Subgroups — C₃×C₅×#2657 #14275691

Order 1920.

1688 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#127	C₂	1	3	yes	maximal, complemented
2	C₂×C₃×C₅×#124	C₂	1	3	yes	maximal, complemented
3	C₃×C₅×#392	C₂	1	8	yes	maximal, complemented
4	C₃×C₅×#376	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#2657	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₅×D₄	C₂²	1	3	yes	—
8	C₂²×C₃×C₅×D₄	C₂²	1	12	yes	—
9	C₃×C₅×#127	—	2	12	no	—
10	C₃×C₅×#124	—	2	12	no	—
11	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	12	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	3	yes	—
13	C₃×C₅×#95	C₂²	1	4	yes	—
14	C₃×#2657	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂×C₅×#127	C₂×C₃	1	3	yes	complemented
16	C₂×C₅×#124	C₂×C₃	1	3	yes	complemented
17	C₅×#392	C₂×C₃	1	8	yes	complemented
18	C₅×#376	C₂×C₃	1	1	yes	characteristic, complemented
19	C₂⁴×C₃×C₅	C₂³	1	6	yes	—
20	C₂×C₃×C₅×Q₈	—	2	6	no	—
21	C₂×C₃×C₅×D₄	—	2	6	no	—
22	C₂×C₃×C₅×D₄	—	2	24	no	—
23	C₂×C₃×C₅×D₄	—	2	12	no	—
24	C₂×C₃×C₅×D₄	—	4	24	no	—
25	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
26	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₃×C₅×C₂²⋊C₄	—	2	24	no	—
29	C₃×C₄²×C₅	D₄	1	12	yes	complemented
30	C₂×C₃×#127	C₂×C₅	1	3	yes	complemented
31	C₂×C₃×#124	C₂×C₅	1	3	yes	complemented
32	C₃×#392	C₂×C₅	1	8	yes	complemented
33	C₃×#376	C₂×C₅	1	1	yes	characteristic, complemented
34	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
35	C₂²×C₅×D₄	C₂²×C₃	1	3	yes	—
36	C₂²×C₅×D₄	C₂²×C₃	1	12	yes	—
37	C₅×#127	—	2	12	no	—
38	C₅×#124	—	2	12	no	—
39	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
40	C₂×C₄²×C₅	C₂²×C₃	1	3	yes	—
41	C₅×#95	C₂²×C₃	1	4	yes	—
42	#2657	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
43	C₂³×C₃×C₅	—	2	6	no	—
44	C₂³×C₃×C₅	—	2	12	no	—
45	C₂³×C₃×C₅	—	2	24	no	—
46	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
47	C₃×C₅×Q₈	—	4	4	no	—
48	C₃×C₅×D₄	—	4	48	no	—
49	C₃×C₅×D₄	—	4	12	no	—
50	C₂×C₃×C₄×C₅	—	2	12	no	—
51	C₂×C₃×C₄×C₅	—	2	6	no	—
52	C₂×C₃×C₄×C₅	C₂×D₄	1	12	yes	—
53	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
54	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
55	C₂²×C₃×D₄	C₂²×C₅	1	3	yes	—
56	C₂²×C₃×D₄	C₂²×C₅	1	12	yes	—
57	C₃×#127	—	2	12	no	—
58	C₃×#124	—	2	12	no	—
59	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	12	yes	—
60	C₂×C₃×C₄²	C₂²×C₅	1	3	yes	—
61	C₃×#95	C₂²×C₅	1	4	yes	—
62	C₂⁴×C₅	C₂³×C₃	1	6	yes	—
63	C₂×C₅×Q₈	—	2	6	no	—
64	C₂×C₅×D₄	—	2	6	no	—
65	C₂×C₅×D₄	—	2	24	no	—
66	C₂×C₅×D₄	—	2	12	no	—
