Subgroups — C₆₁×#33871 #29537304

Order 15616.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3342	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3287	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2240	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2234	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2225	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#2217	C₂	1	2	yes	maximal, complemented
7	C₆₁×#2207	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#2189	C₂	1	2	yes	maximal, complemented
9	C₄×Q₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#2138	C₂	1	2	yes	maximal, complemented
12	C₂×#135×C₆₁	C₂²	1	2	yes	—
13	C₆₁×#556	—	2	1	no	—
14	C₆₁×#550	—	2	1	no	—
15	C₂×Q₃₂×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×SD₃₂×C₆₁	C₂²	1	2	yes	—
17	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
18	C₆₁×#500	C₂²	1	1	yes	characteristic
19	C₆₁×#493	C₂²	1	1	yes	characteristic
20	C₆₁×#485	—	2	2	no	—
21	C₆₁×#475	—	2	1	no	—
22	C₆₁×#467	C₂²	1	2	yes	—
23	C₆₁×#466	C₂²	1	2	yes	—
24	C₆₁×#460	—	2	1	no	—
25	C₆₁×#449	—	2	2	no	—
26	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
27	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
28	C₈×D₄×C₆₁	C₂²	1	1	yes	characteristic
29	C₆₁×#427	C₂²	1	2	yes	—
30	C₆₁×#367	C₂²	1	2	yes	—
31	C₆₁×#366	C₂²	1	1	yes	characteristic
32	C₆₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₆₁×#358	C₂²	1	1	yes	characteristic
34	C₆₁×#358	C₂²	1	2	yes	—
35	C₆₁×#358	C₂²	1	1	yes	characteristic
36	C₆₁×#357	C₂²	1	2	yes	—
37	C₆₁×#357	C₂²	1	2	yes	—
38	C₆₁×#357	C₂²	1	2	yes	—
39	C₆₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
41	C₂×#41×C₆₁	—	2	2	no	—
42	#135×C₆₁	—	2	4	no	—
43	#135×C₆₁	—	4	2	no	—
44	C₂×Q₁₆×C₆₁	C₂³	1	1	yes	characteristic
45	C₂×Q₁₆×C₆₁	C₂³	1	1	yes	characteristic
46	C₂×SD₁₆×C₆₁	—	2	2	no	—
47	C₂×D₈×C₆₁	—	2	1	no	—
48	C₂×D₈×C₆₁	C₂³	1	2	yes	—
49	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₆₁	—	2	1	no	—
51	#127×C₆₁	—	2	1	no	—
52	#127×C₆₁	—	4	1	no	—
53	#126×C₆₁	—	4	2	no	—
54	#124×C₆₁	—	4	1	no	—
55	C₂²⋊Q₈×C₆₁	—	2	2	no	—
56	#121×C₆₁	—	2	2	no	—
57	#121×C₆₁	—	4	2	no	—
58	C₄×Q₈×C₆₁	—	2	1	no	—
59	C₄×Q₈×C₆₁	—	2	1	no	—
60	C₄×Q₈×C₆₁	—	4	1	no	—
61	C₄×D₄×C₆₁	—	4	1	no	—
62	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
63	#117×C₆₁	—	2	2	no	—
64	Q₃₂×C₆₁	—	2	2	no	—
65	SD₃₂×C₆₁	—	4	2	no	—
66	C₂×C₁₆×C₆₁	—	4	2	no	—
67	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
69	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
70	#107×C₆₁	—	2	1	no	—
71	#107×C₆₁	C₂³	1	1	yes	characteristic
72	#107×C₆₁	C₂³	1	2	yes	—
73	#107×C₆₁	C₂³	1	1	yes	characteristic
74	#106×C₆₁	—	2	2	no	—
75	C₄⋊C₈×C₆₁	D₄	1	1	yes	characteristic
76	#103×C₆₁	—	2	1	no	—
77	#103×C₆₁	—	2	1	no	—
78	#103×C₆₁	—	2	2	no	—
79	#102×C₆₁	—	2	2	no	—
80	C₂²⋊C₈×C₆₁	D₄	1	2	yes	—
81	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
