Subgroups — C₆₁×#13202 #29516635

Order 15616.

614 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×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#357	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1845	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈×C₆₁	—	2	1	no	—
6	C₂²×D₈×C₆₁	C₂²	1	2	yes	—
7	C₂×#127×C₆₁	—	2	1	no	—
8	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
9	C₆₁×#493	—	2	2	no	—
10	C₆₁×#493	C₄	1	4	yes	complemented
11	C₂×#107×C₆₁	—	2	1	no	—
12	C₂×#107×C₆₁	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#384	—	2	1	no	—
15	C₆₁×#357	—	2	8	no	—
16	C₄×C₁₆×C₆₁	—	2	2	no	—
17	C₂²×D₄×C₆₁	—	2	2	no	—
18	C₂²×D₄×C₆₁	—	4	2	no	—
19	C₂×D₈×C₆₁	—	2	4	no	—
20	C₂×D₈×C₆₁	—	2	4	no	—
21	C₂×D₈×C₆₁	—	2	2	no	—
22	C₂×D₈×C₆₁	—	2	4	no	—
23	C₂×D₈×C₆₁	—	4	2	no	—
24	C₂×D₈×C₆₁	—	4	2	no	—
25	C₂×D₈×C₆₁	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
28	#127×C₆₁	—	2	4	no	—
29	#127×C₆₁	—	4	2	no	—
30	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₆₁	—	4	2	no	—
32	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
33	C₂×C₁₆×C₆₁	—	2	2	no	—
34	C₂×C₁₆×C₆₁	—	2	2	no	—
35	C₂×C₁₆×C₆₁	D₄	1	4	yes	complemented
36	#107×C₆₁	—	2	4	no	—
37	#107×C₆₁	—	4	2	no	—
38	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₆₁	D₄	1	2	yes	—
40	C₂⁴×C₆₁	—	4	2	no	—
41	C₂×D₄×C₆₁	—	2	4	no	—
42	C₂×D₄×C₆₁	—	4	8	no	—
43	C₂×D₄×C₆₁	—	4	2	no	—
44	C₂×D₄×C₆₁	—	4	2	no	—
45	C₂×D₄×C₆₁	—	4	2	no	—
46	C₂×D₄×C₆₁	—	4	2	no	—
47	C₂×D₄×C₆₁	—	4	2	no	—
48	C₂×D₄×C₆₁	—	8	4	no	—
49	C₂×D₄×C₆₁	—	8	4	no	—
50	C₂²×C₄×C₆₁	—	4	2	no	—
51	C₂²×C₄×C₆₁	D₈	1	2	yes	—
52	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
53	D₈×C₆₁	—	2	8	no	—
54	D₈×C₆₁	—	4	8	no	—
55	D₈×C₆₁	—	4	4	no	—
56	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₆₁	D₈	1	2	yes	—
58	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
59	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₆₁	#41	1	2	yes	—
63	C₄⋊C₄×C₆₁	—	4	2	no	—
64	C₄⋊C₄×C₆₁	—	4	2	no	—
65	C₄⋊C₄×C₆₁	—	8	4	no	—
66	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₆₁	SD₁₆	1	2	yes	—
68	C₁₆×C₆₁	—	2	4	no	—
69	C₂³×C₆₁	—	4	4	no	—
70	C₂³×C₆₁	—	8	2	no	—
71	C₂³×C₆₁	—	8	2	no	—
72	C₂³×C₆₁	—	8	4	no	—
73	C₂³×C₆₁	—	8	4	no	—
74	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
75	D₄×C₆₁	—	4	4	no	—
76	D₄×C₆₁	—	4	4	no	—
77	D₄×C₆₁	—	8	16	no	—
78	D₄×C₆₁	—	8	2	no	—
79	D₄×C₆₁	—	8	2	no	—
80	C₂×C₄×C₆₁	—	8	2	no	—
81	C₂×C₄×C₆₁	—	8	4	no	—
82	C₂×C₄×C₆₁	#102	1	4	yes	—
83	C₂×C₄×C₆₁	D₁₆	1	4	yes	—
84	C₂×C₄×C₆₁	SD₃₂	1	4	yes	—
85	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
87	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
