Subgroups — C₆₁×#3172 #21443213

Order 7808.

364 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#529	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#501	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#497	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#491	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#489	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#470	C₂	1	2	yes	maximal, complemented
8	C₆₁×#468	C₂	1	2	yes	maximal
9	C₄×Q₁₆×C₆₁	C₂	1	2	yes	maximal, complemented
10	C₄×SD₁₆×C₆₁	C₂	1	2	yes	maximal, complemented
11	C₄×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×C₆₁	C₂²	1	2	yes	—
13	C₂×SD₁₆×C₆₁	C₂²	1	2	yes	—
14	C₂×M₁₆×C₆₁	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
17	#126×C₆₁	—	2	1	no	—
18	#125×C₆₁	—	2	1	no	—
19	#125×C₆₁	C₂²	1	1	yes	characteristic
20	#124×C₆₁	C₂²	1	1	yes	characteristic
21	#123×C₆₁	—	2	1	no	—
22	C₂²⋊Q₈×C₆₁	—	2	1	no	—
23	C₂²⋊Q₈×C₆₁	C₂²	1	2	yes	—
24	C₂²⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
25	#121×C₆₁	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₆₁	—	2	1	no	—
27	C₄×Q₈×C₆₁	C₂²	1	2	yes	—
28	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
29	C₄×D₄×C₆₁	—	2	1	no	—
30	C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
31	#117×C₆₁	—	2	1	no	—
32	#117×C₆₁	—	2	1	no	—
33	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
34	#107×C₆₁	C₂²	1	2	yes	—
35	#106×C₆₁	C₂²	1	2	yes	—
36	#103×C₆₁	C₂²	1	2	yes	—
37	#103×C₆₁	C₂²	1	2	yes	—
38	#103×C₆₁	C₂²	1	2	yes	—
39	#102×C₆₁	C₂²	1	2	yes	—
40	#97×C₆₁	C₂²	1	1	yes	characteristic
41	#97×C₆₁	C₂²	1	1	yes	characteristic
42	C₄×C₈×C₆₁	C₂²	1	2	yes	—
43	C₂×Q₈×C₆₁	—	2	1	no	—
44	C₂×Q₈×C₆₁	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₆₁	C₂³	1	2	yes	—
46	C₂×D₄×C₆₁	—	2	1	no	—
47	C₂×D₄×C₆₁	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	—	2	1	no	—
49	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
52	Q₁₆×C₆₁	—	2	4	no	—
53	SD₁₆×C₆₁	—	2	4	no	—
54	M₁₆×C₆₁	—	2	4	no	—
55	C₂×C₈×C₆₁	C₂³	1	2	yes	—
56	C₂×C₈×C₆₁	C₂³	1	2	yes	—
57	C₄⋊C₄×C₆₁	—	2	1	no	—
58	C₄⋊C₄×C₆₁	—	2	1	no	—
59	C₄⋊C₄×C₆₁	—	2	1	no	—
60	C₄⋊C₄×C₆₁	—	2	1	no	—
61	C₄⋊C₄×C₆₁	—	2	1	no	—
62	C₄⋊C₄×C₆₁	—	2	1	no	—
63	C₄⋊C₄×C₆₁	—	2	2	no	—
64	C₄⋊C₄×C₆₁	—	2	2	no	—
65	C₄⋊C₄×C₆₁	—	2	1	no	—
66	C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
67	C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
69	C₂²⋊C₄×C₆₁	—	2	2	no	—
70	C₂²⋊C₄×C₆₁	—	2	1	no	—
71	C₂²⋊C₄×C₆₁	—	2	1	no	—
72	C₂²⋊C₄×C₆₁	—	2	1	no	—
73	C₂²⋊C₄×C₆₁	—	2	1	no	—
74	C₂²⋊C₄×C₆₁	—	2	1	no	—
75	C₄²×C₆₁	—	2	1	no	—
76	C₄²×C₆₁	—	2	2	no	—
77	C₄²×C₆₁	D₄	1	1	yes	characteristic
78	C₄²×C₆₁	D₄	1	1	yes	characteristic
79	C₄²×C₆₁	C₂³	1	1	yes	characteristic
80	C₄²×C₆₁	C₂³	1	1	yes	characteristic
81	C₂³×C₆₁	—	2	1	no	—
82	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
83	Q₈×C₆₁	—	2	2	no	—
84	Q₈×C₆₁	—	2	2	no	—
