Subgroups — C₆₁×#33826 #29537259

Order 15616.

488 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3177	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3176	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2229	C₂	1	2	yes	maximal, complemented
4	C₆₁×#2228	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2220	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2217	C₂	1	2	yes	maximal, complemented
7	C₆₁×#2208	C₂	1	2	yes	maximal, complemented
8	C₆₁×#2207	C₂	1	2	yes	maximal, complemented
9	C₆₁×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#541	—	2	1	no	—
11	C₆₁×#540	—	2	1	no	—
12	C₂×SD₃₂×C₆₁	C₂²	1	2	yes	—
13	C₂×SD₃₂×C₆₁	C₂²	1	2	yes	—
14	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#499	C₂²	1	1	yes	characteristic
17	C₆₁×#499	C₂²	1	1	yes	characteristic
18	C₆₁×#494	C₂²	1	1	yes	characteristic
19	C₆₁×#493	C₂²	1	1	yes	characteristic
20	C₆₁×#491	—	2	1	no	—
21	C₆₁×#490	—	2	1	no	—
22	C₆₁×#470	—	2	1	no	—
23	C₆₁×#469	—	2	1	no	—
24	C₆₁×#467	C₂²	1	2	yes	—
25	C₆₁×#466	C₂²	1	2	yes	—
26	C₆₁×#442	—	2	1	no	—
27	C₆₁×#441	—	2	1	no	—
28	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
29	C₄×D₈×C₆₁	C₂²	1	2	yes	—
30	C₆₁×#405	C₂²	1	1	yes	characteristic
31	C₆₁×#367	C₂²	1	2	yes	—
32	C₆₁×#366	C₂²	1	2	yes	—
33	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
34	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
35	C₆₁×#358	C₂²	1	2	yes	—
36	C₆₁×#358	C₂²	1	2	yes	—
37	C₆₁×#357	C₂²	1	2	yes	—
38	C₆₁×#357	C₂²	1	2	yes	—
39	C₂×Q₁₆×C₆₁	—	2	1	no	—
40	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
41	C₂×D₈×C₆₁	—	2	1	no	—
42	C₂×D₈×C₆₁	C₂³	1	2	yes	—
43	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
45	C₄⋊Q₈×C₆₁	—	2	1	no	—
46	#127×C₆₁	—	2	1	no	—
47	#126×C₆₁	—	4	2	no	—
48	#125×C₆₁	—	2	1	no	—
49	#125×C₆₁	—	2	1	no	—
50	#125×C₆₁	—	4	2	no	—
51	#123×C₆₁	—	4	2	no	—
52	C₂²⋊Q₈×C₆₁	—	2	2	no	—
53	#121×C₆₁	—	2	2	no	—
54	#121×C₆₁	—	4	2	no	—
55	C₄×Q₈×C₆₁	—	2	2	no	—
56	C₄×D₄×C₆₁	—	2	2	no	—
57	#117×C₆₁	D₄	1	1	yes	characteristic
58	SD₃₂×C₆₁	—	4	2	no	—
59	SD₃₂×C₆₁	—	4	2	no	—
60	M₃₂×C₆₁	—	4	1	no	—
61	C₂×C₁₆×C₆₁	—	4	1	no	—
62	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
63	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
64	#107×C₆₁	C₂³	1	2	yes	—
65	#107×C₆₁	C₂³	1	2	yes	—
66	#106×C₆₁	—	2	1	no	—
67	#106×C₆₁	—	2	1	no	—
68	#103×C₆₁	—	2	2	no	—
69	#103×C₆₁	—	2	2	no	—
70	#102×C₆₁	—	2	2	no	—
71	#102×C₆₁	—	2	2	no	—
72	#97×C₆₁	D₄	1	1	yes	characteristic
73	#97×C₆₁	D₄	1	1	yes	characteristic
74	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
76	C₂×Q₈×C₆₁	—	2	2	no	—
77	C₂×D₄×C₆₁	—	2	2	no	—
78	C₂×D₄×C₆₁	—	4	2	no	—
79	C₂×D₄×C₆₁	—	4	2	no	—
80	C₂²×C₄×C₆₁	—	4	2	no	—
81	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
82	Q₁₆×C₆₁	—	2	2	no	—
83	Q₁₆×C₆₁	—	2	2	no	—
84	Q₁₆×C₆₁	—	4	2	no	—
85	D₈×C₆₁	—	2	2	no	—
86	D₈×C₆₁	—	2	2	no	—
87	D₈×C₆₁	—	4	2	no	—
88	M₁₆×C₆₁	—	4	1	no	—
89	C₂×C₈×C₆₁	—	2	1	no	—
90	C₂×C₈×C₆₁	—	2	1	no	—
91	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₆₁	—	2	2	no	—
96	C₄⋊C₄×C₆₁	—	2	2	no	—
97	C₄⋊C₄×C₆₁	—	4	2	no	—
98	C₄⋊C₄×C₆₁	—	4	2	no	—
99	C₄⋊C₄×C₆₁	—	4	2	no	—
100	C₄⋊C₄×C₆₁	—	4	2	no	—
101	C₄⋊C₄×C₆₁	—	4	2	no	—
102	C₄⋊C₄×C₆₁	D₈	1	2	yes	—
103	C₂²⋊C₄×C₆₁	—	4	2	no	—
104	C₂²⋊C₄×C₆₁	—	4	2	no	—
105	C₂²⋊C₄×C₆₁	—	4	2	no	—
