Subgroups — C₆₁×#2947 #21442988

Order 7808.

470 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#137×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#520	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#519	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#441	C₂	1	2	yes	maximal, complemented
5	C₆₁×#440	C₂	1	2	yes	maximal, complemented
6	C₄×Q₁₆×C₆₁	C₂	1	2	yes	maximal, complemented
7	C₄×SD₁₆×C₆₁	C₂	1	2	yes	maximal, complemented
8	C₆₁×#428	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#416	C₂	1	1	yes	maximal, characteristic
10	C₂×#103×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₆₁	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₆₁	C₂²	1	1	yes	characteristic
14	#137×C₆₁	C₄	1	8	yes	complemented
15	C₂×Q₁₆×C₆₁	C₂²	1	2	yes	—
16	C₂×SD₁₆×C₆₁	C₂²	1	2	yes	—
17	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₆₁	—	2	1	no	—
20	C₄×Q₈×C₆₁	—	2	1	no	—
21	C₄×Q₈×C₆₁	C₂²	1	2	yes	—
22	C₄×Q₈×C₆₁	C₂²	1	2	yes	—
23	C₄×Q₈×C₆₁	C₂²	1	2	yes	—
24	C₄×D₄×C₆₁	—	2	2	no	—
25	C₄×D₄×C₆₁	C₂²	1	2	yes	—
26	#117×C₆₁	—	2	1	no	—
27	#117×C₆₁	—	2	2	no	—
28	#117×C₆₁	C₂²	1	1	yes	characteristic
29	#117×C₆₁	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
31	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
32	#107×C₆₁	C₂²	1	2	yes	—
33	#106×C₆₁	C₂²	1	2	yes	—
34	#103×C₆₁	C₂²	1	2	yes	—
35	#103×C₆₁	C₂²	1	2	yes	—
36	#103×C₆₁	C₂²	1	1	yes	characteristic
37	#103×C₆₁	C₂²	1	1	yes	characteristic
38	#102×C₆₁	C₂²	1	1	yes	characteristic
39	#102×C₆₁	C₂²	1	1	yes	characteristic
40	#97×C₆₁	C₂²	1	2	yes	—
41	C₄×C₈×C₆₁	C₂²	1	2	yes	—
42	#41×C₆₁	—	2	2	no	—
43	#41×C₆₁	C₂×C₄	1	4	yes	—
44	C₂×Q₈×C₆₁	—	2	1	no	—
45	C₂×Q₈×C₆₁	—	2	2	no	—
46	C₂×Q₈×C₆₁	C₂×C₄	1	4	yes	complemented
47	C₂×Q₈×C₆₁	C₂³	1	2	yes	—
48	C₂×Q₈×C₆₁	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₆₁	—	2	1	no	—
50	C₂×D₄×C₆₁	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	—	2	1	no	—
52	C₂²×C₄×C₆₁	—	2	1	no	—
53	C₂²×C₄×C₆₁	—	2	1	no	—
54	C₂²×C₄×C₆₁	—	2	2	no	—
55	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
56	Q₁₆×C₆₁	C₂×C₄	1	8	yes	complemented
57	SD₁₆×C₆₁	C₂×C₄	1	8	yes	complemented
58	M₁₆×C₆₁	C₂×C₄	1	4	yes	complemented
59	C₂×C₈×C₆₁	—	2	2	no	—
60	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₆₁	C₂³	1	2	yes	—
63	C₄⋊C₄×C₆₁	—	2	1	no	—
64	C₄⋊C₄×C₆₁	—	2	2	no	—
65	C₄⋊C₄×C₆₁	—	2	2	no	—
66	C₄⋊C₄×C₆₁	—	2	2	no	—
67	C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
69	C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
72	C₂²⋊C₄×C₆₁	—	2	1	no	—
73	C₂²⋊C₄×C₆₁	—	2	2	no	—
74	C₂²⋊C₄×C₆₁	D₄	1	2	yes	—
75	C₄²×C₆₁	—	2	2	no	—
76	C₄²×C₆₁	—	2	2	no	—
77	C₄²×C₆₁	—	2	2	no	—
78	C₄²×C₆₁	—	2	1	no	—
79	C₄²×C₆₁	C₂³	1	2	yes	—
80	C₂³×C₆₁	—	2	1	no	—
81	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
82	Q₈×C₆₁	—	2	4	no	—
83	Q₈×C₆₁	—	2	2	no	—
84	Q₈×C₆₁	—	2	1	no	—
85	Q₈×C₆₁	C₂²×C₄	1	4	yes	—
86	Q₈×C₆₁	C₂²×C₄	1	2	yes	—
87	D₄×C₆₁	—	2	4	no	—
88	D₄×C₆₁	—	2	1	no	—
89	D₄×C₆₁	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₆₁	—	2	4	no	—
91	C₂×C₄×C₆₁	—	2	1	no	—
92	C₂×C₄×C₆₁	—	2	2	no	—
93	C₂×C₄×C₆₁	—	2	4	no	—
94	C₂×C₄×C₆₁	—	2	1	no	—
