Subgroups — C₃×#10601732 #14291452

Order 1920.

376 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#65316	C₂	1	2	yes	maximal, complemented
3	C₃×#64984	C₂	1	1	yes	maximal, characteristic
4	C₃×#64705	C₂	1	2	yes	maximal, complemented
5	C₃×#64678	C₂	1	1	yes	maximal, characteristic, complemented
6	#10601732	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×#688	—	2	2	no	—
9	C₂×C₃×D₄₀	—	2	1	no	—
10	C₂×C₃×D₄₀	—	2	1	no	—
11	C₃×#4020	C₂²	1	2	yes	—
12	C₃×C₅×C₄⋊C₈	—	2	1	no	—
13	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₅×C₈	—	2	1	no	—
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₃×#3941	—	2	2	no	—
17	C₃×C₄⋊#168	C₂²	1	2	yes	—
18	C₃×#1668	—	5	1	no	maximal, Hall
19	C₅×C₈×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
20	#65316	C₂×C₃	1	2	yes	complemented
21	#64984	C₂×C₃	1	1	yes	characteristic
22	#64705	C₂×C₃	1	2	yes	complemented
23	#64678	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
25	C₂×C₃×D₂₀	—	2	2	no	—
26	C₂×C₃×D₂₀	—	2	2	no	—
27	C₂×C₃×D₂₀	—	2	2	no	—
28	C₂×C₃×C₅×C₈	—	2	1	no	—
29	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
31	C₃×C₅×C₄⋊C₄	—	2	1	no	—
32	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
33	C₃×C₄²×C₅	—	2	1	no	—
34	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₃×#688	—	4	4	no	—
36	C₂×C₃×#168	—	2	2	no	—
37	C₃×D₄₀	—	4	2	no	—
38	C₃×D₄₀	—	4	2	no	—
39	C₃×#493	—	5	1	no	—
40	C₃×#460	—	5	2	no	—
41	C₃×C₈×Q₈	D₅	1	1	yes	characteristic, complemented
42	C₃×#335	—	5	1	no	—
43	C₃×#331	—	5	2	no	—
44	C₄×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
45	C₂×#688	—	2	2	no	—
46	C₂×D₄₀	—	2	1	no	—
47	C₂×D₄₀	—	2	1	no	—
48	#4020	C₂²×C₃	1	2	yes	—
49	C₅×C₄⋊C₈	—	2	1	no	—
50	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
51	C₄×C₅×C₈	—	2	1	no	—
52	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
53	#3941	—	2	2	no	—
54	C₄⋊#168	C₂²×C₃	1	2	yes	—
55	#1668	—	5	1	no	Sylow
56	C₂²×C₃×D₅	—	4	2	no	—
57	C₃×C₅×Q₈	—	2	1	no	—
58	C₃×C₅×Q₈	D₈	1	2	yes	complemented
59	C₂×C₃×C₄×C₅	—	2	1	no	—
60	C₂×C₃×C₄×C₅	—	2	1	no	—
61	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
64	C₃×D₂₀	—	4	2	no	—
65	C₃×D₂₀	—	4	2	no	—
66	C₃×D₂₀	—	4	2	no	—
67	C₃×D₂₀	—	4	2	no	—
68	C₃×D₂₀	—	8	2	no	—
69	C₃×C₅×C₈	—	2	1	no	—
70	C₃×C₅×C₈	—	4	1	no	—
71	C₃×C₅×C₈	SD₁₆	1	2	yes	complemented
72	C₃×#168	—	4	2	no	—
73	C₂×C₃×SD₁₆	—	10	2	no	—
74	C₂×C₃×D₈	—	10	1	no	—
75	C₂×C₃×D₈	—	10	1	no	—
76	C₃×#127	—	5	2	no	—
77	C₃×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
78	C₃×C₄⋊C₈	—	2	1	no	—
79	C₃×C₄⋊C₈	—	5	2	no	—
80	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
81	C₃×#102	—	10	2	no	—
82	C₃×C₄×C₈	—	2	1	no	—
83	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
84	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
85	C₂×D₂₀	—	2	2	no	—
86	C₂×D₂₀	—	2	2	no	—
87	C₂×D₂₀	—	2	2	no	—
88	C₂×C₅×C₈	—	2	1	no	—
89	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
91	C₅×C₄⋊C₄	—	2	1	no	—
92	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
93	C₄²×C₅	—	2	1	no	—
94	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
