Subgroups — C₆₁×#20817 #29524250

Order 15616.

982 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2285	C₂	1	2	yes	maximal
3	C₆₁×#1845	C₂	1	2	yes	maximal, complemented
4	C₆₁×#1834	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1808	C₂	1	4	yes	maximal
6	C₄×#107×C₆₁	C₂	1	4	yes	maximal, complemented
7	C₂³×C₈×C₆₁	C₂²	1	2	yes	—
8	C₂×#117×C₆₁	—	2	2	no	—
9	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×#107×C₆₁	C₂²	1	8	yes	—
11	C₂×C₄×C₈×C₆₁	C₄	1	8	yes	complemented
12	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#384	C₂²	1	4	yes	—
16	C₆₁×#382	—	2	2	no	—
17	C₆₁×#380	—	2	2	no	—
18	C₆₁×#380	C₂²	1	4	yes	—
19	C₄×C₄⋊C₄×C₆₁	—	2	2	no	—
20	C₄×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
21	C₆₁×#340	C₂²	1	8	yes	—
22	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₆₁	—	2	8	no	—
26	C₂²×C₈×C₆₁	C₂×C₄	1	8	yes	—
27	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
28	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
29	#117×C₆₁	—	4	8	no	—
30	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
31	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
32	C₂×C₄⋊C₄×C₆₁	C₂³	1	8	yes	—
33	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
34	C₂×C₄²×C₆₁	—	2	4	no	—
35	C₂×C₄²×C₆₁	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₁	Q₈	1	4	yes	—
39	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
41	#107×C₆₁	—	2	16	no	—
42	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₆₁	C₂×C₄	1	8	yes	—
44	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
45	#95×C₆₁	—	2	4	no	—
46	#95×C₆₁	—	2	4	no	—
47	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	—	2	4	no	—
49	C₂²×C₄×C₆₁	—	2	4	no	—
50	C₂²×C₄×C₆₁	—	2	8	no	—
51	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₆₁	D₈	1	4	yes	—
54	C₂²×C₄×C₆₁	Q₁₆	1	4	yes	—
55	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
63	C₂×C₈×C₆₁	—	2	4	no	—
64	C₂×C₈×C₆₁	—	2	16	no	—
65	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
66	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
67	C₂×C₈×C₆₁	#41	1	8	yes	—
68	C₄⋊C₄×C₆₁	—	2	16	no	—
69	C₄⋊C₄×C₆₁	—	4	8	no	—
70	C₄⋊C₄×C₆₁	—	4	8	no	—
71	C₂²⋊C₄×C₆₁	—	4	8	no	—
72	C₄²×C₆₁	—	4	8	no	—
73	C₄²×C₆₁	C₄⋊C₄	1	4	yes	—
74	C₄²×C₆₁	C₄⋊C₄	1	4	yes	—
75	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
76	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
77	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
78	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
79	C₂³×C₆₁	—	2	2	no	—
80	C₂³×C₆₁	—	2	2	no	—
81	C₂³×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
82	C₂³×C₆₁	C₂×D₈	1	2	yes	—
83	C₂³×C₆₁	C₂×Q₁₆	1	2	yes	—
84	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
85	C₂×C₄×C₆₁	—	2	2	no	—
86	C₂×C₄×C₆₁	—	2	2	no	—
87	C₂×C₄×C₆₁	—	2	4	no	—
88	C₂×C₄×C₆₁	—	2	4	no	—
89	C₂×C₄×C₆₁	—	2	8	no	—
90	C₂×C₄×C₆₁	—	4	8	no	—
91	C₂×C₄×C₆₁	—	4	8	no	—
92	C₂×C₄×C₆₁	—	4	8	no	—
93	C₂×C₄×C₆₁	#107	1	16	yes	complemented
94	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
100	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
101	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
