Subgroups — C₃×#10603869 #14314369

Order 1920.

320 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3990	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×D₅⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃×#64997	C₂	1	1	yes	maximal, characteristic
4	C₃×#64888	C₂	1	2	yes	maximal, complemented
5	C₃×C₄⋊#674	C₂	1	2	yes	maximal, complemented
6	#10603869	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×#676	C₄	1	1	yes	characteristic
8	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
9	C₃×#4018	C₄	1	1	yes	characteristic
10	C₂×C₃×#674	C₂²	1	2	yes	—
11	C₂×C₃×#674	C₂²	1	2	yes	—
12	C₃×#3990	—	2	2	no	—
13	C₃×D₅⋊C₁₆	—	2	2	no	—
14	C₃×C₅×#97	C₄	1	1	yes	characteristic
15	C₃×#3947	C₄	1	1	yes	characteristic
16	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
17	C₃×C₄×#168	C₂²	1	1	yes	characteristic
18	C₃×#2151	—	5	1	no	maximal, Hall
19	C₂×#3990	C₂×C₃	1	1	yes	characteristic
20	C₂×D₅⋊C₁₆	C₂×C₃	1	1	yes	characteristic
21	#64997	C₂×C₃	1	1	yes	characteristic
22	#64888	C₂×C₃	1	2	yes	complemented
23	C₄⋊#674	C₂×C₃	1	2	yes	complemented
24	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₃×#685	C₈	1	2	yes	—
29	C₃×#683	C₈	1	2	yes	—
30	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×#168	C₂³	1	1	yes	characteristic
33	C₃×#676	—	2	2	no	—
34	C₃×C₈×D₅	D₄	1	2	yes	—
35	C₃×C₈×D₅	Q₈	1	2	yes	—
36	C₃×#674	—	2	2	no	—
37	C₃×#674	—	2	2	no	—
38	C₂×C₃×M₃₂	—	5	1	no	—
39	C₂²×C₃×C₁₆	—	5	1	no	—
40	C₃×#405	—	5	1	no	—
41	C₃×C₄⋊C₁₆	—	5	2	no	—
42	C₃×C₄⋊C₁₆	—	5	2	no	—
43	C₂×#676	C₃×C₄	1	1	yes	characteristic
44	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
45	#4018	C₃×C₄	1	1	yes	characteristic
46	C₂×#674	C₂²×C₃	1	2	yes	—
47	C₂×#674	C₂²×C₃	1	2	yes	—
48	#3990	—	2	2	no	—
49	D₅⋊C₁₆	—	2	2	no	—
50	C₅×#97	C₃×C₄	1	1	yes	characteristic
51	#3947	C₃×C₄	1	1	yes	characteristic
52	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
53	C₄×#168	C₂²×C₃	1	1	yes	characteristic
54	#2151	—	5	1	no	Sylow
55	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
57	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃×Dic₅	C₂×C₈	1	2	yes	—
59	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
60	C₃×C₄×D₅	C₄⋊C₄	1	2	yes	—
61	C₃×C₄×D₅	M₁₆	1	2	yes	—
62	C₃×C₅×C₈	—	2	1	no	—
63	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
64	C₃×#168	—	2	1	no	—
65	C₃×#168	C₂×D₄	1	1	yes	characteristic
66	C₃×#168	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₃×M₁₆	—	5	1	no	—
68	C₂²×C₃×C₈	—	5	1	no	—
69	C₃×#117	—	5	1	no	—
70	C₃×M₃₂	—	10	2	no	—
71	C₂×C₃×C₁₆	—	5	2	no	—
72	C₂×C₃×C₁₆	—	5	2	no	—
73	C₂×C₃×C₁₆	—	10	2	no	—
74	C₃×#97	—	5	1	no	—
75	C₃×#97	AGL(1,𝔽₅)	1	1	yes	characteristic
76	C₃×C₄×C₈	—	5	1	no	—
