Subgroups — C₆₁×#33763 #29537196

Order 15616.

592 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3050	C₂	1	2	yes	maximal, complemented
2	C₆₁×#2237	C₂	1	2	yes	maximal, complemented
3	C₆₁×#2234	C₂	1	2	yes	maximal, complemented
4	C₆₁×#2220	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2219	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#2218	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2189	C₂	1	2	yes	maximal, complemented
8	C₆₁×#2186	C₂	1	2	yes	maximal, complemented
9	C₆₁×#2112	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#135×C₆₁	C₂²	1	2	yes	—
11	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#525	—	2	2	no	—
13	C₂×Q₃₂×C₆₁	C₂²	1	1	yes	characteristic, complemented
14	C₂×SD₃₂×C₆₁	C₂²	1	2	yes	complemented
15	C₂×D₁₆×C₆₁	C₂²	1	1	yes	characteristic, complemented
16	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#467	C₂²	1	2	yes	—
19	C₆₁×#467	C₂²	1	2	yes	—
20	C₆₁×#466	C₂²	1	2	yes	—
21	C₆₁×#466	C₂²	1	2	yes	—
22	C₆₁×#465	—	2	2	no	—
23	C₆₁×#465	—	2	2	no	—
24	C₆₁×#427	C₂²	1	2	yes	—
25	C₆₁×#416	—	2	2	no	—
26	C₆₁×#367	C₂²	1	1	yes	characteristic, complemented
27	C₆₁×#367	C₂²	1	1	yes	characteristic, complemented
28	C₆₁×#366	C₂²	1	2	yes	complemented
29	C₆₁×#358	C₂²	1	2	yes	complemented
30	C₆₁×#358	C₂²	1	2	yes	complemented
31	C₆₁×#357	C₂²	1	2	yes	complemented
32	C₆₁×#357	C₂²	1	2	yes	complemented
33	C₆₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
34	C₆₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
35	C₂×#41×C₆₁	—	2	2	no	—
36	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
37	#135×C₆₁	—	4	2	no	—
38	#135×C₆₁	—	4	2	no	—
39	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
40	C₂×SD₁₆×C₆₁	—	2	2	no	—
41	C₂×D₈×C₆₁	C₂³	1	2	yes	—
42	C₂²×C₈×C₆₁	—	4	1	no	—
43	C₂²×C₈×C₆₁	—	4	1	no	—
44	C₂²×C₈×C₆₁	D₄	1	2	yes	—
45	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
48	#123×C₆₁	—	4	2	no	—
49	C₂²⋊Q₈×C₆₁	—	2	2	no	—
50	C₂²⋊Q₈×C₆₁	—	2	2	no	—
51	#121×C₆₁	—	2	2	no	—
52	#121×C₆₁	—	2	2	no	—
53	#120×C₆₁	—	4	2	no	—
54	C₄×D₄×C₆₁	—	4	2	no	—
55	C₄×D₄×C₆₁	—	4	2	no	—
56	#117×C₆₁	—	2	2	no	—
57	Q₃₂×C₆₁	—	4	1	no	—
58	SD₃₂×C₆₁	—	4	2	no	—
59	D₁₆×C₆₁	—	4	1	no	—
60	M₃₂×C₆₁	—	4	1	no	—
61	M₃₂×C₆₁	—	4	1	no	—
62	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
64	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
65	#107×C₆₁	C₂³	1	2	yes	—
66	#107×C₆₁	C₂³	1	2	yes	—
67	#106×C₆₁	—	2	2	no	—
68	#103×C₆₁	—	2	2	no	—
69	#103×C₆₁	—	2	2	no	—
70	#102×C₆₁	—	2	2	no	—
71	#102×C₆₁	—	2	2	no	—
72	C₂⁴×C₆₁	D₈	1	1	yes	characteristic
73	#41×C₆₁	—	4	4	no	—
74	#41×C₆₁	—	4	4	no	—
75	C₂×Q₈×C₆₁	—	2	2	no	—
76	C₂×D₄×C₆₁	—	2	2	no	—
77	C₂×D₄×C₆₁	—	4	2	no	—