85	Q₈×C₆₁	—	2	2	no	—
86	Q₈×C₆₁	—	4	2	no	—
87	D₄×C₆₁	—	2	2	no	—
88	D₄×C₆₁	—	4	2	no	—
89	C₂×C₄×C₆₁	—	2	2	no	—
90	C₂×C₄×C₆₁	—	2	2	no	—
91	C₂×C₄×C₆₁	—	2	1	no	—
92	C₂×C₄×C₆₁	—	2	2	no	—
93	C₂×C₄×C₆₁	—	2	1	no	—
94	C₂×C₄×C₆₁	—	2	2	no	—
95	C₂×C₄×C₆₁	—	2	1	no	—
96	C₂×C₄×C₆₁	—	4	1	no	—
97	C₂×C₄×C₆₁	—	4	2	no	—
98	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
104	C₈×C₆₁	—	2	2	no	—
105	C₈×C₆₁	#41	1	4	yes	—
106	C₂²×C₆₁	—	2	1	no	—
107	C₂²×C₆₁	—	2	1	no	—
108	C₂²×C₆₁	—	2	1	no	—
109	C₂²×C₆₁	—	4	1	no	—
110	C₂²×C₆₁	—	4	1	no	—
111	C₂²×C₆₁	—	4	1	no	—
112	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
113	C₄×C₆₁	—	2	1	no	—
114	C₄×C₆₁	—	2	1	no	—
115	C₄×C₆₁	—	2	1	no	—
116	C₄×C₆₁	—	2	1	no	—
117	C₄×C₆₁	—	4	1	no	—
118	C₄×C₆₁	—	4	2	no	—
119	C₄×C₆₁	—	4	1	no	—
120	C₄×C₆₁	—	4	1	no	—
121	C₄×C₆₁	—	4	2	no	—
122	C₄×C₆₁	#137	1	2	yes	—
123	C₄×C₆₁	C₂×#41	1	2	yes	—
124	#3172	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂×C₆₁	—	4	1	no	—
126	C₂×C₆₁	—	8	1	no	—
127	C₂×C₆₁	#532	1	1	yes	characteristic
128	C₂×C₆₁	C₂×#137	1	1	yes	characteristic
129	C₂×C₆₁	#575	1	1	yes	characteristic
130	#533	C₂×C₆₁	1	1	yes	characteristic, complemented
131	#529	C₂×C₆₁	1	1	yes	characteristic
132	#501	C₂×C₆₁	1	1	yes	characteristic, complemented
133	#497	C₂×C₆₁	1	1	yes	characteristic, complemented
134	#491	C₂×C₆₁	1	1	yes	characteristic, complemented
135	#489	C₂×C₆₁	1	1	yes	characteristic, complemented
136	#470	C₂×C₆₁	1	2	yes	complemented
137	#468	C₂×C₆₁	1	2	yes	—
138	C₄×Q₁₆	C₂×C₆₁	1	2	yes	complemented
139	C₄×SD₁₆	C₂×C₆₁	1	2	yes	complemented
140	C₄×M₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
141	C₆₁	#3172	1	1	yes	characteristic, direct product factor, Sylow
142	C₂×Q₁₆	C₂²×C₆₁	1	2	yes	—
143	C₂×SD₁₆	C₂²×C₆₁	1	2	yes	—
144	C₂×M₁₆	C₂²×C₆₁	1	2	yes	—
145	C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
146	C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
147	#126	—	2	1	no	—
148	#125	—	2	1	no	—
149	#125	C₂²×C₆₁	1	1	yes	characteristic
150	#124	C₂²×C₆₁	1	1	yes	characteristic
151	#123	—	2	1	no	—
152	C₂²⋊Q₈	—	2	1	no	—
153	C₂²⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
154	C₂²⋊Q₈	C₂²×C₆₁	1	2	yes	—
155	#121	C₂²×C₆₁	1	1	yes	characteristic
156	C₄×Q₈	—	2	1	no	—
157	C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
158	C₄×Q₈	C₂²×C₆₁	1	2	yes	—
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
161	#117	—	2	1	no	—
162	#117	—	2	1	no	—
163	C₂×C₄²	C₂²×C₆₁	1	1	yes	characteristic
164	#107	C₂²×C₆₁	1	2	yes	—
165	#106	C₂²×C₆₁	1	2	yes	—
166	#103	C₂²×C₆₁	1	2	yes	—
167	#103	C₂²×C₆₁	1	2	yes	—
168	#103	C₂²×C₆₁	1	2	yes	—
169	#102	C₂²×C₆₁	1	2	yes	—
170	#97	C₂²×C₆₁	1	1	yes	characteristic
171	#97	C₂²×C₆₁	1	1	yes	characteristic
172	C₄×C₈	C₂²×C₆₁	1	2	yes	—
173	C₂×Q₈	—	2	1	no	—
174	C₂×Q₈	C₂³×C₆₁	1	1	yes	characteristic
175	C₂×Q₈	C₂³×C₆₁	1	2	yes	—
176	C₂×D₄	—	2	1	no	—
