Subgroups — D₆₁×#1370 #29591579

Order 15616.

286 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₁₆×D₆₁	C₂	1	2	yes	maximal
2	C₈²×D₆₁	C₂	1	1	yes	maximal, characteristic
3	#21443889	C₂	1	1	yes	maximal, characteristic, complemented
4	#21443888	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1370	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×D₆₁	—	2	2	no	—
7	C₄×C₈×D₆₁	C₄	1	1	yes	characteristic
8	C₄×C₈×D₆₁	C₄	1	1	yes	characteristic
9	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
10	C₄⋊#11758185	C₂²	1	2	yes	—
11	#16839580	—	2	2	no	—
12	C₈×#92418	C₄	1	1	yes	characteristic
13	C₈×#92418	C₄	1	1	yes	characteristic
14	C₈×#92418	C₂²	1	1	yes	characteristic
15	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₈²×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₆₁	—	2	1	no	—
18	C₂×C₈×D₆₁	C₈	1	1	yes	characteristic
19	C₂×C₈×D₆₁	C₈	1	1	yes	characteristic
20	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
21	C₂×C₈×D₆₁	Q₈	1	1	yes	characteristic
22	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂×#11758185	—	2	2	no	—
24	C₁₆×D₆₁	—	4	4	no	—
25	C₄×#92418	C₂×C₄	1	1	yes	characteristic
26	C₄×#92418	C₂×C₄	1	1	yes	characteristic
27	C₄×#92418	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₆₁	—	2	1	no	—
29	C₈×Dic₆₁	C₈	1	1	yes	characteristic
30	C₈×Dic₆₁	C₈	1	1	yes	characteristic
31	C₈×Dic₆₁	D₄	1	1	yes	characteristic
32	C₈×Dic₆₁	Q₈	1	1	yes	characteristic
33	C₂×C₁₆×C₆₁	—	2	2	no	—
34	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
37	C₂×C₄×D₆₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
39	C₂×C₄×D₆₁	M₁₆	1	1	yes	characteristic
40	C₂×C₈×C₆₁	—	2	1	no	—
41	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₆₁	C₂×C₈	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	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
47	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
48	C₄×Dic₆₁	M₁₆	1	1	yes	characteristic
49	C₂×#92418	—	2	1	no	—
50	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
51	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
52	C₂×#92418	C₂×C₈	1	1	yes	characteristic
53	C₂×#92418	C₂×C₈	1	1	yes	characteristic
54	C₈×D₆₁	—	2	4	no	—
55	C₈×D₆₁	—	2	2	no	—
56	C₈×D₆₁	—	2	2	no	—
57	C₈×D₆₁	C₁₆	1	4	yes	complemented
58	C₈×D₆₁	D₈	1	2	yes	—
59	C₈×D₆₁	Q₁₆	1	2	yes	—
60	C₁₆×C₆₁	—	4	2	no	—
61	#11758185	—	4	2	no	—
62	C₂²×D₆₁	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
66	C₂×Dic₆₁	C₄⋊C₈	1	1	yes	characteristic
67	C₂×Dic₆₁	C₄⋊C₈	1	1	yes	characteristic
68	C₂×Dic₆₁	C₄⋊C₈	1	1	yes	characteristic
69	C₄×D₆₁	—	2	2	no	—
70	C₄×D₆₁	#107	1	2	yes	—
71	C₄×D₆₁	#108	1	2	yes	—
72	C₄×D₆₁	C₂×C₁₆	1	2	yes	—
73	C₄×D₆₁	M₃₂	1	2	yes	—
74	C₈×C₆₁	—	2	2	no	—
75	C₈×C₆₁	—	2	1	no	—
76	C₈×C₆₁	—	2	1	no	—
77	C₈×C₆₁	C₂×C₁₆	1	2	yes	complemented
78	C₈×C₆₁	C₂×D₈	1	1	yes	characteristic
79	C₈×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
80	#92418	—	2	2	no	—
81	#92418	—	2	1	no	—
82	#92418	—	2	1	no	—
83	#92418	#107	1	2	yes	—
84	#92418	C₂×C₁₆	1	2	yes	complemented
85	C₂×#1370	—	61	1	no	maximal, Sylow
86	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₂×D₆₁	#335	1	2	yes	—
88	C₂×D₆₁	C₄⋊C₁₆	1	2	yes	—
89	C₂×D₆₁	#364	1	2	yes	—
90	C₄×C₆₁	—	2	1	no	—
91	C₄×C₆₁	C₂×#107	1	1	yes	characteristic
92	C₄×C₆₁	C₂×#108	1	1	yes	characteristic
93	C₄×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
94	C₄×C₆₁	C₂×M₃₂	1	1	yes	characteristic
95	Dic₆₁	—	2	1	no	—
96	Dic₆₁	#335	1	2	yes	—
97	Dic₆₁	C₄⋊C₁₆	1	2	yes	—
98	C₂×C₄⋊C₁₆	—	61	2	no	—
99	C₂×C₈²	—	61	1	no	—
100	#1370	—	61	2	no	—
101	#1370	—	61	1	no	—
102	#1370	D₆₁	1	1	yes	characteristic, direct product factor
103	C₂×C₆₁	C₂×#335	1	1	yes	characteristic
104	C₂×C₆₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
105	C₂×C₆₁	C₂×#364	1	1	yes	characteristic
