Subgroups — C₂×C₆₁×#1453 #29520754

Order 15616.

866 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#431	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1453	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₆₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₆₁×#322	C₂	1	2	yes	maximal, complemented
6	C₂×C₈²×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×M₁₆×C₆₁	C₄	1	2	yes	—
8	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
9	C₆₁×#431	C₄	1	8	yes	—
10	C₆₁×#431	C₂²	1	8	yes	—
11	C₂×C₄⋊C₈×C₆₁	C₄	1	4	yes	—
12	C₂×C₂²⋊C₈×C₆₁	C₄	1	4	yes	—
13	C₄×M₁₆×C₆₁	C₄	1	4	yes	—
14	C₄×M₁₆×C₆₁	C₂²	1	4	yes	—
15	C₂×#97×C₆₁	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
20	C₆₁×#322	C₂²	1	4	yes	complemented
21	C₆₁×#322	C₂²	1	8	yes	complemented
22	C₈²×C₆₁	C₂²	1	4	yes	complemented
23	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₆₁	C₂×C₄	1	2	yes	—
25	C₂×M₁₆×C₆₁	—	2	8	no	—
26	C₂×M₁₆×C₆₁	C₂×C₄	1	8	yes	—
27	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₆₁	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
36	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₆₁	C₂×C₄	1	16	yes	—
38	C₂²⋊C₈×C₆₁	C₂×C₄	1	16	yes	—
39	#97×C₆₁	C₂×C₄	1	8	yes	—
40	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
44	C₄×C₈×C₆₁	C₂³	1	2	yes	—
45	C₄×C₈×C₆₁	C₂³	1	4	yes	—
46	C₄×C₈×C₆₁	C₂³	1	2	yes	—
47	C₄×C₈×C₆₁	C₂³	1	4	yes	—
48	C₂⁴×C₆₁	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	—	2	4	no	—
50	C₂²×C₄×C₆₁	—	2	8	no	—
51	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
53	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
54	C₂²×C₄×C₆₁	C₄²	1	4	yes	—
55	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
59	M₁₆×C₆₁	—	2	16	no	—
60	C₂×C₈×C₆₁	—	2	4	no	—
61	C₂×C₈×C₆₁	—	2	8	no	—
62	C₂×C₈×C₆₁	—	2	8	no	—
63	C₂×C₈×C₆₁	M₁₆	1	8	yes	—
64	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₆₁	C₂²×C₄	1	8	yes	—
66	C₂×C₈×C₆₁	C₂²×C₄	1	8	yes	—
67	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
68	C₄²×C₆₁	C₄²	1	2	yes	—
69	C₄²×C₆₁	C₄²	1	4	yes	—
70	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
71	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
73	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
74	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
75	C₂³×C₆₁	—	2	2	no	—
76	C₂³×C₆₁	—	2	4	no	—
77	C₂³×C₆₁	C₂×C₄²	1	2	yes	—
78	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₆₁	—	2	2	no	—
80	C₂×C₄×C₆₁	—	2	4	no	—
81	C₂×C₄×C₆₁	—	2	4	no	—
82	C₂×C₄×C₆₁	—	2	8	no	—
83	C₂×C₄×C₆₁	—	2	4	no	—
84	C₂×C₄×C₆₁	—	2	8	no	—
85	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
86	C₂×C₄×C₆₁	C₂×C₄²	1	4	yes	—
87	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
89	C₂×C₄×C₆₁	C₂×M₁₆	1	4	yes	—
90	C₂×C₄×C₆₁	#131	1	8	yes	—
