Subgroups — C₃×#10600369 #14299056

Order 1920.

560 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	C₃×C₅×#384	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3940	C₂	1	4	yes	maximal
3	C₃×#65224	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic
5	#10600369	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
7	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄⋊Dic₅	—	2	2	no	—
10	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
11	C₂²×C₃×#168	C₂²	1	2	yes	—
12	C₃×#3940	—	2	8	no	—
13	C₃×C₄×#168	C₄	1	4	yes	complemented
14	C₃×#1844	—	5	1	no	maximal, Hall
15	C₅×#384	C₂×C₃	1	1	yes	characteristic
16	C₂×#3940	C₂×C₃	1	4	yes	—
17	#65224	C₂×C₃	1	1	yes	characteristic
18	C₂×C₄×#168	C₂×C₃	1	1	yes	characteristic
19	C₂²×C₃×C₄×C₅	—	2	2	no	—
20	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
21	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂²×C₃×Dic₅	—	2	2	no	—
23	C₃×C₅×C₄⋊C₄	—	2	4	no	—
24	C₃×C₅×C₄⋊C₄	—	4	4	no	—
25	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
26	C₃×C₄²×C₅	Q₈	1	2	yes	—
27	C₃×C₄⋊Dic₅	—	2	4	no	—
28	C₃×C₄⋊Dic₅	—	4	4	no	—
29	C₂×C₃×#168	C₂×C₄	1	4	yes	—
30	C₂×C₃×#168	D₄	1	4	yes	—
31	C₂×C₃×#168	Q₈	1	4	yes	—
32	C₂×C₃×#106	—	5	4	no	—
33	C₂×C₃×C₄×C₈	—	5	1	no	—
34	C₃×#384	—	5	1	no	—
35	C₃×#384	D₅	1	1	yes	characteristic
36	C₂×C₅×C₄⋊C₄	—	2	2	no	—
37	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
38	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
39	C₂×C₄⋊Dic₅	—	2	2	no	—
40	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
41	C₂²×#168	C₂²×C₃	1	2	yes	—
42	#3940	—	2	8	no	—
43	C₄×#168	C₃×C₄	1	4	yes	complemented
44	#1844	—	5	1	no	Sylow
45	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
46	C₂×C₃×C₄×C₅	—	4	2	no	—
47	C₂×C₃×C₄×C₅	—	4	4	no	—
48	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
49	C₂×C₃×C₄×C₅	SD₁₆	1	4	yes	—
50	C₂×C₃×C₄×C₅	SD₁₆	1	4	yes	—
51	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
53	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₃×Dic₅	—	4	2	no	—
56	C₂×C₃×Dic₅	—	4	4	no	—
57	C₃×#168	C₄⋊C₄	1	8	yes	complemented
58	C₂²×C₃×C₈	—	5	2	no	—
59	C₂×C₃×C₄⋊C₄	—	2	2	no	—
60	C₂×C₃×C₄⋊C₄	—	5	2	no	—
61	C₂×C₃×C₄⋊C₄	—	10	2	no	—
62	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
63	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
64	C₃×#106	—	10	8	no	—
65	C₃×C₄×C₈	—	5	4	no	—
66	C₂²×C₄×C₅	—	2	2	no	—
67	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
68	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
69	C₂²×Dic₅	—	2	2	no	—
70	C₅×C₄⋊C₄	—	2	4	no	—
71	C₅×C₄⋊C₄	—	4	4	no	—
72	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
73	C₄²×C₅	C₃×Q₈	1	2	yes	—
74	C₄⋊Dic₅	—	2	4	no	—
75	C₄⋊Dic₅	—	4	4	no	—
76	C₂×#168	C₂×C₃×C₄	1	4	yes	—
77	C₂×#168	C₃×D₄	1	4	yes	—
78	C₂×#168	C₃×Q₈	1	4	yes	—
79	C₂×#106	—	5	4	no	—
