Subgroups — C₆₁×#3455 #21443496

Order 7808.

1040 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#542	C₂	1	4	yes	maximal, complemented
2	C₆₁×#541	C₂	1	2	yes	maximal, complemented
3	C₆₁×#540	C₂	1	2	yes	maximal, complemented
4	C₆₁×#539	C₂	1	4	yes	maximal, complemented
5	C₆₁×#538	C₂	1	4	yes	maximal, complemented
6	C₆₁×#533	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#532	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#529	C₂	1	8	yes	maximal, complemented
9	C₆₁×#525	C₂	1	2	yes	maximal, complemented
10	C₄×#41×C₆₁	C₂	1	2	yes	maximal, complemented
11	C₂×#117×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₆₁	C₂²	1	2	yes	—
13	C₂³×C₄×C₆₁	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic, complemented
15	C₄⋊Q₈×C₆₁	C₂²	1	2	yes	complemented
16	#127×C₆₁	C₂²	1	1	yes	characteristic, complemented
17	#126×C₆₁	C₂²	1	8	yes	—
18	#126×C₆₁	C₂²	1	8	yes	complemented
19	#125×C₆₁	C₂²	1	2	yes	complemented
20	#125×C₆₁	C₂²	1	8	yes	complemented
21	#124×C₆₁	C₂²	1	8	yes	—
22	#124×C₆₁	C₂²	1	2	yes	complemented
23	#123×C₆₁	C₂²	1	8	yes	—
24	#123×C₆₁	C₂²	1	4	yes	—
25	#123×C₆₁	C₂²	1	8	yes	complemented
26	C₂²⋊Q₈×C₆₁	C₂²	1	4	yes	—
27	C₂²⋊Q₈×C₆₁	C₂²	1	4	yes	—
28	C₂²⋊Q₈×C₆₁	C₂²	1	8	yes	complemented
29	#121×C₆₁	C₂²	1	4	yes	—
30	#121×C₆₁	C₂²	1	8	yes	—
31	#121×C₆₁	C₂²	1	4	yes	—
32	#120×C₆₁	C₂²	1	4	yes	—
33	C₄×Q₈×C₆₁	C₂²	1	4	yes	complemented
34	C₄×Q₈×C₆₁	C₂²	1	4	yes	complemented
35	C₄×D₄×C₆₁	C₂²	1	8	yes	—
36	C₄×D₄×C₆₁	C₂²	1	8	yes	—
37	C₄×D₄×C₆₁	C₂²	1	4	yes	complemented
38	C₄×D₄×C₆₁	C₂²	1	4	yes	complemented
39	#117×C₆₁	C₂²	1	2	yes	—
40	#117×C₆₁	C₂²	1	2	yes	—
41	#117×C₆₁	C₂²	1	2	yes	—
42	#117×C₆₁	C₂²	1	4	yes	—
43	#117×C₆₁	C₂²	1	4	yes	complemented
44	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
45	C₂×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
46	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
47	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
48	C₂×C₄²×C₆₁	C₂²	1	4	yes	complemented
49	C₂⁴×C₆₁	C₂³	1	1	yes	characteristic
50	#41×C₆₁	—	2	8	no	—
51	C₂×Q₈×C₆₁	C₂³	1	2	yes	—
52	C₂×Q₈×C₆₁	C₂³	1	4	yes	—
53	C₂×D₄×C₆₁	C₂³	1	4	yes	—
54	C₂×D₄×C₆₁	C₂³	1	2	yes	—
55	C₂×D₄×C₆₁	C₂³	1	8	yes	—
56	C₂×D₄×C₆₁	C₂³	1	4	yes	—
57	C₂²×C₄×C₆₁	—	2	4	no	—
58	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
60	C₂²×C₄×C₆₁	C₂³	1	4	yes	—
61	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
62	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
63	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
64	C₂²×C₄×C₆₁	C₂³	1	8	yes	—
65	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
67	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
68	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
69	C₄⋊C₄×C₆₁	C₂³	1	8	yes	—
70	C₄⋊C₄×C₆₁	C₂³	1	8	yes	—