91	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
93	C₈×C₆₁	—	2	4	no	—
94	C₈×C₆₁	—	2	8	no	—
95	C₈×C₆₁	—	2	8	no	—
96	C₈×C₆₁	C₂×M₁₆	1	8	yes	—
97	C₂×#1453	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂²×C₆₁	—	2	2	no	—
99	C₂²×C₆₁	—	2	4	no	—
100	C₂²×C₆₁	—	4	4	no	—
101	C₂²×C₆₁	C₄×M₁₆	1	2	yes	—
102	C₂²×C₆₁	#405	1	4	yes	—
103	C₂²×C₆₁	C₂²×C₄²	1	1	yes	characteristic
104	C₄×C₆₁	—	2	2	no	—
105	C₄×C₆₁	—	2	4	no	—
106	C₄×C₆₁	C₄×M₁₆	1	2	yes	—
107	C₄×C₆₁	#405	1	4	yes	—
108	C₄×C₆₁	C₂²×M₁₆	1	2	yes	—
109	C₄×C₆₁	C₂×#131	1	4	yes	—
110	C₂×#431	C₂×C₆₁	1	2	yes	—
111	C₂×C₄×M₁₆	C₂×C₆₁	1	1	yes	characteristic
112	#1453	C₂×C₆₁	1	8	yes	direct product factor
113	C₂×#322	C₂×C₆₁	1	1	yes	characteristic, complemented
114	C₂×#322	C₂×C₆₁	1	2	yes	complemented
115	C₂×C₈²	C₂×C₆₁	1	1	yes	characteristic, complemented
116	C₂×C₆₁	—	4	2	no	—
117	C₂×C₆₁	#1453	1	4	yes	direct product factor
118	C₂×C₆₁	C₂×C₄×M₁₆	1	1	yes	characteristic
119	C₂×C₆₁	C₂×#405	1	2	yes	—
120	C₂²×M₁₆	C₄×C₆₁	1	2	yes	—
121	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
122	#431	C₄×C₆₁	1	8	yes	—
123	#431	C₂²×C₆₁	1	8	yes	—
124	C₂×C₄⋊C₈	C₄×C₆₁	1	4	yes	—
125	C₂×C₂²⋊C₈	C₄×C₆₁	1	4	yes	—
126	C₄×M₁₆	C₄×C₆₁	1	4	yes	—
127	C₄×M₁₆	C₂²×C₆₁	1	4	yes	—
128	C₂×#97	C₄×C₆₁	1	2	yes	—
129	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
132	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
133	#322	C₂²×C₆₁	1	4	yes	complemented
134	#322	C₂²×C₆₁	1	8	yes	complemented
135	C₈²	C₂²×C₆₁	1	4	yes	complemented
136	C₆₁	C₂×#1453	1	1	yes	characteristic, direct product factor, Sylow
137	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
138	C₂³×C₄	C₂×C₄×C₆₁	1	2	yes	—
139	C₂×M₁₆	—	2	8	no	—
140	C₂×M₁₆	C₂×C₄×C₆₁	1	8	yes	—
141	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
142	C₂²×C₈	C₂×C₄×C₆₁	1	4	yes	—
143	C₂²×C₈	C₂×C₄×C₆₁	1	4	yes	—
144	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
145	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
146	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
147	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
148	C₂×C₄²	C₂×C₄×C₆₁	1	4	yes	—
149	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
150	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
151	C₄⋊C₈	C₂×C₄×C₆₁	1	16	yes	—
152	C₂²⋊C₈	C₂×C₄×C₆₁	1	16	yes	—
153	#97	C₂×C₄×C₆₁	1	8	yes	—
154	C₄×C₈	C₂×C₄×C₆₁	1	2	yes	—
155	C₄×C₈	C₂×C₄×C₆₁	1	2	yes	—
156	C₄×C₈	C₂×C₄×C₆₁	1	4	yes	—
157	C₄×C₈	C₂×C₄×C₆₁	1	4	yes	—
158	C₄×C₈	C₂³×C₆₁	1	2	yes	—
159	C₄×C₈	C₂³×C₆₁	1	4	yes	—
160	C₄×C₈	C₂³×C₆₁	1	2	yes	—
161	C₄×C₈	C₂³×C₆₁	1	4	yes	—
162	C₂⁴	C₄²×C₆₁	1	1	yes	characteristic
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	8	no	—
165	C₂²×C₄	C₄²×C₆₁	1	1	yes	characteristic
166	C₂²×C₄	C₄²×C₆₁	1	2	yes	—
167	C₂²×C₄	C₄²×C₆₁	1	2	yes	—
168	C₂²×C₄	C₄²×C₆₁	1	4	yes	—
