Subgroups — C₃×C₅×#2642 #14275676

Order 1920.

832 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#125	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#400	C₂	1	8	yes	maximal
4	C₃×C₅×#391	C₂	1	4	yes	maximal
5	C₃×C₅×#376	C₂	1	1	yes	maximal, characteristic
6	C₅×#2642	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
8	C₃×C₅×C₄⋊Q₈	—	2	4	no	—
9	C₃×C₅×#125	—	2	4	no	—
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₃×C₅×#95	C₂²	1	4	yes	—
16	C₃×C₅×#95	C₂²	1	16	yes	—
17	C₃×#2642	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
19	C₂×C₅×#125	C₂×C₃	1	1	yes	characteristic
20	C₅×#400	C₂×C₃	1	8	yes	—
21	C₅×#391	C₂×C₃	1	4	yes	—
22	C₅×#376	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×C₅×Q₈	—	2	4	no	—
24	C₂×C₃×C₅×Q₈	—	2	8	no	—
25	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
26	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
29	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₃×C₅×C₄⋊C₄	—	2	8	no	—
31	C₃×C₅×C₄⋊C₄	—	2	4	no	—
32	C₃×C₅×C₄⋊C₄	—	2	8	no	—
33	C₃×C₄²×C₅	—	2	4	no	—
34	C₃×C₄²×C₅	D₄	1	4	yes	—
35	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
36	C₂×C₃×#125	C₂×C₅	1	1	yes	characteristic
37	C₃×#400	C₂×C₅	1	8	yes	—
38	C₃×#391	C₂×C₅	1	4	yes	—
39	C₃×#376	C₂×C₅	1	1	yes	characteristic
40	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
41	C₅×C₄⋊Q₈	—	2	4	no	—
42	C₅×#125	—	2	4	no	—
43	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
44	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
45	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
46	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
47	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
48	C₅×#95	C₂²×C₃	1	4	yes	—
49	C₅×#95	C₂²×C₃	1	16	yes	—
50	#2642	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
51	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
52	C₃×C₅×Q₈	—	4	8	no	—
53	C₂×C₃×C₄×C₅	—	2	4	no	—
54	C₂×C₃×C₄×C₅	—	2	4	no	—
55	C₂×C₃×C₄×C₅	—	2	4	no	—
56	C₂×C₃×C₄×C₅	—	2	4	no	—
57	C₂×C₃×C₄×C₅	—	2	8	no	—
58	C₂×C₃×C₄×C₅	—	2	8	no	—
59	C₂×C₃×C₄×C₅	—	2	8	no	—
60	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
61	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
62	C₂×C₃×C₄×C₅	#41	1	4	yes	—
63	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
64	C₃×C₄⋊Q₈	—	2	4	no	—
65	C₃×#125	—	2	4	no	—
66	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
67	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
68	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
69	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
70	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
71	C₃×#95	C₂²×C₅	1	4	yes	—
72	C₃×#95	C₂²×C₅	1	16	yes	—
73	C₂×C₅×Q₈	—	2	4	no	—
74	C₂×C₅×Q₈	—	2	8	no	—
75	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
76	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
77	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
78	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
79	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
80	C₅×C₄⋊C₄	—	2	8	no	—
81	C₅×C₄⋊C₄	—	2	4	no	—
82	C₅×C₄⋊C₄	—	2	8	no	—
83	C₄²×C₅	—	2	4	no	—
84	C₄²×C₅	C₃×D₄	1	4	yes	—
85	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic
86	C₂×#125	C₂×C₃×C₅	1	1	yes	characteristic
87	#400	C₂×C₃×C₅	1	8	yes	—
88	#391	C₂×C₃×C₅	1	4	yes	—
89	#376	C₂×C₃×C₅	1	1	yes	characteristic
90	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
91	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
92	C₂²×C₃×C₅	2₋¹⁺⁴	1	4	yes	—
93	C₃×C₄×C₅	—	2	4	no	—
94	C₃×C₄×C₅	—	4	4	no	—
95	C₃×C₄×C₅	—	4	4	no	—
96	C₃×C₄×C₅	—	4	4	no	—
97	C₃×C₄×C₅	#124	1	4	yes	—
98	C₂×C₃×Q₈	—	2	4	no	—
99	C₂×C₃×Q₈	—	2	8	no	—
100	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
101	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
102	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
103	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
104	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
