Subgroups — C₆₁×#33957 #29537390

Order 15616.

444 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3383	C₂	1	2	yes	maximal
2	C₆₁×#2255	C₂	1	2	yes	maximal
3	C₆₁×#2253	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#2229	C₂	1	2	yes	maximal
5	C₆₁×#2229	C₂	1	2	yes	maximal
6	C₆₁×#2227	C₂	1	2	yes	maximal
7	C₆₁×#2183	C₂	1	1	yes	maximal, characteristic
8	C₆₁×#2176	C₂	1	2	yes	maximal
9	C₄×Q₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
10	C₆₁×#557	—	2	2	no	—
11	C₂×Q₃₂×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#500	C₂²	1	2	yes	—
13	C₆₁×#499	C₂²	1	2	yes	—
14	C₆₁×#499	C₂²	1	2	yes	—
15	C₆₁×#479	—	2	2	no	—
16	C₆₁×#479	—	2	2	no	—
17	C₆₁×#475	—	2	2	no	—
18	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
20	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
21	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
22	C₆₁×#367	C₂²	1	2	yes	—
23	C₆₁×#367	C₂²	1	2	yes	—
24	C₆₁×#366	C₂²	1	1	yes	characteristic
25	C₆₁×#366	C₂²	1	1	yes	characteristic
26	C₆₁×#366	C₂²	1	2	yes	—
27	C₆₁×#366	C₂²	1	1	yes	characteristic
28	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₆₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₆₁×#358	C₂²	1	2	yes	—
31	C₆₁×#358	C₂²	1	2	yes	—
32	C₆₁×#358	C₂²	1	2	yes	—
33	C₆₁×#346	C₂²	1	2	yes	—
34	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
36	C₄⋊Q₈×C₆₁	—	2	2	no	—
37	C₄⋊Q₈×C₆₁	—	4	2	no	—
38	#125×C₆₁	—	2	2	no	—
39	#125×C₆₁	—	2	2	no	—
40	#125×C₆₁	—	4	2	no	—
41	#125×C₆₁	—	4	2	no	—
42	C₄×Q₈×C₆₁	—	2	2	no	—
43	C₄×Q₈×C₆₁	—	2	2	no	—
44	C₄×Q₈×C₆₁	—	2	2	no	—
45	C₄×Q₈×C₆₁	—	4	2	no	—
46	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
47	Q₃₂×C₆₁	Q₈	1	4	yes	complemented
48	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
50	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
51	#107×C₆₁	—	2	2	no	—
52	#107×C₆₁	C₂³	1	2	yes	—
53	#107×C₆₁	C₂³	1	2	yes	—
54	#107×C₆₁	C₂³	1	2	yes	—
55	#106×C₆₁	—	2	2	no	—
56	#106×C₆₁	—	2	2	no	—
57	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
58	C₄⋊C₈×C₆₁	D₄	1	1	yes	characteristic
59	#103×C₆₁	—	2	2	no	—
60	#103×C₆₁	—	2	2	no	—
61	#103×C₆₁	—	2	2	no	—
62	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₆₁	C₂³	1	2	yes	—
64	C₂×Q₈×C₆₁	—	2	2	no	—
65	C₂×Q₈×C₆₁	—	4	2	no	—
66	C₂×Q₈×C₆₁	D₈	1	1	yes	characteristic
67	Q₁₆×C₆₁	—	2	2	no	—
68	Q₁₆×C₆₁	C₂×Q₈	1	4	yes	—
69	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄×C₆₁	—	2	2	no	—
73	C₄⋊C₄×C₆₁	—	2	2	no	—
74	C₄⋊C₄×C₆₁	—	2	2	no	—
75	C₄⋊C₄×C₆₁	—	4	2	no	—
76	C₄⋊C₄×C₆₁	—	4	2	no	—
77	C₄⋊C₄×C₆₁	—	4	2	no	—
78	C₄⋊C₄×C₆₁	—	4	2	no	—
79	C₄⋊C₄×C₆₁	—	4	2	no	—
80	C₄⋊C₄×C₆₁	—	4	2	no	—
81	C₄⋊C₄×C₆₁	—	4	2	no	—
82	C₄⋊C₄×C₆₁	—	4	2	no	—
83	C₄⋊C₄×C₆₁	D₈	1	2	yes	—
