Subgroups — D₆₁⋊#1370 #30605768

Order 15616.

286 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₆₁⋊(C₄⋊C₁₆)	C₂	1	2	yes	maximal
2	#21460849	C₂	1	2	yes	maximal, complemented
3	#21460848	C₂	1	2	yes	maximal, complemented
4	C₈²×D₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×D₆₁⋊C₁₆	—	2	2	no	—
6	#16840683	—	2	2	no	—
7	C₄⋊#11758187	C₂²	1	4	yes	—
8	C₄×C₈×D₆₁	C₄	1	1	yes	characteristic
9	C₄×C₈×D₆₁	C₄	1	1	yes	characteristic
10	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
11	C₈×#92418	C₄	1	1	yes	characteristic
12	C₈×#92418	C₂²	1	1	yes	characteristic
13	C₈×#92418	C₂²	1	1	yes	characteristic
14	C₈²×C₆₁	C₄	1	1	yes	characteristic
15	D₆₁⋊C₁₆	—	4	4	no	—
16	C₂×#11758187	—	2	4	no	—
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	C₄×#92418	C₂×C₄	1	1	yes	characteristic
24	C₄×#92418	C₂×C₄	1	1	yes	characteristic
25	C₄×#92418	C₂³	1	1	yes	characteristic
26	C₈×Dic₆₁	—	2	1	no	—
27	C₈×Dic₆₁	C₈	1	1	yes	characteristic
28	C₈×Dic₆₁	C₈	1	1	yes	characteristic
29	C₈×Dic₆₁	D₄	1	1	yes	characteristic
30	C₈×Dic₆₁	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₆₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×D₆₁	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₆₁	—	2	1	no	—
38	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
44	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
45	C₄×Dic₆₁	M₁₆	1	1	yes	characteristic
46	C₂×#92418	—	2	1	no	—
47	C₂×#92418	C₂×C₈	1	1	yes	characteristic
48	C₂×#92418	C₂×C₈	1	1	yes	characteristic
49	C₂×#92418	C₂×D₄	1	1	yes	characteristic
50	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
51	C₈×D₆₁	—	2	4	no	—
52	C₈×D₆₁	—	2	2	no	—
53	C₈×D₆₁	—	2	2	no	—
54	C₈×D₆₁	C₁₆	1	4	yes	complemented
55	C₈×D₆₁	D₈	1	2	yes	—
56	C₈×D₆₁	Q₁₆	1	2	yes	—
57	#11758187	—	4	4	no	—
58	C₂²×D₆₁	C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
62	C₂×Dic₆₁	C₄⋊C₈	1	1	yes	characteristic
63	C₂×Dic₆₁	C₄⋊C₈	1	1	yes	characteristic
64	C₂×Dic₆₁	C₄⋊C₈	1	1	yes	characteristic
65	C₄×D₆₁	—	2	2	no	—
66	C₄×D₆₁	#107	1	2	yes	—
67	C₄×D₆₁	#108	1	2	yes	—
68	C₄×D₆₁	C₂×C₁₆	1	2	yes	—
69	C₄×D₆₁	M₃₂	1	2	yes	—
70	C₈×C₆₁	—	2	2	no	—
71	C₈×C₆₁	—	2	1	no	—
72	C₈×C₆₁	—	2	1	no	—
73	C₈×C₆₁	#107	1	2	yes	—
74	C₈×C₆₁	C₂×C₁₆	1	2	yes	complemented
75	#92418	—	2	2	no	—
76	#92418	—	2	1	no	—
77	#92418	—	2	1	no	—
78	#92418	C₂×C₁₆	1	2	yes	complemented
79	#92418	C₂×D₈	1	1	yes	characteristic
80	#92418	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×#1370	—	61	1	no	maximal, Sylow
82	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₂×D₆₁	#335	1	2	yes	—
84	C₂×D₆₁	C₄⋊C₁₆	1	2	yes	—
85	C₂×D₆₁	#364	1	2	yes	—
86	C₄×C₆₁	—	2	1	no	—
87	C₄×C₆₁	C₂×#107	1	1	yes	characteristic
88	C₄×C₆₁	C₂×#108	1	1	yes	characteristic
89	C₄×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
90	C₄×C₆₁	C₂×M₃₂	1	1	yes	characteristic
91	Dic₆₁	—	2	1	no	—
92	Dic₆₁	#335	1	2	yes	—
93	Dic₆₁	C₄⋊C₁₆	1	2	yes	—
94	C₂×C₄⋊C₁₆	—	61	2	no	—
95	C₂×C₈²	—	61	1	no	—
96	#1370	—	61	2	no	—
97	#1370	—	61	2	no	—
98	C₂×C₆₁	C₂×#335	1	1	yes	characteristic
99	C₂×C₆₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
100	C₂×C₆₁	C₂×#364	1	1	yes	characteristic
101	D₆₁	#1370	1	4	yes	complemented
102	C₂²×C₁₆	—	122	2	no	—
103	C₂×C₄×C₈	—	61	1	no	—
104	C₂×C₄×C₈	—	61	1	no	—