71	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
72	C₄⋊C₄×C₆₁	C₂³	1	8	yes	—
73	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
74	C₂²⋊C₄×C₆₁	C₂³	1	4	yes	—
75	C₂²⋊C₄×C₆₁	C₂³	1	4	yes	—
76	C₂²⋊C₄×C₆₁	C₂³	1	4	yes	—
77	C₂²⋊C₄×C₆₁	C₂³	1	8	yes	—
78	C₂²⋊C₄×C₆₁	C₂³	1	8	yes	—
79	C₂²⋊C₄×C₆₁	C₂³	1	8	yes	—
80	C₂²⋊C₄×C₆₁	C₂³	1	8	yes	—
81	C₄²×C₆₁	C₂³	1	2	yes	—
82	C₄²×C₆₁	C₂³	1	1	yes	characteristic
83	C₄²×C₆₁	C₂³	1	1	yes	characteristic
84	C₄²×C₆₁	C₂³	1	8	yes	—
85	C₄²×C₆₁	C₂³	1	2	yes	—
86	C₄²×C₆₁	C₂³	1	2	yes	—
87	C₄²×C₆₁	C₂³	1	4	yes	—
88	C₂³×C₆₁	—	2	2	no	—
89	C₂³×C₆₁	—	4	2	no	—
90	C₂³×C₆₁	C₂⁴	1	2	yes	—
91	C₂³×C₆₁	C₂⁴	1	4	yes	—
92	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
93	Q₈×C₆₁	—	2	4	no	—
94	Q₈×C₆₁	—	4	4	no	—
95	D₄×C₆₁	—	2	8	no	—
96	D₄×C₆₁	—	2	4	no	—
97	D₄×C₆₁	—	4	4	no	—
98	D₄×C₆₁	—	4	8	no	—
99	C₂×C₄×C₆₁	—	2	2	no	—
100	C₂×C₄×C₆₁	—	2	4	no	—
101	C₂×C₄×C₆₁	—	2	4	no	—
102	C₂×C₄×C₆₁	—	2	8	no	—
103	C₂×C₄×C₆₁	—	4	2	no	—
104	C₂×C₄×C₆₁	—	4	8	no	—
105	C₂×C₄×C₆₁	—	4	2	no	—
106	C₂×C₄×C₆₁	#41	1	4	yes	—
107	C₂×C₄×C₆₁	C₂⁴	1	4	yes	—
108	C₂×C₄×C₆₁	C₂⁴	1	2	yes	—
109	C₂×C₄×C₆₁	C₂⁴	1	4	yes	—
110	C₂×C₄×C₆₁	C₂⁴	1	8	yes	—
111	C₂×C₄×C₆₁	C₂⁴	1	2	yes	—
112	C₂×C₄×C₆₁	C₂⁴	1	2	yes	—
113	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
114	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
115	C₂²×C₆₁	—	2	2	no	—
116	C₂²×C₆₁	—	2	2	no	—
117	C₂²×C₆₁	—	2	4	no	—
118	C₂²×C₆₁	—	2	8	no	—
119	C₂²×C₆₁	—	2	4	no	—
120	C₂²×C₆₁	—	4	4	no	—
121	C₂²×C₆₁	C₂×#41	1	2	yes	—
122	C₂²×C₆₁	C₂⁵	1	1	yes	characteristic
123	C₄×C₆₁	—	2	2	no	—
124	C₄×C₆₁	—	2	4	no	—
125	C₄×C₆₁	—	2	4	no	—
126	C₄×C₆₁	—	2	2	no	—
127	C₄×C₆₁	—	2	8	no	—
128	C₄×C₆₁	—	2	2	no	—
129	C₄×C₆₁	C₂×#41	1	2	yes	—
130	C₄×C₆₁	C₂×#41	1	2	yes	—
131	#3455	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂×C₆₁	—	2	2	no	—
133	C₂×C₆₁	—	4	4	no	—
134	C₂×C₆₁	—	4	2	no	—
135	C₂×C₆₁	C₂²×#41	1	1	yes	characteristic
136	C₂×C₆₁	#585	1	2	yes	—
137	#542	C₂×C₆₁	1	4	yes	complemented
138	#541	C₂×C₆₁	1	2	yes	complemented
139	#540	C₂×C₆₁	1	2	yes	complemented
140	#539	C₂×C₆₁	1	4	yes	complemented
141	#538	C₂×C₆₁	1	4	yes	complemented
142	#533	C₂×C₆₁	1	1	yes	characteristic, complemented
143	#532	C₂×C₆₁	1	1	yes	characteristic, complemented
144	#529	C₂×C₆₁	1	8	yes	complemented
145	#525	C₂×C₆₁	1	2	yes	complemented
146	C₄×#41	C₂×C₆₁	1	2	yes	complemented
147	C₂×#117	C₂×C₆₁	1	1	yes	characteristic, complemented
148	C₆₁	#3455	1	1	yes	characteristic, direct product factor, Sylow
149	C₂×#41	C₂²×C₆₁	1	2	yes	—
150	C₂³×C₄	C₂²×C₆₁	1	1	yes	characteristic
151	C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic, complemented
152	C₄⋊Q₈	C₂²×C₆₁	1	2	yes	complemented
153	#127	C₂²×C₆₁	1	1	yes	characteristic, complemented
154	#126	C₂²×C₆₁	1	8	yes	—
155	#126	C₂²×C₆₁	1	8	yes	complemented