177	C₂×D₄	C₂³×C₆₁	1	1	yes	characteristic
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	D₄×C₆₁	1	1	yes	characteristic
180	C₂²×C₄	D₄×C₆₁	1	1	yes	characteristic
181	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
182	Q₁₆	—	2	4	no	—
183	SD₁₆	—	2	4	no	—
184	M₁₆	—	2	4	no	—
185	C₂×C₈	C₂³×C₆₁	1	2	yes	—
186	C₂×C₈	C₂³×C₆₁	1	2	yes	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
197	C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
198	C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
199	C₂²⋊C₄	—	2	2	no	—
200	C₂²⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	2	no	—
207	C₄²	D₄×C₆₁	1	1	yes	characteristic
208	C₄²	D₄×C₆₁	1	1	yes	characteristic
209	C₄²	C₂³×C₆₁	1	1	yes	characteristic
210	C₄²	C₂³×C₆₁	1	1	yes	characteristic
211	C₂³	—	2	1	no	—
212	C₂³	C₂×D₄×C₆₁	1	1	yes	characteristic
213	Q₈	—	2	2	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	2	2	no	—
216	Q₈	—	4	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	4	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
229	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
230	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
231	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
232	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
233	C₂×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
234	C₈	—	2	2	no	—
235	C₈	#41×C₆₁	1	4	yes	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	4	1	no	—
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	C₂²×D₄×C₆₁	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	4	1	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	1	no	—
251	C₄	—	4	2	no	—
252	C₄	#137×C₆₁	1	2	yes	—
253	C₄	C₂×#41×C₆₁	1	2	yes	—
254	C₂	—	4	1	no	—
255	C₂	—	8	1	no	—
256	C₂	C₆₁×#532	1	1	yes	characteristic
257	C₂	C₂×#137×C₆₁	1	1	yes	characteristic
258	C₂	C₆₁×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₆₁
socle	C₂²×C₆₁	C₂²×D₄
prime radical	#121×C₆₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#533	C₂	semidirect
3. C₆₁×#501	C₂	semidirect
4. C₆₁×#497	C₂	semidirect
5. C₆₁×#491	C₂	semidirect
6. C₆₁×#489	C₂	semidirect
7. C₆₁×#470	C₂	semidirect
9. C₄×Q₁₆×C₆₁	C₂	semidirect
10. C₄×SD₁₆×C₆₁	C₂	semidirect
11. C₄×M₁₆×C₆₁	C₂	semidirect
124. #3172	C₆₁	direct (N × H)
130. #533	C₂×C₆₁	semidirect
132. #501	C₂×C₆₁	semidirect
133. #497	C₂×C₆₁	semidirect
134. #491	C₂×C₆₁	semidirect
135. #489	C₂×C₆₁	semidirect
136. #470	C₂×C₆₁	semidirect
138. C₄×Q₁₆	C₂×C₆₁	semidirect
139. C₄×SD₁₆	C₂×C₆₁	semidirect
140. C₄×M₁₆	C₂×C₆₁	semidirect
141. C₆₁	#3172	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×#533	2
C₆₁×#529	2
C₆₁×#501	2
C₆₁×#497	2
C₆₁×#491	2
C₆₁×#489	2
C₆₁×#470	2
C₆₁×#468	2
C₄×Q₁₆×C₆₁	2
C₄×SD₁₆×C₆₁	2
C₄×M₁₆×C₆₁	2
#3172	61

Subgroups — C61×#3172 #21443213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#3172 #21443213