106	C₂²⋊C₄×C₆₁	D₈	1	2	yes	—
107	C₄²×C₆₁	—	4	2	no	—
108	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
110	C₁₆×C₆₁	—	2	2	no	—
111	C₁₆×C₆₁	—	2	2	no	—
112	C₂³×C₆₁	—	4	2	no	—
113	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
114	Q₈×C₆₁	—	4	2	no	—
115	Q₈×C₆₁	—	4	2	no	—
116	D₄×C₆₁	—	4	2	no	—
117	D₄×C₆₁	—	4	2	no	—
118	D₄×C₆₁	—	16	2	no	—
119	D₄×C₆₁	—	16	2	no	—
120	C₂×C₄×C₆₁	—	2	1	no	—
121	C₂×C₄×C₆₁	—	2	1	no	—
122	C₂×C₄×C₆₁	—	4	2	no	—
123	C₂×C₄×C₆₁	—	4	2	no	—
124	C₂×C₄×C₆₁	—	4	2	no	—
125	C₂×C₄×C₆₁	—	16	2	no	—
126	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
127	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₆₁	—	2	1	no	—
131	C₈×C₆₁	—	2	1	no	—
132	C₈×C₆₁	—	2	1	no	—
133	C₈×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
134	C₈×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
135	#33826	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×C₆₁	—	2	2	no	—
137	C₂²×C₆₁	—	2	1	no	—
138	C₂²×C₆₁	—	8	2	no	—
139	C₂²×C₆₁	—	8	2	no	—
140	C₂²×C₆₁	—	8	2	no	—
141	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
142	C₄×C₆₁	—	2	2	no	—
143	C₄×C₆₁	—	2	2	no	—
144	C₄×C₆₁	—	2	1	no	—
145	C₄×C₆₁	—	8	2	no	—
146	C₄×C₆₁	—	8	2	no	—
147	C₄×C₆₁	—	8	2	no	—
148	C₄×C₆₁	#537	1	1	yes	characteristic
149	C₄×C₆₁	#575	1	1	yes	characteristic
150	#3177	C₂×C₆₁	1	1	yes	characteristic, complemented
151	#3176	C₂×C₆₁	1	1	yes	characteristic
152	#2229	C₂×C₆₁	1	2	yes	complemented
153	#2228	C₂×C₆₁	1	2	yes	complemented
154	#2220	C₂×C₆₁	1	2	yes	complemented
155	#2217	C₂×C₆₁	1	2	yes	complemented
156	#2208	C₂×C₆₁	1	2	yes	complemented
157	#2207	C₂×C₆₁	1	2	yes	complemented
158	#2151	C₂×C₆₁	1	1	yes	characteristic, complemented
159	C₂×C₆₁	—	4	1	no	—
160	C₂×C₆₁	—	16	2	no	—
161	C₂×C₆₁	#3245	1	1	yes	characteristic
162	C₂×C₆₁	#3416	1	2	yes	—
163	#541	—	2	1	no	—
164	#540	—	2	1	no	—
165	C₂×SD₃₂	C₂²×C₆₁	1	2	yes	—
166	C₂×SD₃₂	C₂²×C₆₁	1	2	yes	—
167	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
168	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
169	#499	C₂²×C₆₁	1	1	yes	characteristic
170	#499	C₂²×C₆₁	1	1	yes	characteristic
171	#494	C₂²×C₆₁	1	1	yes	characteristic
172	#493	C₂²×C₆₁	1	1	yes	characteristic
173	#491	—	2	1	no	—
174	#490	—	2	1	no	—
175	#470	—	2	1	no	—
176	#469	—	2	1	no	—
177	#467	C₂²×C₆₁	1	2	yes	—
178	#466	C₂²×C₆₁	1	2	yes	—
179	#442	—	2	1	no	—
180	#441	—	2	1	no	—
181	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
182	C₄×D₈	C₂²×C₆₁	1	2	yes	—
183	#405	C₂²×C₆₁	1	1	yes	characteristic
184	#367	C₂²×C₆₁	1	2	yes	—
185	#366	C₂²×C₆₁	1	2	yes	—
186	C₄⋊C₁₆	C₂²×C₆₁	1	2	yes	—
187	C₄⋊C₁₆	C₂²×C₆₁	1	2	yes	—
188	#358	C₂²×C₆₁	1	2	yes	—
189	#358	C₂²×C₆₁	1	2	yes	—
190	#357	C₂²×C₆₁	1	2	yes	—
191	#357	C₂²×C₆₁	1	2	yes	—
192	C₆₁	#33826	1	1	yes	characteristic, direct product factor, Sylow
193	C₂×Q₁₆	—	2	1	no	—
194	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
195	C₂×D₈	—	2	1	no	—
196	C₂×D₈	C₂³×C₆₁	1	2	yes	—
197	C₂×M₁₆	D₄×C₆₁	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
199	C₄⋊Q₈	—	2	1	no	—
200	#127	—	2	1	no	—
201	#126	—	4	2	no	—
202	#125	—	2	1	no	—
203	#125	—	2	1	no	—
204	#125	—	4	2	no	—
205	#123	—	4	2	no	—
206	C₂²⋊Q₈	—	2	2	no	—
207	#121	—	2	2	no	—
208	#121	—	4	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×D₄	—	2	2	no	—