105	C₃×C₄⋊C₄	—	2	8	no	—
106	C₃×C₄⋊C₄	—	2	4	no	—
107	C₃×C₄⋊C₄	—	2	8	no	—
108	C₃×C₄²	—	2	4	no	—
109	C₃×C₄²	C₅×D₄	1	4	yes	—
110	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
111	C₅×Q₈	—	4	8	no	—
112	C₂×C₄×C₅	—	2	4	no	—
113	C₂×C₄×C₅	—	2	4	no	—
114	C₂×C₄×C₅	—	2	4	no	—
115	C₂×C₄×C₅	—	2	4	no	—
116	C₂×C₄×C₅	—	2	8	no	—
117	C₂×C₄×C₅	—	2	8	no	—
118	C₂×C₄×C₅	—	2	8	no	—
119	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
120	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
121	C₂×C₄×C₅	C₃×#41	1	4	yes	—
122	C₂²×Q₈	C₂²×C₃×C₅	1	2	yes	—
123	C₄⋊Q₈	—	2	4	no	—
124	#125	—	2	4	no	—
125	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
126	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
127	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
128	C₂×C₄²	C₂²×C₃×C₅	1	2	yes	—
129	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
130	#95	C₂²×C₃×C₅	1	4	yes	—
131	#95	C₂²×C₃×C₅	1	16	yes	—
132	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
133	C₂×C₃×C₅	#536	1	2	yes	—
134	C₂×C₃×C₅	#541	1	4	yes	—
135	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
136	C₃×Q₈	—	4	8	no	—
137	C₂×C₃×C₄	—	2	4	no	—
138	C₂×C₃×C₄	—	2	4	no	—
139	C₂×C₃×C₄	—	2	4	no	—
140	C₂×C₃×C₄	—	2	4	no	—
141	C₂×C₃×C₄	—	2	8	no	—
142	C₂×C₃×C₄	—	2	8	no	—
143	C₂×C₃×C₄	—	2	8	no	—
144	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
145	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
146	C₂×C₃×C₄	C₅×#41	1	4	yes	—
147	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
148	C₂²×C₅	C₂×C₃×#41	1	2	yes	—
149	C₂²×C₅	C₃×2₋¹⁺⁴	1	4	yes	—
150	C₄×C₅	—	2	4	no	—
151	C₄×C₅	—	4	4	no	—
152	C₄×C₅	—	4	4	no	—
153	C₄×C₅	—	4	4	no	—
154	C₄×C₅	C₃×#124	1	4	yes	—
155	C₂×Q₈	—	2	4	no	—
156	C₂×Q₈	—	2	8	no	—
157	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
158	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
159	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
160	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
161	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
162	C₄⋊C₄	—	2	8	no	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	8	no	—
165	C₄²	—	2	4	no	—
166	C₄²	C₃×C₅×D₄	1	4	yes	—
167	C₃×C₅	#2642	1	1	yes	characteristic, direct product factor, Hall
168	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
169	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
170	C₂²×C₃	C₅×2₋¹⁺⁴	1	4	yes	—
171	C₃×C₄	—	2	4	no	—
172	C₃×C₄	—	4	4	no	—
173	C₃×C₄	—	4	4	no	—
174	C₃×C₄	—	4	4	no	—
175	C₃×C₄	C₅×#124	1	4	yes	—
176	C₂×C₅	C₂×C₃×#124	1	1	yes	characteristic
177	C₂×C₅	C₃×#536	1	2	yes	—
178	C₂×C₅	C₃×#541	1	4	yes	—
179	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
180	Q₈	—	4	8	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	—	2	8	no	—
187	C₂×C₄	—	2	8	no	—
188	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
189	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
190	C₂×C₄	C₃×C₅×#41	1	4	yes	—
191	C₂×C₃	C₂×C₅×#124	1	1	yes	characteristic
192	C₂×C₃	C₅×#536	1	2	yes	—
193	C₂×C₃	C₅×#541	1	4	yes	—
194	C₅	C₃×#2642	1	1	yes	characteristic, direct product factor, Sylow
195	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
196	C₂²	C₂×C₃×C₅×#41	1	2	yes	—
197	C₂²	C₃×C₅×2₋¹⁺⁴	1	4	yes	—
198	C₄	—	2	4	no	—
199	C₄	—	4	4	no	—
200	C₄	—	4	4	no	—
201	C₄	—	4	4	no	—
202	C₄	C₃×C₅×#124	1	4	yes	—
203	C₃	C₅×#2642	1	1	yes	characteristic, direct product factor, Sylow
204	C₂	C₂×C₃×C₅×#124	1	1	yes	characteristic
205	C₂	C₃×C₅×#536	1	2	yes	—
206	C₂	C₃×C₅×#541	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
6. C₅×#2642	C₃	direct (N × H)
17. C₃×#2642	C₅	direct (N × H)
50. #2642	C₃×C₅	direct (N × H)
167. C₃×C₅	#2642	direct (N × H)
194. C₅	C₃×#2642	direct (N × H)
203. C₃	C₅×#2642	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×C₅×C₄⋊Q₈	2
C₂×C₃×C₅×#125	2
C₃×C₅×#400	2
C₃×C₅×#391	2
C₃×C₅×#376	2
C₅×#2642	3
C₃×#2642	5

Subgroups — C3×C5×#2642 #14275676

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2642 #14275676