Subgroups — C₆₁×#13162 #29516595

Order 15616.

290 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#364	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2151	C₂	1	2	yes	maximal
3	C₂×C₆₁×#346	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1772	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1379	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
7	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
8	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
9	C₂×#108×C₆₁	C₄	1	1	yes	characteristic
10	C₆₁×#427	C₄	1	1	yes	characteristic
11	C₆₁×#405	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×#364	—	2	2	no	—
14	C₆₁×C₄⋊C₁₆	—	2	4	no	—
15	C₆₁×#346	—	2	2	no	—
16	C₆₁×#339	C₄	1	2	yes	—
17	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
20	#117×C₆₁	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
22	M₃₂×C₆₁	—	2	2	no	—
23	M₃₂×C₆₁	—	2	2	no	—
24	C₂×C₁₆×C₆₁	—	2	1	no	—
25	C₂×C₁₆×C₆₁	—	2	1	no	—
26	C₂×C₁₆×C₆₁	—	2	1	no	—
27	C₂×C₁₆×C₆₁	—	2	1	no	—
28	C₂×C₁₆×C₆₁	—	2	2	no	—
29	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
30	C₂×C₁₆×C₆₁	Q₈	1	2	yes	—
31	#108×C₆₁	—	2	2	no	—
32	#107×C₆₁	C₈	1	2	yes	—
33	#106×C₆₁	C₈	1	2	yes	—
34	#97×C₆₁	—	2	2	no	—
35	C₄×C₈×C₆₁	—	2	2	no	—
36	C₄×C₈×C₆₁	D₄	1	2	yes	—
37	C₄×C₈×C₆₁	Q₈	1	2	yes	—
38	C₂²×C₄×C₆₁	—	2	1	no	—
39	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	M₁₆×C₆₁	—	2	2	no	—
41	M₁₆×C₆₁	—	2	2	no	—
42	C₂×C₈×C₆₁	—	2	2	no	—
43	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
46	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
47	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
52	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
53	C₄⋊C₄×C₆₁	C₂×C₈	1	4	yes	—
54	C₂²⋊C₄×C₆₁	—	2	2	no	—
55	C₄²×C₆₁	—	2	2	no	—
56	C₄²×C₆₁	—	2	1	no	—
57	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
58	C₁₆×C₆₁	—	2	2	no	—
59	C₁₆×C₆₁	—	2	2	no	—
60	C₁₆×C₆₁	—	4	2	no	—
61	C₂³×C₆₁	#131	1	1	yes	characteristic
62	C₂×C₄×C₆₁	—	2	2	no	—
63	C₂×C₄×C₆₁	—	2	4	no	—
64	C₂×C₄×C₆₁	—	4	2	no	—
65	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
71	C₈×C₆₁	—	4	2	no	—
72	C₈×C₆₁	C₄⋊C₈	1	2	yes	—
73	C₈×C₆₁	C₄⋊C₈	1	2	yes	—
74	C₈×C₆₁	#121	1	1	yes	characteristic
75	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
76	C₈×C₆₁	#125	1	1	yes	characteristic
77	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
78	#13162	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂²×C₆₁	—	2	2	no	—
80	C₂²×C₆₁	#424	1	1	yes	characteristic
81	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
82	C₂²×C₆₁	#446	1	1	yes	characteristic
83	C₄×C₆₁	—	4	2	no	—
84	C₄×C₆₁	—	4	4	no	—
85	C₄×C₆₁	#389	1	1	yes	characteristic
86	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₄×C₆₁	#436	1	1	yes	characteristic
88	C₄×C₆₁	C₈×Q₈	1	1	yes	characteristic
89	C₂×#364	C₂×C₆₁	1	1	yes	characteristic
90	#2151	C₂×C₆₁	1	2	yes	—
91	C₂×#346	C₂×C₆₁	1	1	yes	characteristic
92	#1772	C₂×C₆₁	1	1	yes	characteristic
93	#1379	C₂×C₆₁	1	2	yes	—
94	C₂×C₆₁	—	2	1	no	—
95	C₂×C₆₁	—	2	1	no	—
96	C₂×C₆₁	#1939	1	1	yes	characteristic
97	C₂×C₆₁	#2171	1	2	yes	—
98	C₂×M₃₂	C₂²×C₆₁	1	2	yes	—
99	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
100	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
101	C₂×#108	C₄×C₆₁	1	1	yes	characteristic
