Subgroups — C₃×#10598673 #14278800

Order 1920.

272 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65718	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×D₅⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃×D₅×M₃₂	C₂	1	1	yes	maximal, characteristic
4	#10598673	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃×#4130	C₄	1	1	yes	characteristic
6	C₃×D₅⋊M₁₆	C₄	1	1	yes	characteristic
7	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
8	C₂×C₃×#674	C₄	1	2	yes	complemented
9	C₃×#3992	—	2	1	no	—
10	C₃×C₈×AGL(1,𝔽₅)	—	2	1	no	—
11	C₃×D₅⋊C₁₆	—	2	1	no	—
12	C₃×D₅⋊C₁₆	—	2	1	no	—
13	C₃×C₅×M₃₂	C₄	1	1	yes	characteristic, complemented
14	C₃×#3944	C₄	1	1	yes	characteristic, complemented
15	C₃×#3930	—	2	1	no	—
16	C₃×D₅×C₁₆	—	2	1	no	—
17	C₃×#1379	—	5	1	no	maximal, Hall
18	#65718	C₂×C₃	1	1	yes	characteristic
19	C₂×D₅⋊C₁₆	C₂×C₃	1	1	yes	characteristic
20	D₅×M₃₂	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
22	C₃×C₂²⋊AGL(1,𝔽₅)	—	2	1	no	—
23	C₃×#704	—	2	1	no	—
24	C₃×C₄⋊AGL(1,𝔽₅)	C₈	1	2	yes	—
25	C₃×C₄×AGL(1,𝔽₅)	—	2	1	no	—
26	C₃×#700	C₈	1	2	yes	—
27	C₃×D₅⋊C₈	—	2	1	no	—
28	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
30	C₃×C₈×D₅	D₄	1	1	yes	characteristic
31	C₃×C₈×D₅	D₄	1	1	yes	characteristic
32	C₃×C₈×D₅	D₄	1	1	yes	characteristic
33	C₃×C₈×D₅	Q₈	1	1	yes	characteristic
34	C₃×#674	—	2	2	no	—
35	C₃×C₅×C₁₆	—	2	1	no	—
36	C₃×#672	—	2	1	no	—
37	C₂×C₃×M₃₂	—	5	1	no	—
38	C₂²×C₃×C₁₆	—	5	1	no	—
39	C₃×#405	—	5	1	no	—
40	#4130	C₃×C₄	1	1	yes	characteristic
41	D₅⋊M₁₆	C₃×C₄	1	1	yes	characteristic
42	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
43	C₂×#674	C₃×C₄	1	2	yes	complemented
44	#3992	—	2	1	no	—
45	C₈×AGL(1,𝔽₅)	—	2	1	no	—
46	D₅⋊C₁₆	—	2	1	no	—
47	D₅⋊C₁₆	—	2	1	no	—
48	C₅×M₃₂	C₃×C₄	1	1	yes	characteristic, complemented
49	#3944	C₃×C₄	1	1	yes	characteristic, complemented
50	#3930	—	2	1	no	—
51	D₅×C₁₆	—	2	1	no	—
52	#1379	—	5	1	no	Sylow
53	C₂²×C₃×D₅	M₁₆	1	1	yes	characteristic
54	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
55	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
56	C₂×C₃×Dic₅	M₁₆	1	1	yes	characteristic
57	C₃×C₄×D₅	C₂²⋊C₄	1	1	yes	characteristic
58	C₃×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
59	C₃×C₄×D₅	C₂×C₈	1	1	yes	characteristic
60	C₃×C₄×D₅	C₂×C₈	1	1	yes	characteristic
61	C₃×#170	—	2	2	no	—
62	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
63	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
64	C₃×#168	C₂²⋊C₄	1	1	yes	characteristic
65	C₃×#168	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₃×M₁₆	—	5	1	no	—
67	C₂²×C₃×C₈	—	5	1	no	—
68	C₃×#117	—	5	1	no	—
69	C₃×M₃₂	—	5	1	no	—
70	C₃×M₃₂	—	10	1	no	—
71	C₃×M₃₂	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
72	C₂×C₃×C₁₆	—	5	2	no	—
73	C₂×C₃×C₁₆	—	10	1	no	—
74	C₂×C₃×C₁₆	—	10	1	no	—
75	C₂×C₃×C₁₆	—	10	1	no	—
76	C₃×#97	—	10	1	no	—
77	C₃×C₄×C₈	—	10	1	no	—
78	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
79	C₂²⋊AGL(1,𝔽₅)	—	2	1	no	—
80	#704	—	2	1	no	—
81	C₄⋊AGL(1,𝔽₅)	C₃×C₈	1	2	yes	—
