Subgroups — C₃×C₅×#1707 #14274741

Order 1920.

360 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#500	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#498	C₂	1	2	yes	maximal
3	C₃×C₅×#494	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#487	C₂	1	2	yes	maximal
5	C₃×C₅×C₈²	C₂	1	1	yes	maximal, characteristic
6	C₅×#1707	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×Q₁₆	—	2	2	no	—
8	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₃×C₅×#107	—	2	2	no	—
11	C₃×C₅×#106	—	2	4	no	—
12	C₃×C₅×#103	—	2	4	no	—
13	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₃×#1707	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#500	C₂×C₃	1	1	yes	characteristic
18	C₅×#498	C₂×C₃	1	2	yes	—
19	C₅×#494	C₂×C₃	1	1	yes	characteristic
20	C₅×#487	C₂×C₃	1	2	yes	—
21	C₅×C₈²	C₂×C₃	1	1	yes	characteristic
22	C₂×C₃×C₅×Q₈	—	2	2	no	—
23	C₂×C₃×C₅×Q₈	—	2	2	no	—
24	C₃×C₅×Q₁₆	—	4	4	no	—
25	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
26	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
27	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
28	C₃×C₅×C₄⋊C₄	—	2	2	no	—
29	C₃×C₅×C₄⋊C₄	—	2	2	no	—
30	C₃×C₅×C₄⋊C₄	—	2	2	no	—
31	C₃×C₅×C₄⋊C₄	—	2	2	no	—
32	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
33	C₃×#500	C₂×C₅	1	1	yes	characteristic
34	C₃×#498	C₂×C₅	1	2	yes	—
35	C₃×#494	C₂×C₅	1	1	yes	characteristic
36	C₃×#487	C₂×C₅	1	2	yes	—
37	C₃×C₈²	C₂×C₅	1	1	yes	characteristic
38	C₂×C₅×Q₁₆	—	2	2	no	—
39	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
40	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
41	C₅×#107	—	2	2	no	—
42	C₅×#106	—	2	4	no	—
43	C₅×#103	—	2	4	no	—
44	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
45	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
46	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
47	#1707	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
48	C₃×C₅×Q₈	—	4	4	no	—
49	C₃×C₅×Q₈	—	4	4	no	—
50	C₂×C₃×C₄×C₅	—	4	2	no	—
51	C₂×C₃×C₄×C₅	—	4	2	no	—
52	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₃×C₅×C₈	—	2	2	no	—
56	C₃×C₅×C₈	SD₁₆	1	4	yes	—
57	C₃×C₅×C₈	Q₁₆	1	4	yes	complemented
58	C₂×C₃×Q₁₆	—	2	2	no	—
59	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
60	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
61	C₃×#107	—	2	2	no	—
62	C₃×#106	—	2	4	no	—
63	C₃×#103	—	2	4	no	—
64	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
65	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
66	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
67	C₂×C₅×Q₈	—	2	2	no	—
68	C₂×C₅×Q₈	—	2	2	no	—
69	C₅×Q₁₆	—	4	4	no	—
70	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
71	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
72	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
73	C₅×C₄⋊C₄	—	2	2	no	—
74	C₅×C₄⋊C₄	—	2	2	no	—
75	C₅×C₄⋊C₄	—	2	2	no	—
76	C₅×C₄⋊C₄	—	2	2	no	—
77	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
78	#500	C₂×C₃×C₅	1	1	yes	characteristic
79	#498	C₂×C₃×C₅	1	2	yes	—
80	#494	C₂×C₃×C₅	1	1	yes	characteristic
81	#487	C₂×C₃×C₅	1	2	yes	—
82	C₈²	C₂×C₃×C₅	1	1	yes	characteristic
83	C₂²×C₃×C₅	#127	1	1	yes	characteristic
84	C₃×C₄×C₅	—	8	2	no	—
85	C₃×C₄×C₅	—	8	2	no	—
86	C₃×C₄×C₅	C₂×SD₁₆	1	2	yes	—
87	C₃×C₄×C₅	C₂×Q₁₆	1	2	yes	—
88	C₃×C₄×C₅	#135	1	2	yes	—
89	C₂×C₃×Q₈	—	2	2	no	—
90	C₂×C₃×Q₈	—	2	2	no	—
91	C₃×Q₁₆	—	4	4	no	—
92	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
93	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
94	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
