Subgroups — C₃×C₅×#1558 #14274592

Order 1920.

456 conjugacy classes of subgroups.

The subgroup lattice has 196 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	C₃×C₅×#432	C₂	1	2	yes	maximal
3	C₃×C₅×#333	C₂	1	4	yes	maximal, complemented
4	C₅×#1558	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₅×Q₈	C₄	1	2	yes	—
6	C₃×C₅×C₄⋊Q₈	—	2	2	no	—
7	C₃×C₅×C₄⋊Q₈	C₄	1	2	yes	—
8	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₃×C₅×C₄⋊C₈	C₂²	1	4	yes	—
12	C₃×C₅×C₂²⋊C₈	—	2	2	no	—
13	C₃×C₅×#97	—	2	2	no	—
14	C₃×#1558	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
16	C₅×#432	C₂×C₃	1	2	yes	—
17	C₅×#333	C₂×C₃	1	4	yes	complemented
18	C₂×C₃×C₅×Q₈	—	2	2	no	—
19	C₂×C₃×C₅×Q₈	—	4	4	no	—
20	C₂×C₃×C₅×Q₈	C₂×C₄	1	4	yes	—
21	C₂²×C₃×C₄×C₅	—	2	2	no	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
24	C₂×C₃×C₅×C₈	—	2	4	no	—
25	C₃×C₅×C₄⋊C₄	—	2	4	no	—
26	C₃×C₅×C₄⋊C₄	—	2	4	no	—
27	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₃×C₄²×C₅	D₄	1	2	yes	—
29	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
30	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
31	C₃×#432	C₂×C₅	1	2	yes	—
32	C₃×#333	C₂×C₅	1	4	yes	complemented
33	C₂²×C₅×Q₈	C₃×C₄	1	2	yes	—
34	C₅×C₄⋊Q₈	—	2	2	no	—
35	C₅×C₄⋊Q₈	C₃×C₄	1	2	yes	—
36	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
37	C₂×C₅×C₄⋊C₄	—	2	2	no	—
38	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
39	C₅×C₄⋊C₈	C₂²×C₃	1	4	yes	—
40	C₅×C₂²⋊C₈	—	2	2	no	—
41	C₅×#97	—	2	2	no	—
42	#1558	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
43	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
44	C₃×C₅×Q₈	—	4	4	no	—
45	C₃×C₅×Q₈	—	4	4	no	—
46	C₂×C₃×C₄×C₅	—	2	2	no	—
47	C₂×C₃×C₄×C₅	—	2	4	no	—
48	C₂×C₃×C₄×C₅	—	2	4	no	—
49	C₂×C₃×C₄×C₅	—	4	4	no	—
50	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
54	C₃×C₅×C₈	—	4	4	no	—
55	C₂²×C₃×Q₈	C₄×C₅	1	2	yes	—
56	C₃×C₄⋊Q₈	—	2	2	no	—
57	C₃×C₄⋊Q₈	C₄×C₅	1	2	yes	—
58	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
59	C₂×C₃×C₄⋊C₄	—	2	2	no	—
60	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
61	C₃×C₄⋊C₈	C₂²×C₅	1	4	yes	—
62	C₃×C₂²⋊C₈	—	2	2	no	—
63	C₃×#97	—	2	2	no	—
64	C₂×C₅×Q₈	—	2	2	no	—
65	C₂×C₅×Q₈	—	4	4	no	—
66	C₂×C₅×Q₈	C₂×C₃×C₄	1	4	yes	—
67	C₂²×C₄×C₅	—	2	2	no	—
68	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
70	C₂×C₅×C₈	—	2	4	no	—
71	C₅×C₄⋊C₄	—	2	4	no	—
72	C₅×C₄⋊C₄	—	2	4	no	—
73	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
74	C₄²×C₅	C₃×D₄	1	2	yes	—
75	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
76	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic
77	#432	C₂×C₃×C₅	1	2	yes	—
78	#333	C₂×C₃×C₅	1	4	yes	complemented
79	C₂²×C₃×C₅	—	2	2	no	—
80	C₂²×C₃×C₅	#100	1	2	yes	—
81	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₃×C₄×C₅	—	2	1	no	—
83	C₃×C₄×C₅	—	2	1	no	—
84	C₃×C₄×C₅	—	2	2	no	—
85	C₃×C₄×C₅	—	4	4	no	—
86	C₃×C₄×C₅	#137	1	4	yes	—
87	C₂×C₃×Q₈	—	2	2	no	—
88	C₂×C₃×Q₈	—	4	4	no	—
89	C₂×C₃×Q₈	C₂×C₄×C₅	1	4	yes	—
90	C₂²×C₃×C₄	—	2	2	no	—
91	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
92	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
93	C₂×C₃×C₈	—	2	4	no	—
94	C₃×C₄⋊C₄	—	2	4	no	—
95	C₃×C₄⋊C₄	—	2	4	no	—
96	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
97	C₃×C₄²	C₅×D₄	1	2	yes	—
98	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
99	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
100	C₅×Q₈	—	4	4	no	—
