Subgroups — D₆₁⋊#2258 #30624529

Order 15616.

470 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462243	C₂	1	6	yes	maximal, complemented
2	#21462242	C₂	1	2	yes	maximal, complemented
3	D₆₁⋊M₆₄	C₂	1	3	yes	maximal, complemented
4	C₂×D₆₁⋊C₃₂	C₂	1	3	yes	maximal, complemented
5	#504×D₆₁	C₂	1	1	yes	maximal, characteristic
6	#16840695	C₂²	1	6	yes	complemented
7	#16840694	C₂²	1	6	yes	complemented
8	D₆₁⋊C₃₂	C₂²	1	1	yes	characteristic
9	D₆₁⋊C₃₂	C₂²	1	3	yes	—
10	D₆₁⋊C₃₂	C₂²	1	6	yes	complemented
11	C₂×#14519827	C₂²	1	6	yes	complemented
12	#131×D₆₁	C₄	1	1	yes	characteristic
13	#16840180	C₂²	1	3	yes	—
14	#16840179	C₂²	1	1	yes	characteristic
15	#16840178	C₄	1	3	yes	—
16	M₃₂×D₆₁	C₄	1	3	yes	—
17	C₂×C₁₆×D₆₁	C₂²	1	3	yes	—
18	C₆₁×#504	C₄	1	1	yes	characteristic
19	#14519827	C₂³	1	2	yes	—
20	#14519827	C₂³	1	6	yes	—
21	#41×D₆₁	C₈	1	1	yes	characteristic
22	#14519785	C₈	1	3	yes	—
23	#14519784	C₈	1	1	yes	characteristic
24	#14519783	C₂×C₄	1	3	yes	—
25	M₁₆×D₆₁	C₈	1	3	yes	—
26	C₂×C₈×D₆₁	C₂×C₄	1	3	yes	—
27	#14519717	C₂×C₄	1	3	yes	—
28	#14519716	C₂×C₄	1	6	yes	—
29	C₂×#11758185	C₂³	1	3	yes	—
30	C₁₆×D₆₁	C₂×C₄	1	3	yes	—
31	C₁₆×D₆₁	C₂×C₄	1	3	yes	—
32	C₁₆×D₆₁	C₂³	1	1	yes	characteristic
33	C₁₆×D₆₁	C₂³	1	3	yes	—
34	#131×C₆₁	C₂×C₄	1	1	yes	characteristic
35	M₃₂×C₆₁	C₂×C₄	1	3	yes	—
36	C₂×C₁₆×C₆₁	C₂×C₄	1	3	yes	—
37	#41×C₆₁	C₂×C₈	1	1	yes	characteristic
38	#11758226	C₁₆	1	1	yes	characteristic
39	Q₈×D₆₁	C₁₆	1	1	yes	characteristic
40	#11758224	C₁₆	1	3	yes	—
41	D₄×D₆₁	C₁₆	1	3	yes	—
42	#11758222	C₂×C₈	1	3	yes	—
43	C₂×C₄×D₆₁	C₂×C₈	1	3	yes	—
44	M₁₆×C₆₁	C₂×C₈	1	3	yes	—
45	C₂×C₈×C₆₁	C₂²×C₄	1	3	yes	—
46	#11758194	C₂×C₈	1	3	yes	—
47	C₂×#92418	C₂×C₈	1	3	yes	—
48	#11758189	C₂×C₈	1	6	yes	—
49	C₈×D₆₁	C₂×C₈	1	3	yes	—
50	C₈×D₆₁	C₂×C₈	1	3	yes	—
51	C₈×D₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₈×D₆₁	C₂²×C₄	1	3	yes	—
53	C₁₆×C₆₁	C₂²×C₄	1	3	yes	—
54	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	#11758185	C₂²×C₄	1	3	yes	—
56	#11758185	C₂⁴	1	1	yes	characteristic
57	C₂²×D₆₁	C₂×C₁₆	1	3	yes	—
58	Q₈×C₆₁	C₂×C₁₆	1	1	yes	characteristic
59	D₄×C₆₁	C₂×C₁₆	1	3	yes	—
60	C₂×C₄×C₆₁	C₂²×C₈	1	3	yes	—
61	C₂²⋊D₆₁	C₂×C₁₆	1	6	yes	—
62	C₂×Dic₆₁	C₂×C₁₆	1	3	yes	—
63	D₂₄₄	C₂×C₁₆	1	3	yes	—
64	C₄×D₆₁	C₂×C₁₆	1	3	yes	—
65	C₄×D₆₁	C₂×C₁₆	1	3	yes	—
66	C₄×D₆₁	C₂²×C₈	1	1	yes	characteristic
67	C₄×D₆₁	C₂²×C₈	1	3	yes	—
68	Dic₁₂₂	C₂×C₁₆	1	3	yes	—
69	C₈×C₆₁	C₂²×C₈	1	3	yes	—
70	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
71	#92418	C₂²×C₈	1	1	yes	characteristic
72	#92418	C₂²×C₈	1	3	yes	—
73	C₂×#2258	—	61	1	no	maximal, Sylow
74	C₂²×C₆₁	C₂²×C₁₆	1	3	yes	—
75	C₂×D₆₁	—	2	6	no	—
76	C₂×D₆₁	C₂²×C₁₆	1	3	yes	—
77	C₂×D₆₁	C₂²×C₁₆	1	1	yes	characteristic
78	C₄×C₆₁	C₂²×C₁₆	1	3	yes	—
79	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
80	Dic₆₁	C₂²×C₁₆	1	3	yes	—
81	Dic₆₁	C₂²×C₁₆	1	1	yes	characteristic
82	C₂×#504	—	61	1	no	—
83	#2258	—	61	2	no	—
84	#2258	—	61	6	no	—
85	C₂×M₆₄	—	61	3	no	—
86	C₂²×C₃₂	—	61	3	no	—
87	C₂×C₆₁	—	2	3	no	—
88	C₂×C₆₁	C₂³×C₁₆	1	1	yes	characteristic