84	C₄⋊C₄×C₆₁	D₈	1	1	yes	characteristic
85	C₄²×C₆₁	—	4	2	no	—
86	C₄²×C₆₁	—	4	2	no	—
87	C₄²×C₆₁	—	4	2	no	—
88	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₆₁	C₂×D₄	1	2	yes	—
90	C₁₆×C₆₁	—	2	1	no	—
91	C₁₆×C₆₁	—	2	2	no	—
92	C₁₆×C₆₁	C₂×Q₈	1	2	yes	—
93	Q₈×C₆₁	—	2	4	no	—
94	Q₈×C₆₁	—	4	1	no	—
95	Q₈×C₆₁	—	4	2	no	—
96	Q₈×C₆₁	—	8	4	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	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
103	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
106	C₈×C₆₁	—	2	1	no	—
107	C₈×C₆₁	—	2	2	no	—
108	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
109	C₈×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
110	#33957	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
112	C₄×C₆₁	—	2	1	no	—
113	C₄×C₆₁	—	2	2	no	—
114	C₄×C₆₁	—	2	1	no	—
115	C₄×C₆₁	—	2	2	no	—
116	C₄×C₆₁	—	4	4	no	—
117	C₄×C₆₁	—	8	2	no	—
118	C₄×C₆₁	—	8	2	no	—
119	C₄×C₆₁	—	8	2	no	—
120	C₄×C₆₁	D₄×Q₈	1	1	yes	characteristic
121	C₄×C₆₁	#578	1	1	yes	characteristic
122	#3383	C₂×C₆₁	1	2	yes	—
123	#2255	C₂×C₆₁	1	2	yes	—
124	#2253	C₂×C₆₁	1	1	yes	characteristic
125	#2229	C₂×C₆₁	1	2	yes	—
126	#2229	C₂×C₆₁	1	2	yes	—
127	#2227	C₂×C₆₁	1	2	yes	—
128	#2183	C₂×C₆₁	1	1	yes	characteristic
129	#2176	C₂×C₆₁	1	2	yes	—
130	C₄×Q₃₂	C₂×C₆₁	1	1	yes	characteristic
131	C₂×C₆₁	Q₈×D₈	1	1	yes	characteristic
132	C₂×C₆₁	#3414	1	1	yes	characteristic
133	C₂×C₆₁	#3417	1	1	yes	characteristic
134	#557	—	2	2	no	—
135	C₂×Q₃₂	C₂²×C₆₁	1	1	yes	characteristic
136	#500	C₂²×C₆₁	1	2	yes	—
137	#499	C₂²×C₆₁	1	2	yes	—
138	#499	C₂²×C₆₁	1	2	yes	—
139	#479	—	2	2	no	—
140	#479	—	2	2	no	—
141	#475	—	2	2	no	—
142	C₈×Q₈	C₂²×C₆₁	1	1	yes	characteristic
143	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
144	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
145	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
146	#367	C₂²×C₆₁	1	2	yes	—
147	#367	C₂²×C₆₁	1	2	yes	—
148	#366	C₂²×C₆₁	1	1	yes	characteristic
149	#366	C₂²×C₆₁	1	1	yes	characteristic
150	#366	C₂²×C₆₁	1	2	yes	—
151	#366	C₂²×C₆₁	1	1	yes	characteristic
152	C₄⋊C₁₆	C₂²×C₆₁	1	2	yes	—
153	C₄⋊C₁₆	C₂²×C₆₁	1	1	yes	characteristic
154	#358	C₂²×C₆₁	1	2	yes	—
155	#358	C₂²×C₆₁	1	2	yes	—
156	#358	C₂²×C₆₁	1	2	yes	—
157	#346	C₂²×C₆₁	1	2	yes	—
158	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
159	C₆₁	#33957	1	1	yes	characteristic, direct product factor, Sylow
160	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
161	C₄⋊Q₈	—	2	2	no	—
162	C₄⋊Q₈	—	4	2	no	—
163	#125	—	2	2	no	—
164	#125	—	2	2	no	—
165	#125	—	4	2	no	—
166	#125	—	4	2	no	—
167	C₄×Q₈	—	2	2	no	—
168	C₄×Q₈	—	2	2	no	—
169	C₄×Q₈	—	2	2	no	—
170	C₄×Q₈	—	4	2	no	—
171	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