82	C₄×AGL(1,𝔽₅)	—	2	1	no	—
83	#700	C₃×C₈	1	2	yes	—
84	D₅⋊C₈	—	2	1	no	—
85	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
86	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
87	C₈×D₅	C₃×D₄	1	1	yes	characteristic
88	C₈×D₅	C₃×D₄	1	1	yes	characteristic
89	C₈×D₅	C₃×D₄	1	1	yes	characteristic
90	C₈×D₅	C₃×Q₈	1	1	yes	characteristic
91	#674	—	2	2	no	—
92	C₅×C₁₆	—	2	1	no	—
93	#672	—	2	1	no	—
94	C₂×M₃₂	—	5	1	no	—
95	C₂²×C₁₆	—	5	1	no	—
96	#405	—	5	1	no	—
97	C₂²×C₃×C₅	#97	1	1	yes	characteristic
98	C₂×C₃×D₅	—	2	2	no	—
99	C₂×C₃×D₅	C₂²⋊C₈	1	1	yes	characteristic
100	C₂×C₃×D₅	C₄⋊C₈	1	1	yes	characteristic
101	C₃×AGL(1,𝔽₅)	—	4	2	no	—
102	C₃×C₄×C₅	#95	1	1	yes	characteristic
103	C₃×C₄×C₅	C₄×C₈	1	1	yes	characteristic
104	C₃×Dic₅	C₂²⋊C₈	1	1	yes	characteristic
105	C₃×Dic₅	C₄⋊C₈	1	1	yes	characteristic
106	C₂²×C₃×C₄	—	5	1	no	—
107	C₃×M₁₆	—	5	2	no	—
108	C₃×M₁₆	—	10	1	no	—
109	C₂×C₃×C₈	—	5	1	no	—
110	C₂×C₃×C₈	—	5	1	no	—
111	C₂×C₃×C₈	—	5	1	no	—
112	C₂×C₃×C₈	—	5	1	no	—
113	C₂×C₃×C₈	—	5	1	no	—
114	C₂×C₃×C₈	—	10	1	no	—
115	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
116	C₃×C₄⋊C₄	—	5	2	no	—
117	C₃×C₂²⋊C₄	—	10	1	no	—
118	C₃×C₄²	—	10	1	no	—
119	C₃×C₁₆	—	2	1	no	—
120	C₃×C₁₆	—	10	2	no	—
121	C₃×C₁₆	—	10	1	no	—
122	C₂²×D₅	C₃×M₁₆	1	1	yes	characteristic
123	C₂×AGL(1,𝔽₅)	—	2	2	no	—
124	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
125	C₂×Dic₅	C₃×M₁₆	1	1	yes	characteristic
126	C₄×D₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
127	C₄×D₅	C₃×C₄⋊C₄	1	1	yes	characteristic
128	C₄×D₅	C₂×C₃×C₈	1	1	yes	characteristic
129	C₄×D₅	C₂×C₃×C₈	1	1	yes	characteristic
130	#170	—	2	2	no	—
131	C₅×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
132	C₅×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
133	#168	C₃×C₂²⋊C₄	1	1	yes	characteristic
134	#168	C₃×C₄⋊C₄	1	1	yes	characteristic
135	C₂×M₁₆	—	5	1	no	—
136	C₂²×C₈	—	5	1	no	—
137	#117	—	5	1	no	—
138	M₃₂	—	5	1	no	—
139	M₃₂	—	10	1	no	—
140	M₃₂	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
141	C₂×C₁₆	—	5	2	no	—
142	C₂×C₁₆	—	10	1	no	—
143	C₂×C₁₆	—	10	1	no	—
144	C₂×C₁₆	—	10	1	no	—
145	#97	—	10	1	no	—
146	C₄×C₈	—	10	1	no	—
147	C₂×C₃×C₅	—	2	1	no	—
148	C₂×C₃×C₅	#336	1	1	yes	characteristic
149	C₃×D₅	—	2	1	no	—
150	C₃×D₅	#350	1	2	yes	—
151	C₂³×C₃	—	5	1	no	—
152	C₂×C₃×C₄	—	5	1	no	—
153	C₂×C₃×C₄	—	5	1	no	—
154	C₂×C₃×C₄	—	5	1	no	—
155	C₂×C₃×C₄	—	5	1	no	—
156	C₂×C₃×C₄	—	5	1	no	—
157	C₂×C₃×C₄	—	10	2	no	—
158	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
159	C₃×C₈	—	5	1	no	—
160	C₃×C₈	—	5	1	no	—
161	C₃×C₈	—	10	2	no	—
162	C₃×C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
163	C₃×C₈	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
164	C₂²×C₅	C₃×#97	1	1	yes	characteristic
165	C₂×D₅	—	2	2	no	—
166	C₂×D₅	C₃×C₂²⋊C₈	1	1	yes	characteristic
167	C₂×D₅	C₃×C₄⋊C₈	1	1	yes	characteristic
168	AGL(1,𝔽₅)	—	4	2	no	—
169	C₄×C₅	C₃×#95	1	1	yes	characteristic
170	C₄×C₅	C₃×C₄×C₈	1	1	yes	characteristic
171	Dic₅	C₃×C₂²⋊C₈	1	1	yes	characteristic