169	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
170	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
171	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
172	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
173	M₁₆	—	2	16	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	2	8	no	—
176	C₂×C₈	—	2	8	no	—
177	C₂×C₈	M₁₆×C₆₁	1	8	yes	—
178	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
179	C₂×C₈	C₂²×C₄×C₆₁	1	8	yes	—
180	C₂×C₈	C₂²×C₄×C₆₁	1	8	yes	—
181	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
182	C₄²	C₄²×C₆₁	1	2	yes	—
183	C₄²	C₄²×C₆₁	1	4	yes	—
184	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
185	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
186	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
187	C₄²	C₂²×C₄×C₆₁	1	4	yes	—
188	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
189	C₂³	—	2	2	no	—
190	C₂³	—	2	4	no	—
191	C₂³	C₂×C₄²×C₆₁	1	2	yes	—
192	C₂³	C₂×C₄²×C₆₁	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	8	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	C₂×C₄²×C₆₁	1	2	yes	—
200	C₂×C₄	C₂×C₄²×C₆₁	1	4	yes	—
201	C₂×C₄	C₂×C₄²×C₆₁	1	1	yes	characteristic
202	C₂×C₄	C₂×C₄²×C₆₁	1	2	yes	—
203	C₂×C₄	C₂×M₁₆×C₆₁	1	4	yes	—
204	C₂×C₄	#131×C₆₁	1	8	yes	—
205	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
206	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
207	C₈	—	2	4	no	—
208	C₈	—	2	8	no	—
209	C₈	—	2	8	no	—
210	C₈	C₂×M₁₆×C₆₁	1	8	yes	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	4	no	—
213	C₂²	—	4	4	no	—
214	C₂²	C₄×M₁₆×C₆₁	1	2	yes	—
215	C₂²	C₆₁×#405	1	4	yes	—
216	C₂²	C₂²×C₄²×C₆₁	1	1	yes	characteristic
217	C₄	—	2	2	no	—
218	C₄	—	2	4	no	—
219	C₄	C₄×M₁₆×C₆₁	1	2	yes	—
220	C₄	C₆₁×#405	1	4	yes	—
221	C₄	C₂²×M₁₆×C₆₁	1	2	yes	—
222	C₄	C₂×#131×C₆₁	1	4	yes	—
223	C₂	—	4	2	no	—
224	C₂	C₆₁×#1453	1	4	yes	direct product factor
225	C₂	C₂×C₄×M₁₆×C₆₁	1	1	yes	characteristic
226	C₂	C₂×C₆₁×#405	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₆₁×#1453	C₂	direct (N × H)
4. C₂×C₆₁×#322	C₂	semidirect
5. C₂×C₆₁×#322	C₂	semidirect
6. C₂×C₈²×C₆₁	C₂	semidirect
20. C₆₁×#322	C₂²	semidirect
21. C₆₁×#322	C₂²	semidirect
22. C₈²×C₆₁	C₂²	semidirect
97. C₂×#1453	C₆₁	direct (N × H)
112. #1453	C₂×C₆₁	direct (N × H)
113. C₂×#322	C₂×C₆₁	semidirect
114. C₂×#322	C₂×C₆₁	semidirect
115. C₂×C₈²	C₂×C₆₁	semidirect
117. C₂×C₆₁	#1453	direct (N × H)
133. #322	C₂²×C₆₁	semidirect
134. #322	C₂²×C₆₁	semidirect
135. C₈²	C₂²×C₆₁	semidirect
136. C₆₁	C₂×#1453	direct (N × H)
224. C₂	C₆₁×#1453	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×#431	2
C₂×C₄×M₁₆×C₆₁	2
C₆₁×#1453	2
C₂×C₆₁×#322	2
C₂×C₈²×C₆₁	2
C₂×#1453	61

Subgroups — C2×C61×#1453 #29520754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#1453 #29520754