Subgroups — C₆₁×#13168 #29516601

Order 15616.

318 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#367	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#366	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#346	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1942	C₂	1	2	yes	maximal
5	C₆₁×#1380	C₂	1	2	yes	maximal
6	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
8	C₂×#107×C₆₁	C₂²	1	2	yes	—
9	C₂×#107×C₆₁	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₆₁×#389	—	2	2	no	—
12	C₆₁×#367	—	2	2	no	—
13	C₆₁×#366	C₄	1	4	yes	complemented
14	C₆₁×#346	—	2	2	no	—
15	C₆₁×#340	—	2	2	no	—
16	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
19	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
20	C₂×C₄⋊C₄×C₆₁	—	4	2	no	—
21	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₆₁	—	2	1	no	—
23	C₂×C₁₆×C₆₁	—	2	1	no	—
24	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
25	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
26	C₂×C₁₆×C₆₁	Q₈	1	2	yes	—
27	C₂×C₁₆×C₆₁	Q₈	1	2	yes	—
28	#107×C₆₁	—	2	2	no	—
29	#107×C₆₁	—	2	2	no	—
30	#107×C₆₁	—	2	2	no	—
31	#107×C₆₁	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₆₁	—	2	1	no	—
33	C₄×C₈×C₆₁	—	2	1	no	—
34	#95×C₆₁	—	4	2	no	—
35	C₂²×C₄×C₆₁	—	4	2	no	—
36	C₂²×C₄×C₆₁	—	4	2	no	—
37	C₂²×C₄×C₆₁	D₈	1	2	yes	—
38	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₆₁	—	2	1	no	—
40	C₂×C₈×C₆₁	—	2	1	no	—
41	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
42	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
47	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
48	C₄⋊C₄×C₆₁	—	2	4	no	—
49	C₄⋊C₄×C₆₁	—	4	2	no	—
50	C₄⋊C₄×C₆₁	—	4	2	no	—
51	C₄⋊C₄×C₆₁	—	4	2	no	—
52	C₄⋊C₄×C₆₁	—	8	4	no	—
53	C₄²×C₆₁	—	2	1	no	—
54	C₄²×C₆₁	—	2	1	no	—
55	C₁₆×C₆₁	—	2	2	no	—
56	C₁₆×C₆₁	C₄⋊C₄	1	4	yes	complemented
57	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₆₁	—	2	2	no	—
59	C₂×C₄×C₆₁	—	2	2	no	—
60	C₂×C₄×C₆₁	—	2	2	no	—
61	C₂×C₄×C₆₁	—	4	4	no	—
62	C₂×C₄×C₆₁	—	8	2	no	—
63	C₂×C₄×C₆₁	—	8	2	no	—
64	C₂×C₄×C₆₁	—	8	2	no	—
65	C₂×C₄×C₆₁	—	8	2	no	—
66	C₂×C₄×C₆₁	—	8	2	no	—
67	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
72	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
73	C₈×C₆₁	—	2	2	no	—
74	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₆₁	C₄×Q₈	1	1	yes	characteristic
76	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
77	C₈×C₆₁	#125	1	1	yes	characteristic
78	#13168	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂²×C₆₁	C₄×D₈	1	1	yes	characteristic
80	C₂²×C₆₁	#466	1	1	yes	characteristic
81	C₂²×C₆₁	#500	1	1	yes	characteristic
82	C₂²×C₆₁	#509	1	2	yes	—
83	C₂²×C₆₁	#510	1	2	yes	—
84	C₄×C₆₁	—	4	2	no	—
85	C₄×C₆₁	—	8	4	no	—
86	C₄×C₆₁	—	16	2	no	—
87	C₄×C₆₁	#389	1	1	yes	characteristic
88	C₄×C₆₁	C₄×Q₁₆	1	1	yes	characteristic
89	C₄×C₆₁	#467	1	1	yes	characteristic
90	C₄×C₆₁	#499	1	1	yes	characteristic
91	C₂×#367	C₂×C₆₁	1	1	yes	characteristic
92	C₂×#366	C₂×C₆₁	1	1	yes	characteristic
93	C₂×#346	C₂×C₆₁	1	1	yes	characteristic
94	#1942	C₂×C₆₁	1	2	yes	—
95	#1380	C₂×C₆₁	1	2	yes	—
96	C₂×C₆₁	#1942	1	1	yes	characteristic
97	C₂×C₆₁	#2176	1	1	yes	characteristic
