Subgroups — C₆₁×#33822 #29537255

Order 15616.

496 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3168	C₂	1	2	yes	maximal, complemented
2	C₆₁×#2229	C₂	1	2	yes	maximal, complemented
3	C₆₁×#2228	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
7	C₆₁×#2211	C₂	1	2	yes	maximal, complemented
8	C₆₁×#2210	C₂	1	2	yes	maximal, complemented
9	C₆₁×#2150	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#529	—	2	2	no	—
11	C₂×Q₃₂×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×SD₃₂×C₆₁	C₂²	1	2	yes	—
13	C₂×D₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#499	C₂²	1	2	yes	—
17	C₆₁×#495	C₂²	1	2	yes	—
18	C₆₁×#488	—	2	2	no	—
19	C₆₁×#467	C₂²	1	2	yes	—
20	C₆₁×#466	C₂²	1	2	yes	—
21	C₆₁×#465	—	2	2	no	—
22	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
23	C₄×SD₁₆×C₆₁	—	2	2	no	—
24	C₄×D₈×C₆₁	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
26	C₆₁×#367	C₂²	1	1	yes	characteristic
27	C₆₁×#367	C₂²	1	1	yes	characteristic
28	C₆₁×#366	C₂²	1	2	yes	—
29	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₆₁×C₄⋊C₁₆	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	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
36	C₂×SD₁₆×C₆₁	—	2	2	no	—
37	C₂×D₈×C₆₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
40	#126×C₆₁	—	4	2	no	—
41	#125×C₆₁	—	2	2	no	—
42	#124×C₆₁	—	2	2	no	—
43	#123×C₆₁	—	4	2	no	—
44	C₂²⋊Q₈×C₆₁	—	2	2	no	—
45	#121×C₆₁	—	2	2	no	—
46	C₄×Q₈×C₆₁	—	2	2	no	—
47	C₄×D₄×C₆₁	—	2	2	no	—
48	C₄×D₄×C₆₁	—	4	2	no	—
49	#117×C₆₁	—	4	2	no	—
50	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
51	Q₃₂×C₆₁	—	4	1	no	—
52	SD₃₂×C₆₁	—	4	2	no	—
53	D₁₆×C₆₁	—	4	1	no	—
54	M₃₂×C₆₁	—	4	1	no	—
55	M₃₂×C₆₁	—	4	1	no	—
56	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
57	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
58	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
59	#107×C₆₁	C₂³	1	2	yes	—
60	#107×C₆₁	C₂³	1	2	yes	—
61	#106×C₆₁	—	2	2	no	—
62	#103×C₆₁	—	2	2	no	—
63	#103×C₆₁	—	2	2	no	—
64	#102×C₆₁	—	2	2	no	—
65	#102×C₆₁	—	2	2	no	—
66	C₄×C₈×C₆₁	D₄	1	2	yes	—
67	C₄×C₈×C₆₁	C₂³	1	2	yes	—
68	C₂×Q₈×C₆₁	—	2	2	no	—
69	C₂×D₄×C₆₁	—	2	2	no	—
70	C₂×D₄×C₆₁	—	4	2	no	—
71	C₂²×C₄×C₆₁	—	4	2	no	—
72	C₂²×C₄×C₆₁	D₈	1	2	yes	—
73	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
74	Q₁₆×C₆₁	—	2	2	no	—
75	Q₁₆×C₆₁	—	2	2	no	—
76	SD₁₆×C₆₁	—	4	4	no	—
77	D₈×C₆₁	—	2	2	no	—
78	D₈×C₆₁	—	2	2	no	—
79	C₂×C₈×C₆₁	—	2	1	no	—
80	C₂×C₈×C₆₁	—	2	1	no	—
81	C₂×C₈×C₆₁	—	4	1	no	—
82	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₆₁	—	2	2	no	—
86	C₄⋊C₄×C₆₁	—	2	2	no	—
87	C₄⋊C₄×C₆₁	—	4	2	no	—
88	C₄⋊C₄×C₆₁	—	4	2	no	—
89	C₄⋊C₄×C₆₁	—	4	2	no	—
90	C₄⋊C₄×C₆₁	—	4	2	no	—
91	C₂²⋊C₄×C₆₁	—	4	2	no	—
92	C₂²⋊C₄×C₆₁	—	4	2	no	—
93	C₂²⋊C₄×C₆₁	—	4	2	no	—
94	C₂²⋊C₄×C₆₁	—	4	2	no	—
95	C₂²⋊C₄×C₆₁	—	4	2	no	—
96	C₄²×C₆₁	—	4	2	no	—
97	C₄²×C₆₁	D₈	1	1	yes	characteristic
98	C₄²×C₆₁	D₈	1	1	yes	characteristic
99	C₄²×C₆₁	C₂×D₄	1	2	yes	—
100	C₁₆×C₆₁	—	2	1	no	—
101	C₁₆×C₆₁	—	2	2	no	—
102	C₁₆×C₆₁	—	2	1	no	—
103	C₂³×C₆₁	—	4	2	no	—
