Subgroups — C₆₁×#33739 #29537172

Order 15616.

508 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3175	C₂	1	2	yes	maximal, complemented
2	C₆₁×#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#2249	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2243	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2220	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2216	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2215	C₂	1	1	yes	maximal, characteristic
8	C₆₁×#2175	C₂	1	2	yes	maximal, complemented
9	C₄×Q₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#542	—	2	2	no	—
13	C₂×Q₃₂×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×SD₃₂×C₆₁	C₂²	1	2	yes	—
15	C₂×D₁₆×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#500	C₂²	1	2	yes	—
19	C₆₁×#495	C₂²	1	2	yes	—
20	C₆₁×#489	—	2	2	no	—
21	C₆₁×#468	—	2	2	no	—
22	C₆₁×#467	C₂²	1	2	yes	—
23	C₆₁×#466	C₂²	1	2	yes	—
24	C₆₁×#440	—	2	2	no	—
25	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
26	C₄×D₈×C₆₁	C₂²	1	2	yes	—
27	C₆₁×#405	C₂²	1	1	yes	characteristic
28	C₆₁×#366	C₂²	1	1	yes	characteristic
29	C₆₁×#366	C₂²	1	1	yes	characteristic
30	C₆₁×#366	C₂²	1	2	yes	—
31	C₆₁×#358	C₂²	1	2	yes	—
32	C₆₁×#358	C₂²	1	2	yes	—
33	C₆₁×#357	C₂²	1	2	yes	—
34	C₆₁×#357	C₂²	1	2	yes	—
35	C₆₁×#346	C₂²	1	1	yes	characteristic
36	C₆₁×#346	C₂²	1	1	yes	characteristic
37	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
40	C₂×SD₁₆×C₆₁	—	2	2	no	—
41	C₂×D₈×C₆₁	C₂³	1	2	yes	—
42	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
44	C₄⋊Q₈×C₆₁	—	2	2	no	—
45	#126×C₆₁	—	4	2	no	—
46	#124×C₆₁	—	2	2	no	—
47	#124×C₆₁	—	4	2	no	—
48	#123×C₆₁	—	4	2	no	—
49	C₂²⋊Q₈×C₆₁	—	2	2	no	—
50	C₂²⋊Q₈×C₆₁	—	4	2	no	—
51	#121×C₆₁	—	2	2	no	—
52	C₄×Q₈×C₆₁	—	2	2	no	—
53	C₄×D₄×C₆₁	—	2	2	no	—
54	#117×C₆₁	D₄	1	1	yes	characteristic
55	Q₃₂×C₆₁	—	2	2	no	—
56	SD₃₂×C₆₁	—	2	4	no	—
57	D₁₆×C₆₁	—	2	2	no	—
58	M₃₂×C₆₁	—	2	2	no	—
59	C₂×C₁₆×C₆₁	—	2	2	no	—
60	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
61	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
64	#107×C₆₁	—	2	2	no	—
65	#107×C₆₁	C₂³	1	2	yes	—
66	#107×C₆₁	C₂³	1	2	yes	—
67	#103×C₆₁	—	2	2	no	—
68	#103×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	C₂×Q₈×C₆₁	—	2	2	no	—
76	C₂×Q₈×C₆₁	—	4	2	no	—
77	C₂×D₄×C₆₁	—	2	2	no	—
78	C₂×D₄×C₆₁	—	4	2	no	—
79	C₂²×C₄×C₆₁	—	4	2	no	—
80	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
81	Q₁₆×C₆₁	—	2	2	no	—
82	Q₁₆×C₆₁	—	2	2	no	—
83	SD₁₆×C₆₁	—	4	4	no	—
84	D₈×C₆₁	—	2	2	no	—
85	D₈×C₆₁	—	2	2	no	—
86	M₁₆×C₆₁	—	2	2	no	—
87	C₂×C₈×C₆₁	—	2	1	no	—
88	C₂×C₈×C₆₁	—	2	1	no	—
89	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₆₁	—	2	2	no	—
94	C₄⋊C₄×C₆₁	—	2	2	no	—
95	C₄⋊C₄×C₆₁	—	4	2	no	—
96	C₄⋊C₄×C₆₁	—	4	2	no	—
97	C₄⋊C₄×C₆₁	—	4	2	no	—
98	C₄⋊C₄×C₆₁	D₈	1	1	yes	characteristic
99	C₄⋊C₄×C₆₁	D₈	1	1	yes	characteristic
100	C₂²⋊C₄×C₆₁	—	4	2	no	—
101	C₂²⋊C₄×C₆₁	—	4	2	no	—
102	C₂²⋊C₄×C₆₁	—	4	2	no	—
103	C₂²⋊C₄×C₆₁	—	4	2	no	—
104	C₂²⋊C₄×C₆₁	—	4	2	no	—
105	C₂²⋊C₄×C₆₁	D₈	1	1	yes	characteristic
106	C₂²⋊C₄×C₆₁	D₈	1	1	yes	characteristic
107	C₄²×C₆₁	—	4	2	no	—
108	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
110	C₁₆×C₆₁	—	2	1	no	—
111	C₁₆×C₆₁	—	2	1	no	—
112	C₁₆×C₆₁	#41	1	2	yes	—
113	C₁₆×C₆₁	#41	1	2	yes	—
114	C₂³×C₆₁	—	4	2	no	—
115	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
116	Q₈×C₆₁	—	4	2	no	—
117	Q₈×C₆₁	—	4	2	no	—
118	Q₈×C₆₁	—	16	2	no	—
119	D₄×C₆₁	—	4	2	no	—
120	D₄×C₆₁	—	4	2	no	—
