Subgroups — C₃×C₅×#2658 #14275692

Order 1920.

1224 conjugacy classes of subgroups.

The subgroup lattice has 176 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	3	yes	maximal, complemented
2	C₂×C₃×C₅×#124	C₂	1	3	yes	maximal
3	C₃×C₅×#399	C₂	1	8	yes	maximal, complemented
4	C₃×C₅×#376	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#2658	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	C₂²	1	3	yes	—
7	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
8	C₃×C₅×C₄⋊Q₈	—	2	12	no	—
9	C₃×C₅×#124	—	2	12	no	—
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	12	yes	—
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×#2658	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	3	yes	complemented
16	C₂×C₅×#124	C₂×C₃	1	3	yes	—
17	C₅×#399	C₂×C₃	1	8	yes	complemented
18	C₅×#376	C₂×C₃	1	1	yes	characteristic, complemented
19	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
20	C₂×C₃×C₅×Q₈	—	2	6	no	—
21	C₂×C₃×C₅×Q₈	—	2	12	no	—
22	C₂×C₃×C₅×D₄	—	2	6	no	—
23	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₃×C₅×C₄⋊C₄	—	2	24	no	—
27	C₃×C₅×C₂²⋊C₄	—	2	24	no	—
28	C₃×C₄²×C₅	D₄	1	12	yes	—
29	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	3	yes	complemented
30	C₂×C₃×#124	C₂×C₅	1	3	yes	—
31	C₃×#399	C₂×C₅	1	8	yes	complemented
32	C₃×#376	C₂×C₅	1	1	yes	characteristic, complemented
33	C₂²×C₅×Q₈	C₂²×C₃	1	3	yes	—
34	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
35	C₅×C₄⋊Q₈	—	2	12	no	—
36	C₅×#124	—	2	12	no	—
37	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	12	yes	—
38	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
39	C₂×C₄²×C₅	C₂²×C₃	1	3	yes	—
40	C₅×#95	C₂²×C₃	1	4	yes	—
41	#2658	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
42	C₂³×C₃×C₅	—	2	6	no	—
43	C₂³×C₃×C₅	—	2	8	no	—
44	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
45	C₃×C₅×Q₈	—	4	12	no	—
46	C₃×C₅×D₄	—	4	4	no	—
47	C₂×C₃×C₄×C₅	—	2	12	no	—
48	C₂×C₃×C₄×C₅	—	2	6	no	—
49	C₂×C₃×C₄×C₅	—	2	12	no	—
50	C₂×C₃×C₄×C₅	C₂×D₄	1	12	yes	—
51	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
52	C₂²×C₃×Q₈	C₂²×C₅	1	3	yes	—
53	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
54	C₃×C₄⋊Q₈	—	2	12	no	—
55	C₃×#124	—	2	12	no	—
56	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	12	yes	—
57	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	12	yes	—
58	C₂×C₃×C₄²	C₂²×C₅	1	3	yes	—
59	C₃×#95	C₂²×C₅	1	4	yes	—
60	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
61	C₂×C₅×Q₈	—	2	6	no	—
62	C₂×C₅×Q₈	—	2	12	no	—
63	C₂×C₅×D₄	—	2	6	no	—
64	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
65	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
66	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
67	C₅×C₄⋊C₄	—	2	24	no	—
68	C₅×C₂²⋊C₄	—	2	24	no	—
69	C₄²×C₅	C₃×D₄	1	12	yes	—
70	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	3	yes	complemented
71	C₂×#124	C₂×C₃×C₅	1	3	yes	—
72	#399	C₂×C₃×C₅	1	8	yes	complemented
73	#376	C₂×C₃×C₅	1	1	yes	characteristic, complemented
74	C₂²×C₃×C₅	—	4	2	no	—
75	C₂²×C₃×C₅	—	4	12	no	—
76	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
77	C₂²×C₃×C₅	2₋¹⁺⁴	1	4	yes	—
78	C₃×C₄×C₅	—	2	12	no	—
79	C₃×C₄×C₅	—	4	6	no	—
80	C₃×C₄×C₅	#127	1	4	yes	—
81	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
82	C₂×C₃×Q₈	—	2	6	no	—
83	C₂×C₃×Q₈	—	2	12	no	—
84	C₂×C₃×D₄	—	2	6	no	—
85	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
86	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
87	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
88	C₃×C₄⋊C₄	—	2	24	no	—
89	C₃×C₂²⋊C₄	—	2	24	no	—
90	C₃×C₄²	C₅×D₄	1	12	yes	—
91	C₂³×C₅	—	2	6	no	—
92	C₂³×C₅	—	2	8	no	—
93	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
94	C₅×Q₈	—	4	12	no	—
95	C₅×D₄	—	4	4	no	—
96	C₂×C₄×C₅	—	2	12	no	—
97	C₂×C₄×C₅	—	2	6	no	—
98	C₂×C₄×C₅	—	2	12	no	—
99	C₂×C₄×C₅	C₂×C₃×D₄	1	12	yes	—
100	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
101	C₂²×Q₈	C₂²×C₃×C₅	1	3	yes	—
102	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
103	C₄⋊Q₈	—	2	12	no	—
104	#124	—	2	12	no	—
105	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	12	yes	—
106	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	12	yes	—
107	C₂×C₄²	C₂²×C₃×C₅	1	3	yes	—
108	#95	C₂²×C₃×C₅	1	4	yes	—
109	C₂×C₃×C₅	—	8	2	no	—
110	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
111	C₂×C₃×C₅	#536	1	6	yes	—
112	C₂³×C₃	—	2	6	no	—
113	C₂³×C₃	—	2	8	no	—
114	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
115	C₃×Q₈	—	4	12	no	—
116	C₃×D₄	—	4	4	no	—
117	C₂×C₃×C₄	—	2	12	no	—
118	C₂×C₃×C₄	—	2	6	no	—
119	C₂×C₃×C₄	—	2	12	no	—
120	C₂×C₃×C₄	C₂×C₅×D₄	1	12	yes	—
121	C₂×C₃×C₄	C₂×C₅×D₄	1	6	yes	—
122	C₂²×C₅	—	4	2	no	—
123	C₂²×C₅	—	4	12	no	—
124	C₂²×C₅	C₂²×C₃×D₄	1	3	yes	—
125	C₂²×C₅	C₃×2₋¹⁺⁴	1	4	yes	—
126	C₄×C₅	—	2	12	no	—
127	C₄×C₅	—	4	6	no	—
128	C₄×C₅	C₃×#127	1	4	yes	—
129	C₂⁴	C₂³×C₃×C₅	1	2	yes	—
130	C₂×Q₈	—	2	6	no	—
131	C₂×Q₈	—	2	12	no	—
132	C₂×D₄	—	2	6	no	—
133	C₂²×C₄	C₂³×C₃×C₅	1	6	yes	—
134	C₂²×C₄	C₂³×C₃×C₅	1	6	yes	—
135	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
136	C₄⋊C₄	—	2	24	no	—
137	C₂²⋊C₄	—	2	24	no	—
138	C₄²	C₃×C₅×D₄	1	12	yes	—
139	C₃×C₅	#2658	1	1	yes	characteristic, direct product factor, Hall
140	C₂²×C₃	—	4	2	no	—
141	C₂²×C₃	—	4	12	no	—
142	C₂²×C₃	C₂²×C₅×D₄	1	3	yes	—
143	C₂²×C₃	C₅×2₋¹⁺⁴	1	4	yes	—
144	C₃×C₄	—	2	12	no	—
145	C₃×C₄	—	4	6	no	—
146	C₃×C₄	C₅×#127	1	4	yes	—
147	C₂×C₅	—	8	2	no	—
148	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
149	C₂×C₅	C₃×#536	1	6	yes	—
150	C₂³	—	2	6	no	—
151	C₂³	—	2	8	no	—
152	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
153	Q₈	—	4	12	no	—
154	D₄	—	4	4	no	—
155	C₂×C₄	—	2	12	no	—
156	C₂×C₄	—	2	6	no	—
157	C₂×C₄	—	2	12	no	—
158	C₂×C₄	C₂×C₃×C₅×D₄	1	12	yes	—
159	C₂×C₄	C₂×C₃×C₅×D₄	1	6	yes	—
160	C₂×C₃	—	8	2	no	—
161	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
162	C₂×C₃	C₅×#536	1	6	yes	—
163	C₅	C₃×#2658	1	1	yes	characteristic, direct product factor, Sylow
164	C₂²	—	4	2	no	—
165	C₂²	—	4	12	no	—
166	C₂²	C₂²×C₃×C₅×D₄	1	3	yes	—
167	C₂²	C₃×C₅×2₋¹⁺⁴	1	4	yes	—
168	C₄	—	2	12	no	—
169	C₄	—	4	6	no	—
170	C₄	C₃×C₅×#127	1	4	yes	—
171	C₃	C₅×#2658	1	1	yes	characteristic, direct product factor, Sylow
172	C₂	—	8	2	no	—
173	C₂	C₂×C₃×C₅×#127	1	1	yes	characteristic
174	C₂	C₃×C₅×#536	1	6	yes	—

