Subgroups — #29571181

Order 15616.

390 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14519687	C₂	1	1	yes	maximal, characteristic
2	#21444567	C₂	1	1	yes	maximal, characteristic
3	#21444103	C₂	1	2	yes	maximal
4	#21443651	C₂	1	2	yes	maximal
5	C₆₁×#1564	C₂	1	1	yes	maximal, characteristic
6	#16839815	C₂²	1	2	yes	—
7	C₂×#14519687	C₄	1	2	yes	—
8	C₂×#14519687	C₄	1	2	yes	—
9	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
10	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
11	#16839515	C₄	1	2	yes	complemented
12	#16839513	C₄	1	2	yes	complemented
13	#16839484	—	2	2	no	—
14	C₈×#92418	—	2	2	no	—
15	C₆₁×#431	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#335	C₄	1	2	yes	complemented
18	C₆₁×#334	C₄	1	2	yes	complemented
19	C₂²×#92418	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₆₁	—	2	1	no	—
21	C₂×C₄×Dic₆₁	—	2	1	no	—
22	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
23	C₄⋊#92418	C₂×C₄	1	4	yes	—
24	C₂²⋊#92418	—	2	2	no	—
25	#14519687	—	2	4	no	—
26	#14519687	C₈	1	8	yes	complemented
27	C₄×#92418	—	2	2	no	—
28	C₄×#92418	D₄	1	2	yes	—
29	C₄×#92418	Q₈	1	2	yes	—
30	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
34	C₂²⋊C₈×C₆₁	—	2	2	no	—
35	C₄×C₈×C₆₁	—	2	2	no	—
36	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
43	C₂²×Dic₆₁	—	2	2	no	—
44	C₂×C₈×C₆₁	—	2	2	no	—
45	C₂×C₈×C₆₁	—	2	4	no	—
46	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
51	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
55	C₄×Dic₆₁	—	2	2	no	—
56	C₄×Dic₆₁	—	4	2	no	—
57	C₄×Dic₆₁	—	4	2	no	—
58	C₄×Dic₆₁	C₂×C₈	1	4	yes	—
59	C₂×#92418	—	2	4	no	—
60	C₂×#92418	C₄⋊C₄	1	4	yes	—
61	C₂×#92418	C₂×C₈	1	4	yes	—
62	C₂×#92418	M₁₆	1	4	yes	—
63	C₂³×C₆₁	#117	1	1	yes	characteristic
64	C₂×C₄×C₆₁	—	2	2	no	—
65	C₂×C₄×C₆₁	—	2	2	no	—
66	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
73	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
74	C₂×Dic₆₁	—	2	4	no	—
75	C₂×Dic₆₁	—	4	2	no	—
76	C₂×Dic₆₁	—	4	2	no	—
77	C₈×C₆₁	—	2	1	no	—
78	C₈×C₆₁	—	2	1	no	—
79	C₈×C₆₁	—	4	4	no	—
80	C₈×C₆₁	C₄×C₈	1	2	yes	—
81	C₈×C₆₁	C₄×C₈	1	2	yes	—
82	#92418	—	4	4	no	—
83	#92418	C₄⋊C₈	1	8	yes	—
84	#8268	—	61	1	no	maximal, Sylow
85	C₂²×C₆₁	—	2	1	no	—
86	C₂²×C₆₁	—	2	1	no	—
87	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₆₁	#431	1	2	yes	—
89	C₄×C₆₁	—	2	1	no	—
90	C₄×C₆₁	—	2	1	no	—
91	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
92	C₄×C₆₁	C₄×M₁₆	1	1	yes	characteristic
93	C₄×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
94	C₄×C₆₁	#444	1	2	yes	—
95	C₄×C₆₁	#444	1	2	yes	—
96	Dic₆₁	—	4	4	no	—
97	C₄×#97	—	61	1	no	—
98	#1564	—	61	1	no	—
99	#1564	D₆₁	1	1	yes	characteristic
100	#1450	—	61	2	no	—
101	#1276	—	61	2	no	—
102	C₂×C₆₁	—	2	2	no	—
103	C₂×C₆₁	C₄×C₄⋊C₈	1	1	yes	characteristic
104	C₂×C₆₁	#1773	1	1	yes	characteristic
105	C₂×C₆₁	#1773	1	1	yes	characteristic
106	#431	—	61	2	no	—
107	#431	C₂×D₆₁	1	2	yes	—
108	C₂×#97	—	61	2	no	—
109	C₂×#97	—	61	2	no	—
110	C₂×C₄×C₈	—	61	1	no	—
111	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
112	C₄³	—	61	1	no	—
113	#335	—	61	2	no	—
114	#335	Dic₆₁	1	2	yes	complemented
115	#334	—	61	2	no	—
116	#334	Dic₆₁	1	2	yes	complemented
117	#322	—	122	2	no	—
118	C₈²	—	122	2	no	—
119	C₆₁	#8268	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₈	—	61	2	no	—
121	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