102	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
103	C₂×C₄×C₆₁	#135	1	4	yes	—
104	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₂×#41	1	4	yes	—
107	C₈×C₆₁	—	2	4	no	—
108	C₈×C₆₁	#117	1	8	yes	complemented
109	#20817	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂²×C₆₁	—	2	2	no	—
111	C₂²×C₆₁	—	2	4	no	—
112	C₂²×C₆₁	—	2	4	no	—
113	C₂²×C₆₁	—	2	2	no	—
114	C₂²×C₆₁	C₂×#107	1	4	yes	—
115	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₆₁	#532	1	2	yes	—
117	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
118	C₂²×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
119	C₂²×C₆₁	C₂×#135	1	2	yes	—
120	C₄×C₆₁	—	2	2	no	—
121	C₄×C₆₁	—	2	2	no	—
122	C₄×C₆₁	—	8	8	no	—
123	C₄×C₆₁	C₂×#107	1	4	yes	—
124	C₄×C₆₁	C₂×#107	1	4	yes	—
125	C₄×C₆₁	#427	1	4	yes	—
126	C₄×C₆₁	C₂×#117	1	2	yes	—
127	C₄×C₆₁	#533	1	2	yes	—
128	C₂²×C₄×C₈	C₂×C₆₁	1	1	yes	characteristic
129	#2285	C₂×C₆₁	1	2	yes	—
130	#1845	C₂×C₆₁	1	2	yes	complemented
131	#1834	C₂×C₆₁	1	2	yes	complemented
132	#1808	C₂×C₆₁	1	4	yes	—
133	C₄×#107	C₂×C₆₁	1	4	yes	complemented
134	C₂×C₆₁	—	2	4	no	—
135	C₂×C₆₁	#2285	1	1	yes	characteristic
136	C₂×C₆₁	C₂²×#107	1	1	yes	characteristic
137	C₂×C₆₁	C₂×#427	1	1	yes	characteristic
138	C₂×C₆₁	#3169	1	2	yes	—
139	C₂×C₆₁	#3170	1	2	yes	—
140	C₂³×C₈	C₂²×C₆₁	1	2	yes	—
141	C₂×#117	—	2	2	no	—
142	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
143	C₂×#107	C₂²×C₆₁	1	8	yes	—
144	C₂×C₄×C₈	C₄×C₆₁	1	8	yes	complemented
145	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
148	#384	C₂²×C₆₁	1	4	yes	—
149	#382	—	2	2	no	—
150	#380	—	2	2	no	—
151	#380	C₂²×C₆₁	1	4	yes	—
152	C₄×C₄⋊C₄	—	2	2	no	—
153	C₄×C₄⋊C₄	C₂²×C₆₁	1	4	yes	—
154	#340	C₂²×C₆₁	1	8	yes	—
155	C₆₁	#20817	1	1	yes	characteristic, direct product factor, Sylow
156	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
157	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
158	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
159	C₂²×C₈	—	2	8	no	—
160	C₂²×C₈	C₂×C₄×C₆₁	1	8	yes	—
161	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
162	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
163	#117	—	4	8	no	—
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄⋊C₄	—	2	4	no	—
166	C₂×C₄⋊C₄	C₂³×C₆₁	1	8	yes	—
167	C₂×C₂²⋊C₄	—	2	4	no	—
168	C₂×C₄²	—	2	4	no	—
169	C₂×C₄²	C₂×C₄×C₆₁	1	4	yes	—
170	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
171	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
172	C₂×C₄²	Q₈×C₆₁	1	4	yes	—
173	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
174	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
175	#107	—	2	16	no	—
176	C₄×C₈	C₂×C₄×C₆₁	1	4	yes	—
177	C₄×C₈	C₂×C₄×C₆₁	1	8	yes	—
178	C₄×C₈	C₂×C₄×C₆₁	1	4	yes	—
179	#95	—	2	4	no	—
180	#95	—	2	4	no	—
181	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	C₄⋊C₄×C₆₁	1	4	yes	—
186	C₂²×C₄	C₄⋊C₄×C₆₁	1	4	yes	—
187	C₂²×C₄	D₈×C₆₁	1	4	yes	—
188	C₂²×C₄	Q₁₆×C₆₁	1	4	yes	—
189	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
190	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
195	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