211	#117	D₄×C₆₁	1	1	yes	characteristic
212	SD₃₂	—	4	2	no	—
213	SD₃₂	—	4	2	no	—
214	M₃₂	—	4	1	no	—
215	C₂×C₁₆	—	4	1	no	—
216	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
217	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
218	#107	C₂³×C₆₁	1	2	yes	—
219	#107	C₂³×C₆₁	1	2	yes	—
220	#106	—	2	1	no	—
221	#106	—	2	1	no	—
222	#103	—	2	2	no	—
223	#103	—	2	2	no	—
224	#102	—	2	2	no	—
225	#102	—	2	2	no	—
226	#97	D₄×C₆₁	1	1	yes	characteristic
227	#97	D₄×C₆₁	1	1	yes	characteristic
228	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
229	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
230	C₂×Q₈	—	2	2	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
236	Q₁₆	—	2	2	no	—
237	Q₁₆	—	2	2	no	—
238	Q₁₆	—	4	2	no	—
239	D₈	—	2	2	no	—
240	D₈	—	2	2	no	—
241	D₈	—	4	2	no	—
242	M₁₆	—	4	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
246	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
247	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
248	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	D₈×C₆₁	1	2	yes	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₂²⋊C₄	—	4	2	no	—
259	C₂²⋊C₄	—	4	2	no	—
260	C₂²⋊C₄	D₈×C₆₁	1	2	yes	—
261	C₄²	—	4	2	no	—
262	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
263	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
264	C₁₆	—	2	2	no	—
265	C₁₆	—	2	2	no	—
266	C₂³	—	4	2	no	—
267	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
268	Q₈	—	4	2	no	—
269	Q₈	—	4	2	no	—
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	16	2	no	—
273	D₄	—	16	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	16	2	no	—
280	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
281	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
282	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
283	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
288	C₈	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
289	C₂²	—	2	2	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	8	2	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	8	2	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	2	no	—
301	C₄	C₆₁×#537	1	1	yes	characteristic
302	C₄	C₆₁×#575	1	1	yes	characteristic
303	C₂	—	4	1	no	—
304	C₂	—	16	2	no	—
305	C₂	C₆₁×#3245	1	1	yes	characteristic
306	C₂	C₆₁×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
prime radical	C₆₁×#3176	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ #117×C₆₁ ◁ C₆₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3177	C₂	semidirect
3. C₆₁×#2229	C₂	semidirect
4. C₆₁×#2228	C₂	semidirect
5. C₆₁×#2220	C₂	semidirect
6. C₆₁×#2217	C₂	semidirect
7. C₆₁×#2208	C₂	semidirect
8. C₆₁×#2207	C₂	semidirect
9. C₆₁×#2151	C₂	semidirect
135. #33826	C₆₁	direct (N × H)
150. #3177	C₂×C₆₁	semidirect
152. #2229	C₂×C₆₁	semidirect
153. #2228	C₂×C₆₁	semidirect
154. #2220	C₂×C₆₁	semidirect
155. #2217	C₂×C₆₁	semidirect
156. #2208	C₂×C₆₁	semidirect
157. #2207	C₂×C₆₁	semidirect
158. #2151	C₂×C₆₁	semidirect
192. C₆₁	#33826	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×#3177	2
C₆₁×#3176	2
C₆₁×#2229	2
C₆₁×#2228	2
C₆₁×#2220	2
C₆₁×#2217	2
C₆₁×#2208	2
C₆₁×#2207	2
C₆₁×#2151	2
#33826	61

Subgroups — C61×#33826 #29537259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33826 #29537259