89	D₆₁	—	2	3	no	—
90	D₆₁	#2258	1	2	yes	complemented
91	C₂×#131	—	61	1	no	—
92	#504	—	61	1	no	—
93	#504	—	61	3	no	—
94	#504	—	61	3	no	—
95	#504	#6923	1	1	yes	characteristic
96	C₂×M₃₂	—	61	3	no	—
97	C₂²×C₁₆	—	61	3	no	—
98	M₆₄	—	61	6	no	—
99	M₆₄	—	61	6	no	—
100	C₂×C₃₂	—	61	1	no	—
101	C₂×C₃₂	—	61	3	no	—
102	C₂×C₃₂	—	61	6	no	—
103	C₂×C₃₂	—	61	6	no	—
104	C₆₁	C₂×#2258	1	1	yes	characteristic, complemented, Sylow
105	C₂×#41	—	61	1	no	—
106	#131	—	61	1	no	—
107	#131	—	61	3	no	—
108	#131	—	61	3	no	—
109	#131	C₂×#6923	1	1	yes	characteristic
110	C₂×M₁₆	—	61	3	no	—
111	C₂²×C₈	—	61	3	no	—
112	M₃₂	—	61	3	no	—
113	M₃₂	—	61	6	no	—
114	M₃₂	C₂×#6923	1	3	yes	—
115	C₂×C₁₆	—	61	3	no	—
116	C₂×C₁₆	—	61	3	no	—
117	C₂×C₁₆	—	61	3	no	—
118	C₂×C₁₆	—	61	1	no	—
119	C₂×C₁₆	—	61	3	no	—
120	C₂×C₁₆	C₂×#6923	1	3	yes	—
121	C₃₂	—	61	2	no	—
122	C₃₂	—	61	6	no	—
123	#41	—	61	1	no	—
124	#41	—	61	3	no	—
125	#41	—	61	3	no	—
126	#41	D₆₁⋊C₈	1	1	yes	characteristic
127	C₂×Q₈	—	61	1	no	—
128	C₂×D₄	—	61	3	no	—
129	C₂²×C₄	—	61	3	no	—
130	M₁₆	—	61	3	no	—
131	M₁₆	—	61	6	no	—
132	M₁₆	D₆₁⋊C₈	1	3	yes	—
133	C₂×C₈	—	61	3	no	—
134	C₂×C₈	—	61	3	no	—
135	C₂×C₈	—	61	3	no	—
136	C₂×C₈	—	61	1	no	—
137	C₂×C₈	—	61	3	no	—
138	C₂×C₈	C₂²×#6923	1	3	yes	—
139	C₁₆	—	61	1	no	—
140	C₁₆	—	61	3	no	—
141	C₁₆	C₂²×#6923	1	3	yes	—
142	C₁₆	C₂²×#6923	1	1	yes	characteristic
143	C₂³	—	61	3	no	—
144	Q₈	—	61	3	no	—
145	Q₈	D₆₁⋊C₁₆	1	1	yes	characteristic
146	D₄	—	61	6	no	—
147	D₄	—	61	3	no	—
148	D₄	D₆₁⋊C₁₆	1	3	yes	—
149	C₂×C₄	—	61	3	no	—
150	C₂×C₄	—	61	3	no	—
151	C₂×C₄	—	61	3	no	—
152	C₂×C₄	—	61	1	no	—
153	C₂×C₄	—	61	3	no	—
154	C₂×C₄	C₂×D₆₁⋊C₈	1	3	yes	—
155	C₈	—	61	1	no	—
156	C₈	—	61	3	no	—
157	C₈	C₂×D₆₁⋊C₈	1	3	yes	—
158	C₈	C₂³×#6923	1	1	yes	characteristic
159	C₂²	—	61	1	no	—
160	C₂²	—	61	3	no	—
161	C₂²	—	122	6	no	—
162	C₂²	C₂×D₆₁⋊C₁₆	1	3	yes	—
163	C₄	—	61	1	no	—
164	C₄	—	61	3	no	—
165	C₄	C₂×D₆₁⋊C₁₆	1	3	yes	—
166	C₄	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
167	C₂	—	2	3	no	—
168	C₂	—	61	2	no	—
169	C₂	—	122	3	no	—
170	C₂	C₂²×D₆₁⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²×#6923
commutator, socle	C₂×C₆₁	C₂³×C₁₆
Frattini	C₈	C₂³×#6923
Fitting	C₆₁×#504	C₄
prime radical	#41×D₆₁	C₈
cosocle	#11758185	C₂⁴
hypercenter	#504	#6923
hypocenter	C₆₁	C₂×#2258
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₁₆ ◁ #504

Complements

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

N	H ≅ G/N	type
1. #21462243	C₂	semidirect
2. #21462242	C₂	semidirect
3. D₆₁⋊M₆₄	C₂	semidirect
4. C₂×D₆₁⋊C₃₂	C₂	semidirect
6. #16840695	C₂²	semidirect
7. #16840694	C₂²	semidirect
10. D₆₁⋊C₃₂	C₂²	semidirect
11. C₂×#14519827	C₂²	semidirect
90. D₆₁	#2258	semidirect
104. C₆₁	C₂×#2258	semidirect

Maximal subgroups

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

subgroup	index
#21462243	2
#21462242	2
D₆₁⋊M₆₄	2
C₂×D₆₁⋊C₃₂	2
#504×D₆₁	2
C₂×#2258	61

Subgroups — D61⋊#2258 #30624529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₆₁⋊#2258 #30624529