Subgroups — C₆₁×#13063 #29516496

Order 15616.

508 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3050	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#359	C₂	1	2	yes	maximal, complemented
4	C₆₁×#1810	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1382	C₂	1	2	yes	maximal, complemented
6	C₂×#135×C₆₁	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
8	C₆₁×#525	—	2	1	no	—
9	C₂²×C₁₆×C₆₁	—	2	2	no	—
10	C₂²×C₁₆×C₆₁	—	2	4	no	—
11	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
12	C₆₁×#467	C₄	1	2	yes	—
13	C₆₁×#466	C₄	1	2	yes	—
14	C₆₁×#465	—	2	2	no	—
15	C₂×#108×C₆₁	C₂²	1	2	yes	—
16	C₆₁×#427	C₂²	1	1	yes	characteristic
17	C₆₁×#416	—	2	1	no	—
18	C₆₁×#407	—	2	1	no	—
19	C₆₁×#359	—	2	4	no	—
20	C₆₁×#341	—	2	1	no	—
21	C₂×#41×C₆₁	—	2	1	no	—
22	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
23	#135×C₆₁	—	2	2	no	—
24	#135×C₆₁	—	2	2	no	—
25	C₂×Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂×SD₁₆×C₆₁	—	2	1	no	—
27	C₂×D₈×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆×C₆₁	—	2	2	no	—
29	C₂²×C₈×C₆₁	—	2	2	no	—
30	C₂²×C₈×C₆₁	—	2	2	no	—
31	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₁	D₄	1	2	yes	—
34	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
35	#123×C₆₁	—	4	1	no	—
36	C₂²⋊Q₈×C₆₁	—	2	2	no	—
37	#121×C₆₁	—	2	2	no	—
38	#120×C₆₁	—	4	1	no	—
39	C₄×D₄×C₆₁	—	4	2	no	—
40	#117×C₆₁	—	2	1	no	—
41	C₂×C₁₆×C₆₁	—	2	8	no	—
42	C₂×C₁₆×C₆₁	—	2	2	no	—
43	C₂×C₁₆×C₆₁	—	2	2	no	—
44	C₂×C₁₆×C₆₁	D₄	1	4	yes	complemented
45	#108×C₆₁	—	2	2	no	—
46	#108×C₆₁	—	2	2	no	—
47	#107×C₆₁	C₂×C₄	1	2	yes	—
48	#106×C₆₁	—	2	1	no	—
49	#103×C₆₁	—	2	2	no	—
50	#102×C₆₁	—	2	2	no	—
51	C₂²⋊C₈×C₆₁	—	4	2	no	—
52	C₂⁴×C₆₁	SD₁₆	1	1	yes	characteristic
53	#41×C₆₁	—	4	2	no	—
54	#41×C₆₁	—	4	2	no	—
55	C₂×Q₈×C₆₁	—	2	1	no	—
56	C₂×D₄×C₆₁	—	2	1	no	—
57	C₂×D₄×C₆₁	—	4	2	no	—
58	C₂×D₄×C₆₁	—	4	1	no	—
59	C₂²×C₄×C₆₁	—	2	2	no	—
60	C₂²×C₄×C₆₁	—	2	2	no	—
61	C₂²×C₄×C₆₁	—	4	1	no	—
62	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₆₁	D₈	1	2	yes	—
64	C₂²×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
66	Q₁₆×C₆₁	—	2	2	no	—
67	SD₁₆×C₆₁	—	4	1	no	—
68	SD₁₆×C₆₁	—	4	1	no	—
69	D₈×C₆₁	—	2	2	no	—
70	M₁₆×C₆₁	—	4	2	no	—
71	M₁₆×C₆₁	—	4	2	no	—
72	C₂×C₈×C₆₁	—	2	2	no	—
73	C₂×C₈×C₆₁	—	2	4	no	—
74	C₂×C₈×C₆₁	—	2	2	no	—
75	C₂×C₈×C₆₁	—	2	2	no	—
76	C₂×C₈×C₆₁	—	4	2	no	—
77	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₆₁	#41	1	2	yes	—
82	C₄⋊C₄×C₆₁	—	2	2	no	—
83	C₄⋊C₄×C₆₁	—	4	2	no	—
84	C₂²⋊C₄×C₆₁	—	4	2	no	—
85	C₂²⋊C₄×C₆₁	—	4	1	no	—
86	C₂²⋊C₄×C₆₁	—	4	2	no	—
87	C₄²×C₆₁	—	4	1	no	—
88	C₁₆×C₆₁	—	2	4	no	—
89	C₂³×C₆₁	—	2	4	no	—
90	C₂³×C₆₁	—	2	2	no	—
91	C₂³×C₆₁	—	4	1	no	—
92	C₂³×C₆₁	#102	1	2	yes	—
93	C₂³×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
94	Q₈×C₆₁	—	4	1	no	—
95	Q₈×C₆₁	—	4	1	no	—
96	D₄×C₆₁	—	4	1	no	—
97	D₄×C₆₁	—	4	1	no	—
98	D₄×C₆₁	—	8	4	no	—