106	D₆₁	#1370	1	4	yes	direct product factor
107	C₂²×C₁₆	—	122	2	no	—
108	C₂×C₄×C₈	—	61	1	no	—
109	C₂×C₄×C₈	—	61	1	no	—
110	C₂×C₄×C₈	—	61	1	no	—
111	C₄⋊C₁₆	—	61	2	no	—
112	C₄⋊C₁₆	—	122	2	no	—
113	C₄⋊C₁₆	C₂×D₆₁	1	2	yes	—
114	C₈²	—	61	1	no	—
115	C₈²	—	61	1	no	—
116	C₈²	—	61	1	no	—
117	C₈²	C₂×D₆₁	1	1	yes	characteristic
118	C₆₁	C₂×#1370	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₈	—	61	1	no	—
120	C₂²×C₈	—	61	1	no	—
121	C₂²×C₈	—	61	1	no	—
122	C₂²×C₈	—	61	1	no	—
123	C₂²×C₈	—	122	1	no	—
124	C₂×C₄²	—	61	1	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	122	2	no	—
127	C₂×C₁₆	—	244	4	no	—
128	C₄×C₈	—	61	1	no	—
129	C₄×C₈	—	61	1	no	—
130	C₄×C₈	—	61	1	no	—
131	C₄×C₈	—	61	1	no	—
132	C₄×C₈	—	61	1	no	—
133	C₄×C₈	—	61	1	no	—
134	C₄×C₈	—	61	1	no	—
135	C₄×C₈	—	122	1	no	—
136	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
137	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
138	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
139	C₂²×C₄	—	61	1	no	—
140	C₂²×C₄	—	61	1	no	—
141	C₂²×C₄	—	61	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	61	1	no	—
144	C₂×C₈	—	61	1	no	—
145	C₂×C₈	—	61	4	no	—
146	C₂×C₈	—	61	2	no	—
147	C₂×C₈	—	61	2	no	—
148	C₂×C₈	—	61	1	no	—
149	C₂×C₈	—	61	1	no	—
150	C₂×C₈	—	122	2	no	—
151	C₂×C₈	—	122	2	no	—
152	C₂×C₈	—	122	4	no	—
153	C₂×C₈	—	122	1	no	—
154	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
155	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
156	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
157	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
158	C₄²	—	61	1	no	—
159	C₄²	—	61	1	no	—
160	C₄²	—	61	1	no	—
161	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
162	C₁₆	—	4	2	no	—
163	C₁₆	—	244	2	no	—
164	C₂³	—	61	1	no	—
165	C₂×C₄	—	61	1	no	—
166	C₂×C₄	—	61	1	no	—
167	C₂×C₄	—	61	2	no	—
168	C₂×C₄	—	61	2	no	—
169	C₂×C₄	—	61	2	no	—
170	C₂×C₄	—	61	2	no	—
171	C₂×C₄	—	61	1	no	—
172	C₂×C₄	—	122	2	no	—
173	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
174	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
175	C₂×C₄	M₁₆×D₆₁	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	2	1	no	—
178	C₈	—	2	1	no	—
179	C₈	—	61	2	no	—
180	C₈	—	61	2	no	—
181	C₈	—	122	2	no	—
182	C₈	—	122	1	no	—
183	C₈	—	122	1	no	—
184	C₈	C₁₆×D₆₁	1	2	yes	complemented
185	C₈	D₈×D₆₁	1	1	yes	characteristic
186	C₈	Q₁₆×D₆₁	1	1	yes	characteristic
187	C₂²	—	61	2	no	—
188	C₂²	—	61	2	no	—
189	C₂²	—	61	2	no	—
190	C₂²	C₄⋊C₈×D₆₁	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	61	2	no	—
193	C₄	—	61	2	no	—
194	C₄	—	122	1	no	—
195	C₄	#107×D₆₁	1	1	yes	characteristic
196	C₄	#108×D₆₁	1	1	yes	characteristic
197	C₄	C₂×C₁₆×D₆₁	1	1	yes	characteristic
198	C₄	M₃₂×D₆₁	1	1	yes	characteristic
199	C₂	—	61	4	no	—
200	C₂	#335×D₆₁	1	1	yes	characteristic
201	C₂	C₄⋊C₁₆×D₆₁	1	1	yes	characteristic
202	C₂	#364×D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₆₁
commutator	C₄×C₆₁	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1370	C₂
socle	C₂²×C₆₁	C₂×C₄⋊C₈
prime radical	C₂²×D₆₁	C₄⋊C₈
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	#1370	D₆₁
hypocenter	C₆₁	C₂×#1370
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

Complements

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

N	H ≅ G/N	type
3. #21443889	C₂	semidirect
4. #21443888	C₂	semidirect
5. C₆₁×#1370	C₂	semidirect
57. C₈×D₆₁	C₁₆	semidirect
77. C₈×C₆₁	C₂×C₁₆	semidirect
84. #92418	C₂×C₁₆	semidirect
102. #1370	D₆₁	direct (N × H)
106. D₆₁	#1370	direct (N × H)
118. C₆₁	C₂×#1370	semidirect
184. C₈	C₁₆×D₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₁₆×D₆₁	2
C₈²×D₆₁	2
#21443889	2
#21443888	2
C₆₁×#1370	2
C₂×#1370	61

Subgroups — D61×#1370 #29591579

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₆₁×#1370 #29591579