Subgroups — C₃×C₅×#2399 #14275433

Order 1920.

1008 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×C₄×C₅×Q₈	C₂	1	2	yes	maximal
2	C₃×C₅×#398	C₂	1	4	yes	maximal
3	C₃×C₅×#395	C₂	1	4	yes	maximal
4	C₃×C₄×C₅×C₄⋊C₄	C₂	1	4	yes	maximal
5	C₃×C₅×#376	C₂	1	1	yes	maximal, characteristic
6	C₅×#2399	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
8	C₃×C₄×C₅×Q₈	—	2	8	no	—
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	8	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	4	yes	—
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
15	C₃×C₅×#95	C₂²	1	4	yes	—
16	C₃×C₅×#95	C₂²	1	4	yes	—
17	C₃×C₅×#95	C₂²	1	4	yes	—
18	C₃×#2399	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₄×C₅×Q₈	C₂×C₃	1	2	yes	—
20	C₅×#398	C₂×C₃	1	4	yes	—
21	C₅×#395	C₂×C₃	1	4	yes	—
22	C₄×C₅×C₄⋊C₄	C₂×C₃	1	4	yes	—
23	C₅×#376	C₂×C₃	1	1	yes	characteristic
24	C₂×C₃×C₅×Q₈	—	2	4	no	—
25	C₂×C₃×C₅×Q₈	—	2	8	no	—
26	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
29	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₃×C₅×C₄⋊C₄	—	2	8	no	—
32	C₃×C₅×C₄⋊C₄	—	2	4	no	—
33	C₃×C₅×C₄⋊C₄	—	2	16	no	—
34	C₃×C₄²×C₅	—	2	8	no	—
35	C₃×C₄²×C₅	D₄	1	4	yes	—
36	C₃×C₄²×C₅	Q₈	1	8	yes	complemented
37	C₂×C₃×C₄×Q₈	C₂×C₅	1	2	yes	—
38	C₃×#398	C₂×C₅	1	4	yes	—
39	C₃×#395	C₂×C₅	1	4	yes	—
40	C₃×C₄×C₄⋊C₄	C₂×C₅	1	4	yes	—
41	C₃×#376	C₂×C₅	1	1	yes	characteristic
42	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
43	C₄×C₅×Q₈	—	2	8	no	—
44	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	8	yes	—
45	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
46	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
47	C₂×C₄²×C₅	C₂²×C₃	1	4	yes	—
48	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
50	C₅×#95	C₂²×C₃	1	4	yes	—
51	C₅×#95	C₂²×C₃	1	4	yes	—
52	C₅×#95	C₂²×C₃	1	4	yes	—
53	#2399	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
54	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
55	C₃×C₅×Q₈	—	2	16	no	—
56	C₂×C₃×C₄×C₅	—	2	4	no	—
57	C₂×C₃×C₄×C₅	—	2	8	no	—
58	C₂×C₃×C₄×C₅	—	2	4	no	—
59	C₂×C₃×C₄×C₅	—	2	4	no	—
60	C₂×C₃×C₄×C₅	—	2	4	no	—
61	C₂×C₃×C₄×C₅	—	2	8	no	—
62	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
63	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂×Q₈	1	8	yes	—
65	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
66	C₂×C₃×C₄×C₅	#41	1	8	yes	—
67	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
68	C₃×C₄×Q₈	—	2	8	no	—
69	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	8	yes	—
70	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
71	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
72	C₂×C₃×C₄²	C₂²×C₅	1	4	yes	—
73	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
74	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
75	C₃×#95	C₂²×C₅	1	4	yes	—
76	C₃×#95	C₂²×C₅	1	4	yes	—
77	C₃×#95	C₂²×C₅	1	4	yes	—
78	C₂×C₅×Q₈	—	2	4	no	—
79	C₂×C₅×Q₈	—	2	8	no	—
80	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
81	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
82	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
83	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
84	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
85	C₅×C₄⋊C₄	—	2	8	no	—
86	C₅×C₄⋊C₄	—	2	4	no	—
87	C₅×C₄⋊C₄	—	2	16	no	—
88	C₄²×C₅	—	2	8	no	—
89	C₄²×C₅	C₃×D₄	1	4	yes	—
90	C₄²×C₅	C₃×Q₈	1	8	yes	complemented
91	C₂×C₄×Q₈	C₂×C₃×C₅	1	2	yes	—
92	#398	C₂×C₃×C₅	1	4	yes	—
93	#395	C₂×C₃×C₅	1	4	yes	—
94	C₄×C₄⋊C₄	C₂×C₃×C₅	1	4	yes	—
95	#376	C₂×C₃×C₅	1	1	yes	characteristic
96	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
97	C₂²×C₃×C₅	C₂²×Q₈	1	2	yes	—
98	C₂²×C₃×C₅	C₂×#41	1	4	yes	—
99	C₃×C₄×C₅	—	2	4	no	—
100	C₃×C₄×C₅	—	2	8	no	—
101	C₃×C₄×C₅	—	2	8	no	—
102	C₃×C₄×C₅	—	4	4	no	—
103	C₃×C₄×C₅	C₄⋊Q₈	1	4	yes	—
104	C₂×C₃×Q₈	—	2	4	no	—
105	C₂×C₃×Q₈	—	2	8	no	—
106	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
107	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
108	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
109	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
110	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
111	C₃×C₄⋊C₄	—	2	8	no	—
112	C₃×C₄⋊C₄	—	2	4	no	—
113	C₃×C₄⋊C₄	—	2	16	no	—
114	C₃×C₄²	—	2	8	no	—
115	C₃×C₄²	C₅×D₄	1	4	yes	—
116	C₃×C₄²	C₅×Q₈	1	8	yes	complemented
117	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