156	#125	C₂²×C₆₁	1	8	yes	complemented
157	#125	C₂²×C₆₁	1	2	yes	complemented
158	#124	C₂²×C₆₁	1	8	yes	—
159	#124	C₂²×C₆₁	1	2	yes	complemented
160	#123	C₂²×C₆₁	1	4	yes	—
161	#123	C₂²×C₆₁	1	8	yes	—
162	#123	C₂²×C₆₁	1	8	yes	complemented
163	C₂²⋊Q₈	C₂²×C₆₁	1	4	yes	—
164	C₂²⋊Q₈	C₂²×C₆₁	1	4	yes	—
165	C₂²⋊Q₈	C₂²×C₆₁	1	8	yes	complemented
166	#121	C₂²×C₆₁	1	4	yes	—
167	#121	C₂²×C₆₁	1	8	yes	—
168	#121	C₂²×C₆₁	1	4	yes	—
169	#120	C₂²×C₆₁	1	4	yes	—
170	C₄×Q₈	C₂²×C₆₁	1	4	yes	complemented
171	C₄×Q₈	C₂²×C₆₁	1	4	yes	complemented
172	C₄×D₄	C₂²×C₆₁	1	8	yes	—
173	C₄×D₄	C₂²×C₆₁	1	8	yes	—
174	C₄×D₄	C₂²×C₆₁	1	4	yes	complemented
175	C₄×D₄	C₂²×C₆₁	1	4	yes	complemented
176	#117	C₂²×C₆₁	1	4	yes	—
177	#117	C₂²×C₆₁	1	2	yes	—
178	#117	C₂²×C₆₁	1	2	yes	—
179	#117	C₂²×C₆₁	1	2	yes	—
180	#117	C₂²×C₆₁	1	4	yes	complemented
181	C₂×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
182	C₂×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
183	C₂×C₄²	C₂²×C₆₁	1	1	yes	characteristic
184	C₂×C₄²	C₂²×C₆₁	1	1	yes	characteristic
185	C₂×C₄²	C₂²×C₆₁	1	4	yes	complemented
186	C₂⁴	C₂³×C₆₁	1	1	yes	characteristic
187	#41	—	2	8	no	—
188	C₂×Q₈	C₂³×C₆₁	1	2	yes	—
189	C₂×Q₈	C₂³×C₆₁	1	4	yes	—
190	C₂×D₄	C₂³×C₆₁	1	4	yes	—
191	C₂×D₄	C₂³×C₆₁	1	2	yes	—
192	C₂×D₄	C₂³×C₆₁	1	8	yes	—
193	C₂×D₄	C₂³×C₆₁	1	4	yes	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	C₂³×C₆₁	1	4	yes	—
196	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
197	C₂²×C₄	C₂³×C₆₁	1	2	yes	—
198	C₂²×C₄	C₂³×C₆₁	1	2	yes	—
199	C₂²×C₄	C₂³×C₆₁	1	2	yes	—
200	C₂²×C₄	C₂³×C₆₁	1	2	yes	—
201	C₂²×C₄	C₂³×C₆₁	1	8	yes	—
202	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
203	C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
204	C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
205	C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
206	C₄⋊C₄	C₂³×C₆₁	1	8	yes	—
207	C₄⋊C₄	C₂³×C₆₁	1	8	yes	—
208	C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
209	C₄⋊C₄	C₂³×C₆₁	1	8	yes	—
210	C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
211	C₂²⋊C₄	C₂³×C₆₁	1	4	yes	—
212	C₂²⋊C₄	C₂³×C₆₁	1	4	yes	—
213	C₂²⋊C₄	C₂³×C₆₁	1	4	yes	—
214	C₂²⋊C₄	C₂³×C₆₁	1	8	yes	—
215	C₂²⋊C₄	C₂³×C₆₁	1	8	yes	—
216	C₂²⋊C₄	C₂³×C₆₁	1	8	yes	—
217	C₂²⋊C₄	C₂³×C₆₁	1	8	yes	—
218	C₄²	C₂³×C₆₁	1	2	yes	—
219	C₄²	C₂³×C₆₁	1	2	yes	—
220	C₄²	C₂³×C₆₁	1	2	yes	—
221	C₄²	C₂³×C₆₁	1	1	yes	characteristic
222	C₄²	C₂³×C₆₁	1	1	yes	characteristic
223	C₄²	C₂³×C₆₁	1	4	yes	—
224	C₄²	C₂³×C₆₁	1	8	yes	—
225	C₂³	—	2	2	no	—
226	C₂³	—	4	2	no	—
227	C₂³	C₂⁴×C₆₁	1	2	yes	—
228	C₂³	C₂⁴×C₆₁	1	4	yes	—
229	C₂³	C₂⁴×C₆₁	1	1	yes	characteristic
230	Q₈	—	2	4	no	—
231	Q₈	—	4	4	no	—
232	D₄	—	2	8	no	—
233	D₄	—	2	4	no	—
234	D₄	—	4	4	no	—
235	D₄	—	4	8	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	8	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	#41×C₆₁	1	4	yes	—
