Subgroups — C₆₁×#9727 #29513160

Order 15616.

392 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	C₆₁×#3170	C₂	1	1	yes	maximal, characteristic
2	C₄×#108×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1526	C₂	1	2	yes	maximal
4	C₆₁×#1485	C₂	1	2	yes	maximal
5	C₆₁×#1451	C₂	1	1	yes	maximal, characteristic
6	C₆₁×#533	C₂²	1	2	yes	—
7	C₆₁×#500	C₄	1	1	yes	characteristic
8	C₆₁×#494	C₄	1	1	yes	characteristic
9	C₆₁×#487	C₄	1	2	yes	—
10	C₆₁×#467	—	2	1	no	—
11	C₆₁×#467	—	2	1	no	—
12	C₄×Q₁₆×C₆₁	—	2	1	no	—
13	C₄×Q₁₆×C₆₁	—	2	1	no	—
14	C₆₁×#431	C₂²	1	1	yes	characteristic
15	C₂×#108×C₆₁	—	2	1	no	—
16	C₄×M₁₆×C₆₁	C₂²	1	2	yes	—
17	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₆₁×#341	—	2	1	no	—
20	C₆₁×#330	—	2	4	no	—
21	C₆₁×Q₈⋊C₈	—	2	4	no	—
22	C₆₁×#322	—	2	1	no	—
23	C₆₁×#322	—	2	1	no	—
24	C₂×Q₁₆×C₆₁	—	2	1	no	—
25	C₂×Q₁₆×C₆₁	—	2	1	no	—
26	C₂×M₁₆×C₆₁	—	2	1	no	—
27	C₂×M₁₆×C₆₁	—	2	2	no	—
28	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
30	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
31	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
32	#125×C₆₁	—	2	2	no	—
33	C₂²⋊Q₈×C₆₁	—	2	2	no	—
34	C₂²⋊Q₈×C₆₁	—	2	2	no	—
35	C₄×Q₈×C₆₁	—	2	2	no	—
36	C₄×Q₈×C₆₁	—	2	2	no	—
37	#117×C₆₁	—	2	2	no	—
38	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
39	#108×C₆₁	—	2	4	no	—
40	#107×C₆₁	—	2	1	no	—
41	#107×C₆₁	—	2	1	no	—
42	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
43	#103×C₆₁	—	2	2	no	—
44	#103×C₆₁	—	2	2	no	—
45	C₂²⋊C₈×C₆₁	—	2	1	no	—
46	#97×C₆₁	—	2	2	no	—
47	#97×C₆₁	D₄	1	2	yes	—
48	C₄×C₈×C₆₁	—	2	1	no	—
49	C₄×C₈×C₆₁	—	2	1	no	—
50	C₄×C₈×C₆₁	—	2	2	no	—
51	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
54	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
55	C₂×Q₈×C₆₁	—	2	2	no	—
56	C₂×Q₈×C₆₁	—	2	2	no	—
57	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
60	Q₁₆×C₆₁	—	4	2	no	—
61	Q₁₆×C₆₁	—	4	2	no	—
62	M₁₆×C₆₁	—	2	4	no	—
63	M₁₆×C₆₁	—	4	2	no	—
64	M₁₆×C₆₁	—	4	2	no	—
65	C₂×C₈×C₆₁	—	2	2	no	—
66	C₂×C₈×C₆₁	—	2	2	no	—
67	C₂×C₈×C₆₁	—	2	2	no	—
68	C₂×C₈×C₆₁	—	4	1	no	—
69	C₂×C₈×C₆₁	—	4	2	no	—
70	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₄⋊C₄×C₆₁	—	2	2	no	—
75	C₄⋊C₄×C₆₁	—	2	2	no	—
76	C₄⋊C₄×C₆₁	—	2	2	no	—
77	C₄⋊C₄×C₆₁	—	2	2	no	—
78	C₄⋊C₄×C₆₁	—	4	2	no	—
79	C₄⋊C₄×C₆₁	—	4	2	no	—
80	C₂²⋊C₄×C₆₁	—	4	2	no	—
81	C₄²×C₆₁	—	4	2	no	—
82	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
86	C₂³×C₆₁	#124	1	1	yes	characteristic
87	Q₈×C₆₁	—	4	2	no	—
88	Q₈×C₆₁	—	4	2	no	—
89	Q₈×C₆₁	—	4	4	no	—
90	C₂×C₄×C₆₁	—	2	1	no	—
91	C₂×C₄×C₆₁	—	2	1	no	—
92	C₂×C₄×C₆₁	—	2	2	no	—
93	C₂×C₄×C₆₁	—	2	2	no	—
94	C₂×C₄×C₆₁	—	4	2	no	—
95	C₂×C₄×C₆₁	—	4	2	no	—
96	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
100	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
101	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
102	C₈×C₆₁	—	2	1	no	—
103	C₈×C₆₁	—	2	1	no	—
104	C₈×C₆₁	—	2	1	no	—
105	C₈×C₆₁	—	4	2	no	—
106	C₈×C₆₁	—	4	4	no	—
107	C₈×C₆₁	—	4	2	no	—
108	C₈×C₆₁	#104	1	2	yes	—
109	#9727	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂²×C₆₁	—	2	1	no	—
111	C₂²×C₆₁	—	2	1	no	—
112	C₂²×C₆₁	—	2	1	no	—
113	C₂²×C₆₁	#390	1	1	yes	characteristic
114	C₄×C₆₁	—	2	1	no	—
115	C₄×C₆₁	—	2	1	no	—
116	C₄×C₆₁	—	2	1	no	—
117	C₄×C₆₁	—	8	2	no	—
118	C₄×C₆₁	—	8	2	no	—
119	C₄×C₆₁	C₂×#104	1	1	yes	characteristic
120	C₄×C₆₁	#421	1	1	yes	characteristic
121	C₄×C₆₁	#444	1	2	yes	—
122	C₄×C₆₁	#473	1	2	yes	—
123	#3170	C₂×C₆₁	1	1	yes	characteristic
124	C₄×#108	C₂×C₆₁	1	1	yes	characteristic
125	#1526	C₂×C₆₁	1	2	yes	—
126	#1485	C₂×C₆₁	1	2	yes	—
127	#1451	C₂×C₆₁	1	1	yes	characteristic
128	C₂×C₆₁	—	4	1	no	—
129	C₂×C₆₁	#1975	1	1	yes	characteristic
130	C₂×C₆₁	#1979	1	1	yes	characteristic
131	C₂×C₆₁	#1980	1	1	yes	characteristic
132	#533	C₂²×C₆₁	1	2	yes	—