102	#427	C₄×C₆₁	1	1	yes	characteristic
103	#405	C₂²×C₆₁	1	2	yes	—
104	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
105	#364	—	2	2	no	—
106	C₄⋊C₁₆	—	2	4	no	—
107	#346	—	2	2	no	—
108	#339	C₄×C₆₁	1	2	yes	—
109	C₆₁	#13162	1	1	yes	characteristic, direct product factor, Sylow
110	C₂×M₁₆	C₂×C₄×C₆₁	1	2	yes	—
111	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
112	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
113	#117	C₂×C₄×C₆₁	1	2	yes	—
114	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
115	M₃₂	—	2	2	no	—
116	M₃₂	—	2	2	no	—
117	C₂×C₁₆	—	2	1	no	—
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	—	2	1	no	—
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	D₄×C₆₁	1	2	yes	—
123	C₂×C₁₆	Q₈×C₆₁	1	2	yes	—
124	#108	—	2	2	no	—
125	#107	C₈×C₆₁	1	2	yes	—
126	#106	C₈×C₆₁	1	2	yes	—
127	#97	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	D₄×C₆₁	1	2	yes	—
130	C₄×C₈	Q₈×C₆₁	1	2	yes	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
133	M₁₆	—	2	2	no	—
134	M₁₆	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	C₄⋊C₄×C₆₁	1	2	yes	—
137	C₂×C₈	C₂×C₈×C₆₁	1	1	yes	characteristic
138	C₂×C₈	C₂×C₈×C₆₁	1	1	yes	characteristic
139	C₂×C₈	M₁₆×C₆₁	1	2	yes	—
140	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
141	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
142	C₂×C₈	C₂×Q₈×C₆₁	1	1	yes	characteristic
143	C₂×C₈	C₂×Q₈×C₆₁	1	1	yes	characteristic
144	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
145	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
146	C₄⋊C₄	C₂×C₈×C₆₁	1	4	yes	—
147	C₂²⋊C₄	—	2	2	no	—
148	C₄²	—	2	2	no	—
149	C₄²	—	2	1	no	—
150	C₄²	C₄⋊C₄×C₆₁	1	2	yes	—
151	C₁₆	—	2	2	no	—
152	C₁₆	—	2	2	no	—
153	C₁₆	—	4	2	no	—
154	C₂³	#131×C₆₁	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
159	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
160	C₂×C₄	C₄×Q₈×C₆₁	1	1	yes	characteristic
161	C₂×C₄	C₂²×C₈×C₆₁	1	1	yes	characteristic
162	C₂×C₄	C₂×M₁₆×C₆₁	1	1	yes	characteristic
163	C₂×C₄	#131×C₆₁	1	1	yes	characteristic
164	C₈	—	4	2	no	—
165	C₈	C₄⋊C₈×C₆₁	1	2	yes	—
166	C₈	C₄⋊C₈×C₆₁	1	2	yes	—
167	C₈	#121×C₆₁	1	1	yes	characteristic
168	C₈	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
169	C₈	#125×C₆₁	1	1	yes	characteristic
170	C₈	C₄⋊Q₈×C₆₁	1	1	yes	characteristic
171	C₂²	—	2	2	no	—
172	C₂²	C₆₁×#424	1	1	yes	characteristic
173	C₂²	C₈×D₄×C₆₁	1	1	yes	characteristic
174	C₂²	C₆₁×#446	1	1	yes	characteristic
175	C₄	—	4	2	no	—
176	C₄	—	4	4	no	—
177	C₄	C₆₁×#389	1	1	yes	characteristic
178	C₄	C₂×C₄⋊C₈×C₆₁	1	1	yes	characteristic
179	C₄	C₆₁×#436	1	1	yes	characteristic
180	C₄	C₈×Q₈×C₆₁	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	C₆₁×#1939	1	1	yes	characteristic
184	C₂	C₆₁×#2171	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₆₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
socle	C₂²×C₆₁	C₈×D₄
prime radical	C₂³×C₆₁	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₁ ◁ C₂²×C₁₆×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
78. #13162	C₆₁	direct (N × H)
109. C₆₁	#13162	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×#364	2
C₆₁×#2151	2
C₂×C₆₁×#346	2
C₆₁×#1772	2
C₆₁×#1379	2
#13162	61

Subgroups — C61×#13162 #29516595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#13162 #29516595