82	#41×C₆₁	—	4	4	no	—
83	#41×C₆₁	—	8	2	no	—
84	C₂×Q₈×C₆₁	—	2	1	no	—
85	C₂×Q₈×C₆₁	—	2	1	no	—
86	C₂×Q₈×C₆₁	—	4	1	no	—
87	C₂×D₄×C₆₁	—	2	2	no	—
88	C₂×D₄×C₆₁	—	4	2	no	—
89	C₂×D₄×C₆₁	—	4	2	no	—
90	C₂×D₄×C₆₁	—	4	1	no	—
91	C₂×D₄×C₆₁	—	4	1	no	—
92	C₂×D₄×C₆₁	D₈	1	1	yes	characteristic
93	C₂²×C₄×C₆₁	—	4	2	no	—
94	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
95	Q₁₆×C₆₁	—	2	1	no	—
96	Q₁₆×C₆₁	—	2	1	no	—
97	Q₁₆×C₆₁	—	2	1	no	—
98	Q₁₆×C₆₁	#41	1	2	yes	—
99	SD₁₆×C₆₁	—	4	4	no	—
100	D₈×C₆₁	—	2	2	no	—
101	D₈×C₆₁	—	2	2	no	—
102	D₈×C₆₁	—	8	1	no	—
103	C₂×C₈×C₆₁	—	2	2	no	—
104	C₂×C₈×C₆₁	—	2	2	no	—
105	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄×C₆₁	—	2	1	no	—
109	C₄⋊C₄×C₆₁	—	2	1	no	—
110	C₄⋊C₄×C₆₁	—	2	2	no	—
111	C₄⋊C₄×C₆₁	—	4	1	no	—
112	C₄⋊C₄×C₆₁	—	4	1	no	—
113	C₄⋊C₄×C₆₁	—	4	2	no	—
114	C₄⋊C₄×C₆₁	—	4	1	no	—
115	C₄⋊C₄×C₆₁	D₈	1	1	yes	characteristic
116	C₂²⋊C₄×C₆₁	—	4	2	no	—
117	C₂²⋊C₄×C₆₁	—	4	2	no	—
118	C₂²⋊C₄×C₆₁	—	4	2	no	—
119	C₂²⋊C₄×C₆₁	D₈	1	2	yes	—
120	C₄²×C₆₁	—	4	2	no	—
121	C₄²×C₆₁	—	4	1	no	—
122	C₄²×C₆₁	—	4	1	no	—
123	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
124	C₁₆×C₆₁	—	2	1	no	—
125	C₁₆×C₆₁	—	2	1	no	—
126	C₁₆×C₆₁	—	2	2	no	—
127	C₂³×C₆₁	—	4	2	no	—
128	C₂³×C₆₁	C₂×D₈	1	2	yes	—
129	Q₈×C₆₁	—	2	2	no	—
130	Q₈×C₆₁	—	4	1	no	—
131	Q₈×C₆₁	—	4	1	no	—
132	Q₈×C₆₁	—	4	1	no	—
133	Q₈×C₆₁	—	8	2	no	—
134	D₄×C₆₁	—	2	2	no	—
135	D₄×C₆₁	—	4	2	no	—
136	D₄×C₆₁	—	4	2	no	—
137	D₄×C₆₁	—	8	4	no	—
138	D₄×C₆₁	—	8	2	no	—
139	D₄×C₆₁	—	16	1	no	—
140	D₄×C₆₁	—	16	2	no	—
141	C₂×C₄×C₆₁	—	2	2	no	—
142	C₂×C₄×C₆₁	—	2	2	no	—
143	C₂×C₄×C₆₁	—	4	1	no	—
144	C₂×C₄×C₆₁	—	4	1	no	—
145	C₂×C₄×C₆₁	—	4	2	no	—
146	C₂×C₄×C₆₁	—	4	1	no	—
147	C₂×C₄×C₆₁	—	4	1	no	—
148	C₂×C₄×C₆₁	—	8	4	no	—
149	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
150	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
153	C₈×C₆₁	—	2	2	no	—
154	C₈×C₆₁	—	2	1	no	—
155	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
156	C₈×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
157	#33871	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂²×C₆₁	—	2	2	no	—
159	C₂²×C₆₁	—	2	2	no	—
160	C₂²×C₆₁	—	2	2	no	—
161	C₂²×C₆₁	—	8	2	no	—
162	C₂²×C₆₁	—	8	2	no	—
163	C₂²×C₆₁	—	8	2	no	—
164	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
165	C₄×C₆₁	—	2	1	no	—
166	C₄×C₆₁	—	2	2	no	—
167	C₄×C₆₁	—	4	2	no	—
168	C₄×C₆₁	—	8	2	no	—
169	C₄×C₆₁	—	8	1	no	—
170	C₄×C₆₁	—	8	1	no	—
171	C₄×C₆₁	—	8	1	no	—
172	C₄×C₆₁	#508	1	2	yes	—
173	C₄×C₆₁	#546	1	1	yes	characteristic
174	C₄×C₆₁	#577	1	1	yes	characteristic
175	#3342	C₂×C₆₁	1	1	yes	characteristic, complemented
176	#3287	C₂×C₆₁	1	1	yes	characteristic
177	#2240	C₂×C₆₁	1	1	yes	characteristic, complemented
178	#2234	C₂×C₆₁	1	2	yes	complemented
179	#2225	C₂×C₆₁	1	1	yes	characteristic, complemented
180	#2217	C₂×C₆₁	1	2	yes	complemented
181	#2207	C₂×C₆₁	1	1	yes	characteristic, complemented
182	#2189	C₂×C₆₁	1	2	yes	complemented