244	C₂×C₄	C₂⁴×C₆₁	1	4	yes	—
245	C₂×C₄	C₂⁴×C₆₁	1	2	yes	—
246	C₂×C₄	C₂⁴×C₆₁	1	4	yes	—
247	C₂×C₄	C₂⁴×C₆₁	1	8	yes	—
248	C₂×C₄	C₂⁴×C₆₁	1	2	yes	—
249	C₂×C₄	C₂⁴×C₆₁	1	2	yes	—
250	C₂×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
251	C₂×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	8	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	4	4	no	—
258	C₂²	C₂×#41×C₆₁	1	2	yes	—
259	C₂²	C₂⁵×C₆₁	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	2	4	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	8	no	—
265	C₄	—	2	2	no	—
266	C₄	C₂×#41×C₆₁	1	2	yes	—
267	C₄	C₂×#41×C₆₁	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	4	2	no	—
270	C₂	—	4	4	no	—
271	C₂	C₂²×#41×C₆₁	1	1	yes	characteristic
272	C₂	C₆₁×#585	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₆₁
socle	C₂²×C₆₁	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#542	C₂	semidirect
2. C₆₁×#541	C₂	semidirect
3. C₆₁×#540	C₂	semidirect
4. C₆₁×#539	C₂	semidirect
5. C₆₁×#538	C₂	semidirect
6. C₆₁×#533	C₂	semidirect
7. C₆₁×#532	C₂	semidirect
8. C₆₁×#529	C₂	semidirect
9. C₆₁×#525	C₂	semidirect
10. C₄×#41×C₆₁	C₂	semidirect
11. C₂×#117×C₆₁	C₂	semidirect
14. C₄⋊Q₈×C₆₁	C₂²	semidirect
15. C₄⋊Q₈×C₆₁	C₂²	semidirect
16. #127×C₆₁	C₂²	semidirect
18. #126×C₆₁	C₂²	semidirect
19. #125×C₆₁	C₂²	semidirect
20. #125×C₆₁	C₂²	semidirect
22. #124×C₆₁	C₂²	semidirect
25. #123×C₆₁	C₂²	semidirect
28. C₂²⋊Q₈×C₆₁	C₂²	semidirect
33. C₄×Q₈×C₆₁	C₂²	semidirect
34. C₄×Q₈×C₆₁	C₂²	semidirect
37. C₄×D₄×C₆₁	C₂²	semidirect
38. C₄×D₄×C₆₁	C₂²	semidirect
43. #117×C₆₁	C₂²	semidirect
48. C₂×C₄²×C₆₁	C₂²	semidirect
131. #3455	C₆₁	direct (N × H)
137. #542	C₂×C₆₁	semidirect
138. #541	C₂×C₆₁	semidirect
139. #540	C₂×C₆₁	semidirect
140. #539	C₂×C₆₁	semidirect
141. #538	C₂×C₆₁	semidirect
142. #533	C₂×C₆₁	semidirect
143. #532	C₂×C₆₁	semidirect
144. #529	C₂×C₆₁	semidirect
145. #525	C₂×C₆₁	semidirect
146. C₄×#41	C₂×C₆₁	semidirect
147. C₂×#117	C₂×C₆₁	semidirect
148. C₆₁	#3455	direct (N × H)
151. C₄⋊Q₈	C₂²×C₆₁	semidirect
152. C₄⋊Q₈	C₂²×C₆₁	semidirect
153. #127	C₂²×C₆₁	semidirect
155. #126	C₂²×C₆₁	semidirect
156. #125	C₂²×C₆₁	semidirect
157. #125	C₂²×C₆₁	semidirect
159. #124	C₂²×C₆₁	semidirect
162. #123	C₂²×C₆₁	semidirect
165. C₂²⋊Q₈	C₂²×C₆₁	semidirect
170. C₄×Q₈	C₂²×C₆₁	semidirect
171. C₄×Q₈	C₂²×C₆₁	semidirect
174. C₄×D₄	C₂²×C₆₁	semidirect
175. C₄×D₄	C₂²×C₆₁	semidirect
180. #117	C₂²×C₆₁	semidirect
185. C₂×C₄²	C₂²×C₆₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₆₁×#542	2
C₆₁×#541	2
C₆₁×#540	2
C₆₁×#539	2
C₆₁×#538	2
C₆₁×#533	2
C₆₁×#532	2
C₆₁×#529	2
C₆₁×#525	2
C₄×#41×C₆₁	2
C₂×#117×C₆₁	2
#3455	61

Subgroups — C61×#3455 #21443496

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#3455 #21443496