104	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
105	Q₈×C₆₁	—	4	2	no	—
106	Q₈×C₆₁	—	4	2	no	—
107	D₄×C₆₁	—	4	2	no	—
108	D₄×C₆₁	—	4	2	no	—
109	D₄×C₆₁	—	8	4	no	—
110	C₂×C₄×C₆₁	—	2	1	no	—
111	C₂×C₄×C₆₁	—	2	1	no	—
112	C₂×C₄×C₆₁	—	4	2	no	—
113	C₂×C₄×C₆₁	—	4	2	no	—
114	C₂×C₄×C₆₁	—	4	2	no	—
115	C₂×C₄×C₆₁	—	4	2	no	—
116	C₂×C₄×C₆₁	—	8	4	no	—
117	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
121	C₈×C₆₁	—	2	1	no	—
122	C₈×C₆₁	—	2	2	no	—
123	C₈×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
124	C₈×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
125	#33822	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂²×C₆₁	—	2	2	no	—
127	C₂²×C₆₁	—	2	1	no	—
128	C₂²×C₆₁	—	8	2	no	—
129	C₂²×C₆₁	—	8	2	no	—
130	C₂²×C₆₁	—	8	2	no	—
131	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
132	C₄×C₆₁	—	2	2	no	—
133	C₄×C₆₁	—	2	2	no	—
134	C₄×C₆₁	—	2	1	no	—
135	C₄×C₆₁	—	8	2	no	—
136	C₄×C₆₁	—	8	2	no	—
137	C₄×C₆₁	—	8	2	no	—
138	C₄×C₆₁	#537	1	1	yes	characteristic
139	C₄×C₆₁	#575	1	1	yes	characteristic
140	#3168	C₂×C₆₁	1	2	yes	complemented
141	#2229	C₂×C₆₁	1	2	yes	complemented
142	#2228	C₂×C₆₁	1	2	yes	complemented
143	#2220	C₂×C₆₁	1	2	yes	complemented
144	#2219	C₂×C₆₁	1	1	yes	characteristic, complemented
145	#2218	C₂×C₆₁	1	1	yes	characteristic
146	#2211	C₂×C₆₁	1	2	yes	complemented
147	#2210	C₂×C₆₁	1	2	yes	complemented
148	#2150	C₂×C₆₁	1	1	yes	characteristic, complemented
149	C₂×C₆₁	—	4	1	no	—
150	C₂×C₆₁	—	16	2	no	—
151	C₂×C₆₁	#3245	1	1	yes	characteristic
152	C₂×C₆₁	#3414	1	2	yes	—
153	#529	—	2	2	no	—
154	C₂×Q₃₂	C₂²×C₆₁	1	1	yes	characteristic
155	C₂×SD₃₂	C₂²×C₆₁	1	2	yes	—
156	C₂×D₁₆	C₂²×C₆₁	1	1	yes	characteristic
157	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
158	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
159	#499	C₂²×C₆₁	1	2	yes	—
160	#495	C₂²×C₆₁	1	2	yes	—
161	#488	—	2	2	no	—
162	#467	C₂²×C₆₁	1	2	yes	—
163	#466	C₂²×C₆₁	1	2	yes	—
164	#465	—	2	2	no	—
165	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
166	C₄×SD₁₆	—	2	2	no	—
167	C₄×D₈	C₂²×C₆₁	1	2	yes	—
168	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
169	#367	C₂²×C₆₁	1	1	yes	characteristic
170	#367	C₂²×C₆₁	1	1	yes	characteristic
171	#366	C₂²×C₆₁	1	2	yes	—
172	C₄⋊C₁₆	C₂²×C₆₁	1	2	yes	—
173	C₄⋊C₁₆	C₂²×C₆₁	1	2	yes	—
174	#358	C₂²×C₆₁	1	2	yes	—
175	#358	C₂²×C₆₁	1	2	yes	—
176	#357	C₂²×C₆₁	1	2	yes	—
177	#357	C₂²×C₆₁	1	2	yes	—
178	C₆₁	#33822	1	1	yes	characteristic, direct product factor, Sylow
179	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
180	C₂×SD₁₆	—	2	2	no	—
181	C₂×D₈	C₂³×C₆₁	1	2	yes	—
182	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
183	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
184	#126	—	4	2	no	—
185	#125	—	2	2	no	—
186	#124	—	2	2	no	—
187	#123	—	4	2	no	—
188	C₂²⋊Q₈	—	2	2	no	—
189	#121	—	2	2	no	—
190	C₄×Q₈	—	2	2	no	—
191	C₄×D₄	—	2	2	no	—
192	C₄×D₄	—	4	2	no	—
193	#117	—	4	2	no	—
194	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
195	Q₃₂	—	4	1	no	—
196	SD₃₂	—	4	2	no	—
197	D₁₆	—	4	1	no	—
198	M₃₂	—	4	1	no	—