105	C₂×C₄×C₈	—	61	1	no	—
106	C₄⋊C₁₆	—	61	4	no	—
107	C₄⋊C₁₆	—	122	2	no	—
108	C₈²	—	61	1	no	—
109	C₈²	—	61	1	no	—
110	C₈²	—	61	1	no	—
111	C₈²	#6923	1	1	yes	characteristic
112	C₆₁	C₂×#1370	1	1	yes	characteristic, complemented, Sylow
113	C₂²×C₈	—	61	1	no	—
114	C₂²×C₈	—	61	1	no	—
115	C₂²×C₈	—	61	1	no	—
116	C₂²×C₈	—	61	1	no	—
117	C₂²×C₈	—	122	1	no	—
118	C₂×C₄²	—	61	1	no	—
119	C₂×C₁₆	—	122	4	no	—
120	C₂×C₁₆	—	244	4	no	—
121	C₄×C₈	—	61	1	no	—
122	C₄×C₈	—	61	1	no	—
123	C₄×C₈	—	61	1	no	—
124	C₄×C₈	—	61	1	no	—
125	C₄×C₈	—	61	1	no	—
126	C₄×C₈	—	61	1	no	—
127	C₄×C₈	—	61	1	no	—
128	C₄×C₈	—	122	1	no	—
129	C₄×C₈	C₂×#6923	1	1	yes	characteristic
130	C₄×C₈	C₂×#6923	1	1	yes	characteristic
131	C₄×C₈	C₂×#6923	1	1	yes	characteristic
132	C₂²×C₄	—	61	1	no	—
133	C₂²×C₄	—	61	1	no	—
134	C₂²×C₄	—	61	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	61	1	no	—
137	C₂×C₈	—	61	1	no	—
138	C₂×C₈	—	61	2	no	—
139	C₂×C₈	—	61	2	no	—
140	C₂×C₈	—	61	4	no	—
141	C₂×C₈	—	61	1	no	—
142	C₂×C₈	—	61	1	no	—
143	C₂×C₈	—	122	2	no	—
144	C₂×C₈	—	122	2	no	—
145	C₂×C₈	—	122	4	no	—
146	C₂×C₈	—	122	1	no	—
147	C₂×C₈	D₆₁⋊C₈	1	1	yes	characteristic
148	C₂×C₈	D₆₁⋊C₈	1	1	yes	characteristic
149	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
150	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
151	C₄²	—	61	1	no	—
152	C₄²	—	61	1	no	—
153	C₄²	—	61	1	no	—
154	C₄²	C₂²×#6923	1	1	yes	characteristic
155	C₁₆	—	244	4	no	—
156	C₂³	—	61	1	no	—
157	C₂×C₄	—	61	1	no	—
158	C₂×C₄	—	61	1	no	—
159	C₂×C₄	—	61	2	no	—
160	C₂×C₄	—	61	2	no	—
161	C₂×C₄	—	61	2	no	—
162	C₂×C₄	—	61	2	no	—
163	C₂×C₄	—	61	1	no	—
164	C₂×C₄	—	122	2	no	—
165	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
166	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
167	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	61	1	no	—
172	C₈	—	61	1	no	—
173	C₈	—	61	2	no	—
174	C₈	—	122	2	no	—
175	C₈	—	122	1	no	—
176	C₈	—	122	1	no	—
177	C₈	#14519846	1	2	yes	—
178	C₈	D₆₁⋊C₁₆	1	2	yes	complemented
179	C₂²	—	61	2	no	—
180	C₂²	—	61	2	no	—
181	C₂²	—	61	2	no	—
182	C₂²	D₆₁⋊(C₄⋊C₈)	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	61	2	no	—
185	C₄	—	61	2	no	—
186	C₄	—	122	1	no	—
187	C₄	C₂×#14519846	1	1	yes	characteristic
188	C₄	D₆₁⋊#108	1	1	yes	characteristic
189	C₄	C₂×D₆₁⋊C₁₆	1	1	yes	characteristic
190	C₄	D₆₁⋊M₃₂	1	1	yes	characteristic
191	C₂	—	61	4	no	—
192	C₂	D₆₁⋊#335	1	1	yes	characteristic
193	C₂	D₆₁⋊(C₄⋊C₁₆)	1	1	yes	characteristic
194	C₂	D₆₁⋊#364	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄⋊#6923
commutator	C₄×C₆₁	C₂²×C₁₆
Frattini	C₄²	C₂²×#6923
Fitting	C₈²×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₄⋊C₈
prime radical	C₂²×D₆₁	C₄⋊C₈
cosocle	C₄×#92418	C₂³
hypercenter	C₈²	#6923
hypocenter	C₆₁	C₂×#1370
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. #21460849	C₂	semidirect
3. #21460848	C₂	semidirect
54. C₈×D₆₁	C₁₆	semidirect
74. C₈×C₆₁	C₂×C₁₆	semidirect
78. #92418	C₂×C₁₆	semidirect
101. D₆₁	#1370	semidirect
112. C₆₁	C₂×#1370	semidirect
178. C₈	D₆₁⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
D₆₁⋊(C₄⋊C₁₆)	2
#21460849	2
#21460848	2
C₈²×D₆₁	2
C₂×#1370	61

Subgroups — D61⋊#1370 #30605768

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₆₁⋊#1370 #30605768