98	C₂×C₆₁	#2177	1	1	yes	characteristic
99	C₂×C₆₁	#2220	1	2	yes	—
100	C₂×C₆₁	#2255	1	2	yes	—
101	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
102	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
103	C₂×#107	C₂²×C₆₁	1	2	yes	—
104	C₂×#107	C₂²×C₆₁	1	2	yes	—
105	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
106	#389	—	2	2	no	—
107	#367	—	2	2	no	—
108	#366	C₄×C₆₁	1	4	yes	complemented
109	#346	—	2	2	no	—
110	#340	—	2	2	no	—
111	C₆₁	#13168	1	1	yes	characteristic, direct product factor, Sylow
112	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
113	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	4	2	no	—
117	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	C₂×C₄×C₆₁	1	2	yes	—
121	C₂×C₁₆	D₄×C₆₁	1	2	yes	—
122	C₂×C₁₆	Q₈×C₆₁	1	2	yes	—
123	C₂×C₁₆	Q₈×C₆₁	1	2	yes	—
124	#107	—	2	2	no	—
125	#107	—	2	2	no	—
126	#107	—	2	2	no	—
127	#107	C₂×C₄×C₆₁	1	4	yes	—
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	—	2	1	no	—
130	#95	—	4	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	D₈×C₆₁	1	2	yes	—
134	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	Q₁₆×C₆₁	1	2	yes	—
138	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
139	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
140	C₂×C₈	C₂×Q₈×C₆₁	1	1	yes	characteristic
141	C₂×C₈	C₂×Q₈×C₆₁	1	1	yes	characteristic
142	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
143	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	8	4	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₁₆	—	2	2	no	—
152	C₁₆	C₄⋊C₄×C₆₁	1	4	yes	complemented
153	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
164	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
165	C₂×C₄	C₄⋊Q₈×C₆₁	1	1	yes	characteristic
166	C₂×C₄	C₂×Q₁₆×C₆₁	1	1	yes	characteristic
167	C₂×C₄	#135×C₆₁	1	1	yes	characteristic
168	C₂×C₄	#135×C₆₁	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
171	C₈	C₄×Q₈×C₆₁	1	1	yes	characteristic
172	C₈	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
173	C₈	#125×C₆₁	1	1	yes	characteristic
174	C₂²	C₄×D₈×C₆₁	1	1	yes	characteristic
175	C₂²	C₆₁×#466	1	1	yes	characteristic
176	C₂²	C₆₁×#500	1	1	yes	characteristic
177	C₂²	C₆₁×#509	1	2	yes	—
178	C₂²	C₆₁×#510	1	2	yes	—
179	C₄	—	4	2	no	—
180	C₄	—	8	4	no	—
181	C₄	—	16	2	no	—
182	C₄	C₆₁×#389	1	1	yes	characteristic
183	C₄	C₄×Q₁₆×C₆₁	1	1	yes	characteristic
184	C₄	C₆₁×#467	1	1	yes	characteristic
185	C₄	C₆₁×#499	1	1	yes	characteristic
186	C₂	C₆₁×#1942	1	1	yes	characteristic
187	C₂	C₆₁×#2176	1	1	yes	characteristic
188	C₂	C₆₁×#2177	1	1	yes	characteristic
189	C₂	C₆₁×#2220	1	2	yes	—
190	C₂	C₆₁×#2255	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
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
13. C₆₁×#366	C₄	semidirect
56. C₁₆×C₆₁	C₄⋊C₄	semidirect
78. #13168	C₆₁	direct (N × H)
108. #366	C₄×C₆₁	semidirect
111. C₆₁	#13168	direct (N × H)
152. C₁₆	C₄⋊C₄×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×#367	2
C₂×C₆₁×#366	2
C₂×C₆₁×#346	2
C₆₁×#1942	2
C₆₁×#1380	2
#13168	61

Subgroups — C61×#13168 #29516601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#13168 #29516601