Subgroups — #367×Dic₆₁ #29595418

Order 15616.

366 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839590	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	#107×Dic₆₁	C₂	1	2	yes	maximal
4	#21443910	C₂	1	2	yes	maximal
5	C₂×C₆₁×#367	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758185	C₂²	1	1	yes	characteristic
7	C₂×#14519711	C₂²	1	2	yes	—
8	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×Dic₆₁	—	2	2	no	—
10	#16839590	C₄	1	4	yes	complemented
11	C₁₆×Dic₆₁	C₄	1	4	yes	complemented
12	#16839524	—	2	2	no	—
13	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#107×C₆₁	C₂²	1	2	yes	—
15	C₆₁×#367	C₄	1	4	yes	complemented
16	C₂²×#92418	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₆₁	—	2	2	no	—
18	C₂×C₄×Dic₆₁	—	4	2	no	—
19	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
20	C₂×#11758185	—	2	1	no	—
21	C₂×#11758185	—	2	1	no	—
22	C₂×#11758185	C₂×C₄	1	2	yes	—
23	#14519711	—	2	2	no	—
24	#14519711	C₂×C₄	1	4	yes	—
25	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
26	C₈×Dic₆₁	Q₈	1	2	yes	—
27	#14519684	—	4	2	no	—
28	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
30	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
33	#107×C₆₁	—	2	2	no	—
34	#107×C₆₁	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₆₁	—	4	2	no	—
36	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₆₁	—	4	2	no	—
38	C₂²×Dic₆₁	D₈	1	1	yes	characteristic
39	C₂²×Dic₆₁	D₈	1	1	yes	characteristic
40	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
45	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
46	C₄⋊C₄×C₆₁	—	2	4	no	—
47	C₄⋊C₄×C₆₁	—	4	2	no	—
48	C₄⋊Dic₆₁	—	2	4	no	—
49	C₄⋊Dic₆₁	—	4	2	no	—
50	C₄×Dic₆₁	—	8	4	no	—
51	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
52	C₄×Dic₆₁	Q₁₆	1	2	yes	—
53	C₂×#92418	—	2	1	no	—
54	C₂×#92418	—	2	1	no	—
55	C₂×#92418	C₄⋊C₄	1	2	yes	—
56	C₁₆×C₆₁	C₄²	1	4	yes	complemented
57	#11758185	—	2	2	no	—
58	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₆₁	—	4	4	no	—
60	C₂×C₄×C₆₁	—	8	2	no	—
61	C₂×C₄×C₆₁	—	8	2	no	—
62	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
67	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
68	C₂×Dic₆₁	—	2	2	no	—
69	C₂×Dic₆₁	—	4	4	no	—
70	C₂×Dic₆₁	—	8	2	no	—
71	C₂×Dic₆₁	—	8	2	no	—
72	C₂×Dic₆₁	#107	1	2	yes	—
73	C₂×Dic₆₁	#107	1	2	yes	—
74	C₂×Dic₆₁	SD₃₂	1	4	yes	—
75	C₈×C₆₁	C₂×C₄²	1	1	yes	characteristic
76	C₈×C₆₁	#117	1	1	yes	characteristic
77	C₈×C₆₁	C₄×Q₈	1	1	yes	characteristic
78	C₈×C₆₁	C₄×Q₈	1	1	yes	characteristic
79	#92418	—	2	2	no	—
80	C₄×#367	—	61	1	no	maximal, Sylow
81	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
82	C₂²×C₆₁	C₄×D₈	1	1	yes	characteristic
83	C₂²×C₆₁	C₄×D₈	1	1	yes	characteristic
84	C₂²×C₆₁	C₂×SD₃₂	1	2	yes	—
85	C₂²×C₆₁	#508	1	2	yes	—
86	C₄×C₆₁	—	8	4	no	—
87	C₄×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
88	C₄×C₆₁	#427	1	1	yes	characteristic
89	C₄×C₆₁	C₄×Q₁₆	1	1	yes	characteristic
90	C₄×C₆₁	C₄×Q₁₆	1	1	yes	characteristic
91	Dic₆₁	—	2	2	no	—
92	Dic₆₁	—	8	4	no	—
93	Dic₆₁	#367	1	4	yes	direct product factor
94	C₂×#367	—	61	1	no	—
95	C₂×#367	D₆₁	1	1	yes	characteristic
96	C₂×C₄×C₁₆	—	61	1	no	—
97	C₄×#107	—	61	2	no	—
98	#1380	—	61	2	no	—
99	C₂×C₆₁	C₄×#107	1	1	yes	characteristic
100	C₂×C₆₁	C₂×#367	1	1	yes	characteristic
101	C₂×C₆₁	#2158	1	1	yes	characteristic
102	C₂×C₆₁	C₄×SD₃₂	1	2	yes	—
103	C₂×C₆₁	C₄×SD₃₂	1	2	yes	—
104	C₂²×C₁₆	—	61	1	no	—
105	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
106	C₂×#107	—	61	2	no	—
107	C₂×#107	C₂×D₆₁	1	2	yes	—
108	C₂×C₄×C₈	—	61	1	no	—
109	C₄×C₄⋊C₄	—	122	2	no	—