99	D₄×C₆₁	—	8	2	no	—
100	C₂×C₄×C₆₁	—	2	2	no	—
101	C₂×C₄×C₆₁	—	2	2	no	—
102	C₂×C₄×C₆₁	—	2	2	no	—
103	C₂×C₄×C₆₁	—	2	4	no	—
104	C₂×C₄×C₆₁	—	4	2	no	—
105	C₂×C₄×C₆₁	—	4	1	no	—
106	C₂×C₄×C₆₁	—	8	2	no	—
107	C₂×C₄×C₆₁	#102	1	2	yes	—
108	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₆₁	#121	1	2	yes	—
110	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₆₁	#135	1	2	yes	—
112	C₈×C₆₁	—	2	2	no	—
113	C₈×C₆₁	—	8	2	no	—
114	C₈×C₆₁	C₄×D₄	1	2	yes	—
115	C₈×C₆₁	#120	1	1	yes	characteristic
116	C₈×C₆₁	#123	1	1	yes	characteristic
117	#13063	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂²×C₆₁	—	2	8	no	—
119	C₂²×C₆₁	—	2	2	no	—
120	C₂²×C₆₁	—	2	2	no	—
121	C₂²×C₆₁	—	2	2	no	—
122	C₂²×C₆₁	—	8	2	no	—
123	C₂²×C₆₁	—	8	1	no	—
124	C₂²×C₆₁	#359	1	4	yes	complemented
125	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
126	C₂²×C₆₁	#465	1	2	yes	—
127	C₄×C₆₁	—	2	2	no	—
128	C₄×C₆₁	—	8	2	no	—
129	C₄×C₆₁	—	8	1	no	—
130	C₄×C₆₁	#386	1	1	yes	characteristic
131	C₄×C₆₁	#416	1	1	yes	characteristic
132	C₄×C₆₁	#466	1	2	yes	—
133	C₂³×C₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
134	#3050	C₂×C₆₁	1	1	yes	characteristic
135	C₂×#359	C₂×C₆₁	1	2	yes	complemented
136	#1810	C₂×C₆₁	1	1	yes	characteristic, complemented
137	#1382	C₂×C₆₁	1	2	yes	complemented
138	C₂×C₆₁	—	2	4	no	—
139	C₂×C₆₁	—	2	2	no	—
140	C₂×C₆₁	—	16	1	no	—
141	C₂×C₆₁	#1893	1	1	yes	characteristic
142	C₂×C₆₁	C₂×#359	1	2	yes	—
143	C₂×#135	C₂²×C₆₁	1	1	yes	characteristic
144	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
145	#525	—	2	1	no	—
146	C₂²×C₁₆	—	2	2	no	—
147	C₂²×C₁₆	—	2	4	no	—
148	C₂²×C₁₆	C₂²×C₆₁	1	2	yes	—
149	#467	C₄×C₆₁	1	2	yes	—
150	#466	C₄×C₆₁	1	2	yes	—
151	#465	—	2	2	no	—
152	C₂×#108	C₂²×C₆₁	1	2	yes	—
153	#427	C₂²×C₆₁	1	1	yes	characteristic
154	#416	—	2	1	no	—
155	#407	—	2	1	no	—
156	#359	—	2	4	no	—
157	#341	—	2	1	no	—
158	C₆₁	#13063	1	1	yes	characteristic, direct product factor, Sylow
159	C₂×#41	—	2	1	no	—
160	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
161	#135	—	2	2	no	—
162	#135	—	2	2	no	—
163	C₂×Q₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
164	C₂×SD₁₆	—	2	1	no	—
165	C₂×D₈	C₂×C₄×C₆₁	1	1	yes	characteristic
166	C₂×M₁₆	—	2	2	no	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
170	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
171	C₂²×C₈	D₄×C₆₁	1	2	yes	—
172	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
173	#123	—	4	1	no	—
174	C₂²⋊Q₈	—	2	2	no	—
175	#121	—	2	2	no	—
176	#120	—	4	1	no	—
177	C₄×D₄	—	4	2	no	—
178	#117	—	2	1	no	—
179	C₂×C₁₆	—	2	8	no	—
180	C₂×C₁₆	—	2	2	no	—
181	C₂×C₁₆	—	2	2	no	—
182	C₂×C₁₆	D₄×C₆₁	1	4	yes	complemented
183	#108	—	2	2	no	—
184	#108	—	2	2	no	—
185	#107	C₂×C₄×C₆₁	1	2	yes	—
186	#106	—	2	1	no	—
187	#103	—	2	2	no	—
188	#102	—	2	2	no	—
189	C₂²⋊C₈	—	4	2	no	—
190	C₂⁴	SD₁₆×C₆₁	1	1	yes	characteristic
