Subgroups — #29833375

Order 15616.

718 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454671	C₂	1	2	yes	maximal, complemented
2	#21447695	C₂	1	4	yes	maximal, complemented
3	D₁₆×Dic₆₁	C₂	1	4	yes	maximal, complemented
4	#21447603	C₂	1	1	yes	maximal, characteristic, complemented
5	#21447535	C₂	1	2	yes	maximal, complemented
6	#21447415	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
8	#16840237	—	2	2	no	—
9	C₂×#14519717	C₂²	1	1	yes	characteristic
10	#16839874	—	2	4	no	—
11	D₈×Dic₆₁	C₂²	1	8	yes	complemented
12	#16839800	C₂²	1	2	yes	—
13	#16839767	—	2	2	no	—
14	#16839717	C₂²	1	1	yes	characteristic
15	#16839590	C₂²	1	2	yes	complemented
16	#16839588	C₂²	1	2	yes	complemented
17	#16839563	C₂²	1	8	yes	complemented
18	#16839537	C₂²	1	2	yes	complemented
19	C₁₆×Dic₆₁	C₂²	1	2	yes	complemented
20	C₂²×D₈×C₆₁	C₂²	1	2	yes	—
21	C₂×D₁₆×C₆₁	C₄	1	8	yes	complemented
22	C₂×D₁₆×C₆₁	C₂²	1	2	yes	—
23	C₂×D₁₆×C₆₁	C₂²	1	2	yes	—
24	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×#11758194	D₄	1	1	yes	characteristic
26	D₄×Dic₆₁	—	2	8	no	—
27	D₄×Dic₆₁	—	4	4	no	—
28	#14519736	—	2	2	no	—
29	#14519736	D₄	1	1	yes	characteristic
30	C₂×C₂²⋊Dic₆₁	—	4	4	no	—
31	#14519717	—	2	2	no	—
32	C₂×#11758185	C₂³	1	2	yes	—
33	#14519711	C₂³	1	4	yes	—
34	#14519709	—	2	2	no	—
35	#14519698	—	2	8	no	—
36	#14519687	D₄	1	2	yes	—
37	C₈×Dic₆₁	C₂³	1	2	yes	—
38	C₂²×D₄×C₆₁	—	2	2	no	—
39	C₂×D₈×C₆₁	—	2	2	no	—
40	C₂×D₈×C₆₁	—	2	4	no	—
41	C₂×D₈×C₆₁	C₂×C₄	1	8	yes	complemented
42	C₂×D₈×C₆₁	C₂³	1	4	yes	—
43	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
44	D₁₆×C₆₁	C₂×C₄	1	8	yes	complemented
45	D₁₆×C₆₁	C₂×C₄	1	8	yes	complemented
46	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	complemented
47	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
49	C₂⁴×C₆₁	—	4	2	no	—
50	C₂×D₄×C₆₁	—	2	8	no	—
51	C₂×D₄×C₆₁	—	2	4	no	—
52	C₂×D₄×C₆₁	—	4	2	no	—
53	C₂×D₄×C₆₁	—	4	4	no	—
54	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₆₁	—	4	8	no	—
56	D₈×C₆₁	—	2	8	no	—
57	D₈×C₆₁	—	2	4	no	—
58	D₈×C₆₁	C₂²×C₄	1	8	yes	—
59	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₆₁	#41	1	2	yes	—
62	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
63	C₂²⋊Dic₆₁	—	4	2	no	—
64	C₂²⋊Dic₆₁	—	4	8	no	—
65	C₂²⋊Dic₆₁	D₈	1	2	yes	—
66	C₄⋊Dic₆₁	—	2	4	no	—
67	C₄⋊Dic₆₁	D₈	1	2	yes	—
68	C₄×Dic₆₁	—	4	2	no	—
69	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
70	#11758194	—	2	2	no	—
71	C₂×#92418	C₂×D₄	1	2	yes	—
72	C₁₆×C₆₁	C₂²×C₄	1	2	yes	—
73	C₁₆×C₆₁	C₂²×C₄	1	2	yes	—
74	#11758185	—	2	2	no	—
75	C₂³×C₆₁	—	4	4	no	—
76	C₂³×C₆₁	—	4	8	no	—
77	C₂³×C₆₁	—	16	4	no	—
78	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
79	D₄×C₆₁	—	2	8	no	—
80	D₄×C₆₁	—	4	4	no	—
81	D₄×C₆₁	—	4	8	no	—
82	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
83	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₆₁	#135	1	2	yes	—
85	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
86	C₂×Dic₆₁	—	4	4	no	—
87	C₂×Dic₆₁	—	16	8	no	—
88	C₂×Dic₆₁	C₂×D₈	1	4	yes	—
89	C₈×C₆₁	C₄×D₄	1	2	yes	—
90	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
91	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
92	#92418	—	2	2	no	—
93	#33456	—	61	1	no	maximal, Sylow
94	C₂²×C₆₁	—	2	2	no	—
95	C₂²×C₆₁	—	4	8	no	—
96	C₂²×C₆₁	—	8	8	no	—
97	C₂²×C₆₁	—	16	8	no	—
98	C₂²×C₆₁	C₄×D₈	1	2	yes	—
99	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
100	C₄×C₆₁	C₄×D₈	1	2	yes	—
101	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
102	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
103	Dic₆₁	—	2	4	no	—
104	Dic₆₁	—	8	4	no	—
105	C₂²×D₁₆	D₆₁	1	1	yes	characteristic