121	D₄×C₆₁	—	16	2	no	—
122	C₂×C₄×C₆₁	—	2	1	no	—
123	C₂×C₄×C₆₁	—	2	1	no	—
124	C₂×C₄×C₆₁	—	4	2	no	—
125	C₂×C₄×C₆₁	—	4	2	no	—
126	C₂×C₄×C₆₁	—	4	2	no	—
127	C₂×C₄×C₆₁	—	16	2	no	—
128	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
134	C₈×C₆₁	—	2	1	no	—
135	C₈×C₆₁	—	2	1	no	—
136	C₈×C₆₁	#135	1	2	yes	—
137	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
138	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
139	#33739	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂²×C₆₁	—	2	1	no	—
141	C₂²×C₆₁	—	2	1	no	—
142	C₂²×C₆₁	—	2	1	no	—
143	C₂²×C₆₁	—	8	2	no	—
144	C₂²×C₆₁	—	8	2	no	—
145	C₂²×C₆₁	—	8	2	no	—
146	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
147	C₄×C₆₁	—	2	1	no	—
148	C₄×C₆₁	—	2	1	no	—
149	C₄×C₆₁	—	2	1	no	—
150	C₄×C₆₁	—	2	1	no	—
151	C₄×C₆₁	—	2	1	no	—
152	C₄×C₆₁	—	8	2	no	—
153	C₄×C₆₁	—	8	2	no	—
154	C₄×C₆₁	—	8	2	no	—
155	C₄×C₆₁	#532	1	1	yes	characteristic
156	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
157	#3175	C₂×C₆₁	1	2	yes	complemented
158	#2252	C₂×C₆₁	1	1	yes	characteristic, complemented
159	#2249	C₂×C₆₁	1	1	yes	characteristic, complemented
160	#2243	C₂×C₆₁	1	2	yes	complemented
161	#2220	C₂×C₆₁	1	2	yes	complemented
162	#2216	C₂×C₆₁	1	1	yes	characteristic, complemented
163	#2215	C₂×C₆₁	1	1	yes	characteristic
164	#2175	C₂×C₆₁	1	2	yes	complemented
165	C₄×Q₃₂	C₂×C₆₁	1	1	yes	characteristic, complemented
166	C₄×D₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
167	#2108	C₂×C₆₁	1	1	yes	characteristic, complemented
168	C₂×C₆₁	—	4	1	no	—
169	C₂×C₆₁	—	16	2	no	—
170	C₂×C₆₁	#3169	1	1	yes	characteristic
171	C₂×C₆₁	#3415	1	1	yes	characteristic
172	C₂×C₆₁	#3417	1	1	yes	characteristic
173	#542	—	2	2	no	—
174	C₂×Q₃₂	C₂²×C₆₁	1	1	yes	characteristic
175	C₂×SD₃₂	C₂²×C₆₁	1	2	yes	—
176	C₂×D₁₆	C₂²×C₆₁	1	1	yes	characteristic
177	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
178	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
179	#500	C₂²×C₆₁	1	2	yes	—
180	#495	C₂²×C₆₁	1	2	yes	—
181	#489	—	2	2	no	—
182	#468	—	2	2	no	—
183	#467	C₂²×C₆₁	1	2	yes	—
184	#466	C₂²×C₆₁	1	2	yes	—
185	#440	—	2	2	no	—
186	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
187	C₄×D₈	C₂²×C₆₁	1	2	yes	—
188	#405	C₂²×C₆₁	1	1	yes	characteristic
189	#366	C₂²×C₆₁	1	1	yes	characteristic
190	#366	C₂²×C₆₁	1	1	yes	characteristic
191	#366	C₂²×C₆₁	1	2	yes	—
192	#358	C₂²×C₆₁	1	2	yes	—
193	#358	C₂²×C₆₁	1	2	yes	—
194	#357	C₂²×C₆₁	1	2	yes	—
195	#357	C₂²×C₆₁	1	2	yes	—
196	#346	C₂²×C₆₁	1	1	yes	characteristic
197	#346	C₂²×C₆₁	1	1	yes	characteristic
198	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
199	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
200	C₆₁	#33739	1	1	yes	characteristic, direct product factor, Sylow
201	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
202	C₂×SD₁₆	—	2	2	no	—
203	C₂×D₈	C₂³×C₆₁	1	2	yes	—
204	C₂×M₁₆	D₄×C₆₁	1	1	yes	characteristic
205	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
206	C₄⋊Q₈	—	2	2	no	—
207	#126	—	4	2	no	—
208	#124	—	2	2	no	—
209	#124	—	4	2	no	—
210	#123	—	4	2	no	—
211	C₂²⋊Q₈	—	2	2	no	—
212	C₂²⋊Q₈	—	4	2	no	—
213	#121	—	2	2	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×D₄	—	2	2	no	—
216	#117	D₄×C₆₁	1	1	yes	characteristic
217	Q₃₂	—	2	2	no	—
218	SD₃₂	—	2	4	no	—
219	D₁₆	—	2	2	no	—
220	M₃₂	—	2	2	no	—
221	C₂×C₁₆	—	2	2	no	—
222	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
223	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