191	#41	—	4	2	no	—
192	#41	—	4	2	no	—
193	C₂×Q₈	—	2	1	no	—
194	C₂×D₄	—	2	1	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂×D₄	—	4	1	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
201	C₂²×C₄	D₈×C₆₁	1	2	yes	—
202	C₂²×C₄	SD₁₆×C₆₁	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
204	Q₁₆	—	2	2	no	—
205	SD₁₆	—	4	1	no	—
206	SD₁₆	—	4	1	no	—
207	D₈	—	2	2	no	—
208	M₁₆	—	4	2	no	—
209	M₁₆	—	4	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
216	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
217	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
218	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
219	C₂×C₈	#41×C₆₁	1	2	yes	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₂²⋊C₄	—	4	2	no	—
223	C₂²⋊C₄	—	4	1	no	—
224	C₂²⋊C₄	—	4	2	no	—
225	C₄²	—	4	1	no	—
226	C₁₆	—	2	4	no	—
227	C₂³	—	2	4	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	4	1	no	—
230	C₂³	#102×C₆₁	1	2	yes	—
231	C₂³	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
232	Q₈	—	4	1	no	—
233	Q₈	—	4	1	no	—
234	D₄	—	4	1	no	—
235	D₄	—	4	1	no	—
236	D₄	—	8	4	no	—
237	D₄	—	8	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	#102×C₆₁	1	2	yes	—
246	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
247	C₂×C₄	#121×C₆₁	1	2	yes	—
248	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
249	C₂×C₄	#135×C₆₁	1	2	yes	—
250	C₈	—	2	2	no	—
251	C₈	—	8	2	no	—
252	C₈	C₄×D₄×C₆₁	1	2	yes	—
253	C₈	#120×C₆₁	1	1	yes	characteristic
254	C₈	#123×C₆₁	1	1	yes	characteristic
255	C₂²	—	2	8	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	8	2	no	—
260	C₂²	—	8	1	no	—
261	C₂²	C₆₁×#359	1	4	yes	complemented
262	C₂²	C₂×#102×C₆₁	1	1	yes	characteristic
263	C₂²	C₆₁×#465	1	2	yes	—
264	C₄	—	2	2	no	—
265	C₄	—	8	2	no	—
266	C₄	—	8	1	no	—
267	C₄	C₆₁×#386	1	1	yes	characteristic
268	C₄	C₆₁×#416	1	1	yes	characteristic
269	C₄	C₆₁×#466	1	2	yes	—
270	C₂	—	2	4	no	—
271	C₂	—	2	2	no	—
272	C₂	—	16	1	no	—
273	C₂	C₆₁×#1893	1	1	yes	characteristic
274	C₂	C₂×C₆₁×#359	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
socle	C₂²×C₆₁	C₂×#102
prime radical	C₆₁×#3050	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ C₂³×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₁₆×C₆₁	C₂	semidirect
3. C₂×C₆₁×#359	C₂	semidirect
4. C₆₁×#1810	C₂	semidirect
5. C₆₁×#1382	C₂	semidirect
44. C₂×C₁₆×C₆₁	D₄	semidirect
117. #13063	C₆₁	direct (N × H)
124. C₂²×C₆₁	#359	semidirect
133. C₂³×C₁₆	C₂×C₆₁	semidirect
135. C₂×#359	C₂×C₆₁	semidirect
136. #1810	C₂×C₆₁	semidirect
137. #1382	C₂×C₆₁	semidirect
158. C₆₁	#13063	direct (N × H)
182. C₂×C₁₆	D₄×C₆₁	semidirect
261. C₂²	C₆₁×#359	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₆₁	2
C₆₁×#3050	2
C₂×C₆₁×#359	2
C₆₁×#1810	2
C₆₁×#1382	2
#13063	61

Subgroups — C61×#13063 #29516496

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#13063 #29516496