122	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
123	C₂×C₄²	—	61	2	no	—
124	C₂×C₄²	—	122	1	no	—
125	C₂×C₄²	—	122	1	no	—
126	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
127	C₄⋊C₈	—	61	4	no	—
128	C₄⋊C₈	C₂×Dic₆₁	1	4	yes	—
129	C₂²⋊C₈	—	2	2	no	—
130	C₂²⋊C₈	—	122	2	no	—
131	#97	—	61	8	no	—
132	#97	—	122	4	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	61	2	no	—
135	C₄×C₈	—	61	2	no	—
136	C₄×C₈	—	122	2	no	—
137	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
138	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
139	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
140	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
141	C₂²×C₄	—	122	2	no	—
142	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
143	C₂²×C₄	#11758224	1	1	yes	characteristic
144	C₂²×C₄	#11758226	1	1	yes	characteristic
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	4	no	—
147	C₂×C₈	—	61	4	no	—
148	C₂×C₈	—	61	4	no	—
149	C₂×C₈	—	61	4	no	—
150	C₂×C₈	—	122	4	no	—
151	C₂×C₈	C₈×D₆₁	1	4	yes	—
152	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
153	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
154	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
155	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
156	C₄²	—	61	4	no	—
157	C₄²	—	122	2	no	—
158	C₄²	—	244	2	no	—
159	C₄²	—	244	2	no	—
160	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
161	C₄²	D₄×D₆₁	1	1	yes	characteristic
162	C₄²	Q₈×D₆₁	1	1	yes	characteristic
163	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
164	C₂³	#14519735	1	1	yes	characteristic
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	122	4	no	—
168	C₂×C₄	—	244	2	no	—
169	C₂×C₄	—	244	2	no	—
170	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
171	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
172	C₂×C₄	#14519739	1	1	yes	characteristic
173	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
174	C₂×C₄	#14519742	1	1	yes	characteristic
175	C₂×C₄	Q₈×Dic₆₁	1	1	yes	characteristic
176	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
177	C₂×C₄	M₁₆×D₆₁	1	2	yes	—
178	C₈	—	2	1	no	—
179	C₈	—	2	1	no	—
180	C₈	—	4	4	no	—
181	C₈	—	61	8	no	—
182	C₈	—	244	4	no	—
183	C₈	C₈×Dic₆₁	1	2	yes	—
184	C₈	C₈×Dic₆₁	1	2	yes	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	C₄⋊C₄×Dic₆₁	1	1	yes	characteristic
188	C₂²	#16839814	1	2	yes	—
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	—	244	4	no	—
192	C₄	C₂×C₈×Dic₆₁	1	1	yes	characteristic
193	C₄	M₁₆×Dic₆₁	1	1	yes	characteristic
194	C₄	C₄⋊C₈×D₆₁	1	2	yes	—
195	C₄	#16839883	1	2	yes	—
196	C₄	#16839884	1	2	yes	—
197	C₂	—	2	2	no	—
198	C₂	C₄⋊C₈×Dic₆₁	1	1	yes	characteristic
199	C₂	#21445705	1	1	yes	characteristic
200	C₂	#21445706	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₆₁
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1564	C₂
socle	C₂²×C₆₁	C₄×C₄⋊C₄
prime radical	C₂³×C₆₁	#117
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1564	D₆₁
hypocenter	C₆₁	#8268
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
11. #16839515	C₄	semidirect
12. #16839513	C₄	semidirect
17. C₆₁×#335	C₄	semidirect
18. C₆₁×#334	C₄	semidirect
26. #14519687	C₈	semidirect
114. #335	Dic₆₁	semidirect
116. #334	Dic₆₁	semidirect
119. C₆₁	#8268	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14519687	2
#21444567	2
#21444103	2
#21443651	2
C₆₁×#1564	2
#8268	61