Subgroups — C₆₁×#19975 #29523408

Order 15616.

758 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3391	C₂	1	2	yes	maximal, complemented
2	C₆₁×#3287	C₂	1	2	yes	maximal, complemented
3	C₆₁×#1712	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1668	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1667	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1621	C₂	1	4	yes	maximal, complemented
7	C₈×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₆₁	—	2	2	no	—
10	C₆₁×#550	C₂²	1	4	yes	—
11	C₆₁×#493	—	2	2	no	—
12	C₆₁×#493	C₂²	1	2	yes	—
13	C₆₁×#493	C₂²	1	2	yes	—
14	C₆₁×#493	C₂²	1	2	yes	—
15	C₆₁×#466	—	2	2	no	—
16	C₆₁×#460	C₂²	1	4	yes	—
17	C₆₁×#459	—	2	4	no	—
18	C₆₁×#459	C₂²	1	4	yes	—
19	C₆₁×#449	—	2	4	no	—
20	C₈×Q₈×C₆₁	C₂²	1	2	yes	—
21	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
22	C₄×D₈×C₆₁	—	2	2	no	—
23	C₄×D₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₈×D₄×C₆₁	C₂²	1	2	yes	—
25	C₆₁×#335	C₂²	1	1	yes	characteristic
26	C₆₁×#335	C₂²	1	1	yes	characteristic
27	C₆₁×#331	C₂²	1	4	yes	—
28	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	—
29	C₆₁×D₄⋊C₈	C₂²	1	2	yes	—
30	C₈²×C₆₁	C₂²	1	1	yes	characteristic
31	C₂×#41×C₆₁	—	2	4	no	—
32	#135×C₆₁	—	4	8	no	—
33	C₂×Q₁₆×C₆₁	D₄	1	1	yes	characteristic
34	C₂×SD₁₆×C₆₁	—	2	4	no	—
35	C₂×D₈×C₆₁	—	2	4	no	—
36	C₂×D₈×C₆₁	—	2	2	no	—
37	C₂×D₈×C₆₁	—	2	2	no	—
38	C₂×D₈×C₆₁	—	2	2	no	—
39	C₂×D₈×C₆₁	—	2	2	no	—
40	C₂×D₈×C₆₁	—	4	2	no	—
41	C₂×D₈×C₆₁	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₁	—	2	2	no	—
43	#127×C₆₁	—	2	4	no	—
44	#127×C₆₁	C₂³	1	4	yes	—
45	#121×C₆₁	—	2	4	no	—
46	#121×C₆₁	—	4	4	no	—
47	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
48	C₄×D₄×C₆₁	—	2	4	no	—
49	C₄×D₄×C₆₁	C₂³	1	2	yes	—
50	#107×C₆₁	D₄	1	1	yes	characteristic
51	#107×C₆₁	D₄	1	1	yes	characteristic
52	C₄⋊C₈×C₆₁	—	2	2	no	—
53	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
54	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
55	#103×C₆₁	D₄	1	2	yes	—
56	#102×C₆₁	—	2	4	no	—
57	#102×C₆₁	—	2	4	no	—
58	#102×C₆₁	D₄	1	2	yes	—
59	C₂²⋊C₈×C₆₁	—	2	2	no	—
60	C₄×C₈×C₆₁	—	2	2	no	—
61	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
64	#41×C₆₁	—	4	8	no	—
65	#41×C₆₁	—	8	4	no	—
66	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
67	C₂×D₄×C₆₁	—	2	4	no	—
68	C₂×D₄×C₆₁	—	2	4	no	—
69	C₂×D₄×C₆₁	—	2	4	no	—
70	C₂×D₄×C₆₁	—	4	4	no	—
71	C₂×D₄×C₆₁	—	4	4	no	—
72	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₆₁	—	2	2	no	—
74	C₂²×C₄×C₆₁	—	4	4	no	—
75	Q₁₆×C₆₁	D₈	1	4	yes	complemented
76	SD₁₆×C₆₁	—	4	8	no	—
77	D₈×C₆₁	—	2	2	no	—
78	D₈×C₆₁	—	4	4	no	—
79	D₈×C₆₁	—	4	4	no	—
80	D₈×C₆₁	—	8	2	no	—
81	D₈×C₆₁	—	8	2	no	—
82	D₈×C₆₁	—	8	4	no	—
83	D₈×C₆₁	—	8	4	no	—
84	C₂×C₈×C₆₁	—	2	1	no	—
85	C₂×C₈×C₆₁	—	2	2	no	—
86	C₂×C₈×C₆₁	—	2	2	no	—
87	C₂×C₈×C₆₁	—	4	4	no	—
88	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
92	C₄⋊C₄×C₆₁	—	2	2	no	—
93	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
94	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
95	C₂²⋊C₄×C₆₁	—	2	2	no	—
96	C₂²⋊C₄×C₆₁	—	4	4	no	—
97	C₄²×C₆₁	—	2	2	no	—
98	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
99	C₂³×C₆₁	—	2	2	no	—
100	C₂³×C₆₁	—	4	4	no	—
101	Q₈×C₆₁	—	2	2	no	—
102	Q₈×C₆₁	C₂×D₈	1	4	yes	—