88	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
91	C₈×C₆₁	#102	1	4	yes	—
92	C₈×C₆₁	C₄×D₄	1	2	yes	—
93	C₈×C₆₁	#121	1	2	yes	—
94	#13202	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂²×C₆₁	—	8	8	no	—
96	C₂²×C₆₁	—	8	4	no	—
97	C₂²×C₆₁	—	16	4	no	—
98	C₂²×C₆₁	—	16	2	no	—
99	C₂²×C₆₁	—	16	2	no	—
100	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
101	C₂²×C₆₁	#486	1	1	yes	characteristic
102	C₂²×C₆₁	#493	1	1	yes	characteristic
103	C₂²×C₆₁	C₂×D₁₆	1	2	yes	—
104	C₂²×C₆₁	C₂×SD₃₂	1	2	yes	—
105	C₄×C₆₁	—	16	2	no	—
106	C₄×C₆₁	#357	1	8	yes	complemented
107	C₄×C₆₁	#390	1	1	yes	characteristic
108	C₄×C₆₁	C₂×#102	1	1	yes	characteristic
109	C₄×C₆₁	#486	1	1	yes	characteristic
110	C₄×C₆₁	#492	1	1	yes	characteristic
111	C₂×#493	C₂×C₆₁	1	1	yes	characteristic
112	C₂×#357	C₂×C₆₁	1	4	yes	complemented
113	C₂×C₄×C₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
114	#1845	C₂×C₆₁	1	1	yes	characteristic, complemented
115	C₂×C₆₁	—	16	4	no	—
116	C₂×C₆₁	#1957	1	1	yes	characteristic
117	C₂×C₆₁	C₂×#357	1	2	yes	—
118	C₂×C₆₁	#2240	1	2	yes	—
119	C₂×C₆₁	#2246	1	1	yes	characteristic
120	C₂×C₆₁	#2248	1	1	yes	characteristic
121	C₂²×D₈	—	2	1	no	—
122	C₂²×D₈	C₂²×C₆₁	1	2	yes	—
123	C₂×#127	—	2	1	no	—
124	C₂²×C₁₆	C₂²×C₆₁	1	2	yes	—
125	#493	—	2	2	no	—
126	#493	C₄×C₆₁	1	4	yes	complemented
127	C₂×#107	—	2	1	no	—
128	C₂×#107	C₂²×C₆₁	1	2	yes	—
129	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
130	#384	—	2	1	no	—
131	#357	—	2	8	no	—
132	C₄×C₁₆	—	2	2	no	—
133	C₆₁	#13202	1	1	yes	characteristic, direct product factor, Sylow
134	C₂²×D₄	—	2	2	no	—
135	C₂²×D₄	—	4	2	no	—
136	C₂×D₈	—	2	4	no	—
137	C₂×D₈	—	2	4	no	—
138	C₂×D₈	—	2	2	no	—
139	C₂×D₈	—	2	4	no	—
140	C₂×D₈	—	4	2	no	—
141	C₂×D₈	—	4	2	no	—
142	C₂×D₈	C₂×C₄×C₆₁	1	4	yes	—
143	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
144	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
145	#127	—	2	4	no	—
146	#127	—	4	2	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	4	2	no	—
149	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	—	2	2	no	—
152	C₂×C₁₆	D₄×C₆₁	1	4	yes	complemented
153	#107	—	2	4	no	—
154	#107	—	4	2	no	—
155	C₄×C₈	C₂×C₄×C₆₁	1	2	yes	—
156	C₄×C₈	D₄×C₆₁	1	2	yes	—
157	C₂⁴	—	4	2	no	—
158	C₂×D₄	—	2	4	no	—
159	C₂×D₄	—	4	8	no	—
160	C₂×D₄	—	4	2	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	4	2	no	—
163	C₂×D₄	—	4	2	no	—
164	C₂×D₄	—	4	2	no	—
165	C₂×D₄	—	8	4	no	—
166	C₂×D₄	—	8	4	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	D₈×C₆₁	1	2	yes	—
169	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
170	D₈	—	2	8	no	—
171	D₈	—	4	8	no	—
172	D₈	—	4	4	no	—
173	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
