Subgroups — C₆₁×#9890 #29513323

Order 15616.

484 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	C₆₁×#2862	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1767	C₂	1	1	yes	maximal, characteristic
3	C₄×#97×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1295	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
6	C₂×C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
7	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
8	C₂×#97×C₆₁	—	2	1	no	—
9	C₂×#97×C₆₁	C₂²	1	2	yes	—
10	C₂×#97×C₆₁	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₆₁	—	2	1	no	—
12	C₆₁×#391	—	2	1	no	—
13	C₆₁×#391	—	2	1	no	—
14	C₆₁×#391	—	2	1	no	—
15	C₆₁×#391	—	2	1	no	—
16	C₆₁×#382	C₄	1	2	yes	—
17	C₆₁×#382	C₂²	1	2	yes	—
18	C₄³×C₆₁	C₂²	1	1	yes	characteristic
19	C₆₁×#336	—	2	1	no	—
20	C₂²×Q₈×C₆₁	—	2	1	no	—
21	C₂²×Q₈×C₆₁	—	2	1	no	—
22	C₂²×C₈×C₆₁	—	2	2	no	—
23	C₂²×C₈×C₆₁	—	2	2	no	—
24	C₄⋊Q₈×C₆₁	—	4	2	no	—
25	C₄⋊Q₈×C₆₁	—	4	2	no	—
26	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
27	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
28	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₆₁	—	2	4	no	—
38	#97×C₆₁	—	2	4	no	—
39	#97×C₆₁	—	2	2	no	—
40	#97×C₆₁	—	2	4	no	—
41	#97×C₆₁	Q₈	1	4	yes	complemented
42	C₄×C₈×C₆₁	—	2	4	no	—
43	#95×C₆₁	—	2	2	no	—
44	#95×C₆₁	—	2	2	no	—
45	#95×C₆₁	—	2	2	no	—
46	#95×C₆₁	—	2	2	no	—
47	C₂×Q₈×C₆₁	—	4	2	no	—
48	C₂×Q₈×C₆₁	—	4	2	no	—
49	C₂×Q₈×C₆₁	—	4	2	no	—
50	C₂×Q₈×C₆₁	—	4	2	no	—
51	C₂×Q₈×C₆₁	—	4	2	no	—
52	C₂×Q₈×C₆₁	—	4	2	no	—
53	C₂²×C₄×C₆₁	—	2	2	no	—
54	C₂²×C₄×C₆₁	—	2	2	no	—
55	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₆₁	—	2	4	no	—
63	C₂×C₈×C₆₁	—	2	4	no	—
64	C₂×C₈×C₆₁	—	2	4	no	—
65	C₂×C₈×C₆₁	—	2	4	no	—
66	C₂×C₈×C₆₁	—	4	2	no	—
67	C₂×C₈×C₆₁	—	4	2	no	—
68	C₄⋊C₄×C₆₁	—	4	2	no	—
69	C₄⋊C₄×C₆₁	—	4	2	no	—
70	C₄⋊C₄×C₆₁	—	4	2	no	—
71	C₄⋊C₄×C₆₁	—	4	2	no	—
72	C₄²×C₆₁	—	2	2	no	—
73	C₄²×C₆₁	—	2	2	no	—
74	C₄²×C₆₁	—	2	2	no	—
75	C₄²×C₆₁	—	2	2	no	—
76	C₄²×C₆₁	—	2	2	no	—
77	C₄²×C₆₁	—	4	1	no	—
78	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
79	C₄²×C₆₁	#41	1	2	yes	—
80	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	Q₈×C₆₁	—	8	4	no	—
82	Q₈×C₆₁	—	8	4	no	—
83	C₂×C₄×C₆₁	—	2	1	no	—
84	C₂×C₄×C₆₁	—	2	1	no	—
85	C₂×C₄×C₆₁	—	2	1	no	—
86	C₂×C₄×C₆₁	—	2	1	no	—
87	C₂×C₄×C₆₁	—	2	1	no	—
88	C₂×C₄×C₆₁	—	2	1	no	—
89	C₂×C₄×C₆₁	—	2	1	no	—
90	C₂×C₄×C₆₁	—	2	1	no	—
91	C₂×C₄×C₆₁	—	2	1	no	—
92	C₂×C₄×C₆₁	—	2	1	no	—
93	C₂×C₄×C₆₁	—	2	1	no	—
94	C₂×C₄×C₆₁	—	2	1	no	—