77	C₃×C₄×C₈	—	5	1	no	—
78	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
81	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
82	#685	C₃×C₈	1	2	yes	—
83	#683	C₃×C₈	1	2	yes	—
84	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
85	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
86	C₂×#168	C₂³×C₃	1	1	yes	characteristic
87	#676	—	2	2	no	—
88	C₈×D₅	C₃×D₄	1	2	yes	—
89	C₈×D₅	C₃×Q₈	1	2	yes	—
90	#674	—	2	2	no	—
91	#674	—	2	2	no	—
92	C₂×M₃₂	—	5	1	no	—
93	C₂²×C₁₆	—	5	1	no	—
94	#405	—	5	1	no	—
95	C₄⋊C₁₆	—	5	2	no	—
96	C₄⋊C₁₆	—	5	2	no	—
97	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₃×D₅	—	2	2	no	—
99	C₂×C₃×D₅	C₄⋊C₈	1	2	yes	—
100	C₃×C₄×C₅	—	2	2	no	—
101	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
103	C₃×Dic₅	—	2	2	no	—
104	C₃×Dic₅	C₄⋊C₈	1	2	yes	—
105	C₂²×C₃×C₄	—	5	1	no	—
106	C₃×M₁₆	—	10	2	no	—
107	C₂×C₃×C₈	—	5	1	no	—
108	C₂×C₃×C₈	—	5	2	no	—
109	C₂×C₃×C₈	—	5	2	no	—
110	C₂×C₃×C₈	—	5	1	no	—
111	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
112	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
113	C₃×C₄⋊C₄	—	5	2	no	—
114	C₃×C₂²⋊C₄	—	5	2	no	—
115	C₃×C₄²	—	5	1	no	—
116	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
117	C₃×C₁₆	—	10	2	no	—
118	C₃×C₁₆	—	10	2	no	—
119	C₂²×D₅	C₂×C₃×C₈	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
121	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
122	C₂×Dic₅	C₂×C₃×C₈	1	2	yes	—
123	C₂×Dic₅	C₂×C₃×C₈	1	1	yes	characteristic
124	C₄×D₅	C₃×C₄⋊C₄	1	2	yes	—
125	C₄×D₅	C₃×M₁₆	1	2	yes	—
126	C₅×C₈	—	2	1	no	—
127	C₅×C₈	C₃×C₄⋊C₄	1	2	yes	—
128	#168	—	2	1	no	—
129	#168	C₂×C₃×D₄	1	1	yes	characteristic
130	#168	C₂×C₃×Q₈	1	1	yes	characteristic
131	C₂×M₁₆	—	5	1	no	—
132	C₂²×C₈	—	5	1	no	—
133	#117	—	5	1	no	—
134	M₃₂	—	10	2	no	—
135	C₂×C₁₆	—	5	2	no	—
136	C₂×C₁₆	—	5	2	no	—
137	C₂×C₁₆	—	10	2	no	—
138	#97	—	5	1	no	—
139	#97	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
140	C₄×C₈	—	5	1	no	—
141	C₄×C₈	—	5	1	no	—
142	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
143	C₂×C₃×C₅	#504	1	2	yes	—
144	C₃×D₅	—	2	2	no	—
145	C₂³×C₃	—	5	1	no	—
146	C₂×C₃×C₄	—	5	1	no	—
147	C₂×C₃×C₄	—	5	2	no	—
148	C₂×C₃×C₄	—	5	2	no	—
149	C₂×C₃×C₄	—	5	2	no	—
150	C₂×C₃×C₄	D₅⋊C₈	1	2	yes	—
151	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
152	C₃×C₈	—	2	1	no	—
153	C₃×C₈	—	5	1	no	—
154	C₃×C₈	—	5	1	no	—
155	C₃×C₈	—	10	1	no	—
156	C₃×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
157	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
158	C₂×D₅	—	2	2	no	—
159	C₂×D₅	C₃×C₄⋊C₈	1	2	yes	—
160	C₄×C₅	—	2	2	no	—
161	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
162	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
163	Dic₅	—	2	2	no	—