67	C₂×C₅×D₄	—	4	24	no	—
68	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
69	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
70	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
71	C₅×C₂²⋊C₄	—	2	24	no	—
72	C₄²×C₅	C₃×D₄	1	12	yes	complemented
73	C₂×#127	C₂×C₃×C₅	1	3	yes	complemented
74	C₂×#124	C₂×C₃×C₅	1	3	yes	complemented
75	#392	C₂×C₃×C₅	1	8	yes	complemented
76	#376	C₂×C₃×C₅	1	1	yes	characteristic, complemented
77	C₂²×C₃×C₅	—	4	6	no	—
78	C₂²×C₃×C₅	—	4	24	no	—
79	C₂²×C₃×C₅	—	4	12	no	—
80	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
81	C₂²×C₃×C₅	H(5,𝔽₂)	1	4	yes	—
82	C₃×C₄×C₅	—	2	12	no	—
83	C₃×C₄×C₅	—	4	2	no	—
84	C₃×C₄×C₅	#127	1	4	yes	—
85	C₂⁴×C₃	C₂³×C₅	1	6	yes	—
86	C₂×C₃×Q₈	—	2	6	no	—
87	C₂×C₃×D₄	—	2	6	no	—
88	C₂×C₃×D₄	—	2	24	no	—
89	C₂×C₃×D₄	—	2	12	no	—
90	C₂×C₃×D₄	—	4	24	no	—
91	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
92	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
93	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
94	C₃×C₂²⋊C₄	—	2	24	no	—
95	C₃×C₄²	C₅×D₄	1	12	yes	complemented
96	C₂³×C₅	—	2	6	no	—
97	C₂³×C₅	—	2	12	no	—
98	C₂³×C₅	—	2	24	no	—
99	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
100	C₅×Q₈	—	4	4	no	—
101	C₅×D₄	—	4	48	no	—
102	C₅×D₄	—	4	12	no	—
103	C₂×C₄×C₅	—	2	12	no	—
104	C₂×C₄×C₅	—	2	6	no	—
105	C₂×C₄×C₅	C₂×C₃×D₄	1	12	yes	—
106	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
107	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
108	C₂²×D₄	C₂²×C₃×C₅	1	3	yes	—
109	C₂²×D₄	C₂²×C₃×C₅	1	12	yes	—
110	#127	—	2	12	no	—
111	#124	—	2	12	no	—
112	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	12	yes	—
113	C₂×C₄²	C₂²×C₃×C₅	1	3	yes	—
114	#95	C₂²×C₃×C₅	1	4	yes	—
115	C₂×C₃×C₅	—	8	6	no	—
116	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
117	C₂×C₃×C₅	#535	1	6	yes	—
118	C₂³×C₃	—	2	6	no	—
119	C₂³×C₃	—	2	12	no	—
120	C₂³×C₃	—	2	24	no	—
121	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
122	C₃×Q₈	—	4	4	no	—
123	C₃×D₄	—	4	48	no	—
124	C₃×D₄	—	4	12	no	—
125	C₂×C₃×C₄	—	2	12	no	—
126	C₂×C₃×C₄	—	2	6	no	—
127	C₂×C₃×C₄	C₂×C₅×D₄	1	12	yes	—
128	C₂×C₃×C₄	C₂×C₅×D₄	1	6	yes	—
129	C₂²×C₅	—	4	6	no	—
130	C₂²×C₅	—	4	24	no	—
131	C₂²×C₅	—	4	12	no	—
132	C₂²×C₅	C₂²×C₃×D₄	1	3	yes	—
133	C₂²×C₅	C₃×H(5,𝔽₂)	1	4	yes	—
134	C₄×C₅	—	2	12	no	—
135	C₄×C₅	—	4	2	no	—
136	C₄×C₅	C₃×#127	1	4	yes	—
137	C₂⁴	C₂³×C₃×C₅	1	6	yes	—
138	C₂×Q₈	—	2	6	no	—
139	C₂×D₄	—	2	6	no	—
140	C₂×D₄	—	2	24	no	—
141	C₂×D₄	—	2	12	no	—
142	C₂×D₄	—	4	24	no	—
