Subgroups — C₆₁×#21975 #29525408

Order 15616.

798 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#423	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2283	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1939	C₂	1	4	yes	maximal, complemented
5	C₆₁×#1851	C₂	1	4	yes	maximal
6	C₆₁×#1843	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1832	C₂	1	2	yes	maximal, complemented
8	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
9	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
10	C₆₁×#431	—	2	4	no	—
11	C₆₁×#423	—	2	4	no	—
12	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₆₁	C₂²	1	8	yes	—
17	C₂×C₂²⋊C₈×C₆₁	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
19	C₆₁×#383	C₄	1	2	yes	—
20	C₆₁×#382	C₄	1	2	yes	—
21	C₆₁×#381	C₄	1	2	yes	—
22	C₆₁×#380	C₄	1	2	yes	—
23	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
24	C₄×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
25	C₆₁×#376	C₂²	1	2	yes	—
26	C₆₁×#336	C₂²	1	8	yes	—
27	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
30	C₂×M₁₆×C₆₁	—	2	4	no	—
31	C₂×M₁₆×C₆₁	—	4	4	no	—
32	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
33	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
34	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
35	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
36	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	D₄	1	4	yes	—
40	C₂×C₄²×C₆₁	Q₈	1	4	yes	—
41	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
42	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₆₁	—	2	4	no	—
45	C₄⋊C₈×C₆₁	—	2	4	no	—
46	C₄⋊C₈×C₆₁	—	2	2	no	—
47	C₄⋊C₈×C₆₁	—	2	2	no	—
48	C₄⋊C₈×C₆₁	—	4	8	no	—
49	C₂²⋊C₈×C₆₁	—	2	4	no	—
50	C₄×C₈×C₆₁	—	2	4	no	—
51	#95×C₆₁	C₂×C₄	1	4	yes	—
52	#95×C₆₁	C₂×C₄	1	4	yes	—
53	#95×C₆₁	C₂×C₄	1	4	yes	—
54	#95×C₆₁	C₂×C₄	1	4	yes	—
55	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	—	2	4	no	—
57	C₂²×C₄×C₆₁	—	2	2	no	—
58	C₂²×C₄×C₆₁	—	2	2	no	—
59	C₂²×C₄×C₆₁	—	4	2	no	—
60	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₆₁	C₂²×C₄	1	8	yes	—
63	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
71	M₁₆×C₆₁	—	4	8	no	—
72	C₂×C₈×C₆₁	—	2	2	no	—
73	C₂×C₈×C₆₁	—	2	2	no	—
74	C₂×C₈×C₆₁	—	2	4	no	—
75	C₂×C₈×C₆₁	—	2	4	no	—
76	C₂×C₈×C₆₁	—	2	8	no	—
77	C₂×C₈×C₆₁	—	2	4	no	—
78	C₄⋊C₄×C₆₁	—	2	8	no	—
79	C₂²⋊C₄×C₆₁	—	4	4	no	—
80	C₄²×C₆₁	—	2	4	no	—
81	C₄²×C₆₁	—	2	4	no	—
82	C₄²×C₆₁	C₄⋊C₄	1	4	yes	—
83	C₄²×C₆₁	C₄⋊C₄	1	4	yes	—
84	C₄²×C₆₁	C₂×D₄	1	2	yes	—
85	C₄²×C₆₁	C₂×D₄	1	2	yes	—
86	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
87	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
88	C₂³×C₆₁	—	2	2	no	—
89	C₂³×C₆₁	—	2	2	no	—
90	C₂³×C₆₁	—	2	2	no	—
91	C₂³×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
92	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₆₁	—	2	2	no	—
94	C₂×C₄×C₆₁	—	2	2	no	—
95	C₂×C₄×C₆₁	—	2	2	no	—
96	C₂×C₄×C₆₁	—	2	2	no	—
97	C₂×C₄×C₆₁	—	2	2	no	—
98	C₂×C₄×C₆₁	—	2	2	no	—
99	C₂×C₄×C₆₁	—	2	4	no	—