103	D₄×C₆₁	—	2	2	no	—
104	D₄×C₆₁	—	2	2	no	—
105	D₄×C₆₁	—	4	4	no	—
106	D₄×C₆₁	—	4	4	no	—
107	D₄×C₆₁	—	4	4	no	—
108	D₄×C₆₁	—	4	4	no	—
109	D₄×C₆₁	—	8	8	no	—
110	D₄×C₆₁	—	8	4	no	—
111	D₄×C₆₁	—	16	4	no	—
112	C₂×C₄×C₆₁	—	2	2	no	—
113	C₂×C₄×C₆₁	—	2	2	no	—
114	C₂×C₄×C₆₁	—	2	2	no	—
115	C₂×C₄×C₆₁	—	4	2	no	—
116	C₂×C₄×C₆₁	—	4	2	no	—
117	C₂×C₄×C₆₁	—	8	8	no	—
118	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
121	C₈×C₆₁	—	2	2	no	—
122	C₈×C₆₁	—	2	1	no	—
123	C₈×C₆₁	—	2	1	no	—
124	C₈×C₆₁	—	4	2	no	—
125	C₈×C₆₁	—	4	2	no	—
126	C₈×C₆₁	C₂×D₈	1	2	yes	—
127	C₈×C₆₁	#135	1	2	yes	complemented
128	#19975	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂²×C₆₁	—	4	2	no	—
130	C₂²×C₆₁	—	4	2	no	—
131	C₂²×C₆₁	—	4	2	no	—
132	C₂²×C₆₁	—	8	4	no	—
133	C₂²×C₆₁	—	8	4	no	—
134	C₂²×C₆₁	—	8	4	no	—
135	C₂²×C₆₁	D₄²	1	1	yes	characteristic
136	C₄×C₆₁	—	2	4	no	—
137	C₄×C₆₁	—	2	1	no	—
138	C₄×C₆₁	—	4	2	no	—
139	C₄×C₆₁	—	4	2	no	—
140	C₄×C₆₁	C₂²×D₈	1	1	yes	characteristic
141	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
142	C₄×C₆₁	#576	1	1	yes	characteristic
143	C₄×C₆₁	#576	1	1	yes	characteristic
144	#3391	C₂×C₆₁	1	2	yes	complemented
145	#3287	C₂×C₆₁	1	2	yes	complemented
146	#1712	C₂×C₆₁	1	1	yes	characteristic, complemented
147	#1668	C₂×C₆₁	1	2	yes	complemented
148	#1667	C₂×C₆₁	1	2	yes	complemented
149	#1621	C₂×C₆₁	1	4	yes	complemented
150	C₈×Q₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
151	C₈×D₈	C₂×C₆₁	1	1	yes	characteristic, complemented
152	C₂×C₆₁	—	8	2	no	—
153	C₂×C₆₁	—	16	4	no	—
154	C₂×C₆₁	D₄×D₈	1	1	yes	characteristic
155	C₂×C₆₁	#3287	1	1	yes	characteristic
156	C₂×C₆₁	#3292	1	1	yes	characteristic
157	C₂×#135	—	2	2	no	—
158	#550	C₂²×C₆₁	1	4	yes	—
159	#493	—	2	2	no	—
160	#493	C₂²×C₆₁	1	2	yes	—
161	#493	C₂²×C₆₁	1	2	yes	—
162	#493	C₂²×C₆₁	1	2	yes	—
163	#466	—	2	2	no	—
164	#460	C₂²×C₆₁	1	4	yes	—
165	#459	—	2	4	no	—
166	#459	C₂²×C₆₁	1	4	yes	—
167	#449	—	2	4	no	—
168	C₈×Q₈	C₂²×C₆₁	1	2	yes	—
169	C₄×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
170	C₄×D₈	—	2	2	no	—
171	C₄×D₈	C₂²×C₆₁	1	1	yes	characteristic
172	C₈×D₄	C₂²×C₆₁	1	2	yes	—
173	#335	C₂²×C₆₁	1	1	yes	characteristic
174	#335	C₂²×C₆₁	1	1	yes	characteristic
175	#331	C₂²×C₆₁	1	4	yes	—
176	Q₈⋊C₈	C₂²×C₆₁	1	2	yes	—
177	D₄⋊C₈	C₂²×C₆₁	1	2	yes	—
178	C₈²	C₂²×C₆₁	1	1	yes	characteristic
179	C₆₁	#19975	1	1	yes	characteristic, direct product factor, Sylow
180	C₂×#41	—	2	4	no	—
181	#135	—	4	8	no	—
182	C₂×Q₁₆	D₄×C₆₁	1	1	yes	characteristic
183	C₂×SD₁₆	—	2	4	no	—
184	C₂×D₈	—	2	4	no	—
185	C₂×D₈	—	2	2	no	—
186	C₂×D₈	—	2	2	no	—
187	C₂×D₈	—	2	2	no	—
188	C₂×D₈	—	2	2	no	—
189	C₂×D₈	—	4	2	no	—
190	C₂×D₈	D₄×C₆₁	1	1	yes	characteristic
191	C₂²×C₈	—	2	2	no	—
192	#127	—	2	4	no	—
193	#127	C₂³×C₆₁	1	4	yes	—
194	#121	—	2	4	no	—
195	#121	—	4	4	no	—
196	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
197	C₄×D₄	—	2	4	no	—
198	C₄×D₄	C₂³×C₆₁	1	2	yes	—
199	#107	D₄×C₆₁	1	1	yes	characteristic
200	#107	D₄×C₆₁	1	1	yes	characteristic
201	C₄⋊C₈	—	2	2	no	—
202	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
203	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
204	#103	D₄×C₆₁	1	2	yes	—
205	#102	—	2	4	no	—
206	#102	—	2	4	no	—
207	#102	D₄×C₆₁	1	2	yes	—
208	C₂²⋊C₈	—	2	2	no	—