174	C₂×C₈	D₈×C₆₁	1	2	yes	—
175	C₂×C₈	SD₁₆×C₆₁	1	2	yes	—
176	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
177	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
178	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
179	C₂×C₈	#41×C₆₁	1	2	yes	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	8	4	no	—
183	C₄²	C₂²⋊C₄×C₆₁	1	2	yes	—
184	C₄²	SD₁₆×C₆₁	1	2	yes	—
185	C₁₆	—	2	4	no	—
186	C₂³	—	4	4	no	—
187	C₂³	—	8	2	no	—
188	C₂³	—	8	2	no	—
189	C₂³	—	8	4	no	—
190	C₂³	—	8	4	no	—
191	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
192	D₄	—	4	4	no	—
193	D₄	—	4	4	no	—
194	D₄	—	8	16	no	—
195	D₄	—	8	2	no	—
196	D₄	—	8	2	no	—
197	C₂×C₄	—	8	2	no	—
198	C₂×C₄	—	8	4	no	—
199	C₂×C₄	#102×C₆₁	1	4	yes	—
200	C₂×C₄	D₁₆×C₆₁	1	4	yes	—
201	C₂×C₄	SD₃₂×C₆₁	1	4	yes	—
202	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
203	C₂×C₄	#124×C₆₁	1	1	yes	characteristic
204	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
205	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
206	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
207	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
208	C₈	#102×C₆₁	1	4	yes	—
209	C₈	C₄×D₄×C₆₁	1	2	yes	—
210	C₈	#121×C₆₁	1	2	yes	—
211	C₂²	—	8	8	no	—
212	C₂²	—	8	4	no	—
213	C₂²	—	16	4	no	—
214	C₂²	—	16	2	no	—
215	C₂²	—	16	2	no	—
216	C₂²	C₂×#102×C₆₁	1	1	yes	characteristic
217	C₂²	C₆₁×#486	1	1	yes	characteristic
218	C₂²	C₆₁×#493	1	1	yes	characteristic
219	C₂²	C₂×D₁₆×C₆₁	1	2	yes	—
220	C₂²	C₂×SD₃₂×C₆₁	1	2	yes	—
221	C₄	—	16	2	no	—
222	C₄	C₆₁×#357	1	8	yes	complemented
223	C₄	C₆₁×#390	1	1	yes	characteristic
224	C₄	C₂×#102×C₆₁	1	1	yes	characteristic
225	C₄	C₆₁×#486	1	1	yes	characteristic
226	C₄	C₆₁×#492	1	1	yes	characteristic
227	C₂	—	16	4	no	—
228	C₂	C₆₁×#1957	1	1	yes	characteristic
229	C₂	C₂×C₆₁×#357	1	2	yes	—
230	C₂	C₆₁×#2240	1	2	yes	—
231	C₂	C₆₁×#2246	1	1	yes	characteristic
232	C₂	C₆₁×#2248	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
prime radical	C₂×C₆₁×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₆₁×#357	C₂	semidirect
3. C₂×C₄×C₁₆×C₆₁	C₂	semidirect
4. C₆₁×#1845	C₂	semidirect
10. C₆₁×#493	C₄	semidirect
35. C₂×C₁₆×C₆₁	D₄	semidirect
94. #13202	C₆₁	direct (N × H)
106. C₄×C₆₁	#357	semidirect
112. C₂×#357	C₂×C₆₁	semidirect
113. C₂×C₄×C₁₆	C₂×C₆₁	semidirect
114. #1845	C₂×C₆₁	semidirect
126. #493	C₄×C₆₁	semidirect
133. C₆₁	#13202	direct (N × H)
152. C₂×C₁₆	D₄×C₆₁	semidirect
222. C₄	C₆₁×#357	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#493	2
C₂×C₆₁×#357	2
C₂×C₄×C₁₆×C₆₁	2
C₆₁×#1845	2
#13202	61

Subgroups — C61×#13202 #29516635

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#13202 #29516635