Subgroups — C₆₁×#12637 #29516070

Order 15616.

526 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#357	C₂	1	2	yes	maximal
3	C₂×C₆₁×#346	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#107×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1380	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₆₁	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₆₁	—	2	1	no	—
8	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
9	C₄×D₈×C₆₁	—	2	2	no	—
10	C₄×D₈×C₆₁	C₄	1	4	yes	complemented
11	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×#107×C₆₁	C₂²	1	2	yes	—
13	C₂×#102×C₆₁	—	2	1	no	—
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₆₁	—	2	1	no	—
16	C₆₁×#357	C₄	1	8	yes	complemented
17	C₆₁×#346	—	2	2	no	—
18	C₆₁×#340	—	2	1	no	—
19	C₂²×D₄×C₆₁	—	2	1	no	—
20	C₂³×C₄×C₆₁	—	4	1	no	—
21	C₂×D₈×C₆₁	—	2	1	no	—
22	C₂×D₈×C₆₁	—	2	2	no	—
23	C₂×D₈×C₆₁	C₂×C₄	1	4	yes	—
24	C₂×D₈×C₆₁	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
27	C₄×D₄×C₆₁	—	2	4	no	—
28	C₄×D₄×C₆₁	—	4	2	no	—
29	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
30	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
31	C₂×C₂²⋊C₄×C₆₁	—	4	1	no	—
32	C₂×C₄²×C₆₁	—	4	1	no	—
33	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₆₁	—	2	2	no	—
35	C₂×C₁₆×C₆₁	—	2	2	no	—
36	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	complemented
37	#107×C₆₁	—	2	1	no	—
38	#107×C₆₁	—	2	2	no	—
39	#107×C₆₁	C₂×C₄	1	2	yes	—
40	#107×C₆₁	C₂×C₄	1	4	yes	complemented
41	#102×C₆₁	—	2	4	no	—
42	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₆₁	D₄	1	2	yes	—
44	#95×C₆₁	—	4	1	no	—
45	C₂⁴×C₆₁	—	4	1	no	—
46	C₂×D₄×C₆₁	—	2	2	no	—
47	C₂×D₄×C₆₁	—	2	4	no	—
48	C₂×D₄×C₆₁	—	4	1	no	—
49	C₂×D₄×C₆₁	—	4	2	no	—
50	C₂²×C₄×C₆₁	—	4	2	no	—
51	C₂²×C₄×C₆₁	—	4	1	no	—
52	C₂²×C₄×C₆₁	—	4	4	no	—
53	C₂²×C₄×C₆₁	—	8	4	no	—
54	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
57	D₈×C₆₁	—	2	2	no	—
58	D₈×C₆₁	—	2	4	no	—
59	D₈×C₆₁	C₄²	1	4	yes	complemented
60	C₂×C₈×C₆₁	—	2	2	no	—
61	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₆₁	#41	1	2	yes	—
66	C₄⋊C₄×C₆₁	—	2	2	no	—
67	C₄⋊C₄×C₆₁	—	2	4	no	—
68	C₄⋊C₄×C₆₁	—	4	1	no	—
69	C₄⋊C₄×C₆₁	—	4	2	no	—
70	C₂²⋊C₄×C₆₁	—	4	4	no	—
71	C₄²×C₆₁	—	8	2	no	—
72	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₆₁	SD₁₆	1	2	yes	—
74	C₁₆×C₆₁	—	2	4	no	—
75	C₂³×C₆₁	—	4	2	no	—
76	C₂³×C₆₁	—	4	4	no	—
77	C₂³×C₆₁	—	8	2	no	—
78	C₂³×C₆₁	—	8	1	no	—
79	C₂³×C₆₁	—	8	1	no	—
80	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
81	D₄×C₆₁	—	2	4	no	—
82	D₄×C₆₁	—	4	2	no	—
83	D₄×C₆₁	—	4	4	no	—
84	C₂×C₄×C₆₁	—	2	1	no	—
85	C₂×C₄×C₆₁	—	2	2	no	—
86	C₂×C₄×C₆₁	—	2	1	no	—
87	C₂×C₄×C₆₁	—	4	2	no	—
88	C₂×C₄×C₆₁	—	4	4	no	—
89	C₂×C₄×C₆₁	—	8	8	no	—
90	C₂×C₄×C₆₁	—	8	2	no	—
91	C₂×C₄×C₆₁	—	8	2	no	—
92	C₂×C₄×C₆₁	—	8	2	no	—
93	C₂×C₄×C₆₁	#102	1	2	yes	—
94	C₂×C₄×C₆₁	#102	1	2	yes	—
