Subgroups — C₆₁×#10814 #29514247

Order 15616.

658 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#405	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1740	C₂	1	2	yes	maximal
3	C₆₁×#1735	C₂	1	2	yes	maximal
4	C₂×C₆₁×H(3,ℤ/4)	C₂	1	2	yes	maximal
5	C₂²×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
7	C₂×#117×C₆₁	C₂²	1	4	yes	—
8	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
9	C₆₁×#405	—	2	4	no	—
10	C₂×#97×C₆₁	C₄	1	2	yes	complemented
11	C₂×C₄×C₈×C₆₁	C₄	1	2	yes	complemented
12	C₆₁×#342	—	2	4	no	—
13	C₆₁×#340	—	2	4	no	—
14	C₆₁×H(3,ℤ/4)	C₄	1	8	yes	complemented
15	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
16	C₂×M₁₆×C₆₁	—	2	2	no	—
17	C₂×M₁₆×C₆₁	—	2	2	no	—
18	C₂×M₁₆×C₆₁	C₂×C₄	1	4	yes	—
19	C₂²×C₈×C₆₁	—	2	2	no	—
20	C₂²×C₈×C₆₁	—	2	2	no	—
21	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₁	D₄	1	4	yes	—
25	#117×C₆₁	—	2	2	no	—
26	#117×C₆₁	—	2	2	no	—
27	#117×C₆₁	—	2	4	no	—
28	#117×C₆₁	—	2	4	no	—
29	#117×C₆₁	—	2	4	no	—
30	#117×C₆₁	C₂×C₄	1	8	yes	—
31	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
32	C₂×C₄⋊C₄×C₆₁	Q₈	1	2	yes	—
33	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
34	C₂×C₂²⋊C₄×C₆₁	D₄	1	2	yes	—
35	C₂×C₄²×C₆₁	—	2	4	no	—
36	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
37	#97×C₆₁	—	2	4	no	—
38	C₄×C₈×C₆₁	—	2	4	no	—
39	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	—	4	4	no	—
41	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	#41	1	2	yes	—
47	C₂²×C₄×C₆₁	#41	1	2	yes	—
48	C₂²×C₄×C₆₁	#41	1	2	yes	—
49	C₂²×C₄×C₆₁	#41	1	2	yes	—
50	M₁₆×C₆₁	—	2	8	no	—
51	C₂×C₈×C₆₁	—	2	2	no	—
52	C₂×C₈×C₆₁	—	2	1	no	—
53	C₂×C₈×C₆₁	—	2	1	no	—
54	C₂×C₈×C₆₁	—	2	1	no	—
55	C₂×C₈×C₆₁	—	2	1	no	—
56	C₂×C₈×C₆₁	—	2	2	no	—
57	C₂×C₈×C₆₁	—	2	8	no	—
58	C₂×C₈×C₆₁	C₄²	1	4	yes	—
59	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
61	C₄⋊C₄×C₆₁	—	2	4	no	—
62	C₄⋊C₄×C₆₁	—	2	8	no	—
63	C₄⋊C₄×C₆₁	—	4	4	no	—
64	C₂²⋊C₄×C₆₁	—	2	4	no	—
65	C₂²⋊C₄×C₆₁	—	2	8	no	—
66	C₂²⋊C₄×C₆₁	—	4	4	no	—
67	C₄²×C₆₁	—	2	2	no	—
68	C₄²×C₆₁	—	2	2	no	—
69	C₄²×C₆₁	—	2	8	no	—
70	C₄²×C₆₁	—	4	4	no	—
71	C₂³×C₆₁	—	4	2	no	—
72	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂³×C₆₁	C₄×D₄	1	2	yes	—
74	C₂³×C₆₁	#121	1	2	yes	—
75	C₂³×C₆₁	#124	1	2	yes	—
76	C₂×C₄×C₆₁	—	2	4	no	—
77	C₂×C₄×C₆₁	—	4	4	no	—
78	C₂×C₄×C₆₁	—	4	8	no	—
79	C₂×C₄×C₆₁	—	8	8	no	—
80	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
83	C₂×C₄×C₆₁	#117	1	4	yes	—
84	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
85	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
86	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
87	C₂×C₄×C₆₁	C₄×D₄	1	4	yes	—