95	C₂×C₄×C₆₁	—	2	1	no	—
96	C₂×C₄×C₆₁	—	4	2	no	—
97	C₂×C₄×C₆₁	—	4	2	no	—
98	C₂×C₄×C₆₁	—	4	2	no	—
99	C₂×C₄×C₆₁	—	4	2	no	—
100	C₂×C₄×C₆₁	—	4	2	no	—
101	C₂×C₄×C₆₁	—	4	2	no	—
102	C₂×C₄×C₆₁	#100	1	2	yes	—
103	C₂×C₄×C₆₁	#101	1	2	yes	—
104	C₂×C₄×C₆₁	#104	1	2	yes	—
105	C₂×C₄×C₆₁	#104	1	2	yes	—
106	C₂×C₄×C₆₁	#104	1	2	yes	—
107	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
108	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
110	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
111	C₈×C₆₁	—	2	8	no	—
112	C₈×C₆₁	—	4	4	no	—
113	#9890	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₆₁	#391	1	1	yes	characteristic
115	C₂²×C₆₁	C₂×#100	1	1	yes	characteristic
116	C₂²×C₆₁	C₂×#101	1	1	yes	characteristic
117	C₂²×C₆₁	C₂×#104	1	1	yes	characteristic
118	C₂²×C₆₁	C₂×#104	1	1	yes	characteristic
119	C₂²×C₆₁	C₂×#104	1	1	yes	characteristic
120	C₂²×C₆₁	#421	1	1	yes	characteristic
121	C₄×C₆₁	—	2	1	no	—
122	C₄×C₆₁	—	2	1	no	—
123	C₄×C₆₁	—	2	2	no	—
124	C₄×C₆₁	—	2	2	no	—
125	C₄×C₆₁	—	2	2	no	—
126	C₄×C₆₁	—	2	2	no	—
127	C₄×C₆₁	—	4	1	no	—
128	C₄×C₆₁	—	8	2	no	—
129	C₄×C₆₁	—	8	2	no	—
130	C₄×C₆₁	#329	1	4	yes	—
131	#2862	C₂×C₆₁	1	1	yes	characteristic
132	#1767	C₂×C₆₁	1	1	yes	characteristic
133	C₄×#97	C₂×C₆₁	1	1	yes	characteristic
134	#1295	C₂×C₆₁	1	4	yes	—
135	C₂×C₆₁	C₂×#329	1	1	yes	characteristic
136	C₂×C₆₁	#1526	1	1	yes	characteristic
137	C₂×C₆₁	#1527	1	1	yes	characteristic
138	C₂×C₆₁	#1531	1	1	yes	characteristic
139	C₂×C₆₁	#1986	1	1	yes	characteristic
140	C₂×C₆₁	#1994	1	1	yes	characteristic
141	C₂×C₆₁	#1997	1	1	yes	characteristic
142	C₂×C₄⋊Q₈	C₄×C₆₁	1	1	yes	characteristic
143	C₂×C₄⋊Q₈	C₄×C₆₁	1	1	yes	characteristic
144	C₂×C₄⋊C₈	—	2	1	no	—
145	C₂×#97	—	2	1	no	—
146	C₂×#97	C₂²×C₆₁	1	2	yes	—
147	C₂×#97	C₂²×C₆₁	1	2	yes	—
148	C₂×C₄×C₈	—	2	1	no	—
149	#391	—	2	1	no	—
150	#391	—	2	1	no	—
151	#391	—	2	1	no	—
152	#391	—	2	1	no	—
153	#382	C₄×C₆₁	1	2	yes	—
154	#382	C₂²×C₆₁	1	2	yes	—
155	C₄³	C₂²×C₆₁	1	1	yes	characteristic
156	#336	—	2	1	no	—
157	C₆₁	#9890	1	1	yes	characteristic, direct product factor, Sylow
158	C₂²×Q₈	—	2	1	no	—
159	C₂²×Q₈	—	2	1	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₄⋊Q₈	—	4	2	no	—
163	C₄⋊Q₈	—	4	2	no	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
167	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
168	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
169	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
170	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