199	M₃₂	—	4	1	no	—
200	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
201	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
202	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
203	#107	C₂³×C₆₁	1	2	yes	—
204	#107	C₂³×C₆₁	1	2	yes	—
205	#106	—	2	2	no	—
206	#103	—	2	2	no	—
207	#103	—	2	2	no	—
208	#102	—	2	2	no	—
209	#102	—	2	2	no	—
210	C₄×C₈	D₄×C₆₁	1	2	yes	—
211	C₄×C₈	C₂³×C₆₁	1	2	yes	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	D₈×C₆₁	1	2	yes	—
217	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
218	Q₁₆	—	2	2	no	—
219	Q₁₆	—	2	2	no	—
220	SD₁₆	—	4	4	no	—
221	D₈	—	2	2	no	—
222	D₈	—	2	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
227	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
228	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₂²⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	4	2	no	—
237	C₂²⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₄²	—	4	2	no	—
241	C₄²	D₈×C₆₁	1	1	yes	characteristic
242	C₄²	D₈×C₆₁	1	1	yes	characteristic
243	C₄²	C₂×D₄×C₆₁	1	2	yes	—
244	C₁₆	—	2	1	no	—
245	C₁₆	—	2	2	no	—
246	C₁₆	—	2	1	no	—
247	C₂³	—	4	2	no	—
248	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
249	Q₈	—	4	2	no	—
250	Q₈	—	4	2	no	—
251	D₄	—	4	2	no	—
252	D₄	—	4	2	no	—
253	D₄	—	8	4	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	8	4	no	—
261	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
262	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
263	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
264	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
265	C₈	—	2	1	no	—
266	C₈	—	2	2	no	—
267	C₈	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
268	C₈	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
269	C₂²	—	2	2	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	2	no	—
274	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	1	no	—
278	C₄	—	8	2	no	—
279	C₄	—	8	2	no	—
280	C₄	—	8	2	no	—
281	C₄	C₆₁×#537	1	1	yes	characteristic
282	C₄	C₆₁×#575	1	1	yes	characteristic
283	C₂	—	4	1	no	—
284	C₂	—	16	2	no	—
285	C₂	C₆₁×#3245	1	1	yes	characteristic
286	C₂	C₆₁×#3414	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
prime radical	C₆₁×#2218	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3168	C₂	semidirect
2. C₆₁×#2229	C₂	semidirect
3. C₆₁×#2228	C₂	semidirect
4. C₆₁×#2220	C₂	semidirect
5. C₆₁×#2219	C₂	semidirect
7. C₆₁×#2211	C₂	semidirect
8. C₆₁×#2210	C₂	semidirect
9. C₆₁×#2150	C₂	semidirect
125. #33822	C₆₁	direct (N × H)
140. #3168	C₂×C₆₁	semidirect
141. #2229	C₂×C₆₁	semidirect
142. #2228	C₂×C₆₁	semidirect
143. #2220	C₂×C₆₁	semidirect
144. #2219	C₂×C₆₁	semidirect
146. #2211	C₂×C₆₁	semidirect
147. #2210	C₂×C₆₁	semidirect
148. #2150	C₂×C₆₁	semidirect
178. C₆₁	#33822	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×#3168	2
C₆₁×#2229	2
C₆₁×#2228	2
C₆₁×#2220	2
C₆₁×#2219	2
C₆₁×#2218	2
C₆₁×#2211	2
C₆₁×#2210	2
C₆₁×#2150	2
#33822	61

Subgroups — C61×#33822 #29537255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33822 #29537255