106	#2948	—	61	2	no	—
107	#2177	—	61	4	no	—
108	C₄×D₁₆	—	61	4	no	—
109	#2158	—	61	1	no	—
110	#2140	—	61	2	no	—
111	#2108	—	61	1	no	—
112	C₂×C₆₁	—	2	2	no	—
113	C₂×C₆₁	—	8	8	no	—
114	C₂×C₆₁	C₂×C₄×D₈	1	1	yes	characteristic
115	C₂×C₆₁	#3415	1	2	yes	—
116	C₂²×D₈	C₂×D₆₁	1	2	yes	—
117	#518	—	122	2	no	—
118	C₂×D₁₆	Dic₆₁	1	8	yes	complemented
119	C₂×D₁₆	C₂×D₆₁	1	2	yes	—
120	C₂×D₁₆	C₂×D₆₁	1	2	yes	—
121	C₂×M₃₂	—	61	1	no	—
122	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
123	#442	—	122	4	no	—
124	C₄×D₈	—	61	8	no	—
125	#427	—	61	2	no	—
126	#418	—	122	2	no	—
127	#405	—	61	1	no	—
128	#367	—	61	2	no	—
129	#366	—	61	2	no	—
130	#357	—	61	8	no	—
131	#346	—	61	2	no	—
132	C₄×C₁₆	—	61	2	no	—
133	C₆₁	#33456	1	1	yes	characteristic, complemented, Sylow
134	C₂²×D₄	—	2	2	no	—
135	C₂×D₈	—	2	2	no	—
136	C₂×D₈	—	2	4	no	—
137	C₂×D₈	C₂×Dic₆₁	1	8	yes	complemented
138	C₂×D₈	C₂²×D₆₁	1	4	yes	—
139	C₂×M₁₆	—	61	1	no	—
140	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
141	C₄×D₄	—	122	8	no	—
142	C₄×D₄	—	244	4	no	—
143	#117	—	61	1	no	—
144	#117	—	122	2	no	—
145	C₂×C₂²⋊C₄	—	244	4	no	—
146	D₁₆	C₂×Dic₆₁	1	8	yes	complemented
147	D₁₆	C₂×Dic₆₁	1	8	yes	complemented
148	M₃₂	—	122	2	no	—
149	C₂×C₁₆	—	61	2	no	—
150	C₂×C₁₆	C₂×Dic₆₁	1	4	yes	complemented
151	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
152	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
153	#107	—	61	4	no	—
154	#106	—	122	2	no	—
155	#102	—	122	8	no	—
156	#97	—	61	2	no	—
157	C₄×C₈	—	61	2	no	—
158	C₂⁴	—	4	2	no	—
159	C₂×D₄	—	2	4	no	—
160	C₂×D₄	—	2	8	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	4	4	no	—
163	C₂²×C₄	—	244	8	no	—
164	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
165	D₈	—	2	4	no	—
166	D₈	—	2	8	no	—
167	D₈	C₂²×Dic₆₁	1	8	yes	—
168	M₁₆	—	122	2	no	—
169	C₂×C₈	—	61	2	no	—
170	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
171	C₂×C₈	#11758224	1	2	yes	—
172	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
173	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
174	C₄⋊C₄	—	61	2	no	—
175	C₄⋊C₄	—	122	4	no	—
176	C₂²⋊C₄	—	61	2	no	—
177	C₂²⋊C₄	—	244	8	no	—
178	C₂²⋊C₄	—	244	2	no	—
179	C₄²	—	61	2	no	—
180	C₄²	—	244	2	no	—
181	C₁₆	—	122	2	no	—
182	C₁₆	C₂²×Dic₆₁	1	2	yes	—
183	C₁₆	C₂²×Dic₆₁	1	2	yes	—
184	C₂³	—	4	4	no	—
185	C₂³	—	4	8	no	—
186	C₂³	—	16	4	no	—
187	C₂³	D₈×D₆₁	1	1	yes	characteristic
188	D₄	—	2	8	no	—
189	D₄	—	4	4	no	—
190	D₄	—	4	8	no	—
191	C₂×C₄	—	61	4	no	—
192	C₂×C₄	—	976	8	no	—
193	C₂×C₄	—	244	4	no	—
194	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
195	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
196	C₂×C₄	#14519796	1	2	yes	—
197	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
198	C₈	—	122	2	no	—
199	C₈	D₄×Dic₆₁	1	2	yes	—
200	C₈	C₂³×Dic₆₁	1	1	yes	characteristic
201	C₈	C₂×#11758224	1	1	yes	characteristic
202	C₂²	—	2	2	no	—
203	C₂²	—	4	8	no	—
204	C₂²	—	8	8	no	—
205	C₂²	—	16	8	no	—
206	C₂²	D₈×Dic₆₁	1	2	yes	—
207	C₂²	C₂×D₈×D₆₁	1	1	yes	characteristic
208	C₄	—	122	4	no	—
209	C₄	—	488	4	no	—
210	C₄	D₈×Dic₆₁	1	2	yes	—
211	C₄	C₂×D₄×Dic₆₁	1	1	yes	characteristic
212	C₄	C₂×#14519796	1	1	yes	characteristic
213	C₂	—	2	2	no	—
214	C₂	—	8	8	no	—
215	C₂	C₂×D₈×Dic₆₁	1	1	yes	characteristic
216	C₂	#21458956	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₁
commutator	C₈×C₆₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₁
Fitting, prime radical	C₂²×D₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₈
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	C₂²×D₁₆	D₆₁
hypocenter	C₆₁	#33456
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