183	C₄×Q₃₂	C₂×C₆₁	1	1	yes	characteristic, complemented
184	D₄×C₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
185	#2138	C₂×C₆₁	1	2	yes	complemented
186	C₂×C₆₁	—	4	2	no	—
187	C₂×C₆₁	—	16	2	no	—
188	C₂×C₆₁	#3294	1	1	yes	characteristic
189	C₂×C₆₁	C₂×#508	1	1	yes	characteristic
190	C₂×C₆₁	#3416	1	1	yes	characteristic
191	C₂×#135	C₂²×C₆₁	1	2	yes	—
192	#556	—	2	1	no	—
193	#550	—	2	1	no	—
194	C₂×Q₃₂	C₂²×C₆₁	1	1	yes	characteristic
195	C₂×SD₃₂	C₂²×C₆₁	1	2	yes	—
196	C₂²×C₁₆	C₂²×C₆₁	1	2	yes	—
197	#500	C₂²×C₆₁	1	1	yes	characteristic
198	#493	C₂²×C₆₁	1	1	yes	characteristic
199	#485	—	2	2	no	—
200	#475	—	2	1	no	—
201	#467	C₂²×C₆₁	1	2	yes	—
202	#466	C₂²×C₆₁	1	2	yes	—
203	#460	—	2	1	no	—
204	#449	—	2	2	no	—
205	C₄×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
206	C₄×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
207	C₈×D₄	C₂²×C₆₁	1	1	yes	characteristic
208	#427	C₂²×C₆₁	1	2	yes	—
209	#367	C₂²×C₆₁	1	2	yes	—
210	#366	C₂²×C₆₁	1	1	yes	characteristic
211	C₄⋊C₁₆	C₂²×C₆₁	1	1	yes	characteristic
212	#358	C₂²×C₆₁	1	1	yes	characteristic
213	#358	C₂²×C₆₁	1	2	yes	—
214	#358	C₂²×C₆₁	1	1	yes	characteristic
215	#357	C₂²×C₆₁	1	2	yes	—
216	#357	C₂²×C₆₁	1	2	yes	—
217	#357	C₂²×C₆₁	1	2	yes	—
218	C₂²⋊C₁₆	C₂²×C₆₁	1	2	yes	—
219	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
220	C₆₁	#33871	1	1	yes	characteristic, direct product factor, Sylow
221	C₂×#41	—	2	2	no	—
222	#135	—	2	4	no	—
223	#135	—	4	2	no	—
224	C₂×Q₁₆	C₂³×C₆₁	1	1	yes	characteristic
225	C₂×Q₁₆	C₂³×C₆₁	1	1	yes	characteristic
226	C₂×SD₁₆	—	2	2	no	—
227	C₂×D₈	—	2	1	no	—
228	C₂×D₈	C₂³×C₆₁	1	2	yes	—
229	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
230	C₄⋊Q₈	—	2	1	no	—
231	#127	—	2	1	no	—
232	#127	—	4	1	no	—
233	#126	—	4	2	no	—
234	#124	—	4	1	no	—
235	C₂²⋊Q₈	—	2	2	no	—
236	#121	—	2	2	no	—
237	#121	—	4	2	no	—
238	C₄×Q₈	—	2	1	no	—
239	C₄×Q₈	—	2	1	no	—
240	C₄×Q₈	—	4	1	no	—
241	C₄×D₄	—	4	1	no	—
242	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
243	#117	—	2	2	no	—
244	Q₃₂	—	2	2	no	—
245	SD₃₂	—	4	2	no	—
246	C₂×C₁₆	—	4	2	no	—
247	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
248	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
249	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
250	#107	—	2	1	no	—
251	#107	C₂³×C₆₁	1	1	yes	characteristic
252	#107	C₂³×C₆₁	1	2	yes	—
253	#107	C₂³×C₆₁	1	1	yes	characteristic
254	#106	—	2	2	no	—
255	C₄⋊C₈	D₄×C₆₁	1	1	yes	characteristic
256	#103	—	2	1	no	—
257	#103	—	2	1	no	—
258	#103	—	2	2	no	—
259	#102	—	2	2	no	—
260	C₂²⋊C₈	D₄×C₆₁	1	2	yes	—
261	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
262	#41	—	4	4	no	—
263	#41	—	8	2	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×Q₈	—	4	1	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	4	2	no	—
269	C₂×D₄	—	4	2	no	—
270	C₂×D₄	—	4	1	no	—
271	C₂×D₄	—	4	1	no	—
272	C₂×D₄	D₈×C₆₁	1	1	yes	characteristic
