Subgroups — C₆₁×#3219 #21443260

Order 7808.

396 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#529	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#497	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#496	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#491	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#490	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#464	C₂	1	2	yes	maximal, complemented
8	C₆₁×#463	C₂	1	2	yes	maximal, complemented
9	C₆₁×#452	C₂	1	2	yes	maximal, complemented
10	C₆₁×#449	C₂	1	2	yes	maximal, complemented
11	C₆₁×#432	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₆₁	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₆₁	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₆₁	C₂²	1	2	yes	—
15	C₂×SD₁₆×C₆₁	C₂²	1	2	yes	—
16	C₂×D₈×C₆₁	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
18	#127×C₆₁	C₂²	1	1	yes	characteristic
19	#126×C₆₁	—	2	1	no	—
20	#125×C₆₁	C₂²	1	1	yes	characteristic
21	#124×C₆₁	C₂²	1	2	yes	—
22	#124×C₆₁	C₂²	1	1	yes	characteristic
23	#123×C₆₁	—	2	1	no	—
24	C₂²⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
25	#121×C₆₁	—	2	1	no	—
26	#121×C₆₁	—	2	1	no	—
27	#121×C₆₁	C₂²	1	1	yes	characteristic
28	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
29	C₄×D₄×C₆₁	—	2	1	no	—
30	C₄×D₄×C₆₁	—	2	1	no	—
31	C₄×D₄×C₆₁	—	2	1	no	—
32	C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
33	#117×C₆₁	—	2	1	no	—
34	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
36	#103×C₆₁	C₂²	1	2	yes	—
37	#103×C₆₁	C₂²	1	2	yes	—
38	#102×C₆₁	C₂²	1	2	yes	—
39	#102×C₆₁	C₂²	1	2	yes	—
40	C₂²⋊C₈×C₆₁	C₂²	1	2	yes	—
41	#97×C₆₁	C₂²	1	1	yes	characteristic
42	#97×C₆₁	C₂²	1	1	yes	characteristic
43	#41×C₆₁	—	4	2	no	—
44	#41×C₆₁	—	4	2	no	—
45	C₂×Q₈×C₆₁	C₂³	1	2	yes	—
46	C₂×Q₈×C₆₁	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₆₁	—	2	1	no	—
48	C₂×D₄×C₆₁	—	2	2	no	—
49	C₂×D₄×C₆₁	—	2	1	no	—
50	C₂×D₄×C₆₁	—	2	1	no	—
51	C₂×D₄×C₆₁	C₂³	1	2	yes	—
52	C₂×D₄×C₆₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	—	2	1	no	—
54	C₂²×C₄×C₆₁	—	2	2	no	—
55	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
58	Q₁₆×C₆₁	—	4	2	no	—
59	SD₁₆×C₆₁	—	4	2	no	—
60	SD₁₆×C₆₁	—	4	2	no	—
61	D₈×C₆₁	—	4	2	no	—
62	C₂×C₈×C₆₁	C₂³	1	2	yes	—
63	C₂×C₈×C₆₁	C₂³	1	2	yes	—
64	C₄⋊C₄×C₆₁	—	2	1	no	—
65	C₄⋊C₄×C₆₁	—	2	1	no	—
66	C₄⋊C₄×C₆₁	—	2	1	no	—
67	C₄⋊C₄×C₆₁	—	2	1	no	—
68	C₄⋊C₄×C₆₁	—	2	1	no	—
69	C₄⋊C₄×C₆₁	—	2	1	no	—
70	C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
72	C₂²⋊C₄×C₆₁	—	2	1	no	—
73	C₂²⋊C₄×C₆₁	—	2	1	no	—
74	C₂²⋊C₄×C₆₁	—	2	1	no	—
75	C₂²⋊C₄×C₆₁	—	2	2	no	—
76	C₂²⋊C₄×C₆₁	—	2	2	no	—
77	C₂²⋊C₄×C₆₁	—	2	1	no	—
78	C₂²⋊C₄×C₆₁	—	2	1	no	—
79	C₄²×C₆₁	—	2	1	no	—
80	C₄²×C₆₁	D₄	1	1	yes	characteristic
81	C₄²×C₆₁	D₄	1	1	yes	characteristic
82	C₄²×C₆₁	C₂³	1	1	yes	characteristic
83	C₄²×C₆₁	C₂³	1	1	yes	characteristic
84	C₂³×C₆₁	—	2	1	no	—
85	C₂³×C₆₁	—	2	2	no	—
86	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