95	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×C₄×C₆₁	#135	1	2	yes	—
99	C₈×C₆₁	—	2	2	no	—
100	C₈×C₆₁	C₂×C₄²	1	1	yes	characteristic
101	C₈×C₆₁	#117	1	1	yes	characteristic
102	C₈×C₆₁	C₄×D₄	1	2	yes	—
103	#12637	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂²×C₆₁	—	4	4	no	—
105	C₂²×C₆₁	—	8	2	no	—
106	C₂²×C₆₁	—	8	2	no	—
107	C₂²×C₆₁	—	8	4	no	—
108	C₂²×C₆₁	—	8	4	no	—
109	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
110	C₂²×C₆₁	C₄×D₈	1	2	yes	—
111	C₂²×C₆₁	#509	1	1	yes	characteristic
112	C₂²×C₆₁	#509	1	2	yes	—
113	C₂²×C₆₁	#510	1	1	yes	characteristic
114	C₄×C₆₁	—	2	2	no	—
115	C₄×C₆₁	—	2	2	no	—
116	C₄×C₆₁	—	8	2	no	—
117	C₄×C₆₁	—	8	4	no	—
118	C₄×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
119	C₄×C₆₁	#416	1	1	yes	characteristic
120	C₄×C₆₁	C₄×SD₁₆	1	2	yes	—
121	C₂×C₄×D₈	C₂×C₆₁	1	1	yes	characteristic
122	C₂×#357	C₂×C₆₁	1	2	yes	—
123	C₂×#346	C₂×C₆₁	1	1	yes	characteristic, complemented
124	C₄×#107	C₂×C₆₁	1	1	yes	characteristic, complemented
125	#1380	C₂×C₆₁	1	2	yes	complemented
126	C₂×C₆₁	—	8	4	no	—
127	C₂×C₆₁	C₄×#102	1	1	yes	characteristic
128	C₂×C₆₁	#2139	1	1	yes	characteristic
129	C₂×C₆₁	#2140	1	1	yes	characteristic
130	C₂×C₆₁	#2175	1	2	yes	—
131	C₂×C₆₁	#2177	1	2	yes	—
132	C₂²×D₈	C₂²×C₆₁	1	1	yes	characteristic
133	C₂×C₄×D₄	—	2	1	no	—
134	C₂²×C₁₆	C₂²×C₆₁	1	2	yes	—
135	C₄×D₈	—	2	2	no	—
136	C₄×D₈	C₄×C₆₁	1	4	yes	complemented
137	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
138	C₂×#107	C₂²×C₆₁	1	2	yes	—
139	C₂×#102	—	2	1	no	—
140	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
141	C₄×C₄⋊C₄	—	2	1	no	—
142	#357	C₄×C₆₁	1	8	yes	complemented
143	#346	—	2	2	no	—
144	#340	—	2	1	no	—
145	C₆₁	#12637	1	1	yes	characteristic, direct product factor, Sylow
146	C₂²×D₄	—	2	1	no	—
147	C₂³×C₄	—	4	1	no	—
148	C₂×D₈	—	2	1	no	—
149	C₂×D₈	—	2	2	no	—
150	C₂×D₈	C₂×C₄×C₆₁	1	4	yes	—
151	C₂×D₈	C₂×C₄×C₆₁	1	2	yes	—
152	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
153	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
154	C₄×D₄	—	2	4	no	—
155	C₄×D₄	—	4	2	no	—
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₂²⋊C₄	—	4	1	no	—
159	C₂×C₄²	—	4	1	no	—
160	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
161	C₂×C₁₆	—	2	2	no	—
162	C₂×C₁₆	—	2	2	no	—
163	C₂×C₁₆	C₂×C₄×C₆₁	1	4	yes	complemented
164	#107	—	2	1	no	—
165	#107	—	2	2	no	—
166	#107	C₂×C₄×C₆₁	1	2	yes	—
167	#107	C₂×C₄×C₆₁	1	4	yes	complemented
168	#102	—	2	4	no	—
169	C₄×C₈	C₂×C₄×C₆₁	1	2	yes	—
170	C₄×C₈	D₄×C₆₁	1	2	yes	—
171	#95	—	4	1	no	—
172	C₂⁴	—	4	1	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	—	4	1	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	—	4	1	no	—
179	C₂²×C₄	—	4	4	no	—
180	C₂²×C₄	—	8	4	no	—
181	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
182	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
184	D₈	—	2	2	no	—
185	D₈	—	2	4	no	—
186	D₈	C₄²×C₆₁	1	4	yes	complemented
