Subgroups — C₆₁×#33318 #29536751

Order 15616.

806 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#501	C₂	1	2	yes	maximal
2	C₂×C₆₁×#499	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#498	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#2660	C₂	1	2	yes	maximal
5	C₆₁×#1847	C₂	1	4	yes	maximal
6	C₆₁×#1846	C₂	1	2	yes	maximal
7	C₆₁×#1844	C₂	1	2	yes	maximal
8	C₆₁×#1726	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
10	C₂×#125×C₆₁	C₂²	1	2	yes	—
11	C₆₁×#501	—	2	8	no	—
12	C₆₁×#499	—	2	4	no	—
13	C₆₁×#498	—	2	4	no	—
14	C₂×#107×C₆₁	C₂²	1	4	yes	—
15	C₂×#107×C₆₁	C₂²	1	2	yes	—
16	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×#106×C₆₁	C₂²	1	2	yes	—
18	C₂×#106×C₆₁	C₂²	1	4	yes	—
19	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×#106×C₆₁	C₂²	1	2	yes	—
21	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×#97×C₆₁	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
25	C₆₁×#398	—	2	4	no	—
26	C₆₁×#395	—	2	4	no	—
27	C₆₁×#384	C₂²	1	4	yes	—
28	C₆₁×#382	C₂²	1	4	yes	—
29	C₆₁×#376	C₂²	1	1	yes	characteristic
30	C₂²×Q₈×C₆₁	—	2	2	no	—
31	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
34	C₄⋊Q₈×C₆₁	—	2	4	no	—
35	C₄⋊Q₈×C₆₁	—	2	4	no	—
36	#125×C₆₁	—	2	4	no	—
37	#125×C₆₁	—	2	4	no	—
38	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₆₁	—	2	8	no	—
42	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
45	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
47	#107×C₆₁	—	2	4	no	—
48	#107×C₆₁	—	2	8	no	—
49	#106×C₆₁	—	2	4	no	—
50	#106×C₆₁	—	2	4	no	—
51	#106×C₆₁	—	2	2	no	—
52	#106×C₆₁	—	2	2	no	—
53	#106×C₆₁	—	2	8	no	—
54	#97×C₆₁	D₄	1	4	yes	—
55	#97×C₆₁	Q₈	1	8	yes	complemented
56	C₄×C₈×C₆₁	D₄	1	4	yes	—
57	C₄×C₈×C₆₁	Q₈	1	8	yes	complemented
58	#95×C₆₁	—	2	8	no	—
59	#95×C₆₁	D₄	1	4	yes	—
60	C₂×Q₈×C₆₁	—	4	4	no	—
61	C₂×Q₈×C₆₁	—	8	4	no	—
62	C₂²×C₄×C₆₁	—	2	2	no	—
63	C₂²×C₄×C₆₁	—	2	4	no	—
64	C₂²×C₄×C₆₁	—	2	2	no	—
65	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
68	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
71	C₂×C₈×C₆₁	C₂×Q₈	1	4	yes	—
72	C₂×C₈×C₆₁	C₂×Q₈	1	4	yes	—
73	C₂×C₈×C₆₁	C₂×Q₈	1	4	yes	—
74	C₂×C₈×C₆₁	C₂×Q₈	1	4	yes	—
75	C₄⋊C₄×C₆₁	—	2	2	no	—
76	C₄⋊C₄×C₆₁	—	2	4	no	—
77	C₄⋊C₄×C₆₁	—	2	2	no	—
78	C₄⋊C₄×C₆₁	—	2	2	no	—
79	C₄⋊C₄×C₆₁	—	2	4	no	—
80	C₄⋊C₄×C₆₁	—	2	2	no	—
81	C₄⋊C₄×C₆₁	—	4	4	no	—
82	C₄⋊C₄×C₆₁	—	4	4	no	—
83	C₄⋊C₄×C₆₁	—	4	4	no	—
84	C₄⋊C₄×C₆₁	—	8	8	no	—
85	C₄²×C₆₁	C₂×D₄	1	2	yes	—
86	C₄²×C₆₁	C₂×D₄	1	2	yes	—
87	C₄²×C₆₁	C₂×Q₈	1	4	yes	—
88	C₄²×C₆₁	C₂×Q₈	1	4	yes	—
89	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