95	C₂×C₄×C₆₁	—	2	1	no	—
96	C₂×C₄×C₆₁	—	2	2	no	—
97	C₂×C₄×C₆₁	—	8	2	no	—
98	C₂×C₄×C₆₁	—	8	1	no	—
99	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
101	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₆₁	#41	1	2	yes	—
104	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
105	C₈×C₆₁	—	2	1	no	—
106	C₈×C₆₁	—	2	1	no	—
107	C₈×C₆₁	C₂²×C₄	1	4	yes	—
108	C₂²×C₆₁	—	2	1	no	—
109	C₂²×C₆₁	—	2	1	no	—
110	C₂²×C₆₁	—	2	2	no	—
111	C₂²×C₆₁	—	4	1	no	—
112	C₂²×C₆₁	—	4	1	no	—
113	C₂²×C₆₁	C₄×D₄	1	2	yes	—
114	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₆₁	—	2	2	no	—
116	C₄×C₆₁	—	2	4	no	—
117	C₄×C₆₁	—	2	2	no	—
118	C₄×C₆₁	—	2	2	no	—
119	C₄×C₆₁	—	4	1	no	—
120	C₄×C₆₁	—	4	2	no	—
121	C₄×C₆₁	—	4	1	no	—
122	C₄×C₆₁	C₄×D₄	1	2	yes	—
123	C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
124	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
125	#2947	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂×C₆₁	—	2	2	no	—
127	C₂×C₆₁	—	4	2	no	—
128	C₂×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
129	C₂×C₆₁	#577	1	1	yes	characteristic
130	C₂×C₆₁	#578	1	1	yes	characteristic
131	C₂×#137	C₂×C₆₁	1	1	yes	characteristic
132	#520	C₂×C₆₁	1	1	yes	characteristic
133	#519	C₂×C₆₁	1	1	yes	characteristic, complemented
134	#441	C₂×C₆₁	1	2	yes	complemented
135	#440	C₂×C₆₁	1	2	yes	complemented
136	C₄×Q₁₆	C₂×C₆₁	1	2	yes	complemented
137	C₄×SD₁₆	C₂×C₆₁	1	2	yes	complemented
138	#428	C₂×C₆₁	1	1	yes	characteristic, complemented
139	#416	C₂×C₆₁	1	1	yes	characteristic
140	C₂×#103	C₂×C₆₁	1	1	yes	characteristic, complemented
141	#405	C₂×C₆₁	1	1	yes	characteristic, complemented
142	C₆₁	#2947	1	1	yes	characteristic, direct product factor, Sylow
143	C₂×#41	C₂²×C₆₁	1	1	yes	characteristic
144	C₂²×Q₈	C₂²×C₆₁	1	1	yes	characteristic
145	#137	C₄×C₆₁	1	8	yes	complemented
146	C₂×Q₁₆	C₂²×C₆₁	1	2	yes	—
147	C₂×SD₁₆	C₂²×C₆₁	1	2	yes	—
148	C₂×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
149	C₂²×C₈	C₂²×C₆₁	1	1	yes	characteristic
150	C₄×Q₈	—	2	1	no	—
151	C₄×Q₈	—	2	1	no	—
152	C₄×Q₈	C₂²×C₆₁	1	2	yes	—
153	C₄×Q₈	C₂²×C₆₁	1	2	yes	—
154	C₄×Q₈	C₂²×C₆₁	1	2	yes	—
155	C₄×D₄	—	2	2	no	—
156	C₄×D₄	C₂²×C₆₁	1	2	yes	—
157	#117	—	2	1	no	—
158	#117	—	2	2	no	—
159	#117	C₂²×C₆₁	1	1	yes	characteristic
160	#117	C₂²×C₆₁	1	1	yes	characteristic
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
163	#107	C₂²×C₆₁	1	2	yes	—
164	#106	C₂²×C₆₁	1	2	yes	—
165	#103	C₂²×C₆₁	1	2	yes	—
166	#103	C₂²×C₆₁	1	2	yes	—
167	#103	C₂²×C₆₁	1	1	yes	characteristic
168	#103	C₂²×C₆₁	1	1	yes	characteristic
169	#102	C₂²×C₆₁	1	1	yes	characteristic
170	#102	C₂²×C₆₁	1	1	yes	characteristic
171	#97	C₂²×C₆₁	1	2	yes	—
172	C₄×C₈	C₂²×C₆₁	1	2	yes	—
173	#41	—	2	2	no	—
174	#41	C₂×C₄×C₆₁	1	4	yes	—
175	C₂×Q₈	—	2	1	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	C₂×C₄×C₆₁	1	4	yes	complemented
178	C₂×Q₈	C₂³×C₆₁	1	2	yes	—
179	C₂×Q₈	C₂³×C₆₁	1	1	yes	characteristic
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	C₂³×C₆₁	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
187	Q₁₆	C₂×C₄×C₆₁	1	8	yes	complemented
188	SD₁₆	C₂×C₄×C₆₁	1	8	yes	complemented