100	C₂×C₄×C₆₁	—	2	8	no	—
101	C₂×C₄×C₆₁	—	2	8	no	—
102	C₂×C₄×C₆₁	—	2	8	no	—
103	C₂×C₄×C₆₁	—	2	8	no	—
104	C₂×C₄×C₆₁	—	4	4	no	—
105	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₆₁	#131	1	4	yes	—
111	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
114	C₂×C₄×C₆₁	2₋¹⁺⁴	1	2	yes	—
115	C₈×C₆₁	—	4	2	no	—
116	C₈×C₆₁	—	4	4	no	—
117	C₈×C₆₁	—	4	2	no	—
118	#21975	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂²×C₆₁	—	2	2	no	—
120	C₂²×C₆₁	—	2	2	no	—
121	C₂²×C₆₁	—	2	2	no	—
122	C₂²×C₆₁	—	4	4	no	—
123	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×C₆₁	#520	1	2	yes	—
125	C₂²×C₆₁	C₂×#131	1	2	yes	—
126	C₂²×C₆₁	#568	1	2	yes	—
127	C₄×C₆₁	—	2	2	no	—
128	C₄×C₆₁	—	2	2	no	—
129	C₄×C₆₁	—	2	2	no	—
130	C₄×C₆₁	—	2	2	no	—
131	C₄×C₆₁	—	4	8	no	—
132	C₄×C₆₁	#424	1	4	yes	—
133	C₄×C₆₁	#535	1	1	yes	characteristic
134	C₄×C₆₁	#536	1	1	yes	characteristic
135	C₄×C₆₁	#544	1	2	yes	—
136	C₂×#431	C₂×C₆₁	1	1	yes	characteristic
137	C₂×#423	C₂×C₆₁	1	1	yes	characteristic
138	#2283	C₂×C₆₁	1	1	yes	characteristic
139	#1939	C₂×C₆₁	1	4	yes	complemented
140	#1851	C₂×C₆₁	1	4	yes	—
141	#1843	C₂×C₆₁	1	2	yes	complemented
142	#1832	C₂×C₆₁	1	2	yes	complemented
143	C₂×C₆₁	—	4	2	no	—
144	C₂×C₆₁	#2317	1	1	yes	characteristic
145	C₂×C₆₁	C₂×#424	1	1	yes	characteristic
146	C₂×C₆₁	#2905	1	1	yes	characteristic
147	C₂×C₆₁	#2993	1	4	yes	—
148	C₂²×M₁₆	C₂²×C₆₁	1	2	yes	—
149	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
150	#431	—	2	4	no	—
151	#423	—	2	4	no	—
152	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
153	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
154	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
155	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
156	C₂×C₄⋊C₈	C₂²×C₆₁	1	8	yes	—
157	C₂×C₂²⋊C₈	C₂²×C₆₁	1	2	yes	—
158	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
159	#383	C₄×C₆₁	1	2	yes	—
160	#382	C₄×C₆₁	1	2	yes	—
161	#381	C₄×C₆₁	1	2	yes	—
162	#380	C₄×C₆₁	1	2	yes	—
163	C₄×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
164	C₄×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
165	#376	C₂²×C₆₁	1	2	yes	—
166	#336	C₂²×C₆₁	1	8	yes	—
167	C₆₁	#21975	1	1	yes	characteristic, direct product factor, Sylow
168	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
169	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
170	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
171	C₂×M₁₆	—	2	4	no	—
172	C₂×M₁₆	—	4	4	no	—
173	C₂²×C₈	C₂³×C₆₁	1	4	yes	—
174	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
175	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
176	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
177	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
178	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
179	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
180	C₂×C₄²	D₄×C₆₁	1	4	yes	—
181	C₂×C₄²	Q₈×C₆₁	1	4	yes	—
182	C₂×C₄²	C₂³×C₆₁	1	4	yes	—
183	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
185	C₄⋊C₈	—	2	4	no	—
