Subgroups — C₃×#10600229 #14296178

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#376	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊#168	C₂	1	2	yes	maximal
3	C₂×C₃×#3935	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic
5	C₃×#64744	C₂	1	2	yes	maximal
6	#10600229	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂²×C₃×#168	C₂²	1	2	yes	—
11	C₂²×C₃×#168	C₂²	1	2	yes	—
12	C₃×C₅×#95	C₄	1	4	yes	—
13	C₃×C₄⋊#168	C₄	1	8	yes	complemented
14	C₃×#3935	—	2	2	no	—
15	C₃×C₄×#168	—	2	2	no	—
16	C₃×#1768	—	5	1	no	maximal, Hall
17	C₅×#376	C₂×C₃	1	1	yes	characteristic
18	C₂×C₄⋊#168	C₂×C₃	1	2	yes	—
19	C₂×#3935	C₂×C₃	1	1	yes	characteristic
20	C₂×C₄×#168	C₂×C₃	1	1	yes	characteristic
21	#64744	C₂×C₃	1	2	yes	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₃×C₄²×C₅	—	2	2	no	—
27	C₃×C₄²×C₅	C₂×C₄	1	4	yes	—
28	C₃×C₄²×C₅	D₄	1	2	yes	—
29	C₃×C₄²×C₅	Q₈	1	2	yes	—
30	C₂×C₃×#168	—	2	2	no	—
31	C₂×C₃×#168	—	2	2	no	—
32	C₂×C₃×#168	—	2	2	no	—
33	C₂×C₃×#168	—	2	2	no	—
34	C₂×C₃×#168	C₂×C₄	1	4	yes	—
35	C₂×C₃×#168	C₂×C₄	1	4	yes	—
36	C₂×C₃×C₄⋊C₈	—	5	2	no	—
37	C₂×C₃×#97	—	5	1	no	—
38	C₂×C₃×C₄×C₈	—	5	1	no	—
39	C₃×#376	D₅	1	1	yes	characteristic
40	C₃×#336	—	5	2	no	—
41	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
44	C₂²×#168	C₂²×C₃	1	2	yes	—
45	C₂²×#168	C₂²×C₃	1	2	yes	—
46	C₅×#95	C₃×C₄	1	4	yes	—
47	C₄⋊#168	C₃×C₄	1	8	yes	complemented
48	#3935	—	2	2	no	—
49	C₄×#168	—	2	2	no	—
50	#1768	—	5	1	no	Sylow
51	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	—	2	4	no	—
53	C₂×C₃×C₄×C₅	—	2	4	no	—
54	C₂×C₃×C₄×C₅	—	2	2	no	—
55	C₂×C₃×C₄×C₅	—	2	4	no	—
56	C₂×C₃×C₄×C₅	C₄²	1	4	yes	—
57	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
58	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
59	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	#41	1	2	yes	—
62	C₃×#168	—	2	4	no	—
63	C₃×#168	—	2	4	no	—
64	C₂²×C₃×C₈	—	5	2	no	—
65	C₂²×C₃×C₈	—	5	2	no	—
66	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
67	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
68	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
69	C₃×C₄⋊C₈	—	5	8	no	—
70	C₃×#97	—	10	2	no	—
71	C₃×C₄×C₈	—	10	2	no	—
72	C₃×#95	Dic₅	1	4	yes	—
73	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
74	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
75	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
76	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
77	C₄²×C₅	—	2	2	no	—
78	C₄²×C₅	C₂×C₃×C₄	1	4	yes	—
79	C₄²×C₅	C₃×D₄	1	2	yes	—
80	C₄²×C₅	C₃×Q₈	1	2	yes	—
81	C₂×#168	—	2	2	no	—
82	C₂×#168	—	2	2	no	—
83	C₂×#168	—	2	2	no	—
84	C₂×#168	—	2	2	no	—
85	C₂×#168	C₂×C₃×C₄	1	4	yes	—
86	C₂×#168	C₂×C₃×C₄	1	4	yes	—
87	C₂×C₄⋊C₈	—	5	2	no	—
88	C₂×#97	—	5	1	no	—
89	C₂×C₄×C₈	—	5	1	no	—
90	#376	C₃×D₅	1	1	yes	characteristic
91	#336	—	5	2	no	—
92	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
93	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₃×C₅	#117	1	1	yes	characteristic
95	C₂²×C₃×C₅	#131	1	4	yes	—
96	C₃×C₄×C₅	—	2	4	no	—
97	C₃×C₄×C₅	—	2	4	no	—
98	C₃×C₄×C₅	—	4	2	no	—
99	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
100	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
101	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
102	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
103	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
104	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
105	C₂×C₃×C₈	—	5	4	no	—
106	C₂×C₃×C₈	—	5	4	no	—
107	C₂×C₃×C₈	—	10	2	no	—
108	C₂×C₃×C₈	—	10	2	no	—
109	C₂×C₃×C₈	—	10	2	no	—
110	C₂×C₃×C₈	—	10	2	no	—
111	C₃×C₄²	—	2	2	no	—
112	C₃×C₄²	Dic₁₀	1	2	yes	—
113	C₃×C₄²	C₄×D₅	1	4	yes	—
114	C₃×C₄²	D₂₀	1	2	yes	—
115	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
116	C₂×C₄×C₅	—	2	4	no	—
117	C₂×C₄×C₅	—	2	4	no	—
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	4	no	—
120	C₂×C₄×C₅	C₃×C₄²	1	4	yes	—
121	C₂×C₄×C₅	C₃×C₄⋊C₄	1	4	yes	—
122	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
123	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₅	C₃×#41	1	2	yes	—