78	C₂×D₄×C₆₁	—	4	2	no	—
79	C₂×D₄×C₆₁	—	4	2	no	—
80	C₂²×C₄×C₆₁	—	4	1	no	—
81	C₂²×C₄×C₆₁	—	4	1	no	—
82	C₂²×C₄×C₆₁	—	4	2	no	—
83	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₆₁	D₈	1	2	yes	—
85	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
87	Q₁₆×C₆₁	—	2	2	no	—
88	Q₁₆×C₆₁	—	2	2	no	—
89	SD₁₆×C₆₁	—	4	4	no	—
90	D₈×C₆₁	—	2	2	no	—
91	D₈×C₆₁	—	2	2	no	—
92	C₂×C₈×C₆₁	—	2	1	no	—
93	C₂×C₈×C₆₁	—	2	1	no	—
94	C₂×C₈×C₆₁	—	2	2	no	—
95	C₂×C₈×C₆₁	—	2	2	no	—
96	C₂×C₈×C₆₁	—	4	2	no	—
97	C₂×C₈×C₆₁	—	4	1	no	—
98	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₆₁	—	2	2	no	—
102	C₄⋊C₄×C₆₁	—	2	2	no	—
103	C₄⋊C₄×C₆₁	—	4	2	no	—
104	C₄⋊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	—	4	2	no	—
109	C₂²⋊C₄×C₆₁	—	4	2	no	—
110	C₄²×C₆₁	—	4	2	no	—
111	C₁₆×C₆₁	—	2	1	no	—
112	C₁₆×C₆₁	—	2	2	no	—
113	C₁₆×C₆₁	—	2	1	no	—
114	C₂³×C₆₁	—	4	1	no	—
115	C₂³×C₆₁	—	4	2	no	—
116	C₂³×C₆₁	—	4	2	no	—
117	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
118	C₂³×C₆₁	C₂×D₈	1	2	yes	—
119	Q₈×C₆₁	—	4	2	no	—
120	Q₈×C₆₁	—	4	2	no	—
121	D₄×C₆₁	—	4	2	no	—
122	D₄×C₆₁	—	4	2	no	—
123	D₄×C₆₁	—	8	4	no	—
124	D₄×C₆₁	—	8	4	no	—
125	D₄×C₆₁	—	8	4	no	—
126	C₂×C₄×C₆₁	—	2	1	no	—
127	C₂×C₄×C₆₁	—	2	2	no	—
128	C₂×C₄×C₆₁	—	2	1	no	—
129	C₂×C₄×C₆₁	—	2	2	no	—
130	C₂×C₄×C₆₁	—	4	1	no	—
131	C₂×C₄×C₆₁	—	4	2	no	—
132	C₂×C₄×C₆₁	—	4	2	no	—
133	C₂×C₄×C₆₁	—	4	2	no	—
134	C₂×C₄×C₆₁	—	4	2	no	—
135	C₂×C₄×C₆₁	—	8	4	no	—
136	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₆₁	—	2	1	no	—
140	C₈×C₆₁	—	2	2	no	—
141	C₈×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
142	C₈×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
143	#33763	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₆₁	—	2	2	no	—
145	C₂²×C₆₁	—	2	2	no	—
146	C₂²×C₆₁	—	2	2	no	—
147	C₂²×C₆₁	—	2	2	no	—
148	C₂²×C₆₁	—	2	1	no	—
149	C₂²×C₆₁	—	4	4	no	—
150	C₂²×C₆₁	—	8	2	no	—
151	C₂²×C₆₁	—	8	2	no	—
152	C₂²×C₆₁	—	8	2	no	—
153	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
154	C₄×C₆₁	—	2	2	no	—
155	C₄×C₆₁	—	2	1	no	—
156	C₄×C₆₁	—	8	2	no	—
157	C₄×C₆₁	—	8	2	no	—
158	C₄×C₆₁	—	8	2	no	—
159	C₄×C₆₁	#534	1	1	yes	characteristic
160	C₄×C₆₁	#575	1	1	yes	characteristic
161	#3050	C₂×C₆₁	1	2	yes	complemented
162	#2237	C₂×C₆₁	1	2	yes	complemented
163	#2234	C₂×C₆₁	1	2	yes	complemented
164	#2220	C₂×C₆₁	1	2	yes	complemented
165	#2219	C₂×C₆₁	1	1	yes	characteristic, complemented
166	#2218	C₂×C₆₁	1	1	yes	characteristic, complemented
167	#2189	C₂×C₆₁	1	2	yes	complemented
168	#2186	C₂×C₆₁	1	2	yes	complemented
169	#2112	C₂×C₆₁	1	1	yes	characteristic, complemented
170	C₂×C₆₁	—	4	1	no	—
171	C₂×C₆₁	—	4	2	no	—
172	C₂×C₆₁	—	16	2	no	—
173	C₂×C₆₁	#3186	1	1	yes	characteristic