Complements

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

N	H ≅ G/N	type
1. #21454671	C₂	semidirect
2. #21447695	C₂	semidirect
3. D₁₆×Dic₆₁	C₂	semidirect
4. #21447603	C₂	semidirect
5. #21447535	C₂	semidirect
6. #21447415	C₂	semidirect
11. D₈×Dic₆₁	C₂²	semidirect
15. #16839590	C₂²	semidirect
16. #16839588	C₂²	semidirect
17. #16839563	C₂²	semidirect
18. #16839537	C₂²	semidirect
19. C₁₆×Dic₆₁	C₂²	semidirect
21. C₂×D₁₆×C₆₁	C₄	semidirect
41. C₂×D₈×C₆₁	C₂×C₄	semidirect
44. D₁₆×C₆₁	C₂×C₄	semidirect
45. D₁₆×C₆₁	C₂×C₄	semidirect
46. C₂×C₁₆×C₆₁	C₂×C₄	semidirect
118. C₂×D₁₆	Dic₆₁	semidirect
133. C₆₁	#33456	semidirect
137. C₂×D₈	C₂×Dic₆₁	semidirect
146. D₁₆	C₂×Dic₆₁	semidirect
147. D₁₆	C₂×Dic₆₁	semidirect
150. C₂×C₁₆	C₂×Dic₆₁	semidirect

Maximal subgroups

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

subgroup	index
#21454671	2
#21447695	2
D₁₆×Dic₆₁	2
#21447603	2
#21447535	2
#21447415	2
C₂²×D₁₆×C₆₁	2
#33456	61