186	C₄⋊C₈	—	2	4	no	—
187	C₄⋊C₈	—	2	2	no	—
188	C₄⋊C₈	—	2	2	no	—
189	C₄⋊C₈	—	4	8	no	—
190	C₂²⋊C₈	—	2	4	no	—
191	C₄×C₈	—	2	4	no	—
192	#95	C₂×C₄×C₆₁	1	4	yes	—
193	#95	C₂×C₄×C₆₁	1	4	yes	—
194	#95	C₂×C₄×C₆₁	1	4	yes	—
195	#95	C₂×C₄×C₆₁	1	4	yes	—
196	C₂⁴	C₂²×C₄×C₆₁	1	1	yes	characteristic
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	C₄⋊C₄×C₆₁	1	4	yes	—
202	C₂²×C₄	C₄⋊C₄×C₆₁	1	4	yes	—
203	C₂²×C₄	C₂²×C₄×C₆₁	1	8	yes	—
204	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
205	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
206	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
207	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
208	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
210	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
211	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
212	M₁₆	—	4	8	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	8	no	—
218	C₂×C₈	—	2	4	no	—
219	C₄⋊C₄	—	2	8	no	—
220	C₂²⋊C₄	—	4	4	no	—
221	C₄²	—	2	4	no	—
222	C₄²	—	2	4	no	—
223	C₄²	C₄⋊C₄×C₆₁	1	4	yes	—
224	C₄²	C₄⋊C₄×C₆₁	1	4	yes	—
225	C₄²	C₂×D₄×C₆₁	1	2	yes	—
226	C₄²	C₂×D₄×C₆₁	1	2	yes	—
227	C₄²	C₂×Q₈×C₆₁	1	2	yes	—
228	C₄²	C₂×Q₈×C₆₁	1	2	yes	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
233	C₂³	C₂³×C₄×C₆₁	1	1	yes	characteristic
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
247	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
248	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
249	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
250	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
251	C₂×C₄	#131×C₆₁	1	4	yes	—
252	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
253	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
254	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
255	C₂×C₄	2₋¹⁺⁴×C₆₁	1	2	yes	—
256	C₈	—	4	2	no	—
257	C₈	—	4	4	no	—
258	C₈	—	4	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	4	4	no	—
263	C₂²	C₂²×C₄⋊C₄×C₆₁	1	1	yes	characteristic
264	C₂²	C₆₁×#520	1	2	yes	—
265	C₂²	C₂×#131×C₆₁	1	2	yes	—
266	C₂²	C₆₁×#568	1	2	yes	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	4	8	no	—
272	C₄	C₆₁×#424	1	4	yes	—
273	C₄	C₆₁×#535	1	1	yes	characteristic
274	C₄	C₆₁×#536	1	1	yes	characteristic
275	C₄	C₆₁×#544	1	2	yes	—
276	C₂	—	4	2	no	—
277	C₂	C₆₁×#2317	1	1	yes	characteristic
278	C₂	C₂×C₆₁×#424	1	1	yes	characteristic
279	C₂	C₆₁×#2905	1	1	yes	characteristic
280	C₂	C₆₁×#2993	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₆₁×#1939	C₂	semidirect
6. C₆₁×#1843	C₂	semidirect
7. C₆₁×#1832	C₂	semidirect
118. #21975	C₆₁	direct (N × H)
139. #1939	C₂×C₆₁	semidirect
141. #1843	C₂×C₆₁	semidirect
142. #1832	C₂×C₆₁	semidirect
167. C₆₁	#21975	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#431	2
C₂×C₆₁×#423	2
C₆₁×#2283	2
C₆₁×#1939	2
C₆₁×#1851	2
C₆₁×#1843	2
C₆₁×#1832	2
#21975	61

Subgroups — C61×#21975 #29525408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#21975 #29525408