174	C₂×C₆₁	#3414	1	2	yes	—
175	C₂×#135	C₂²×C₆₁	1	2	yes	—
176	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
177	#525	—	2	2	no	—
178	C₂×Q₃₂	C₂²×C₆₁	1	1	yes	characteristic, complemented
179	C₂×SD₃₂	C₂²×C₆₁	1	2	yes	complemented
180	C₂×D₁₆	C₂²×C₆₁	1	1	yes	characteristic, complemented
181	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
182	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
183	#467	C₂²×C₆₁	1	2	yes	—
184	#467	C₂²×C₆₁	1	2	yes	—
185	#466	C₂²×C₆₁	1	2	yes	—
186	#466	C₂²×C₆₁	1	2	yes	—
187	#465	—	2	2	no	—
188	#465	—	2	2	no	—
189	#427	C₂²×C₆₁	1	2	yes	—
190	#416	—	2	2	no	—
191	#367	C₂²×C₆₁	1	1	yes	characteristic, complemented
192	#367	C₂²×C₆₁	1	1	yes	characteristic, complemented
193	#366	C₂²×C₆₁	1	2	yes	complemented
194	#358	C₂²×C₆₁	1	2	yes	complemented
195	#358	C₂²×C₆₁	1	2	yes	complemented
196	#357	C₂²×C₆₁	1	2	yes	complemented
197	#357	C₂²×C₆₁	1	2	yes	complemented
198	C₂²⋊C₁₆	C₂²×C₆₁	1	2	yes	—
199	C₂²⋊C₁₆	C₂²×C₆₁	1	2	yes	—
200	C₆₁	#33763	1	1	yes	characteristic, direct product factor, Sylow
201	C₂×#41	—	2	2	no	—
202	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
203	#135	—	4	2	no	—
204	#135	—	4	2	no	—
205	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
206	C₂×SD₁₆	—	2	2	no	—
207	C₂×D₈	C₂³×C₆₁	1	2	yes	—
208	C₂²×C₈	—	4	1	no	—
209	C₂²×C₈	—	4	1	no	—
210	C₂²×C₈	D₄×C₆₁	1	2	yes	—
211	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
212	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
213	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
214	#123	—	4	2	no	—
215	C₂²⋊Q₈	—	2	2	no	—
216	C₂²⋊Q₈	—	2	2	no	—
217	#121	—	2	2	no	—
218	#121	—	2	2	no	—
219	#120	—	4	2	no	—
220	C₄×D₄	—	4	2	no	—
221	C₄×D₄	—	4	2	no	—
222	#117	—	2	2	no	—
223	Q₃₂	—	4	1	no	—
224	SD₃₂	—	4	2	no	—
225	D₁₆	—	4	1	no	—
226	M₃₂	—	4	1	no	—
227	M₃₂	—	4	1	no	—
228	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
229	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
230	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
231	#107	C₂³×C₆₁	1	2	yes	—
232	#107	C₂³×C₆₁	1	2	yes	—
233	#106	—	2	2	no	—
234	#103	—	2	2	no	—
235	#103	—	2	2	no	—
236	#102	—	2	2	no	—
237	#102	—	2	2	no	—
238	C₂⁴	D₈×C₆₁	1	1	yes	characteristic
239	#41	—	4	4	no	—
240	#41	—	4	4	no	—
241	C₂×Q₈	—	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	2	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	—	4	2	no	—
249	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
250	C₂²×C₄	D₈×C₆₁	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
252	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
253	Q₁₆	—	2	2	no	—
254	Q₁₆	—	2	2	no	—
255	SD₁₆	—	4	4	no	—
256	D₈	—	2	2	no	—