90	Q₈×C₆₁	—	8	8	no	—
91	C₂×C₄×C₆₁	—	2	8	no	—
92	C₂×C₄×C₆₁	—	2	4	no	—
93	C₂×C₄×C₆₁	—	4	2	no	—
94	C₂×C₄×C₆₁	—	4	2	no	—
95	C₂×C₄×C₆₁	—	4	4	no	—
96	C₂×C₄×C₆₁	—	4	4	no	—
97	C₂×C₄×C₆₁	—	4	8	no	—
98	C₂×C₄×C₆₁	—	4	4	no	—
99	C₂×C₄×C₆₁	#127	1	4	yes	—
100	C₂×C₄×C₆₁	C₄⋊Q₈	1	8	yes	—
101	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₂²×Q₈	1	2	yes	—
104	C₂×C₄×C₆₁	C₂²×Q₈	1	2	yes	—
105	C₈×C₆₁	C₄⋊Q₈	1	4	yes	—
106	C₈×C₆₁	C₄⋊Q₈	1	4	yes	—
107	C₈×C₆₁	C₄⋊Q₈	1	8	yes	—
108	#33318	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
110	C₂²×C₆₁	C₂×C₄⋊Q₈	1	2	yes	—
111	C₂²×C₆₁	#577	1	4	yes	—
112	C₄×C₆₁	—	2	4	no	—
113	C₄×C₆₁	—	2	8	no	—
114	C₄×C₆₁	—	8	2	no	—
115	C₄×C₆₁	—	8	2	no	—
116	C₄×C₆₁	—	8	4	no	—
117	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
118	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
119	C₄×C₆₁	C₂×C₄⋊Q₈	1	2	yes	—
120	C₂×#501	C₂×C₆₁	1	2	yes	—
121	C₂×#499	C₂×C₆₁	1	1	yes	characteristic
122	C₂×#498	C₂×C₆₁	1	1	yes	characteristic
123	#2660	C₂×C₆₁	1	2	yes	—
124	#1847	C₂×C₆₁	1	4	yes	—
125	#1846	C₂×C₆₁	1	2	yes	—
126	#1844	C₂×C₆₁	1	2	yes	—
127	#1726	C₂×C₆₁	1	1	yes	characteristic
128	C₂×C₆₁	#2868	1	1	yes	characteristic
129	C₂×C₆₁	#3154	1	2	yes	—
130	C₂×C₆₁	#3163	1	4	yes	—
131	C₂×C₄⋊Q₈	C₂²×C₆₁	1	2	yes	—
132	C₂×#125	C₂²×C₆₁	1	2	yes	—
133	#501	—	2	8	no	—
134	#499	—	2	4	no	—
135	#498	—	2	4	no	—
136	C₂×#107	C₂²×C₆₁	1	4	yes	—
137	C₂×#107	C₂²×C₆₁	1	2	yes	—
138	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
139	C₂×#106	C₂²×C₆₁	1	2	yes	—
140	C₂×#106	C₂²×C₆₁	1	4	yes	—
141	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
142	C₂×#106	C₂²×C₆₁	1	2	yes	—
143	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
144	C₂×#97	C₂²×C₆₁	1	2	yes	—
145	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
147	#398	—	2	4	no	—
148	#395	—	2	4	no	—
149	#384	C₂²×C₆₁	1	4	yes	—
150	#382	C₂²×C₆₁	1	4	yes	—
151	#376	C₂²×C₆₁	1	1	yes	characteristic
152	C₆₁	#33318	1	1	yes	characteristic, direct product factor, Sylow
153	C₂²×Q₈	—	2	2	no	—
154	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
155	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
156	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
157	C₄⋊Q₈	—	2	4	no	—
158	C₄⋊Q₈	—	2	4	no	—
159	#125	—	2	4	no	—
160	#125	—	2	4	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	8	no	—
165	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
166	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
167	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
168	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