80	C₂×C₄×C₈	—	5	1	no	—
81	#384	—	5	1	no	—
82	#384	C₃×D₅	1	1	yes	characteristic
83	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₃×C₅	#127	1	1	yes	characteristic
85	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
86	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
87	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
88	C₃×C₄×C₅	—	8	2	no	—
89	C₃×C₄×C₅	#106	1	8	yes	complemented
90	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
91	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
92	C₃×Dic₅	—	8	2	no	—
93	C₂²×C₃×C₄	—	2	2	no	—
94	C₂²×C₃×C₄	—	10	2	no	—
95	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
96	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
97	C₂×C₃×C₈	—	5	4	no	—
98	C₂×C₃×C₈	—	5	4	no	—
99	C₂×C₃×C₈	—	5	4	no	—
100	C₃×C₄⋊C₄	—	2	4	no	—
101	C₃×C₄⋊C₄	—	4	4	no	—
102	C₃×C₄⋊C₄	—	10	4	no	—
103	C₃×C₄⋊C₄	—	20	4	no	—
104	C₃×C₄²	Dic₁₀	1	2	yes	—
105	C₃×C₄²	C₄×D₅	1	2	yes	—
106	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅	—	4	2	no	—
108	C₂×C₄×C₅	—	4	4	no	—
109	C₂×C₄×C₅	C₃×C₄⋊C₄	1	4	yes	—
110	C₂×C₄×C₅	C₃×SD₁₆	1	4	yes	—
111	C₂×C₄×C₅	C₃×SD₁₆	1	4	yes	—
112	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
114	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
116	C₂×Dic₅	—	4	2	no	—
117	C₂×Dic₅	—	4	4	no	—
118	#168	C₃×C₄⋊C₄	1	8	yes	complemented
119	C₂²×C₈	—	5	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	5	2	no	—
122	C₂×C₄⋊C₄	—	10	2	no	—
123	C₂×C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
124	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
125	#106	—	10	8	no	—
126	C₄×C₈	—	5	4	no	—
127	C₂×C₃×C₅	#384	1	1	yes	characteristic
128	C₂×C₃×C₅	C₂×#106	1	2	yes	—
129	C₂×C₃×C₅	#492	1	1	yes	characteristic
130	C₂×C₃×C₅	#492	1	1	yes	characteristic
131	C₂×C₃×C₅	#498	1	2	yes	—
132	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
133	C₂×C₃×C₄	—	4	4	no	—
134	C₂×C₃×C₄	—	4	2	no	—
135	C₂×C₃×C₄	—	20	2	no	—
136	C₂×C₃×C₄	—	20	4	no	—
137	C₂×C₃×C₄	#683	1	4	yes	—
138	C₂×C₃×C₄	#687	1	4	yes	—
139	C₂×C₃×C₄	#688	1	4	yes	—
140	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
142	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
143	C₂×C₃×C₄	Q₈×D₅	1	2	yes	—
144	C₃×C₈	—	5	8	no	—
145	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
146	C₂²×C₅	C₃×#127	1	1	yes	characteristic
147	C₂²×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
148	C₂²×C₅	C₂×C₃×SD₁₆	1	2	yes	—
149	C₂²×C₅	C₂×C₃×SD₁₆	1	2	yes	—
150	C₄×C₅	—	8	2	no	—
151	C₄×C₅	C₃×#106	1	8	yes	complemented
152	C₄×C₅	C₂×C₃×C₄⋊C₄	1	2	yes	—
153	C₄×C₅	C₃×C₄⋊Q₈	1	2	yes	—
154	Dic₅	—	8	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	10	2	no	—
157	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
158	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
159	C₂×C₈	—	5	4	no	—