257	D₈	—	2	2	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
265	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
266	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₄²	—	4	2	no	—
277	C₁₆	—	2	1	no	—
278	C₁₆	—	2	2	no	—
279	C₁₆	—	2	1	no	—
280	C₂³	—	4	1	no	—
281	C₂³	—	4	2	no	—
282	C₂³	—	4	2	no	—
283	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
284	C₂³	C₂×D₈×C₆₁	1	2	yes	—
285	Q₈	—	4	2	no	—
286	Q₈	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	2	no	—
289	D₄	—	8	4	no	—
290	D₄	—	8	4	no	—
291	D₄	—	8	4	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	4	no	—
302	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
303	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	2	no	—
307	C₈	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
308	C₈	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
319	C₄	—	2	2	no	—
320	C₄	—	2	1	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	C₆₁×#534	1	1	yes	characteristic
325	C₄	C₆₁×#575	1	1	yes	characteristic
326	C₂	—	4	1	no	—
327	C₂	—	4	2	no	—
328	C₂	—	16	2	no	—
329	C₂	C₆₁×#3186	1	1	yes	characteristic
330	C₂	C₆₁×#3414	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 ◁ C₂³×C₄×C₆₁ ◁ C₂³×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3050	C₂	semidirect
2. C₆₁×#2237	C₂	semidirect
3. C₆₁×#2234	C₂	semidirect
4. C₆₁×#2220	C₂	semidirect
5. C₆₁×#2219	C₂	semidirect
6. C₆₁×#2218	C₂	semidirect
7. C₆₁×#2189	C₂	semidirect
8. C₆₁×#2186	C₂	semidirect
9. C₆₁×#2112	C₂	semidirect
13. C₂×Q₃₂×C₆₁	C₂²	semidirect
14. C₂×SD₃₂×C₆₁	C₂²	semidirect
15. C₂×D₁₆×C₆₁	C₂²	semidirect
26. C₆₁×#367	C₂²	semidirect
27. C₆₁×#367	C₂²	semidirect
28. C₆₁×#366	C₂²	semidirect
29. C₆₁×#358	C₂²	semidirect
30. C₆₁×#358	C₂²	semidirect
31. C₆₁×#357	C₂²	semidirect
32. C₆₁×#357	C₂²	semidirect
143. #33763	C₆₁	direct (N × H)
161. #3050	C₂×C₆₁	semidirect
162. #2237	C₂×C₆₁	semidirect
163. #2234	C₂×C₆₁	semidirect
164. #2220	C₂×C₆₁	semidirect
165. #2219	C₂×C₆₁	semidirect
166. #2218	C₂×C₆₁	semidirect
167. #2189	C₂×C₆₁	semidirect
168. #2186	C₂×C₆₁	semidirect
169. #2112	C₂×C₆₁	semidirect
178. C₂×Q₃₂	C₂²×C₆₁	semidirect
179. C₂×SD₃₂	C₂²×C₆₁	semidirect
180. C₂×D₁₆	C₂²×C₆₁	semidirect
191. #367	C₂²×C₆₁	semidirect
192. #367	C₂²×C₆₁	semidirect
193. #366	C₂²×C₆₁	semidirect
194. #358	C₂²×C₆₁	semidirect
195. #358	C₂²×C₆₁	semidirect
196. #357	C₂²×C₆₁	semidirect
197. #357	C₂²×C₆₁	semidirect
200. C₆₁	#33763	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×#3050	2
C₆₁×#2237	2
C₆₁×#2234	2
C₆₁×#2220	2
C₆₁×#2219	2
C₆₁×#2218	2
C₆₁×#2189	2
C₆₁×#2186	2
C₆₁×#2112	2
#33763	61

Subgroups — C61×#33763 #29537196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33763 #29537196