101	C₅×Q₈	—	4	4	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	4	no	—
104	C₂×C₄×C₅	—	2	4	no	—
105	C₂×C₄×C₅	—	4	4	no	—
106	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
107	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
110	C₅×C₈	—	4	4	no	—
111	C₂²×Q₈	C₃×C₄×C₅	1	2	yes	—
112	C₄⋊Q₈	—	2	2	no	—
113	C₄⋊Q₈	C₃×C₄×C₅	1	2	yes	—
114	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
117	C₄⋊C₈	C₂²×C₃×C₅	1	4	yes	—
118	C₂²⋊C₈	—	2	2	no	—
119	#97	—	2	2	no	—
120	C₂×C₃×C₅	—	2	2	no	—
121	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
122	C₂×C₃×C₅	#419	1	2	yes	—
123	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
124	C₃×Q₈	—	4	4	no	—
125	C₃×Q₈	—	4	4	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	4	no	—
128	C₂×C₃×C₄	—	2	4	no	—
129	C₂×C₃×C₄	—	4	4	no	—
130	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
131	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
134	C₃×C₈	—	4	4	no	—
135	C₂²×C₅	—	2	2	no	—
136	C₂²×C₅	C₃×#100	1	2	yes	—
137	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₅	—	2	1	no	—
139	C₄×C₅	—	2	1	no	—
140	C₄×C₅	—	2	2	no	—
141	C₄×C₅	—	4	4	no	—
142	C₄×C₅	C₃×#137	1	4	yes	—
143	C₂×Q₈	—	2	2	no	—
144	C₂×Q₈	—	4	4	no	—
145	C₂×Q₈	C₂×C₃×C₄×C₅	1	4	yes	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
148	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
149	C₂×C₈	—	2	4	no	—
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
153	C₄²	C₃×C₅×D₄	1	2	yes	—
154	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
155	C₃×C₅	#1558	1	1	yes	characteristic, direct product factor, Hall
156	C₂²×C₃	—	2	2	no	—
157	C₂²×C₃	C₅×#100	1	2	yes	—
158	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
159	C₃×C₄	—	2	1	no	—
160	C₃×C₄	—	2	1	no	—
161	C₃×C₄	—	2	2	no	—
162	C₃×C₄	—	4	4	no	—
163	C₃×C₄	C₅×#137	1	4	yes	—
164	C₂×C₅	—	2	2	no	—
165	C₂×C₅	C₂×C₃×#100	1	1	yes	characteristic
166	C₂×C₅	C₃×#419	1	2	yes	—
167	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
168	Q₈	—	4	4	no	—
169	Q₈	—	4	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
175	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
176	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
177	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
178	C₈	—	4	4	no	—
179	C₂×C₃	—	2	2	no	—
180	C₂×C₃	C₂×C₅×#100	1	1	yes	characteristic
181	C₂×C₃	C₅×#419	1	2	yes	—
182	C₅	C₃×#1558	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²	—	2	2	no	—
184	C₂²	C₃×C₅×#100	1	2	yes	—
185	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	2	2	no	—
189	C₄	—	4	4	no	—
190	C₄	C₃×C₅×#137	1	4	yes	—
191	C₃	C₅×#1558	1	1	yes	characteristic, direct product factor, Sylow
192	C₂	—	2	2	no	—
193	C₂	C₂×C₃×C₅×#100	1	1	yes	characteristic
194	C₂	C₃×C₅×#419	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₄²	C₂³×C₃×C₅
prime radical	C₂³×C₃×C₅	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#333	C₂	semidirect
4. C₅×#1558	C₃	direct (N × H)
14. C₃×#1558	C₅	direct (N × H)
17. C₅×#333	C₂×C₃	semidirect
32. C₃×#333	C₂×C₅	semidirect
42. #1558	C₃×C₅	direct (N × H)
78. #333	C₂×C₃×C₅	semidirect
155. C₃×C₅	#1558	direct (N × H)
182. C₅	C₃×#1558	direct (N × H)
191. C₃	C₅×#1558	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
C₃×C₅×#432	2
C₃×C₅×#333	2
C₅×#1558	3
C₃×#1558	5

Subgroups — C3×C5×#1558 #14274592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1558 #14274592