95	#688	—	4	4	no	—
96	C₂×#168	—	2	2	no	—
97	D₄₀	—	4	2	no	—
98	D₄₀	—	4	2	no	—
99	#493	—	5	1	no	—
100	#460	—	5	2	no	—
101	C₈×Q₈	C₃×D₅	1	1	yes	characteristic, complemented
102	#335	—	5	1	no	—
103	#331	—	5	2	no	—
104	C₂²×C₃×C₅	#121	1	1	yes	characteristic
105	C₂×C₃×D₅	—	8	2	no	—
106	C₂×C₃×D₅	—	8	2	no	—
107	C₂×C₃×D₅	—	8	2	no	—
108	C₃×C₄×C₅	—	2	2	no	—
109	C₃×C₄×C₅	—	2	1	no	—
110	C₃×C₄×C₅	—	4	1	no	—
111	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
112	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
113	C₃×C₄×C₅	#135	1	1	yes	characteristic
114	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂×C₃×Q₈	D₂₀	1	1	yes	characteristic
116	C₂×C₃×D₄	—	10	2	no	—
117	C₂×C₃×D₄	—	10	2	no	—
118	C₂×C₃×D₄	—	10	2	no	—
119	C₃×SD₁₆	—	20	4	no	—
120	C₃×D₈	—	20	2	no	—
121	C₃×D₈	—	20	2	no	—
122	C₂×C₃×C₈	—	2	1	no	—
123	C₂×C₃×C₈	—	10	2	no	—
124	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
125	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
126	C₃×C₄⋊C₄	—	2	1	no	—
127	C₃×C₄⋊C₄	D₂₀	1	1	yes	characteristic
128	C₃×C₄²	—	2	1	no	—
129	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
130	C₂²×D₅	—	4	2	no	—
131	C₅×Q₈	—	2	1	no	—
132	C₅×Q₈	C₃×D₈	1	2	yes	complemented
133	C₂×C₄×C₅	—	2	1	no	—
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
138	D₂₀	—	4	2	no	—
139	D₂₀	—	4	2	no	—
140	D₂₀	—	4	2	no	—
141	D₂₀	—	4	2	no	—
142	D₂₀	—	8	2	no	—
143	C₅×C₈	—	2	1	no	—
144	C₅×C₈	—	4	1	no	—
145	C₅×C₈	C₃×SD₁₆	1	2	yes	complemented
146	#168	—	4	2	no	—
147	C₂×SD₁₆	—	10	2	no	—
148	C₂×D₈	—	10	1	no	—
149	C₂×D₈	—	10	1	no	—
150	#127	—	5	2	no	—
151	C₄×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
152	C₄⋊C₈	—	2	1	no	—
153	C₄⋊C₈	—	5	2	no	—
154	C₄⋊C₈	C₂×C₃×D₅	1	1	yes	characteristic
155	#102	—	10	2	no	—
156	C₄×C₈	—	2	1	no	—
157	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
158	C₂×C₃×C₅	#460	1	1	yes	characteristic
159	C₂×C₃×C₅	#466	1	1	yes	characteristic
160	C₂×C₃×C₅	#472	1	1	yes	characteristic
161	C₃×D₅	—	16	2	no	—
162	C₂³×C₃	—	20	2	no	—
163	C₃×Q₈	—	2	1	no	—
164	C₃×Q₈	D₄₀	1	2	yes	complemented
165	C₃×D₄	—	20	2	no	—
166	C₃×D₄	—	20	2	no	—
167	C₃×D₄	—	20	2	no	—
168	C₃×D₄	—	20	2	no	—
169	C₃×D₄	—	40	2	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
173	C₂×C₃×C₄	#709	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
175	C₃×C₈	—	2	1	no	—
176	C₃×C₈	—	4	1	no	—
177	C₃×C₈	—	20	2	no	—
178	C₃×C₈	#688	1	2	yes	complemented
179	C₂²×C₅	C₃×#121	1	1	yes	characteristic
180	C₂×D₅	—	8	2	no	—
181	C₂×D₅	—	8	2	no	—
182	C₂×D₅	—	8	2	no	—
183	C₄×C₅	—	2	2	no	—
184	C₄×C₅	—	2	1	no	—
185	C₄×C₅	—	4	1	no	—
186	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
187	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
188	C₄×C₅	C₃×#135	1	1	yes	characteristic
189	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₂×Q₈	C₃×D₂₀	1	1	yes	characteristic
191	C₂×D₄	—	10	2	no	—
192	C₂×D₄	—	10	2	no	—
193	C₂×D₄	—	10	2	no	—
194	SD₁₆	—	20	4	no	—
195	D₈	—	20	2	no	—
196	D₈	—	20	2	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	10	2	no	—