164	Dic₅	C₃×C₄⋊C₈	1	2	yes	—
165	C₂²×C₄	—	5	1	no	—
166	M₁₆	—	10	2	no	—
167	C₂×C₈	—	5	1	no	—
168	C₂×C₈	—	5	2	no	—
169	C₂×C₈	—	5	2	no	—
170	C₂×C₈	—	5	1	no	—
171	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
172	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
173	C₄⋊C₄	—	5	2	no	—
174	C₂²⋊C₄	—	5	2	no	—
175	C₄²	—	5	1	no	—
176	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
177	C₁₆	—	10	2	no	—
178	C₁₆	—	10	2	no	—
179	C₃×C₅	#2151	1	1	yes	characteristic, complemented, Hall
180	C₂²×C₃	—	5	2	no	—
181	C₂²×C₃	—	10	2	no	—
182	C₂²×C₃	C₂×D₅⋊C₈	1	1	yes	characteristic
183	C₃×C₄	—	2	2	no	—
184	C₃×C₄	—	5	2	no	—
185	C₃×C₄	—	10	2	no	—
186	C₃×C₄	D₅⋊M₁₆	1	1	yes	characteristic
187	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
188	C₂×C₅	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic
189	C₂×C₅	C₃×#504	1	2	yes	—
190	D₅	—	2	2	no	—
191	C₂³	—	5	1	no	—
192	C₂×C₄	—	5	1	no	—
193	C₂×C₄	—	5	2	no	—
194	C₂×C₄	—	5	2	no	—
195	C₂×C₄	—	5	2	no	—
196	C₂×C₄	C₃×D₅⋊C₈	1	2	yes	—
197	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
198	C₈	—	2	1	no	—
199	C₈	—	5	1	no	—
200	C₈	—	5	1	no	—
201	C₈	—	10	1	no	—
202	C₈	C₃×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
203	C₂×C₃	—	10	2	no	—
204	C₂×C₃	D₅⋊(C₄⋊C₈)	1	1	yes	characteristic
205	C₂×C₃	#65725	1	2	yes	—
206	C₅	C₃×#2151	1	1	yes	characteristic, complemented, Sylow
207	C₂²	—	5	2	no	—
208	C₂²	—	10	2	no	—
209	C₂²	C₂×C₃×D₅⋊C₈	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	5	2	no	—
212	C₄	—	10	2	no	—
213	C₄	C₃×D₅⋊M₁₆	1	1	yes	characteristic
214	C₄	C₂×C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
215	C₃	#10603869	1	1	yes	characteristic, direct product factor, Sylow
216	C₂	—	10	2	no	—
217	C₂	C₃×D₅⋊(C₄⋊C₈)	1	1	yes	characteristic
218	C₂	C₃×#65725	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂²×C₃×C₈
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×#97	C₄
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂²×C₃×D₅	C₂×C₈
cosocle	C₂×#168	C₂³×C₃
hypercenter	C₃×#97	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#2151
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×#97

Complements

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

N	H ≅ G/N	type
4. C₃×#64888	C₂	semidirect
5. C₃×C₄⋊#674	C₂	semidirect
6. #10603869	C₃	direct (N × H)
22. #64888	C₂×C₃	semidirect
23. C₄⋊#674	C₂×C₃	semidirect
179. C₃×C₅	#2151	semidirect
206. C₅	C₃×#2151	semidirect
215. C₃	#10603869	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#3990	2
C₂×C₃×D₅⋊C₁₆	2
C₃×#64997	2
C₃×#64888	2
C₃×C₄⋊#674	2
#10603869	3
C₃×#2151	5

Subgroups — C3×#10603869 #14314369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10603869 #14314369