189	M₁₆	C₂×C₄×C₆₁	1	4	yes	complemented
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	C₂³×C₆₁	1	1	yes	characteristic
192	C₂×C₈	C₂³×C₆₁	1	1	yes	characteristic
193	C₂×C₈	C₂³×C₆₁	1	2	yes	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
199	C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
200	C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
201	C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
202	C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	D₄×C₆₁	1	2	yes	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	1	no	—
210	C₄²	C₂³×C₆₁	1	2	yes	—
211	C₂³	—	2	1	no	—
212	C₂³	C₂×D₄×C₆₁	1	1	yes	characteristic
213	Q₈	—	2	4	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	2	1	no	—
216	Q₈	C₂²×C₄×C₆₁	1	4	yes	—
217	Q₈	C₂²×C₄×C₆₁	1	2	yes	—
218	D₄	—	2	4	no	—
219	D₄	—	2	1	no	—
220	D₄	C₂²×C₄×C₆₁	1	2	yes	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	8	1	no	—
230	C₂×C₄	C₂²×C₄×C₆₁	1	2	yes	—
231	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
232	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
233	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
234	C₂×C₄	#41×C₆₁	1	2	yes	—
235	C₂×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	C₂²×C₄×C₆₁	1	4	yes	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	C₄×D₄×C₆₁	1	2	yes	—
245	C₂²	C₂²×D₄×C₆₁	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	4	no	—
250	C₄	—	4	1	no	—
251	C₄	—	4	2	no	—
252	C₄	—	4	1	no	—
253	C₄	C₄×D₄×C₆₁	1	2	yes	—
254	C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
255	C₄	C₂×#41×C₆₁	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	4	2	no	—
258	C₂	C₂×C₄×D₄×C₆₁	1	1	yes	characteristic
259	C₂	C₆₁×#577	1	1	yes	characteristic
260	C₂	C₆₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₄
commutator	C₄	C₂³×C₄×C₆₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₆₁
prime radical	C₂×#41×C₆₁	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ #117×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₁×#519	C₂	semidirect
4. C₆₁×#441	C₂	semidirect
5. C₆₁×#440	C₂	semidirect
6. C₄×Q₁₆×C₆₁	C₂	semidirect
7. C₄×SD₁₆×C₆₁	C₂	semidirect
8. C₆₁×#428	C₂	semidirect
10. C₂×#103×C₆₁	C₂	semidirect
11. C₆₁×#405	C₂	semidirect
14. #137×C₆₁	C₄	semidirect
46. C₂×Q₈×C₆₁	C₂×C₄	semidirect
56. Q₁₆×C₆₁	C₂×C₄	semidirect
57. SD₁₆×C₆₁	C₂×C₄	semidirect
58. M₁₆×C₆₁	C₂×C₄	semidirect
125. #2947	C₆₁	direct (N × H)
133. #519	C₂×C₆₁	semidirect
134. #441	C₂×C₆₁	semidirect
135. #440	C₂×C₆₁	semidirect
136. C₄×Q₁₆	C₂×C₆₁	semidirect
137. C₄×SD₁₆	C₂×C₆₁	semidirect
138. #428	C₂×C₆₁	semidirect
140. C₂×#103	C₂×C₆₁	semidirect
141. #405	C₂×C₆₁	semidirect
142. C₆₁	#2947	direct (N × H)
145. #137	C₄×C₆₁	semidirect
177. C₂×Q₈	C₂×C₄×C₆₁	semidirect
187. Q₁₆	C₂×C₄×C₆₁	semidirect
188. SD₁₆	C₂×C₄×C₆₁	semidirect
189. M₁₆	C₂×C₄×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#137×C₆₁	2
C₆₁×#520	2
C₆₁×#519	2
C₆₁×#441	2
C₆₁×#440	2
C₄×Q₁₆×C₆₁	2
C₄×SD₁₆×C₆₁	2
C₆₁×#428	2
C₆₁×#416	2
C₂×#103×C₆₁	2
C₆₁×#405	2
#2947	61

Subgroups — C61×#2947 #21442988

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#2947 #21442988