143	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
144	C₂²×C₄	C₂³×C₃×C₅	1	6	yes	—
145	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
146	C₂²⋊C₄	—	2	24	no	—
147	C₄²	C₃×C₅×D₄	1	12	yes	complemented
148	C₃×C₅	#2657	1	1	yes	characteristic, direct product factor, Hall
149	C₂²×C₃	—	4	6	no	—
150	C₂²×C₃	—	4	24	no	—
151	C₂²×C₃	—	4	12	no	—
152	C₂²×C₃	C₂²×C₅×D₄	1	3	yes	—
153	C₂²×C₃	C₅×H(5,𝔽₂)	1	4	yes	—
154	C₃×C₄	—	2	12	no	—
155	C₃×C₄	—	4	2	no	—
156	C₃×C₄	C₅×#127	1	4	yes	—
157	C₂×C₅	—	8	6	no	—
158	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
159	C₂×C₅	C₃×#535	1	6	yes	—
160	C₂³	—	2	6	no	—
161	C₂³	—	2	12	no	—
162	C₂³	—	2	24	no	—
163	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
164	Q₈	—	4	4	no	—
165	D₄	—	4	48	no	—
166	D₄	—	4	12	no	—
167	C₂×C₄	—	2	12	no	—
168	C₂×C₄	—	2	6	no	—
169	C₂×C₄	C₂×C₃×C₅×D₄	1	12	yes	—
170	C₂×C₄	C₂×C₃×C₅×D₄	1	6	yes	—
171	C₂×C₃	—	8	6	no	—
172	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
173	C₂×C₃	C₅×#535	1	6	yes	—
174	C₅	C₃×#2657	1	1	yes	characteristic, direct product factor, Sylow
175	C₂²	—	4	6	no	—
176	C₂²	—	4	24	no	—
177	C₂²	—	4	12	no	—
178	C₂²	C₂²×C₃×C₅×D₄	1	3	yes	—
179	C₂²	C₃×C₅×H(5,𝔽₂)	1	4	yes	—
180	C₄	—	2	12	no	—
181	C₄	—	4	2	no	—
182	C₄	C₃×C₅×#127	1	4	yes	—
183	C₃	C₅×#2657	1	1	yes	characteristic, direct product factor, Sylow
184	C₂	—	8	6	no	—
185	C₂	C₂×C₃×C₅×#127	1	1	yes	characteristic
186	C₂	C₃×C₅×#535	1	6	yes	—

	subgroup	G/H
center, socle	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
1. C₂×C₃×C₅×#127	C₂	semidirect
2. C₂×C₃×C₅×#124	C₂	semidirect
3. C₃×C₅×#392	C₂	semidirect
4. C₃×C₅×#376	C₂	semidirect
5. C₅×#2657	C₃	direct (N × H)
14. C₃×#2657	C₅	direct (N × H)
15. C₂×C₅×#127	C₂×C₃	semidirect
16. C₂×C₅×#124	C₂×C₃	semidirect
17. C₅×#392	C₂×C₃	semidirect
18. C₅×#376	C₂×C₃	semidirect
29. C₃×C₄²×C₅	D₄	semidirect
30. C₂×C₃×#127	C₂×C₅	semidirect
31. C₂×C₃×#124	C₂×C₅	semidirect
32. C₃×#392	C₂×C₅	semidirect
33. C₃×#376	C₂×C₅	semidirect
42. #2657	C₃×C₅	direct (N × H)
72. C₄²×C₅	C₃×D₄	semidirect
73. C₂×#127	C₂×C₃×C₅	semidirect
74. C₂×#124	C₂×C₃×C₅	semidirect
75. #392	C₂×C₃×C₅	semidirect
76. #376	C₂×C₃×C₅	semidirect
95. C₃×C₄²	C₅×D₄	semidirect
147. C₄²	C₃×C₅×D₄	semidirect
148. C₃×C₅	#2657	direct (N × H)
174. C₅	C₃×#2657	direct (N × H)
183. C₃	C₅×#2657	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×#127	2
C₂×C₃×C₅×#124	2
C₃×C₅×#392	2
C₃×C₅×#376	2
C₅×#2657	3
C₃×#2657	5

Subgroups — C3×C5×#2657 #14275691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2657 #14275691