126	#168	—	2	4	no	—
127	#168	—	2	4	no	—
128	C₂²×C₈	—	5	2	no	—
129	C₂²×C₈	—	5	2	no	—
130	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
131	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
132	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
133	C₄⋊C₈	—	5	8	no	—
134	#97	—	10	2	no	—
135	C₄×C₈	—	10	2	no	—
136	#95	C₃×Dic₅	1	4	yes	—
137	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₃×C₅	#405	1	2	yes	—
139	C₂×C₃×C₅	#424	1	2	yes	—
140	C₂×C₃×C₅	#433	1	2	yes	—
141	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
142	C₂×C₃×C₄	—	2	4	no	—
143	C₂×C₃×C₄	—	2	4	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	4	no	—
146	C₂×C₃×C₄	C₄×Dic₅	1	4	yes	—
147	C₂×C₃×C₄	C₄⋊Dic₅	1	4	yes	—
148	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
150	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
151	C₂×C₃×C₄	#709	1	2	yes	—
152	C₃×C₈	—	10	4	no	—
153	C₃×C₈	—	10	4	no	—
154	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
155	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
156	C₂²×C₅	C₃×#117	1	1	yes	characteristic
157	C₂²×C₅	C₃×#131	1	4	yes	—
158	C₄×C₅	—	2	4	no	—
159	C₄×C₅	—	2	4	no	—
160	C₄×C₅	—	4	2	no	—
161	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
162	C₄×C₅	C₃×C₄×Q₈	1	2	yes	—
163	C₂²×C₄	C₂×C₃×Dic₅	1	2	yes	—
164	C₂²×C₄	C₂×C₃×Dic₅	1	2	yes	—
165	C₂²×C₄	C₂×C₃×Dic₅	1	2	yes	—
166	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
167	C₂×C₈	—	5	4	no	—
168	C₂×C₈	—	5	4	no	—
169	C₂×C₈	—	10	2	no	—
170	C₂×C₈	—	10	2	no	—
171	C₂×C₈	—	10	2	no	—
172	C₂×C₈	—	10	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	C₃×Dic₁₀	1	2	yes	—
175	C₄²	C₃×C₄×D₅	1	4	yes	—
176	C₄²	C₃×D₂₀	1	2	yes	—
177	C₃×C₅	#1768	1	1	yes	characteristic, complemented, Hall
178	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
179	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
180	C₂²×C₃	#4072	1	1	yes	characteristic
181	C₂²×C₃	#4094	1	4	yes	—
182	C₃×C₄	—	2	4	no	—
183	C₃×C₄	—	2	4	no	—
184	C₃×C₄	—	4	2	no	—
185	C₃×C₄	C₄×Dic₁₀	1	2	yes	—
186	C₃×C₄	C₄×D₂₀	1	2	yes	—
187	C₂×C₅	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic
188	C₂×C₅	C₃×#405	1	2	yes	—
189	C₂×C₅	C₃×#424	1	2	yes	—
190	C₂×C₅	C₃×#433	1	2	yes	—
191	C₂³	C₂²×C₃×Dic₅	1	1	yes	characteristic
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	C₃×C₄×Dic₅	1	4	yes	—
197	C₂×C₄	C₃×C₄⋊Dic₅	1	4	yes	—
198	C₂×C₄	C₂×C₃×Dic₁₀	1	1	yes	characteristic
199	C₂×C₄	C₂×C₃×C₄×D₅	1	2	yes	—
200	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
201	C₂×C₄	C₃×#709	1	2	yes	—
202	C₈	—	10	4	no	—
203	C₈	—	10	4	no	—
204	C₂×C₃	C₄×C₄⋊Dic₅	1	1	yes	characteristic
205	C₂×C₃	#65286	1	2	yes	—
206	C₂×C₃	#65288	1	2	yes	—
207	C₂×C₃	#65289	1	2	yes	—
208	C₅	C₃×#1768	1	1	yes	characteristic, complemented, Sylow
209	C₂²	C₂×C₃×C₄×Dic₅	1	1	yes	characteristic
210	C₂²	C₂×C₃×C₄⋊Dic₅	1	1	yes	characteristic
211	C₂²	C₃×#4072	1	1	yes	characteristic
212	C₂²	C₃×#4094	1	4	yes	—
213	C₄	—	2	4	no	—
214	C₄	—	2	4	no	—
215	C₄	—	4	2	no	—
216	C₄	C₃×C₄×Dic₁₀	1	2	yes	—
217	C₄	C₃×C₄×D₂₀	1	2	yes	—
218	C₃	#10600229	1	1	yes	characteristic, direct product factor, Sylow
219	C₂	C₃×C₄×C₄⋊Dic₅	1	1	yes	characteristic
220	C₂	C₃×#65286	1	2	yes	—
221	C₂	C₃×#65288	1	2	yes	—
222	C₂	C₃×#65289	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²×D₅
commutator	C₂²×C₅	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#376	C₂
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#376	D₅
hypocenter	C₅	C₃×#1768
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₃×#376

Complements

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

N	H ≅ G/N	type
6. #10600229	C₃	direct (N × H)
13. C₃×C₄⋊#168	C₄	semidirect
47. C₄⋊#168	C₃×C₄	semidirect
177. C₃×C₅	#1768	semidirect
208. C₅	C₃×#1768	semidirect
218. C₃	#10600229	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×#376	2
C₂×C₃×C₄⋊#168	2
C₂×C₃×#3935	2
C₂×C₃×C₄×#168	2
C₃×#64744	2
#10600229	3
C₃×#1768	5

Subgroups — C3×#10600229 #14296178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600229 #14296178