196	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	16	no	—
199	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
200	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
201	C₂×C₈	#41×C₆₁	1	8	yes	—
202	C₄⋊C₄	—	2	16	no	—
203	C₄⋊C₄	—	4	8	no	—
204	C₄⋊C₄	—	4	8	no	—
205	C₂²⋊C₄	—	4	8	no	—
206	C₄²	—	4	8	no	—
207	C₄²	C₄⋊C₄×C₆₁	1	4	yes	—
208	C₄²	C₄⋊C₄×C₆₁	1	4	yes	—
209	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
210	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
211	C₄²	C₂×Q₈×C₆₁	1	2	yes	—
212	C₄²	C₂×Q₈×C₆₁	1	2	yes	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
216	C₂³	C₂×D₈×C₆₁	1	2	yes	—
217	C₂³	C₂×Q₁₆×C₆₁	1	2	yes	—
218	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	4	8	no	—
225	C₂×C₄	—	4	8	no	—
226	C₂×C₄	—	4	8	no	—
227	C₂×C₄	#107×C₆₁	1	16	yes	complemented
228	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
229	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
230	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
231	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
232	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
233	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
234	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
235	C₂×C₄	C₂×Q₁₆×C₆₁	1	2	yes	—
236	C₂×C₄	C₂×Q₁₆×C₆₁	1	2	yes	—
237	C₂×C₄	#135×C₆₁	1	4	yes	—
238	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
239	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
240	C₂×C₄	C₂×#41×C₆₁	1	4	yes	—
241	C₈	—	2	4	no	—
242	C₈	#117×C₆₁	1	8	yes	complemented
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	4	no	—
247	C₂²	C₂×#107×C₆₁	1	4	yes	—
248	C₂²	C₂²×C₄⋊C₄×C₆₁	1	1	yes	characteristic
249	C₂²	C₆₁×#532	1	2	yes	—
250	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
251	C₂²	C₂²×Q₁₆×C₆₁	1	1	yes	characteristic
252	C₂²	C₂×#135×C₆₁	1	2	yes	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	8	8	no	—
256	C₄	C₂×#107×C₆₁	1	4	yes	—
257	C₄	C₂×#107×C₆₁	1	4	yes	—
258	C₄	C₆₁×#427	1	4	yes	—
259	C₄	C₂×#117×C₆₁	1	2	yes	—
260	C₄	C₆₁×#533	1	2	yes	—
261	C₂	—	2	4	no	—
262	C₂	C₆₁×#2285	1	1	yes	characteristic
263	C₂	C₂²×#107×C₆₁	1	1	yes	characteristic
264	C₂	C₂×C₆₁×#427	1	1	yes	characteristic
265	C₂	C₆₁×#3169	1	2	yes	—
266	C₂	C₆₁×#3170	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
socle	C₂³×C₆₁	C₂²×D₄
prime radical	C₂⁴×C₆₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₁ ◁ C₂²×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₁×#1845	C₂	semidirect
4. C₆₁×#1834	C₂	semidirect
6. C₄×#107×C₆₁	C₂	semidirect
11. C₂×C₄×C₈×C₆₁	C₄	semidirect
93. C₂×C₄×C₆₁	#107	semidirect
108. C₈×C₆₁	#117	semidirect
109. #20817	C₆₁	direct (N × H)
130. #1845	C₂×C₆₁	semidirect
131. #1834	C₂×C₆₁	semidirect
133. C₄×#107	C₂×C₆₁	semidirect
144. C₂×C₄×C₈	C₄×C₆₁	semidirect
155. C₆₁	#20817	direct (N × H)
227. C₂×C₄	#107×C₆₁	semidirect
242. C₈	#117×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈×C₆₁	2
C₆₁×#2285	2
C₆₁×#1845	2
C₆₁×#1834	2
C₆₁×#1808	2
C₄×#107×C₆₁	2
#20817	61

Subgroups — C61×#20817 #29524250

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#20817 #29524250