87	Q₈×C₆₁	—	2	2	no	—
88	Q₈×C₆₁	—	2	2	no	—
89	Q₈×C₆₁	—	2	2	no	—
90	D₄×C₆₁	—	2	2	no	—
91	D₄×C₆₁	—	2	2	no	—
92	D₄×C₆₁	—	2	2	no	—
93	D₄×C₆₁	—	4	2	no	—
94	D₄×C₆₁	—	4	2	no	—
95	D₄×C₆₁	—	4	2	no	—
96	D₄×C₆₁	—	4	4	no	—
97	C₂×C₄×C₆₁	—	2	2	no	—
98	C₂×C₄×C₆₁	—	2	1	no	—
99	C₂×C₄×C₆₁	—	2	1	no	—
100	C₂×C₄×C₆₁	—	2	1	no	—
101	C₂×C₄×C₆₁	—	2	2	no	—
102	C₂×C₄×C₆₁	—	4	1	no	—
103	C₂×C₄×C₆₁	—	4	1	no	—
104	C₂×C₄×C₆₁	—	4	2	no	—
105	C₂×C₄×C₆₁	—	4	4	no	—
106	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
112	C₈×C₆₁	—	2	2	no	—
113	C₈×C₆₁	—	2	2	no	—
114	C₂²×C₆₁	—	2	1	no	—
115	C₂²×C₆₁	—	2	1	no	—
116	C₂²×C₆₁	—	2	1	no	—
117	C₂²×C₆₁	—	4	2	no	—
118	C₂²×C₆₁	—	4	1	no	—
119	C₂²×C₆₁	—	4	1	no	—
120	C₂²×C₆₁	—	4	2	no	—
121	C₂²×C₆₁	—	4	2	no	—
122	C₂²×C₆₁	—	4	1	no	—
123	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
124	C₄×C₆₁	—	2	1	no	—
125	C₄×C₆₁	—	2	1	no	—
126	C₄×C₆₁	—	2	1	no	—
127	C₄×C₆₁	—	2	1	no	—
128	C₄×C₆₁	—	2	1	no	—
129	C₄×C₆₁	—	4	2	no	—
130	C₄×C₆₁	—	4	1	no	—
131	C₄×C₆₁	—	4	1	no	—
132	C₄×C₆₁	—	4	1	no	—
133	C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
134	#3219	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂×C₆₁	—	4	1	no	—
136	C₂×C₆₁	—	8	2	no	—
137	C₂×C₆₁	—	8	1	no	—
138	C₂×C₆₁	#535	1	1	yes	characteristic
139	C₂×C₆₁	#575	1	1	yes	characteristic
140	C₂×C₆₁	#575	1	1	yes	characteristic
141	#532	C₂×C₆₁	1	1	yes	characteristic, complemented
142	#529	C₂×C₆₁	1	1	yes	characteristic, complemented
143	#497	C₂×C₆₁	1	1	yes	characteristic, complemented
144	#496	C₂×C₆₁	1	1	yes	characteristic, complemented
145	#491	C₂×C₆₁	1	1	yes	characteristic, complemented
146	#490	C₂×C₆₁	1	1	yes	characteristic, complemented
147	#464	C₂×C₆₁	1	2	yes	complemented
148	#463	C₂×C₆₁	1	2	yes	complemented
149	#452	C₂×C₆₁	1	2	yes	complemented
150	#449	C₂×C₆₁	1	2	yes	complemented
151	#432	C₂×C₆₁	1	1	yes	characteristic, complemented
152	C₆₁	#3219	1	1	yes	characteristic, direct product factor, Sylow
153	C₂×#41	C₂²×C₆₁	1	2	yes	—
154	C₂×Q₁₆	C₂²×C₆₁	1	2	yes	—
155	C₂×SD₁₆	C₂²×C₆₁	1	2	yes	—
156	C₂×SD₁₆	C₂²×C₆₁	1	2	yes	—
157	C₂×D₈	C₂²×C₆₁	1	2	yes	—
158	C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
159	#127	C₂²×C₆₁	1	1	yes	characteristic
160	#126	—	2	1	no	—
161	#125	C₂²×C₆₁	1	1	yes	characteristic
162	#124	C₂²×C₆₁	1	1	yes	characteristic
163	#124	C₂²×C₆₁	1	2	yes	—
164	#123	—	2	1	no	—
165	C₂²⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
166	#121	—	2	1	no	—
167	#121	—	2	1	no	—
168	#121	C₂²×C₆₁	1	1	yes	characteristic
169	C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
170	C₄×D₄	—	2	1	no	—
171	C₄×D₄	—	2	1	no	—
172	C₄×D₄	—	2	1	no	—
173	C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
174	#117	—	2	1	no	—
175	C₂×C₄²	C₂²×C₆₁	1	1	yes	characteristic
176	C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
177	#103	C₂²×C₆₁	1	2	yes	—
178	#103	C₂²×C₆₁	1	2	yes	—
179	#102	C₂²×C₆₁	1	2	yes	—
180	#102	C₂²×C₆₁	1	2	yes	—