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
275	Q₁₆	—	2	1	no	—
276	Q₁₆	—	2	1	no	—
277	Q₁₆	—	2	1	no	—
278	Q₁₆	#41×C₆₁	1	2	yes	—
279	SD₁₆	—	4	4	no	—
280	D₈	—	2	2	no	—
281	D₈	—	2	2	no	—
282	D₈	—	8	1	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
286	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
287	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	D₈×C₆₁	1	1	yes	characteristic
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	4	2	no	—
299	C₂²⋊C₄	D₈×C₆₁	1	2	yes	—
300	C₄²	—	4	2	no	—
301	C₄²	—	4	1	no	—
302	C₄²	—	4	1	no	—
303	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
304	C₁₆	—	2	1	no	—
305	C₁₆	—	2	1	no	—
306	C₁₆	—	2	2	no	—
307	C₂³	—	4	2	no	—
308	C₂³	C₂×D₈×C₆₁	1	2	yes	—
309	Q₈	—	2	2	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	4	1	no	—
312	Q₈	—	4	1	no	—
313	Q₈	—	8	2	no	—
314	D₄	—	2	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	8	4	no	—
318	D₄	—	8	2	no	—
319	D₄	—	16	1	no	—
320	D₄	—	16	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	8	4	no	—
329	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
330	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
331	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
332	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	—	2	1	no	—
335	C₈	C₂×#41×C₆₁	1	1	yes	characteristic
336	C₈	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	2	1	no	—
346	C₄	—	4	2	no	—
347	C₄	—	8	1	no	—
348	C₄	—	8	2	no	—
349	C₄	—	8	1	no	—
350	C₄	—	8	1	no	—
351	C₄	C₆₁×#508	1	2	yes	—
352	C₄	C₆₁×#546	1	1	yes	characteristic
353	C₄	C₆₁×#577	1	1	yes	characteristic
354	C₂	—	4	2	no	—
355	C₂	—	16	2	no	—
356	C₂	C₆₁×#3294	1	1	yes	characteristic
357	C₂	C₂×C₆₁×#508	1	1	yes	characteristic
358	C₂	C₆₁×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
prime radical	C₆₁×#3287	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄×D₄×C₆₁ ◁ C₈×D₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3342	C₂	semidirect
3. C₆₁×#2240	C₂	semidirect
4. C₆₁×#2234	C₂	semidirect
5. C₆₁×#2225	C₂	semidirect
6. C₆₁×#2217	C₂	semidirect
7. C₆₁×#2207	C₂	semidirect
8. C₆₁×#2189	C₂	semidirect
9. C₄×Q₃₂×C₆₁	C₂	semidirect
10. D₄×C₁₆×C₆₁	C₂	semidirect
11. C₆₁×#2138	C₂	semidirect
157. #33871	C₆₁	direct (N × H)
175. #3342	C₂×C₆₁	semidirect
177. #2240	C₂×C₆₁	semidirect
178. #2234	C₂×C₆₁	semidirect
179. #2225	C₂×C₆₁	semidirect
180. #2217	C₂×C₆₁	semidirect
181. #2207	C₂×C₆₁	semidirect
182. #2189	C₂×C₆₁	semidirect
183. C₄×Q₃₂	C₂×C₆₁	semidirect
184. D₄×C₁₆	C₂×C₆₁	semidirect
185. #2138	C₂×C₆₁	semidirect
220. C₆₁	#33871	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×#3342	2
C₆₁×#3287	2
C₆₁×#2240	2
C₆₁×#2234	2
C₆₁×#2225	2
C₆₁×#2217	2
C₆₁×#2207	2
C₆₁×#2189	2
C₄×Q₃₂×C₆₁	2
D₄×C₁₆×C₆₁	2
C₆₁×#2138	2
#33871	61

Subgroups — C61×#33871 #29537304

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33871 #29537304