169	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
170	#107	—	2	4	no	—
171	#107	—	2	8	no	—
172	#106	—	2	4	no	—
173	#106	—	2	4	no	—
174	#106	—	2	2	no	—
175	#106	—	2	2	no	—
176	#106	—	2	8	no	—
177	#97	D₄×C₆₁	1	4	yes	—
178	#97	Q₈×C₆₁	1	8	yes	complemented
179	C₄×C₈	D₄×C₆₁	1	4	yes	—
180	C₄×C₈	Q₈×C₆₁	1	8	yes	complemented
181	#95	—	2	8	no	—
182	#95	D₄×C₆₁	1	4	yes	—
183	C₂×Q₈	—	4	4	no	—
184	C₂×Q₈	—	8	4	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
189	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
190	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
191	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
192	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
193	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
194	C₂×C₈	C₂×Q₈×C₆₁	1	4	yes	—
195	C₂×C₈	C₂×Q₈×C₆₁	1	4	yes	—
196	C₂×C₈	C₂×Q₈×C₆₁	1	4	yes	—
197	C₂×C₈	C₂×Q₈×C₆₁	1	4	yes	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄⋊C₄	—	8	8	no	—
208	C₄²	C₂×D₄×C₆₁	1	2	yes	—
209	C₄²	C₂×D₄×C₆₁	1	2	yes	—
210	C₄²	C₂×Q₈×C₆₁	1	4	yes	—
211	C₄²	C₂×Q₈×C₆₁	1	4	yes	—
212	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
213	Q₈	—	8	8	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	8	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	#127×C₆₁	1	4	yes	—
223	C₂×C₄	C₄⋊Q₈×C₆₁	1	8	yes	—
224	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
225	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
226	C₂×C₄	C₂²×Q₈×C₆₁	1	2	yes	—
227	C₂×C₄	C₂²×Q₈×C₆₁	1	2	yes	—
228	C₈	C₄⋊Q₈×C₆₁	1	4	yes	—
229	C₈	C₄⋊Q₈×C₆₁	1	4	yes	—
230	C₈	C₄⋊Q₈×C₆₁	1	8	yes	—
231	C₂²	C₂×#127×C₆₁	1	1	yes	characteristic
232	C₂²	C₂×C₄⋊Q₈×C₆₁	1	2	yes	—
233	C₂²	C₆₁×#577	1	4	yes	—
234	C₄	—	2	4	no	—
235	C₄	—	2	8	no	—
236	C₄	—	8	2	no	—
237	C₄	—	8	2	no	—
238	C₄	—	8	4	no	—
239	C₄	C₂×C₄⋊Q₈×C₆₁	1	1	yes	characteristic
240	C₄	C₂×C₄⋊Q₈×C₆₁	1	1	yes	characteristic
241	C₄	C₂×C₄⋊Q₈×C₆₁	1	2	yes	—
242	C₂	C₆₁×#2868	1	1	yes	characteristic
243	C₂	C₆₁×#3154	1	2	yes	—
244	C₂	C₆₁×#3163	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₆₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
55. #97×C₆₁	Q₈	semidirect
57. C₄×C₈×C₆₁	Q₈	semidirect
108. #33318	C₆₁	direct (N × H)
152. C₆₁	#33318	direct (N × H)
178. #97	Q₈×C₆₁	semidirect
180. C₄×C₈	Q₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#501	2
C₂×C₆₁×#499	2
C₂×C₆₁×#498	2
C₆₁×#2660	2
C₆₁×#1847	2
C₆₁×#1846	2
C₆₁×#1844	2
C₆₁×#1726	2
#33318	61

Subgroups — C61×#33318 #29536751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33318 #29536751