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
prime radical	C₂²×C₃×C₅×D₄	C₂²
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×C₄⋊Q₈	C₂	semidirect
3. C₃×C₅×#399	C₂	semidirect
4. C₃×C₅×#376	C₂	semidirect
5. C₅×#2658	C₃	direct (N × H)
14. C₃×#2658	C₅	direct (N × H)
15. C₂×C₅×C₄⋊Q₈	C₂×C₃	semidirect
17. C₅×#399	C₂×C₃	semidirect
18. C₅×#376	C₂×C₃	semidirect
29. C₂×C₃×C₄⋊Q₈	C₂×C₅	semidirect
31. C₃×#399	C₂×C₅	semidirect
32. C₃×#376	C₂×C₅	semidirect
41. #2658	C₃×C₅	direct (N × H)
70. C₂×C₄⋊Q₈	C₂×C₃×C₅	semidirect
72. #399	C₂×C₃×C₅	semidirect
73. #376	C₂×C₃×C₅	semidirect
139. C₃×C₅	#2658	direct (N × H)
163. C₅	C₃×#2658	direct (N × H)
171. C₃	C₅×#2658	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×#124	2
C₃×C₅×#399	2
C₃×C₅×#376	2
C₅×#2658	3
C₃×#2658	5

Subgroups — C3×C5×#2658 #14275692

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2658 #14275692