181	C₂²⋊C₈	C₂²×C₆₁	1	2	yes	—
182	#97	C₂²×C₆₁	1	1	yes	characteristic
183	#97	C₂²×C₆₁	1	1	yes	characteristic
184	#41	—	4	2	no	—
185	#41	—	4	2	no	—
186	C₂×Q₈	C₂³×C₆₁	1	2	yes	—
187	C₂×Q₈	C₂³×C₆₁	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	C₂³×C₆₁	1	2	yes	—
193	C₂×D₄	C₂³×C₆₁	1	1	yes	characteristic
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	D₄×C₆₁	1	1	yes	characteristic
197	C₂²×C₄	D₄×C₆₁	1	1	yes	characteristic
198	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
199	Q₁₆	—	4	2	no	—
200	SD₁₆	—	4	2	no	—
201	SD₁₆	—	4	2	no	—
202	D₈	—	4	2	no	—
203	C₂×C₈	C₂³×C₆₁	1	2	yes	—
204	C₂×C₈	C₂³×C₆₁	1	2	yes	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
212	C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
213	C₂²⋊C₄	—	2	1	no	—
214	C₂²⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	2	1	no	—
220	C₄²	—	2	1	no	—
221	C₄²	D₄×C₆₁	1	1	yes	characteristic
222	C₄²	D₄×C₆₁	1	1	yes	characteristic
223	C₄²	C₂³×C₆₁	1	1	yes	characteristic
224	C₄²	C₂³×C₆₁	1	1	yes	characteristic
225	C₂³	—	2	1	no	—
226	C₂³	—	2	2	no	—
227	C₂³	C₂×D₄×C₆₁	1	1	yes	characteristic
228	Q₈	—	2	2	no	—
229	Q₈	—	2	2	no	—
230	Q₈	—	2	2	no	—
231	D₄	—	2	2	no	—
232	D₄	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
248	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
249	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
250	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
251	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
252	C₂×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	2	2	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	1	no	—
264	C₂²	C₂²×D₄×C₆₁	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	4	2	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
275	C₂	—	4	1	no	—
276	C₂	—	8	2	no	—
277	C₂	—	8	1	no	—
278	C₂	C₆₁×#535	1	1	yes	characteristic
279	C₂	C₆₁×#575	1	1	yes	characteristic
280	C₂	C₆₁×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₆₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#532	C₂	semidirect
2. C₆₁×#529	C₂	semidirect
3. C₆₁×#497	C₂	semidirect
4. C₆₁×#496	C₂	semidirect
5. C₆₁×#491	C₂	semidirect
6. C₆₁×#490	C₂	semidirect
7. C₆₁×#464	C₂	semidirect
8. C₆₁×#463	C₂	semidirect
9. C₆₁×#452	C₂	semidirect
10. C₆₁×#449	C₂	semidirect
11. C₆₁×#432	C₂	semidirect
134. #3219	C₆₁	direct (N × H)
141. #532	C₂×C₆₁	semidirect
142. #529	C₂×C₆₁	semidirect
143. #497	C₂×C₆₁	semidirect
144. #496	C₂×C₆₁	semidirect
145. #491	C₂×C₆₁	semidirect
146. #490	C₂×C₆₁	semidirect
147. #464	C₂×C₆₁	semidirect
148. #463	C₂×C₆₁	semidirect
149. #452	C₂×C₆₁	semidirect
150. #449	C₂×C₆₁	semidirect
151. #432	C₂×C₆₁	semidirect
152. C₆₁	#3219	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×#532	2
C₆₁×#529	2
C₆₁×#497	2
C₆₁×#496	2
C₆₁×#491	2
C₆₁×#490	2
C₆₁×#464	2
C₆₁×#463	2
C₆₁×#452	2
C₆₁×#449	2
C₆₁×#432	2
#3219	61

Subgroups — C61×#3219 #21443260

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#3219 #21443260