133	#500	C₄×C₆₁	1	1	yes	characteristic
134	#494	C₄×C₆₁	1	1	yes	characteristic
135	#487	C₄×C₆₁	1	2	yes	—
136	#467	—	2	1	no	—
137	#467	—	2	1	no	—
138	C₄×Q₁₆	—	2	1	no	—
139	C₄×Q₁₆	—	2	1	no	—
140	#431	C₂²×C₆₁	1	1	yes	characteristic
141	C₂×#108	—	2	1	no	—
142	C₄×M₁₆	C₂²×C₆₁	1	2	yes	—
143	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
145	#341	—	2	1	no	—
146	#330	—	2	4	no	—
147	Q₈⋊C₈	—	2	4	no	—
148	#322	—	2	1	no	—
149	#322	—	2	1	no	—
150	C₆₁	#9727	1	1	yes	characteristic, direct product factor, Sylow
151	C₂×Q₁₆	—	2	1	no	—
152	C₂×Q₁₆	—	2	1	no	—
153	C₂×M₁₆	—	2	1	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
156	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
157	C₄⋊Q₈	C₂×C₄×C₆₁	1	2	yes	—
158	C₄⋊Q₈	C₂×C₄×C₆₁	1	2	yes	—
159	#125	—	2	2	no	—
160	C₂²⋊Q₈	—	2	2	no	—
161	C₂²⋊Q₈	—	2	2	no	—
162	C₄×Q₈	—	2	2	no	—
163	C₄×Q₈	—	2	2	no	—
164	#117	—	2	2	no	—
165	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
166	#108	—	2	4	no	—
167	#107	—	2	1	no	—
168	#107	—	2	1	no	—
169	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
170	#103	—	2	2	no	—
171	#103	—	2	2	no	—
172	C₂²⋊C₈	—	2	1	no	—
173	#97	—	2	2	no	—
174	#97	D₄×C₆₁	1	2	yes	—
175	C₄×C₈	—	2	1	no	—
176	C₄×C₈	—	2	1	no	—
177	C₄×C₈	—	2	2	no	—
178	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
179	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
180	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
181	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
185	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
186	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
187	Q₁₆	—	4	2	no	—
188	Q₁₆	—	4	2	no	—
189	M₁₆	—	2	4	no	—
190	M₁₆	—	4	2	no	—
191	M₁₆	—	4	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
198	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
199	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
200	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₂²⋊C₄	—	4	2	no	—
208	C₄²	—	4	2	no	—
209	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
210	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
211	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
212	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
213	C₂³	#124×C₆₁	1	1	yes	characteristic
214	Q₈	—	4	2	no	—
215	Q₈	—	4	2	no	—
216	Q₈	—	4	4	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
224	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
225	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
226	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
227	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
228	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
229	C₈	—	2	1	no	—
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	4	2	no	—
233	C₈	—	4	4	no	—
234	C₈	—	4	2	no	—
235	C₈	#104×C₆₁	1	2	yes	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	C₆₁×#390	1	1	yes	characteristic
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	8	2	no	—
244	C₄	—	8	2	no	—
245	C₄	C₂×#104×C₆₁	1	1	yes	characteristic
246	C₄	C₆₁×#421	1	1	yes	characteristic
247	C₄	C₆₁×#444	1	2	yes	—
248	C₄	C₆₁×#473	1	2	yes	—
249	C₂	—	4	1	no	—
250	C₂	C₆₁×#1975	1	1	yes	characteristic
251	C₂	C₆₁×#1979	1	1	yes	characteristic
252	C₂	C₆₁×#1980	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	#121
commutator	C₄²	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle	C₂²×C₆₁	#390
prime radical	C₂³×C₆₁	#124
LCS	G ▷ C₄² ▷ C₂² ▷ 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
109. #9727	C₆₁	direct (N × H)
150. C₆₁	#9727	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×#3170	2
C₄×#108×C₆₁	2
C₆₁×#1526	2
C₆₁×#1485	2
C₆₁×#1451	2
#9727	61

Subgroups — C61×#9727 #29513160

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#9727 #29513160