160	C₂×C₈	—	5	4	no	—
161	C₂×C₈	—	5	4	no	—
162	C₄⋊C₄	—	2	4	no	—
163	C₄⋊C₄	—	4	4	no	—
164	C₄⋊C₄	—	10	4	no	—
165	C₄⋊C₄	—	20	4	no	—
166	C₄²	C₃×Dic₁₀	1	2	yes	—
167	C₄²	C₃×C₄×D₅	1	2	yes	—
168	C₃×C₅	#1844	1	1	yes	characteristic, complemented, Hall
169	C₂²×C₃	C₂×#683	1	1	yes	characteristic
170	C₂²×C₃	C₂×#687	1	2	yes	—
171	C₂²×C₃	#4086	1	1	yes	characteristic
172	C₂²×C₃	C₂×#688	1	2	yes	—
173	C₂²×C₃	#4090	1	1	yes	characteristic
174	C₃×C₄	—	8	2	no	—
175	C₃×C₄	—	40	2	no	—
176	C₃×C₄	#3940	1	8	yes	complemented
177	C₃×C₄	#4034	1	2	yes	—
178	C₃×C₄	D₅×C₄⋊C₄	1	2	yes	—
179	C₂×C₅	C₃×#384	1	1	yes	characteristic
180	C₂×C₅	C₂×C₃×#106	1	2	yes	—
181	C₂×C₅	C₃×#492	1	1	yes	characteristic
182	C₂×C₅	C₃×#492	1	1	yes	characteristic
183	C₂×C₅	C₃×#498	1	2	yes	—
184	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	20	2	no	—
188	C₂×C₄	—	20	4	no	—
189	C₂×C₄	C₃×#683	1	4	yes	—
190	C₂×C₄	C₃×#687	1	4	yes	—
191	C₂×C₄	C₃×#688	1	4	yes	—
192	C₂×C₄	C₂×C₃×Dic₁₀	1	1	yes	characteristic
193	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
194	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
195	C₂×C₄	C₃×Q₈×D₅	1	2	yes	—
196	C₈	—	5	8	no	—
197	C₂×C₃	C₂×#3940	1	2	yes	—
198	C₂×C₃	#65265	1	1	yes	characteristic
199	C₂×C₃	#65365	1	1	yes	characteristic
200	C₂×C₃	#65368	1	2	yes	—
201	C₂×C₃	#65374	1	1	yes	characteristic
202	C₅	C₃×#1844	1	1	yes	characteristic, complemented, Sylow
203	C₂²	C₂×C₃×#683	1	1	yes	characteristic
204	C₂²	C₂×C₃×#687	1	2	yes	—
205	C₂²	C₃×#4086	1	1	yes	characteristic
206	C₂²	C₂×C₃×#688	1	2	yes	—
207	C₂²	C₃×#4090	1	1	yes	characteristic
208	C₄	—	8	2	no	—
209	C₄	—	40	2	no	—
210	C₄	C₃×#3940	1	8	yes	complemented
211	C₄	C₃×#4034	1	2	yes	—
212	C₄	C₃×D₅×C₄⋊C₄	1	2	yes	—
213	C₃	#10600369	1	1	yes	characteristic, direct product factor, Sylow
214	C₂	C₂×C₃×#3940	1	2	yes	—
215	C₂	C₃×#65265	1	1	yes	characteristic
216	C₂	C₃×#65365	1	1	yes	characteristic
217	C₂	C₃×#65368	1	2	yes	—
218	C₂	C₃×#65374	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#384	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#384	D₅
hypocenter	C₅	C₃×#1844
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₃×#384

Complements

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

N	H ≅ G/N	type
5. #10600369	C₃	direct (N × H)
13. C₃×C₄×#168	C₄	semidirect
43. C₄×#168	C₃×C₄	semidirect
57. C₃×#168	C₄⋊C₄	semidirect
89. C₃×C₄×C₅	#106	semidirect
118. #168	C₃×C₄⋊C₄	semidirect
151. C₄×C₅	C₃×#106	semidirect
168. C₃×C₅	#1844	semidirect
176. C₃×C₄	#3940	semidirect
202. C₅	C₃×#1844	semidirect
210. C₄	C₃×#3940	semidirect
213. C₃	#10600369	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×#384	2
C₂×C₃×#3940	2
C₃×#65224	2
C₂×C₃×C₄×#168	2
#10600369	3
C₃×#1844	5

Subgroups — C3×#10600369 #14299056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600369 #14299056