95	C₃×C₄⋊C₄	—	2	2	no	—
96	C₃×C₄⋊C₄	—	2	2	no	—
97	C₃×C₄⋊C₄	—	2	2	no	—
98	C₃×C₄⋊C₄	—	2	2	no	—
99	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
100	C₅×Q₈	—	4	4	no	—
101	C₅×Q₈	—	4	4	no	—
102	C₂×C₄×C₅	—	4	2	no	—
103	C₂×C₄×C₅	—	4	2	no	—
104	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
107	C₅×C₈	—	2	2	no	—
108	C₅×C₈	C₃×SD₁₆	1	4	yes	—
109	C₅×C₈	C₃×Q₁₆	1	4	yes	complemented
110	C₂×Q₁₆	—	2	2	no	—
111	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
112	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
113	#107	—	2	2	no	—
114	#106	—	2	4	no	—
115	#103	—	2	4	no	—
116	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
117	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
118	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
119	C₂×C₃×C₅	#492	1	1	yes	characteristic
120	C₂×C₃×C₅	#494	1	1	yes	characteristic
121	C₂×C₃×C₅	#495	1	1	yes	characteristic
122	C₃×Q₈	—	4	4	no	—
123	C₃×Q₈	—	4	4	no	—
124	C₂×C₃×C₄	—	4	2	no	—
125	C₂×C₃×C₄	—	4	2	no	—
126	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
129	C₃×C₈	—	2	2	no	—
130	C₃×C₈	C₅×SD₁₆	1	4	yes	—
131	C₃×C₈	C₅×Q₁₆	1	4	yes	complemented
132	C₂²×C₅	C₃×#127	1	1	yes	characteristic
133	C₄×C₅	—	8	2	no	—
134	C₄×C₅	—	8	2	no	—
135	C₄×C₅	C₂×C₃×SD₁₆	1	2	yes	—
136	C₄×C₅	C₂×C₃×Q₁₆	1	2	yes	—
137	C₄×C₅	C₃×#135	1	2	yes	—
138	C₂×Q₈	—	2	2	no	—
139	C₂×Q₈	—	2	2	no	—
140	Q₁₆	—	4	4	no	—
141	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
142	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
143	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
149	C₃×C₅	#1707	1	1	yes	characteristic, direct product factor, Hall
150	C₂²×C₃	C₅×#127	1	1	yes	characteristic
151	C₃×C₄	—	8	2	no	—
152	C₃×C₄	—	8	2	no	—
153	C₃×C₄	C₂×C₅×SD₁₆	1	2	yes	—
154	C₃×C₄	C₂×C₅×Q₁₆	1	2	yes	—
155	C₃×C₄	C₅×#135	1	2	yes	—
156	C₂×C₅	C₃×#492	1	1	yes	characteristic
157	C₂×C₅	C₃×#494	1	1	yes	characteristic
158	C₂×C₅	C₃×#495	1	1	yes	characteristic
159	Q₈	—	4	4	no	—
160	Q₈	—	4	4	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
164	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
165	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
166	C₈	—	2	2	no	—
167	C₈	C₃×C₅×SD₁₆	1	4	yes	—
168	C₈	C₃×C₅×Q₁₆	1	4	yes	complemented
169	C₂×C₃	C₅×#492	1	1	yes	characteristic
170	C₂×C₃	C₅×#494	1	1	yes	characteristic
171	C₂×C₃	C₅×#495	1	1	yes	characteristic
172	C₅	C₃×#1707	1	1	yes	characteristic, direct product factor, Sylow
173	C₂²	C₃×C₅×#127	1	1	yes	characteristic
174	C₄	—	8	2	no	—
175	C₄	—	8	2	no	—
176	C₄	C₂×C₃×C₅×SD₁₆	1	2	yes	—
177	C₄	C₂×C₃×C₅×Q₁₆	1	2	yes	—
178	C₄	C₃×C₅×#135	1	2	yes	—
179	C₃	C₅×#1707	1	1	yes	characteristic, direct product factor, Sylow
180	C₂	C₃×C₅×#492	1	1	yes	characteristic
181	C₂	C₃×C₅×#494	1	1	yes	characteristic
182	C₂	C₃×C₅×#495	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃×C₅	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₃×C₅
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×#1707	C₃	direct (N × H)
16. C₃×#1707	C₅	direct (N × H)
47. #1707	C₃×C₅	direct (N × H)
57. C₃×C₅×C₈	Q₁₆	semidirect
109. C₅×C₈	C₃×Q₁₆	semidirect
131. C₃×C₈	C₅×Q₁₆	semidirect
149. C₃×C₅	#1707	direct (N × H)
168. C₈	C₃×C₅×Q₁₆	semidirect
172. C₅	C₃×#1707	direct (N × H)
179. C₃	C₅×#1707	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×#500	2
C₃×C₅×#498	2
C₃×C₅×#494	2
C₃×C₅×#487	2
C₃×C₅×C₈²	2
C₅×#1707	3
C₃×#1707	5

Subgroups — C3×C5×#1707 #14274741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1707 #14274741