118	C₅×Q₈	—	2	16	no	—
119	C₂×C₄×C₅	—	2	4	no	—
120	C₂×C₄×C₅	—	2	8	no	—
121	C₂×C₄×C₅	—	2	4	no	—
122	C₂×C₄×C₅	—	2	4	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	2	8	no	—
125	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
126	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
127	C₂×C₄×C₅	C₂×C₃×Q₈	1	8	yes	—
128	C₂×C₄×C₅	C₂×C₃×Q₈	1	4	yes	—
129	C₂×C₄×C₅	C₃×#41	1	8	yes	—
130	C₂²×Q₈	C₂²×C₃×C₅	1	2	yes	—
131	C₄×Q₈	—	2	8	no	—
132	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
133	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
134	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	8	yes	—
135	C₂×C₄²	C₂²×C₃×C₅	1	4	yes	—
136	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
137	C₂×C₄²	C₂²×C₃×C₅	1	2	yes	—
138	#95	C₂²×C₃×C₅	1	4	yes	—
139	#95	C₂²×C₃×C₅	1	4	yes	—
140	#95	C₂²×C₃×C₅	1	4	yes	—
141	C₂×C₃×C₅	#525	1	2	yes	—
142	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
143	C₂×C₃×C₅	#533	1	4	yes	—
144	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
145	C₃×Q₈	—	2	16	no	—
146	C₂×C₃×C₄	—	2	4	no	—
147	C₂×C₃×C₄	—	2	8	no	—
148	C₂×C₃×C₄	—	2	4	no	—
149	C₂×C₃×C₄	—	2	4	no	—
150	C₂×C₃×C₄	—	2	4	no	—
151	C₂×C₃×C₄	—	2	8	no	—
152	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
153	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
154	C₂×C₃×C₄	C₂×C₅×Q₈	1	8	yes	—
155	C₂×C₃×C₄	C₂×C₅×Q₈	1	4	yes	—
156	C₂×C₃×C₄	C₅×#41	1	8	yes	—
157	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
158	C₂²×C₅	C₂²×C₃×Q₈	1	2	yes	—
159	C₂²×C₅	C₂×C₃×#41	1	4	yes	—
160	C₄×C₅	—	2	4	no	—
161	C₄×C₅	—	2	8	no	—
162	C₄×C₅	—	2	8	no	—
163	C₄×C₅	—	4	4	no	—
164	C₄×C₅	C₃×C₄⋊Q₈	1	4	yes	—
165	C₂×Q₈	—	2	4	no	—
166	C₂×Q₈	—	2	8	no	—
167	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
168	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
169	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
170	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
171	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
172	C₄⋊C₄	—	2	8	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	—	2	16	no	—
175	C₄²	—	2	8	no	—
176	C₄²	C₃×C₅×D₄	1	4	yes	—
177	C₄²	C₃×C₅×Q₈	1	8	yes	complemented
178	C₃×C₅	#2399	1	1	yes	characteristic, direct product factor, Hall
179	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
180	C₂²×C₃	C₂²×C₅×Q₈	1	2	yes	—
181	C₂²×C₃	C₂×C₅×#41	1	4	yes	—
182	C₃×C₄	—	2	4	no	—
183	C₃×C₄	—	2	8	no	—
184	C₃×C₄	—	2	8	no	—
185	C₃×C₄	—	4	4	no	—
186	C₃×C₄	C₅×C₄⋊Q₈	1	4	yes	—
187	C₂×C₅	C₃×#525	1	2	yes	—
188	C₂×C₅	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
189	C₂×C₅	C₃×#533	1	4	yes	—
190	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
191	Q₈	—	2	16	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	8	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
199	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
200	C₂×C₄	C₂×C₃×C₅×Q₈	1	8	yes	—
201	C₂×C₄	C₂×C₃×C₅×Q₈	1	4	yes	—
202	C₂×C₄	C₃×C₅×#41	1	8	yes	—
203	C₂×C₃	C₅×#525	1	2	yes	—
204	C₂×C₃	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
205	C₂×C₃	C₅×#533	1	4	yes	—
206	C₅	C₃×#2399	1	1	yes	characteristic, direct product factor, Sylow
207	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
208	C₂²	C₂²×C₃×C₅×Q₈	1	2	yes	—
209	C₂²	C₂×C₃×C₅×#41	1	4	yes	—
210	C₄	—	2	4	no	—
211	C₄	—	2	8	no	—
212	C₄	—	2	8	no	—
213	C₄	—	4	4	no	—
214	C₄	C₃×C₅×C₄⋊Q₈	1	4	yes	—
215	C₃	C₅×#2399	1	1	yes	characteristic, direct product factor, Sylow
216	C₂	C₃×C₅×#525	1	2	yes	—
217	C₂	C₂×C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
218	C₂	C₃×C₅×#533	1	4	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄×C₅	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
socle, prime radical	C₂³×C₃×C₅	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×#2399	C₃	direct (N × H)
18. C₃×#2399	C₅	direct (N × H)
36. C₃×C₄²×C₅	Q₈	semidirect
53. #2399	C₃×C₅	direct (N × H)
90. C₄²×C₅	C₃×Q₈	semidirect
116. C₃×C₄²	C₅×Q₈	semidirect
177. C₄²	C₃×C₅×Q₈	semidirect
178. C₃×C₅	#2399	direct (N × H)
206. C₅	C₃×#2399	direct (N × H)
215. C₃	C₅×#2399	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×C₄×C₅×Q₈	2
C₃×C₅×#398	2
C₃×C₅×#395	2
C₃×C₄×C₅×C₄⋊C₄	2
C₃×C₅×#376	2
C₅×#2399	3
C₃×#2399	5

Subgroups — C3×C5×#2399 #14275433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2399 #14275433