172	Q₃₂	Q₈×C₆₁	1	4	yes	complemented
173	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
174	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
175	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
176	#107	—	2	2	no	—
177	#107	C₂³×C₆₁	1	2	yes	—
178	#107	C₂³×C₆₁	1	2	yes	—
179	#107	C₂³×C₆₁	1	2	yes	—
180	#106	—	2	2	no	—
181	#106	—	2	2	no	—
182	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
183	C₄⋊C₈	D₄×C₆₁	1	1	yes	characteristic
184	#103	—	2	2	no	—
185	#103	—	2	2	no	—
186	#103	—	2	2	no	—
187	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
188	C₄×C₈	C₂³×C₆₁	1	2	yes	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂×Q₈	D₈×C₆₁	1	1	yes	characteristic
192	Q₁₆	—	2	2	no	—
193	Q₁₆	C₂×Q₈×C₆₁	1	4	yes	—
194	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
195	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
196	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	D₈×C₆₁	1	2	yes	—
209	C₄⋊C₄	D₈×C₆₁	1	1	yes	characteristic
210	C₄²	—	4	2	no	—
211	C₄²	—	4	2	no	—
212	C₄²	—	4	2	no	—
213	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
214	C₄²	C₂×D₄×C₆₁	1	2	yes	—
215	C₁₆	—	2	1	no	—
216	C₁₆	—	2	2	no	—
217	C₁₆	C₂×Q₈×C₆₁	1	2	yes	—
218	Q₈	—	2	4	no	—
219	Q₈	—	4	1	no	—
220	Q₈	—	4	2	no	—
221	Q₈	—	8	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
227	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
228	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
229	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
230	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
231	C₈	—	2	1	no	—
232	C₈	—	2	2	no	—
233	C₈	C₂²×Q₈×C₆₁	1	1	yes	characteristic
234	C₈	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
235	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
236	C₄	—	2	1	no	—
237	C₄	—	2	2	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	2	no	—
240	C₄	—	4	4	no	—
241	C₄	—	8	2	no	—
242	C₄	—	8	2	no	—
243	C₄	—	8	2	no	—
244	C₄	D₄×Q₈×C₆₁	1	1	yes	characteristic
245	C₄	C₆₁×#578	1	1	yes	characteristic
246	C₂	Q₈×D₈×C₆₁	1	1	yes	characteristic
247	C₂	C₆₁×#3414	1	1	yes	characteristic
248	C₂	C₆₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₆₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄×Q₈×C₆₁ ◁ C₈×Q₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
47. Q₃₂×C₆₁	Q₈	semidirect
110. #33957	C₆₁	direct (N × H)
159. C₆₁	#33957	direct (N × H)
172. Q₃₂	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×#3383	2
C₆₁×#2255	2
C₆₁×#2253	2
C₆₁×#2229	2
C₆₁×#2227	2
C₆₁×#2183	2
C₆₁×#2176	2
C₄×Q₃₂×C₆₁	2
#33957	61

Subgroups — C61×#33957 #29537390

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33957 #29537390