199	C₂×C₈	C₃×C₂²⋊D₅	1	1	yes	characteristic
200	C₂×C₈	C₃×C₂²⋊D₅	1	1	yes	characteristic
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	C₃×D₂₀	1	1	yes	characteristic
203	C₄²	—	2	1	no	—
204	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
205	C₃×C₅	#1668	1	1	yes	characteristic, complemented, Hall
206	C₂²×C₃	—	40	2	no	—
207	C₂²×C₃	—	40	2	no	—
208	C₂²×C₃	—	40	2	no	—
209	C₂²×C₃	#4076	1	1	yes	characteristic
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	4	1	no	—
213	C₃×C₄	C₂×D₄₀	1	1	yes	characteristic
214	C₃×C₄	#4050	1	1	yes	characteristic
215	C₃×C₄	C₂×#688	1	1	yes	characteristic
216	C₃×C₄	#4095	1	1	yes	characteristic
217	C₂×C₅	C₃×#460	1	1	yes	characteristic
218	C₂×C₅	C₃×#466	1	1	yes	characteristic
219	C₂×C₅	C₃×#472	1	1	yes	characteristic
220	D₅	—	16	2	no	—
221	C₂³	—	20	2	no	—
222	Q₈	—	2	1	no	—
223	Q₈	C₃×D₄₀	1	2	yes	complemented
224	D₄	—	20	2	no	—
225	D₄	—	20	2	no	—
226	D₄	—	20	2	no	—
227	D₄	—	20	2	no	—
228	D₄	—	40	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
232	C₂×C₄	C₃×#709	1	1	yes	characteristic
233	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	4	1	no	—
236	C₈	—	20	2	no	—
237	C₈	C₃×#688	1	2	yes	complemented
238	C₂×C₃	—	80	2	no	—
239	C₂×C₃	#65316	1	1	yes	characteristic
240	C₂×C₃	#65404	1	1	yes	characteristic
241	C₂×C₃	#65431	1	1	yes	characteristic
242	C₅	C₃×#1668	1	1	yes	characteristic, complemented, Sylow
243	C₂²	—	40	2	no	—
244	C₂²	—	40	2	no	—
245	C₂²	—	40	2	no	—
246	C₂²	C₃×#4076	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	1	no	—
250	C₄	C₂×C₃×D₄₀	1	1	yes	characteristic
251	C₄	C₃×#4050	1	1	yes	characteristic
252	C₄	C₂×C₃×#688	1	1	yes	characteristic
253	C₄	C₃×#4095	1	1	yes	characteristic
254	C₃	#10601732	1	1	yes	characteristic, direct product factor, Sylow
255	C₂	—	80	2	no	—
256	C₂	C₃×#65316	1	1	yes	characteristic
257	C₂	C₃×#65404	1	1	yes	characteristic
258	C₂	C₃×#65431	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	#4076
commutator, cosocle	C₄²×C₅	C₂³×C₃
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₃×C₅×C₈×Q₈	C₂
socle	C₂²×C₃×C₅	#121
prime radical	C₃×#64984	C₂
hypercenter	C₃×C₈×Q₈	D₅
hypocenter	C₅	C₃×#1668
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₈×Q₈	C₂	semidirect
2. C₃×#65316	C₂	semidirect
4. C₃×#64705	C₂	semidirect
5. C₃×#64678	C₂	semidirect
6. #10601732	C₃	direct (N × H)
19. C₅×C₈×Q₈	C₂×C₃	semidirect
20. #65316	C₂×C₃	semidirect
22. #64705	C₂×C₃	semidirect
23. #64678	C₂×C₃	semidirect
41. C₃×C₈×Q₈	D₅	semidirect
58. C₃×C₅×Q₈	D₈	semidirect
71. C₃×C₅×C₈	SD₁₆	semidirect
101. C₈×Q₈	C₃×D₅	semidirect
132. C₅×Q₈	C₃×D₈	semidirect
145. C₅×C₈	C₃×SD₁₆	semidirect
164. C₃×Q₈	D₄₀	semidirect
178. C₃×C₈	#688	semidirect
205. C₃×C₅	#1668	semidirect
223. Q₈	C₃×D₄₀	semidirect
237. C₈	C₃×#688	semidirect
242. C₅	C₃×#1668	semidirect
254. C₃	#10601732	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×C₈×Q₈	2
C₃×#65316	2
C₃×#64984	2
C₃×#64705	2
C₃×#64678	2
#10601732	3
C₃×#1668	5

Subgroups — C3×#10601732 #14291452

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10601732 #14291452