224	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
225	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
226	#107	—	2	2	no	—
227	#107	C₂³×C₆₁	1	2	yes	—
228	#107	C₂³×C₆₁	1	2	yes	—
229	#103	—	2	2	no	—
230	#103	—	2	2	no	—
231	#102	—	2	2	no	—
232	#102	—	2	2	no	—
233	#97	D₄×C₆₁	1	1	yes	characteristic
234	#97	D₄×C₆₁	1	1	yes	characteristic
235	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
236	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	4	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
243	Q₁₆	—	2	2	no	—
244	Q₁₆	—	2	2	no	—
245	SD₁₆	—	4	4	no	—
246	D₈	—	2	2	no	—
247	D₈	—	2	2	no	—
248	M₁₆	—	2	2	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
252	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
253	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
254	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	D₈×C₆₁	1	1	yes	characteristic
261	C₄⋊C₄	D₈×C₆₁	1	1	yes	characteristic
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₂²⋊C₄	—	4	2	no	—
267	C₂²⋊C₄	D₈×C₆₁	1	1	yes	characteristic
268	C₂²⋊C₄	D₈×C₆₁	1	1	yes	characteristic
269	C₄²	—	4	2	no	—
270	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
271	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
272	C₁₆	—	2	1	no	—
273	C₁₆	—	2	1	no	—
274	C₁₆	#41×C₆₁	1	2	yes	—
275	C₁₆	#41×C₆₁	1	2	yes	—
276	C₂³	—	4	2	no	—
277	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
278	Q₈	—	4	2	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	16	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	16	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	16	2	no	—
290	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
291	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
292	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
293	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
294	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
295	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	#135×C₆₁	1	2	yes	—
299	C₈	C₂×#41×C₆₁	1	1	yes	characteristic
300	C₈	C₂×#41×C₆₁	1	1	yes	characteristic
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	C₆₁×#532	1	1	yes	characteristic
317	C₄	C₂×#135×C₆₁	1	1	yes	characteristic
318	C₂	—	4	1	no	—
319	C₂	—	16	2	no	—
320	C₂	C₆₁×#3169	1	1	yes	characteristic
321	C₂	C₆₁×#3415	1	1	yes	characteristic
322	C₂	C₆₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
prime radical	C₆₁×#2215	C₂
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×#3175	C₂	semidirect
2. C₆₁×#2252	C₂	semidirect
3. C₆₁×#2249	C₂	semidirect
4. C₆₁×#2243	C₂	semidirect
5. C₆₁×#2220	C₂	semidirect
6. C₆₁×#2216	C₂	semidirect
8. C₆₁×#2175	C₂	semidirect
9. C₄×Q₃₂×C₆₁	C₂	semidirect
10. C₄×D₁₆×C₆₁	C₂	semidirect
11. C₆₁×#2108	C₂	semidirect
139. #33739	C₆₁	direct (N × H)
157. #3175	C₂×C₆₁	semidirect
158. #2252	C₂×C₆₁	semidirect
159. #2249	C₂×C₆₁	semidirect
160. #2243	C₂×C₆₁	semidirect
161. #2220	C₂×C₆₁	semidirect
162. #2216	C₂×C₆₁	semidirect
164. #2175	C₂×C₆₁	semidirect
165. C₄×Q₃₂	C₂×C₆₁	semidirect
166. C₄×D₁₆	C₂×C₆₁	semidirect
167. #2108	C₂×C₆₁	semidirect
200. C₆₁	#33739	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×#3175	2
C₆₁×#2252	2
C₆₁×#2249	2
C₆₁×#2243	2
C₆₁×#2220	2
C₆₁×#2216	2
C₆₁×#2215	2
C₆₁×#2175	2
C₄×Q₃₂×C₆₁	2
C₄×D₁₆×C₆₁	2
C₆₁×#2108	2
#33739	61

Subgroups — C61×#33739 #29537172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33739 #29537172