209	C₄×C₈	—	2	2	no	—
210	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
211	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
212	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
213	#41	—	4	8	no	—
214	#41	—	8	4	no	—
215	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
216	C₂×D₄	—	2	4	no	—
217	C₂×D₄	—	2	4	no	—
218	C₂×D₄	—	2	4	no	—
219	C₂×D₄	—	4	4	no	—
220	C₂×D₄	—	4	4	no	—
221	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	4	4	no	—
224	Q₁₆	D₈×C₆₁	1	4	yes	complemented
225	SD₁₆	—	4	8	no	—
226	D₈	—	2	2	no	—
227	D₈	—	4	4	no	—
228	D₈	—	4	4	no	—
229	D₈	—	8	2	no	—
230	D₈	—	8	2	no	—
231	D₈	—	8	4	no	—
232	D₈	—	8	4	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	4	4	no	—
237	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
238	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
239	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
240	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
243	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
244	C₂²⋊C₄	—	2	2	no	—
245	C₂²⋊C₄	—	4	4	no	—
246	C₄²	—	2	2	no	—
247	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
248	C₂³	—	2	2	no	—
249	C₂³	—	4	4	no	—
250	Q₈	—	2	2	no	—
251	Q₈	C₂×D₈×C₆₁	1	4	yes	—
252	D₄	—	2	2	no	—
253	D₄	—	2	2	no	—
254	D₄	—	4	4	no	—
255	D₄	—	4	4	no	—
256	D₄	—	4	4	no	—
257	D₄	—	4	4	no	—
258	D₄	—	8	8	no	—
259	D₄	—	8	4	no	—
260	D₄	—	16	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	8	8	no	—
267	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
268	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
269	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	2	1	no	—
272	C₈	—	2	1	no	—
273	C₈	—	4	2	no	—
274	C₈	—	4	2	no	—
275	C₈	C₂×D₈×C₆₁	1	2	yes	—
276	C₈	#135×C₆₁	1	2	yes	complemented
277	C₂²	—	4	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	8	4	no	—
281	C₂²	—	8	4	no	—
282	C₂²	—	8	4	no	—
283	C₂²	D₄²×C₆₁	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	4	no	—
286	C₄	—	4	2	no	—
287	C₄	—	4	2	no	—
288	C₄	C₂²×D₈×C₆₁	1	1	yes	characteristic
289	C₄	C₂×#135×C₆₁	1	1	yes	characteristic
290	C₄	C₆₁×#576	1	1	yes	characteristic
291	C₄	C₆₁×#576	1	1	yes	characteristic
292	C₂	—	8	2	no	—
293	C₂	—	16	4	no	—
294	C₂	D₄×D₈×C₆₁	1	1	yes	characteristic
295	C₂	C₆₁×#3287	1	1	yes	characteristic
296	C₂	C₆₁×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3391	C₂	semidirect
2. C₆₁×#3287	C₂	semidirect
3. C₆₁×#1712	C₂	semidirect
4. C₆₁×#1668	C₂	semidirect
5. C₆₁×#1667	C₂	semidirect
6. C₆₁×#1621	C₂	semidirect
7. C₈×Q₁₆×C₆₁	C₂	semidirect
8. C₈×D₈×C₆₁	C₂	semidirect
75. Q₁₆×C₆₁	D₈	semidirect
127. C₈×C₆₁	#135	semidirect
128. #19975	C₆₁	direct (N × H)
144. #3391	C₂×C₆₁	semidirect
145. #3287	C₂×C₆₁	semidirect
146. #1712	C₂×C₆₁	semidirect
147. #1668	C₂×C₆₁	semidirect
148. #1667	C₂×C₆₁	semidirect
149. #1621	C₂×C₆₁	semidirect
150. C₈×Q₁₆	C₂×C₆₁	semidirect
151. C₈×D₈	C₂×C₆₁	semidirect
179. C₆₁	#19975	direct (N × H)
224. Q₁₆	D₈×C₆₁	semidirect
276. C₈	#135×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3391	2
C₆₁×#3287	2
C₆₁×#1712	2
C₆₁×#1668	2
C₆₁×#1667	2
C₆₁×#1621	2
C₈×Q₁₆×C₆₁	2
C₈×D₈×C₆₁	2
#19975	61

Subgroups — C61×#19975 #29523408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19975 #29523408