	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
119	D₄×C₆₁	—	2	4	no	—
120	D₄×C₆₁	—	4	4	no	—
121	D₄×C₆₁	—	4	2	no	—
122	D₄×C₆₁	—	4	2	no	—
123	D₄×C₆₁	—	4	2	no	—
124	D₄×C₆₁	—	16	2	no	—
125	D₄×C₆₁	—	16	2	no	—
126	C₂×C₄×C₆₁	—	2	1	no	—
127	C₂×C₄×C₆₁	—	2	1	no	—
128	C₂×C₄×C₆₁	—	4	2	no	—
129	C₂×C₄×C₆₁	—	4	2	no	—
130	C₂×C₄×C₆₁	—	16	2	no	—
131	C₂×C₄×C₆₁	—	16	2	no	—
132	C₂×C₄×C₆₁	—	16	2	no	—
133	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
137	C₈×C₆₁	—	2	1	no	—
138	C₈×C₆₁	—	2	1	no	—
139	C₈×C₆₁	—	2	1	no	—
140	C₈×C₆₁	C₂×#41	1	2	yes	—
141	#33682	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂²×C₆₁	—	2	1	no	—
143	C₂²×C₆₁	—	2	2	no	—
144	C₂²×C₆₁	—	4	4	no	—
145	C₂²×C₆₁	—	8	2	no	—
146	C₂²×C₆₁	—	8	2	no	—
147	C₂²×C₆₁	—	8	2	no	—
148	C₂²×C₆₁	—	8	2	no	—
149	C₂²×C₆₁	—	8	2	no	—
150	C₂²×C₆₁	—	16	4	no	—
151	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
152	C₄×C₆₁	—	2	1	no	—
153	C₄×C₆₁	—	2	2	no	—
154	C₄×C₆₁	—	2	2	no	—
155	C₄×C₆₁	—	8	2	no	—
156	C₄×C₆₁	—	8	2	no	—
157	C₄×C₆₁	#525	1	1	yes	characteristic
158	C₄×C₆₁	#575	1	1	yes	characteristic
159	#3176	C₂×C₆₁	1	1	yes	characteristic, complemented
160	#2948	C₂×C₆₁	1	1	yes	characteristic, complemented
161	#2234	C₂×C₆₁	1	2	yes	complemented
162	#2228	C₂×C₆₁	1	2	yes	complemented
163	#2206	C₂×C₆₁	1	2	yes	complemented
164	#2184	C₂×C₆₁	1	2	yes	complemented
165	#2177	C₂×C₆₁	1	2	yes	complemented
166	C₄×D₁₆	C₂×C₆₁	1	2	yes	complemented
167	#2164	C₂×C₆₁	1	1	yes	characteristic, complemented
168	C₂×C₆₁	—	4	1	no	—
169	C₂×C₆₁	—	8	4	no	—
170	C₂×C₆₁	—	16	2	no	—
171	C₂×C₆₁	#3100	1	1	yes	characteristic
172	C₂×C₆₁	#3415	1	2	yes	—
173	C₂²×D₈	C₂²×C₆₁	1	1	yes	characteristic
174	#540	—	2	1	no	—
175	#518	—	2	1	no	—
176	C₂×D₁₆	C₂²×C₆₁	1	2	yes	—
177	C₂×D₁₆	C₂²×C₆₁	1	2	yes	—
178	#499	C₂²×C₆₁	1	1	yes	characteristic, complemented
179	#493	C₂²×C₆₁	1	1	yes	characteristic, complemented
180	#490	—	2	1	no	—
181	#469	—	2	1	no	—
182	#466	C₂²×C₆₁	1	2	yes	—
183	#442	—	2	1	no	—
184	#442	—	2	1	no	—
185	#442	—	2	1	no	—
186	C₄×D₈	C₂²×C₆₁	1	2	yes	—
187	C₄×D₈	C₂²×C₆₁	1	2	yes	—
188	C₄×D₈	C₂²×C₆₁	1	2	yes	complemented
189	#427	C₂²×C₆₁	1	1	yes	characteristic
190	#418	—	2	1	no	—
191	#405	C₂²×C₆₁	1	1	yes	characteristic
192	#367	C₂²×C₆₁	1	2	yes	complemented
193	#366	C₂²×C₆₁	1	2	yes	complemented
194	C₄⋊C₁₆	C₂²×C₆₁	1	2	yes	complemented
195	#357	C₂²×C₆₁	1	2	yes	—
196	#357	C₂²×C₆₁	1	2	yes	—
197	#357	C₂²×C₆₁	1	2	yes	complemented
198	#357	C₂²×C₆₁	1	2	yes	complemented
199	C₂²⋊C₁₆	C₂²×C₆₁	1	2	yes	—
200	#346	C₂²×C₆₁	1	1	yes	characteristic, complemented
201	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic, complemented
202	C₆₁	#33682	1	1	yes	characteristic, direct product factor, Sylow
203	C₂²×D₄	—	2	1	no	—
204	C₂×D₈	—	2	4	no	—
205	C₂×D₈	—	2	1	no	—
206	C₂×D₈	—	2	1	no	—
207	C₂×D₈	C₂³×C₆₁	1	2	yes	—
208	C₂×D₈	C₂³×C₆₁	1	2	yes	—
209	C₂×M₁₆	D₄×C₆₁	1	1	yes	characteristic
210	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
211	#127	—	2	1	no	—
212	#125	—	2	1	no	—
213	#123	—	4	2	no	—
214	#121	—	2	2	no	—
215	#121	—	4	2	no	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	—	2	2	no	—
218	C₄×D₄	—	2	2	no	—
219	C₄×D₄	—	4	2	no	—
220	#117	—	2	1	no	—
221	#117	D₄×C₆₁	1	1	yes	characteristic
222	C₂×C₂²⋊C₄	—	4	2	no	—
223	D₁₆	—	2	4	no	—
224	D₁₆	—	2	4	no	—
225	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
226	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
227	#107	C₂³×C₆₁	1	2	yes	—
228	#107	C₂³×C₆₁	1	2	yes	—
229	#106	—	2	1	no	—
230	#106	—	2	1	no	—
231	#102	—	2	2	no	—
232	#102	—	2	2	no	—
233	#102	—	2	2	no	—