88	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
89	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
90	C₂×C₄×C₆₁	#123	1	4	yes	—
91	C₂×C₄×C₆₁	#125	1	2	yes	—
92	C₂×C₄×C₆₁	#126	1	4	yes	—
93	C₈×C₆₁	—	2	4	no	—
94	C₈×C₆₁	—	2	2	no	—
95	C₈×C₆₁	—	2	2	no	—
96	#10814	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂²×C₆₁	—	2	2	no	—
98	C₂²×C₆₁	—	2	4	no	—
99	C₂²×C₆₁	—	4	2	no	—
100	C₂²×C₆₁	—	4	2	no	—
101	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₆₁	#380	1	1	yes	characteristic
103	C₂²×C₆₁	#382	1	1	yes	characteristic
104	C₂²×C₆₁	#388	1	4	yes	—
105	C₄×C₆₁	—	4	4	no	—
106	C₄×C₆₁	—	8	8	no	—
107	C₄×C₆₁	#376	1	1	yes	characteristic
108	C₄×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
109	C₄×C₆₁	#381	1	1	yes	characteristic
110	C₄×C₆₁	#383	1	1	yes	characteristic
111	C₄×C₆₁	#387	1	4	yes	—
112	C₂×#405	C₂×C₆₁	1	1	yes	characteristic
113	#1740	C₂×C₆₁	1	2	yes	—
114	#1735	C₂×C₆₁	1	2	yes	—
115	C₂×H(3,ℤ/4)	C₂×C₆₁	1	2	yes	—
116	C₂×C₆₁	—	2	4	no	—
117	C₂×C₆₁	—	2	1	no	—
118	C₂×C₆₁	—	2	1	no	—
119	C₂×C₆₁	#1434	1	1	yes	characteristic
120	C₂×C₆₁	#1812	1	2	yes	—
121	C₂²×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
122	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
123	C₂×#117	C₂²×C₆₁	1	4	yes	—
124	C₂×#117	C₂²×C₆₁	1	1	yes	characteristic
125	#405	—	2	4	no	—
126	C₂×#97	C₄×C₆₁	1	2	yes	complemented
127	C₂×C₄×C₈	C₄×C₆₁	1	2	yes	complemented
128	#342	—	2	4	no	—
129	#340	—	2	4	no	—
130	H(3,ℤ/4)	C₄×C₆₁	1	8	yes	complemented
131	C₆₁	#10814	1	1	yes	characteristic, direct product factor, Sylow
132	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
133	C₂×M₁₆	—	2	2	no	—
134	C₂×M₁₆	—	2	2	no	—
135	C₂×M₁₆	C₂×C₄×C₆₁	1	4	yes	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
139	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
140	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
141	C₂²×C₈	D₄×C₆₁	1	4	yes	—
142	#117	—	2	2	no	—
143	#117	—	2	2	no	—
144	#117	—	2	4	no	—
145	#117	—	2	4	no	—
146	#117	—	2	4	no	—
147	#117	C₂×C₄×C₆₁	1	8	yes	—
148	C₂×C₄⋊C₄	—	2	4	no	—
149	C₂×C₄⋊C₄	Q₈×C₆₁	1	2	yes	—
150	C₂×C₂²⋊C₄	—	2	4	no	—
151	C₂×C₂²⋊C₄	D₄×C₆₁	1	2	yes	—
152	C₂×C₄²	—	2	4	no	—
153	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
154	#97	—	2	4	no	—
155	C₄×C₈	—	2	4	no	—
156	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
157	C₂²×C₄	—	4	4	no	—
158	C₂²×C₄	C₄⋊C₄×C₆₁	1	4	yes	—
159	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
160	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
161	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
162	C₂²×C₄	C₂×Q₈×C₆₁	1	1	yes	characteristic
163	C₂²×C₄	#41×C₆₁	1	2	yes	—