110	#367	—	61	4	no	—
111	#367	Dic₆₁	1	4	yes	direct product factor
112	C₄×C₁₆	—	61	4	no	—
113	#340	—	122	2	no	—
114	C₆₁	C₄×#367	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	—	61	1	no	—
116	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	122	2	no	—
119	C₂×C₄²	—	61	1	no	—
120	C₂×C₄²	—	244	2	no	—
121	C₂×C₁₆	—	61	2	no	—
122	C₂×C₁₆	—	122	1	no	—
123	C₂×C₁₆	—	122	1	no	—
124	C₂×C₁₆	C₄×D₆₁	1	2	yes	—
125	C₂×C₁₆	C₄×D₆₁	1	2	yes	—
126	C₂×C₁₆	C₂×Dic₆₁	1	2	yes	—
127	#107	—	2	2	no	—
128	#107	—	61	4	no	—
129	#107	—	122	2	no	—
130	#107	C₂×Dic₆₁	1	4	yes	—
131	C₄×C₈	—	61	2	no	—
132	C₄×C₈	—	61	2	no	—
133	#95	—	244	2	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	61	1	no	—
136	C₂²×C₄	—	61	1	no	—
137	C₂²×C₄	—	244	2	no	—
138	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
139	C₂×C₈	—	61	2	no	—
140	C₂×C₈	—	122	1	no	—
141	C₂×C₈	—	122	1	no	—
142	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
143	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
144	C₂×C₈	#11758224	1	1	yes	characteristic
145	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
146	C₂×C₈	#11758226	1	1	yes	characteristic
147	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	122	4	no	—
151	C₄⋊C₄	—	244	2	no	—
152	C₄²	—	61	2	no	—
153	C₄²	—	61	2	no	—
154	C₄²	—	488	4	no	—
155	C₁₆	—	122	2	no	—
156	C₁₆	C₄×Dic₆₁	1	4	yes	complemented
157	C₂³	D₈×D₆₁	1	1	yes	characteristic
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	61	2	no	—
162	C₂×C₄	—	61	4	no	—
163	C₂×C₄	—	61	2	no	—
164	C₂×C₄	—	122	2	no	—
165	C₂×C₄	—	488	2	no	—
166	C₂×C₄	—	488	2	no	—
167	C₂×C₄	—	244	4	no	—
168	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
169	C₂×C₄	#14519739	1	1	yes	characteristic
170	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
171	C₂×C₄	Q₁₆×D₆₁	1	1	yes	characteristic
172	C₂×C₄	#14519796	1	1	yes	characteristic
173	C₂×C₄	#14519800	1	1	yes	characteristic
174	C₈	—	122	2	no	—
175	C₈	C₂×C₄×Dic₆₁	1	1	yes	characteristic
176	C₈	#14519735	1	1	yes	characteristic
177	C₈	#14519742	1	1	yes	characteristic
178	C₈	Q₈×Dic₆₁	1	1	yes	characteristic
179	C₂²	#107×D₆₁	1	1	yes	characteristic
180	C₂²	#16839844	1	1	yes	characteristic
181	C₂²	D₈×Dic₆₁	1	1	yes	characteristic
182	C₂²	SD₃₂×D₆₁	1	2	yes	—
183	C₂²	#16840193	1	2	yes	—
184	C₄	—	8	4	no	—
185	C₄	—	61	4	no	—
186	C₄	—	122	2	no	—
187	C₄	—	488	4	no	—
188	C₄	C₄⋊C₄×Dic₆₁	1	1	yes	characteristic
189	C₄	#16839799	1	1	yes	characteristic
190	C₄	#16839856	1	1	yes	characteristic
191	C₄	Q₁₆×Dic₆₁	1	1	yes	characteristic
192	C₂	#107×Dic₆₁	1	1	yes	characteristic
193	C₂	#367×D₆₁	1	1	yes	characteristic
194	C₂	#21447602	1	1	yes	characteristic
195	C₂	#21447670	1	2	yes	—
196	C₂	SD₃₂×Dic₆₁	1	2	yes	—

	subgroup	G/H
center	C₂³	D₈×D₆₁
commutator	C₈×C₆₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×#367	C₂
socle, prime radical	C₂³×C₆₁	C₂×D₈
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×#367	D₆₁
hypocenter	C₆₁	C₄×#367
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

Complements

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

N	H ≅ G/N	type
10. #16839590	C₄	semidirect
11. C₁₆×Dic₆₁	C₄	semidirect
15. C₆₁×#367	C₄	semidirect
56. C₁₆×C₆₁	C₄²	semidirect
93. Dic₆₁	#367	direct (N × H)
111. #367	Dic₆₁	direct (N × H)
114. C₆₁	C₄×#367	semidirect
156. C₁₆	C₄×Dic₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839590	2
C₂×C₁₆×Dic₆₁	2
#107×Dic₆₁	2
#21443910	2
C₂×C₆₁×#367	2
C₄×#367	61

Subgroups — #367×Dic61 #29595418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #367×Dic₆₁ #29595418