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
189	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
190	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
191	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
192	C₂×C₈	#41×C₆₁	1	2	yes	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	1	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₂²⋊C₄	—	4	4	no	—
198	C₄²	—	8	2	no	—
199	C₄²	C₂²⋊C₄×C₆₁	1	2	yes	—
200	C₄²	SD₁₆×C₆₁	1	2	yes	—
201	C₁₆	—	2	4	no	—
202	C₂³	—	4	2	no	—
203	C₂³	—	4	4	no	—
204	C₂³	—	8	2	no	—
205	C₂³	—	8	1	no	—
206	C₂³	—	8	1	no	—
207	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
208	D₄	—	2	4	no	—
209	D₄	—	4	2	no	—
210	D₄	—	4	4	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	8	8	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	—	8	2	no	—
220	C₂×C₄	#102×C₆₁	1	2	yes	—
221	C₂×C₄	#102×C₆₁	1	2	yes	—
222	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
223	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
224	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
225	C₂×C₄	#135×C₆₁	1	2	yes	—
226	C₈	—	2	2	no	—
227	C₈	C₂×C₄²×C₆₁	1	1	yes	characteristic
228	C₈	#117×C₆₁	1	1	yes	characteristic
229	C₈	C₄×D₄×C₆₁	1	2	yes	—
230	C₂²	—	4	4	no	—
231	C₂²	—	8	2	no	—
232	C₂²	—	8	2	no	—
233	C₂²	—	8	4	no	—
234	C₂²	—	8	4	no	—
235	C₂²	C₂×#102×C₆₁	1	1	yes	characteristic
236	C₂²	C₄×D₈×C₆₁	1	2	yes	—
237	C₂²	C₆₁×#509	1	1	yes	characteristic
238	C₂²	C₆₁×#509	1	2	yes	—
239	C₂²	C₆₁×#510	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	8	2	no	—
243	C₄	—	8	4	no	—
244	C₄	C₄×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
245	C₄	C₆₁×#416	1	1	yes	characteristic
246	C₄	C₄×SD₁₆×C₆₁	1	2	yes	—
247	C₂	—	8	4	no	—
248	C₂	C₄×#102×C₆₁	1	1	yes	characteristic
249	C₂	C₆₁×#2139	1	1	yes	characteristic
250	C₂	C₆₁×#2140	1	1	yes	characteristic
251	C₂	C₆₁×#2175	1	2	yes	—
252	C₂	C₆₁×#2177	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂×D₈
commutator	C₈	C₂×C₄²×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
prime radical	C₂²×D₈×C₆₁	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₆₁×#346	C₂	semidirect
4. C₄×#107×C₆₁	C₂	semidirect
5. C₆₁×#1380	C₂	semidirect
10. C₄×D₈×C₆₁	C₄	semidirect
16. C₆₁×#357	C₄	semidirect
36. C₂×C₁₆×C₆₁	C₂×C₄	semidirect
40. #107×C₆₁	C₂×C₄	semidirect
59. D₈×C₆₁	C₄²	semidirect
103. #12637	C₆₁	direct (N × H)
123. C₂×#346	C₂×C₆₁	semidirect
124. C₄×#107	C₂×C₆₁	semidirect
125. #1380	C₂×C₆₁	semidirect
136. C₄×D₈	C₄×C₆₁	semidirect
142. #357	C₄×C₆₁	semidirect
145. C₆₁	#12637	direct (N × H)
163. C₂×C₁₆	C₂×C₄×C₆₁	semidirect
167. #107	C₂×C₄×C₆₁	semidirect
186. D₈	C₄²×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₆₁	2
C₂×C₆₁×#357	2
C₂×C₆₁×#346	2
C₄×#107×C₆₁	2
C₆₁×#1380	2
#12637	61

Subgroups — C61×#12637 #29516070

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#12637 #29516070