234	#102	—	2	2	no	—
235	#97	D₄×C₆₁	1	1	yes	characteristic
236	#97	D₄×C₆₁	1	1	yes	characteristic
237	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
238	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
239	C₂⁴	—	4	1	no	—
240	C₂×D₄	—	2	4	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	—	4	2	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
251	D₈	—	2	4	no	—
252	D₈	—	2	2	no	—
253	D₈	—	2	2	no	—
254	D₈	—	2	2	no	—
255	D₈	—	4	2	no	—
256	D₈	#41×C₆₁	1	4	yes	—
257	M₁₆	—	4	1	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
262	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	D₈×C₆₁	1	2	yes	—
267	C₂²⋊C₄	—	4	1	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	D₈×C₆₁	1	2	yes	—
273	C₄²	—	4	1	no	—
274	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
275	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
276	C₁₆	—	2	2	no	—
277	C₁₆	—	2	2	no	—
278	C₂³	—	4	4	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	4	2	no	—
281	C₂³	—	16	2	no	—
282	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
283	D₄	—	2	4	no	—
284	D₄	—	4	4	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	16	2	no	—
289	D₄	—	16	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	16	2	no	—
295	C₂×C₄	—	16	2	no	—
296	C₂×C₄	—	16	2	no	—
297	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
298	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
299	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
300	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	C₂×#41×C₆₁	1	2	yes	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	4	4	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	8	2	no	—
310	C₂²	—	8	2	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	16	4	no	—
314	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	C₆₁×#525	1	1	yes	characteristic
321	C₄	C₆₁×#575	1	1	yes	characteristic
322	C₂	—	4	1	no	—
323	C₂	—	8	4	no	—
324	C₂	—	16	2	no	—
325	C₂	C₆₁×#3100	1	1	yes	characteristic
326	C₂	C₆₁×#3415	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ #117×C₆₁ ◁ C₆₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3176	C₂	semidirect
2. C₆₁×#2948	C₂	semidirect
3. C₆₁×#2234	C₂	semidirect
4. C₆₁×#2228	C₂	semidirect
5. C₆₁×#2206	C₂	semidirect
6. C₆₁×#2184	C₂	semidirect
7. C₆₁×#2177	C₂	semidirect
8. C₄×D₁₆×C₆₁	C₂	semidirect
9. C₆₁×#2164	C₂	semidirect
15. C₆₁×#499	C₂²	semidirect
16. C₆₁×#493	C₂²	semidirect
25. C₄×D₈×C₆₁	C₂²	semidirect
29. C₆₁×#367	C₂²	semidirect
30. C₆₁×#366	C₂²	semidirect
31. C₆₁×C₄⋊C₁₆	C₂²	semidirect
34. C₆₁×#357	C₂²	semidirect
35. C₆₁×#357	C₂²	semidirect
37. C₆₁×#346	C₂²	semidirect
38. C₄×C₁₆×C₆₁	C₂²	semidirect
141. #33682	C₆₁	direct (N × H)
159. #3176	C₂×C₆₁	semidirect
160. #2948	C₂×C₆₁	semidirect
161. #2234	C₂×C₆₁	semidirect
162. #2228	C₂×C₆₁	semidirect
163. #2206	C₂×C₆₁	semidirect
164. #2184	C₂×C₆₁	semidirect
165. #2177	C₂×C₆₁	semidirect
166. C₄×D₁₆	C₂×C₆₁	semidirect
167. #2164	C₂×C₆₁	semidirect
178. #499	C₂²×C₆₁	semidirect
179. #493	C₂²×C₆₁	semidirect
188. C₄×D₈	C₂²×C₆₁	semidirect
192. #367	C₂²×C₆₁	semidirect
193. #366	C₂²×C₆₁	semidirect
194. C₄⋊C₁₆	C₂²×C₆₁	semidirect
197. #357	C₂²×C₆₁	semidirect
198. #357	C₂²×C₆₁	semidirect
200. #346	C₂²×C₆₁	semidirect
201. C₄×C₁₆	C₂²×C₆₁	semidirect
202. C₆₁	#33682	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×#3176	2
C₆₁×#2948	2
C₆₁×#2234	2
C₆₁×#2228	2
C₆₁×#2206	2
C₆₁×#2184	2
C₆₁×#2177	2
C₄×D₁₆×C₆₁	2
C₆₁×#2164	2
#33682	61

Subgroups — C61×#33682 #29537115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33682 #29537115