164	C₂²×C₄	#41×C₆₁	1	2	yes	—
165	C₂²×C₄	#41×C₆₁	1	2	yes	—
166	C₂²×C₄	#41×C₆₁	1	2	yes	—
167	M₁₆	—	2	8	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	8	no	—
175	C₂×C₈	C₄²×C₆₁	1	4	yes	—
176	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
177	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	2	8	no	—
180	C₄⋊C₄	—	4	4	no	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₂²⋊C₄	—	2	8	no	—
183	C₂²⋊C₄	—	4	4	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	8	no	—
187	C₄²	—	4	4	no	—
188	C₂³	—	4	2	no	—
189	C₂³	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
190	C₂³	C₄×D₄×C₆₁	1	2	yes	—
191	C₂³	#121×C₆₁	1	2	yes	—
192	C₂³	#124×C₆₁	1	2	yes	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	4	8	no	—
196	C₂×C₄	—	8	8	no	—
197	C₂×C₄	C₂×C₄²×C₆₁	1	1	yes	characteristic
198	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
199	C₂×C₄	#117×C₆₁	1	1	yes	characteristic
200	C₂×C₄	#117×C₆₁	1	1	yes	characteristic
201	C₂×C₄	#117×C₆₁	1	4	yes	—
202	C₂×C₄	#117×C₆₁	1	1	yes	characteristic
203	C₂×C₄	#117×C₆₁	1	1	yes	characteristic
204	C₂×C₄	C₄×D₄×C₆₁	1	4	yes	—
205	C₂×C₄	C₄×Q₈×C₆₁	1	2	yes	—
206	C₂×C₄	C₂²⋊Q₈×C₆₁	1	2	yes	—
207	C₂×C₄	#123×C₆₁	1	4	yes	—
208	C₂×C₄	#125×C₆₁	1	2	yes	—
209	C₂×C₄	#126×C₆₁	1	4	yes	—
210	C₈	—	2	4	no	—
211	C₈	—	2	2	no	—
212	C₈	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	C₄×C₄⋊C₄×C₆₁	1	1	yes	characteristic
218	C₂²	C₆₁×#380	1	1	yes	characteristic
219	C₂²	C₆₁×#382	1	1	yes	characteristic
220	C₂²	C₆₁×#388	1	4	yes	—
221	C₄	—	4	4	no	—
222	C₄	—	8	8	no	—
223	C₄	C₆₁×#376	1	1	yes	characteristic
224	C₄	C₄×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
225	C₄	C₆₁×#381	1	1	yes	characteristic
226	C₄	C₆₁×#383	1	1	yes	characteristic
227	C₄	C₆₁×#387	1	4	yes	—
228	C₂	—	2	4	no	—
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	C₆₁×#1434	1	1	yes	characteristic
232	C₂	C₆₁×#1812	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₁	#117
commutator	C₂×C₄	C₂×C₄²×C₆₁
Frattini, cosocle	C₂³×C₄	C₂³×C₆₁
socle	C₂²×C₆₁	#380
prime radical	C₂⁴×C₆₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂²×M₁₆×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₂×#97×C₆₁	C₄	semidirect
11. C₂×C₄×C₈×C₆₁	C₄	semidirect
14. C₆₁×H(3,ℤ/4)	C₄	semidirect
96. #10814	C₆₁	direct (N × H)
126. C₂×#97	C₄×C₆₁	semidirect
127. C₂×C₄×C₈	C₄×C₆₁	semidirect
130. H(3,ℤ/4)	C₄×C₆₁	semidirect
131. C₆₁	#10814	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×#405	2
C₆₁×#1740	2
C₆₁×#1735	2
C₂×C₆₁×H(3,ℤ/4)	2
#10814	61

Subgroups — C61×#10814 #29514247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#10814 #29514247