172	Dic₅	C₃×C₄⋊C₈	1	1	yes	characteristic
173	C₂²×C₄	—	5	1	no	—
174	M₁₆	—	5	2	no	—
175	M₁₆	—	10	1	no	—
176	C₂×C₈	—	5	1	no	—
177	C₂×C₈	—	5	1	no	—
178	C₂×C₈	—	5	1	no	—
179	C₂×C₈	—	5	1	no	—
180	C₂×C₈	—	5	1	no	—
181	C₂×C₈	—	10	1	no	—
182	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
183	C₄⋊C₄	—	5	2	no	—
184	C₂²⋊C₄	—	10	1	no	—
185	C₄²	—	10	1	no	—
186	C₁₆	—	2	1	no	—
187	C₁₆	—	10	2	no	—
188	C₁₆	—	10	1	no	—
189	C₃×C₅	#1379	1	1	yes	characteristic, complemented, Hall
190	C₂²×C₃	—	5	1	no	—
191	C₂²×C₃	—	5	1	no	—
192	C₂²×C₃	—	10	2	no	—
193	C₂²×C₃	#3992	1	1	yes	characteristic
194	C₃×C₄	—	5	1	no	—
195	C₃×C₄	—	5	1	no	—
196	C₃×C₄	—	20	2	no	—
197	C₃×C₄	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
198	C₃×C₄	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
199	C₂×C₅	—	2	1	no	—
200	C₂×C₅	C₃×#336	1	1	yes	characteristic
201	D₅	—	2	1	no	—
202	D₅	C₃×#350	1	2	yes	—
203	C₂³	—	5	1	no	—
204	C₂×C₄	—	5	1	no	—
205	C₂×C₄	—	5	1	no	—
206	C₂×C₄	—	5	1	no	—
207	C₂×C₄	—	5	1	no	—
208	C₂×C₄	—	5	1	no	—
209	C₂×C₄	—	10	2	no	—
210	C₂×C₄	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
211	C₈	—	5	1	no	—
212	C₈	—	5	1	no	—
213	C₈	—	10	2	no	—
214	C₈	C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
215	C₈	C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
216	C₂×C₃	—	2	1	no	—
217	C₂×C₃	—	5	2	no	—
218	C₂×C₃	—	10	1	no	—
219	C₂×C₃	(C₂×AGL(1,𝔽₅))⋊C₈	1	1	yes	characteristic
220	C₅	C₃×#1379	1	1	yes	characteristic, complemented, Sylow
221	C₂²	—	5	1	no	—
222	C₂²	—	5	1	no	—
223	C₂²	—	10	2	no	—
224	C₂²	C₃×#3992	1	1	yes	characteristic
225	C₄	—	5	1	no	—
226	C₄	—	5	1	no	—
227	C₄	—	20	2	no	—
228	C₄	C₃×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
229	C₄	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
230	C₃	#10598673	1	1	yes	characteristic, direct product factor, Sylow
231	C₂	—	2	1	no	—
232	C₂	—	5	2	no	—
233	C₂	—	10	1	no	—
234	C₂	C₃×(C₂×AGL(1,𝔽₅))⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	C₂²⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₃×C₄×C₈
Frattini	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×M₃₂	C₄
socle	C₂×C₃×C₅	#336
prime radical	C₂²×C₃×D₅	M₁₆
cosocle	C₂×C₈×D₅	C₂²×C₃
hypercenter	C₃×M₃₂	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#1379
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₈ ◁ C₂×C₃×C₈ ◁ C₃×M₃₂

Complements

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

N	H ≅ G/N	type
4. #10598673	C₃	direct (N × H)
8. C₂×C₃×#674	C₄	semidirect
13. C₃×C₅×M₃₂	C₄	semidirect
14. C₃×#3944	C₄	semidirect
43. C₂×#674	C₃×C₄	semidirect
48. C₅×M₃₂	C₃×C₄	semidirect
49. #3944	C₃×C₄	semidirect
71. C₃×M₃₂	AGL(1,𝔽₅)	semidirect
140. M₃₂	C₃×AGL(1,𝔽₅)	semidirect
189. C₃×C₅	#1379	semidirect
220. C₅	C₃×#1379	semidirect
230. C₃	#10598673	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×#65718	2
C₂×C₃×D₅⋊C₁₆	2
C₃×D₅×M₃₂	2
#10598673	3
C₃×#1379	5

Subgroups — C3×#10598673 #14278800

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10598673 #14278800