171	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
172	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
173	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
174	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
175	C₄⋊C₈	—	2	4	no	—
176	#97	—	2	4	no	—
177	#97	—	2	2	no	—
178	#97	—	2	4	no	—
179	#97	Q₈×C₆₁	1	4	yes	complemented
180	C₄×C₈	—	2	4	no	—
181	#95	—	2	2	no	—
182	#95	—	2	2	no	—
183	#95	—	2	2	no	—
184	#95	—	2	2	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂×Q₈	—	4	2	no	—
188	C₂×Q₈	—	4	2	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
194	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
195	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
196	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
197	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	4	1	no	—
216	C₄²	C₂×Q₈×C₆₁	1	2	yes	—
217	C₄²	#41×C₆₁	1	2	yes	—
218	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
219	Q₈	—	8	4	no	—
220	Q₈	—	8	4	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	#100×C₆₁	1	2	yes	—
241	C₂×C₄	#101×C₆₁	1	2	yes	—
242	C₂×C₄	#104×C₆₁	1	2	yes	—
243	C₂×C₄	#104×C₆₁	1	2	yes	—
244	C₂×C₄	#104×C₆₁	1	2	yes	—
245	C₂×C₄	C₄×Q₈×C₆₁	1	2	yes	—
246	C₂×C₄	C₂²⋊Q₈×C₆₁	1	2	yes	—
247	C₂×C₄	#124×C₆₁	1	1	yes	characteristic
248	C₂×C₄	C₄⋊Q₈×C₆₁	1	1	yes	characteristic
249	C₈	—	2	8	no	—
250	C₈	—	4	4	no	—
251	C₂²	C₆₁×#391	1	1	yes	characteristic
252	C₂²	C₂×#100×C₆₁	1	1	yes	characteristic
253	C₂²	C₂×#101×C₆₁	1	1	yes	characteristic
254	C₂²	C₂×#104×C₆₁	1	1	yes	characteristic
255	C₂²	C₂×#104×C₆₁	1	1	yes	characteristic
256	C₂²	C₂×#104×C₆₁	1	1	yes	characteristic
257	C₂²	C₆₁×#421	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	8	2	no	—
266	C₄	—	8	2	no	—
267	C₄	C₆₁×#329	1	4	yes	—
268	C₂	C₂×C₆₁×#329	1	1	yes	characteristic
269	C₂	C₆₁×#1526	1	1	yes	characteristic
270	C₂	C₆₁×#1527	1	1	yes	characteristic
271	C₂	C₆₁×#1531	1	1	yes	characteristic
272	C₂	C₆₁×#1986	1	1	yes	characteristic
273	C₂	C₆₁×#1994	1	1	yes	characteristic
274	C₂	C₆₁×#1997	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₄³×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
41. #97×C₆₁	Q₈	semidirect
113. #9890	C₆₁	direct (N × H)
157. C₆₁	#9890	direct (N × H)
179. #97	Q₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#2862	2
C₆₁×#1767	2
C₄×#97×C₆₁	2
C₆₁×#1295	2
#9890	61

Subgroups — C61×#9890 #29513323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#9890 #29513323