Subgroups — C₃×C₅×#2614 #14275648

Order 1920.

1564 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×#123	C₂	1	3	yes	maximal
3	C₃×C₅×#395	C₂	1	4	yes	maximal, complemented
4	C₃×C₅×#393	C₂	1	4	yes	maximal, complemented
5	C₃×C₅×#381	C₂	1	3	yes	maximal, complemented
6	C₅×#2614	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×Q₈	—	2	12	no	—
8	C₂²×C₃×C₅×Q₈	C₂²	1	4	yes	—
9	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
10	C₂³×C₃×C₄×C₅	C₂²	1	3	yes	—
11	C₃×C₅×#123	—	2	12	no	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	6	yes	—
13	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
14	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	6	yes	—
15	C₃×C₅×#95	C₂²	1	12	yes	—
16	C₃×#2614	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂³×C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
18	C₂×C₅×#123	C₂×C₃	1	3	yes	—
19	C₅×#395	C₂×C₃	1	4	yes	complemented
20	C₅×#393	C₂×C₃	1	4	yes	complemented
21	C₅×#381	C₂×C₃	1	3	yes	complemented
22	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
23	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
24	C₂×C₃×C₅×Q₈	—	2	32	no	—
25	C₂×C₃×C₅×Q₈	—	2	24	no	—
26	C₂×C₃×C₅×D₄	—	2	6	no	—
27	C₂²×C₃×C₄×C₅	—	2	12	no	—
28	C₂²×C₃×C₄×C₅	D₄	1	12	yes	—
29	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
32	C₃×C₅×C₄⋊C₄	—	2	12	no	—
33	C₃×C₅×C₂²⋊C₄	—	2	6	no	—
34	C₃×C₅×C₂²⋊C₄	—	2	12	no	—
35	C₂³×C₃×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
36	C₂×C₃×#123	C₂×C₅	1	3	yes	—
37	C₃×#395	C₂×C₅	1	4	yes	complemented
38	C₃×#393	C₂×C₅	1	4	yes	complemented
39	C₃×#381	C₂×C₅	1	3	yes	complemented
40	C₂²×C₅×Q₈	—	2	12	no	—
41	C₂²×C₅×Q₈	C₂²×C₃	1	4	yes	—
42	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
43	C₂³×C₄×C₅	C₂²×C₃	1	3	yes	—
44	C₅×#123	—	2	12	no	—
45	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	6	yes	—
46	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
47	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	6	yes	—
48	C₅×#95	C₂²×C₃	1	12	yes	—
49	#2614	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
50	C₂³×C₃×C₅	—	2	4	no	—
51	C₂³×C₃×C₅	—	2	3	no	—
52	C₂³×C₃×C₅	—	2	4	no	—
53	C₂³×C₃×C₅	C₂×D₄	1	6	yes	—
54	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
55	C₃×C₅×Q₈	—	2	32	no	—
56	C₃×C₅×D₄	—	4	4	no	—
57	C₂×C₃×C₄×C₅	—	2	24	no	—
58	C₂×C₃×C₄×C₅	—	2	12	no	—
59	C₂×C₃×C₄×C₅	—	2	6	no	—
60	C₂×C₃×C₄×C₅	—	2	3	no	—
61	C₂×C₃×C₄×C₅	—	2	12	no	—
62	C₂×C₃×C₄×C₅	C₂×D₄	1	12	yes	—
63	C₂²×C₃×Q₈	—	2	12	no	—
64	C₂²×C₃×Q₈	C₂²×C₅	1	4	yes	—
65	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
66	C₂³×C₃×C₄	C₂²×C₅	1	3	yes	—
67	C₃×#123	—	2	12	no	—
68	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	6	yes	—
69	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	3	yes	—
70	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	6	yes	—
71	C₃×#95	C₂²×C₅	1	12	yes	—
72	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
73	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
74	C₂×C₅×Q₈	—	2	32	no	—
75	C₂×C₅×Q₈	—	2	24	no	—
76	C₂×C₅×D₄	—	2	6	no	—
77	C₂²×C₄×C₅	—	2	12	no	—
78	C₂²×C₄×C₅	C₃×D₄	1	12	yes	—
79	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
80	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
82	C₅×C₄⋊C₄	—	2	12	no	—
83	C₅×C₂²⋊C₄	—	2	6	no	—
84	C₅×C₂²⋊C₄	—	2	12	no	—
85	C₂³×Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
86	C₂×#123	C₂×C₃×C₅	1	3	yes	—
87	#395	C₂×C₃×C₅	1	4	yes	complemented
88	#393	C₂×C₃×C₅	1	4	yes	complemented
89	#381	C₂×C₃×C₅	1	3	yes	complemented
90	C₂²×C₃×C₅	—	2	12	no	—
91	C₂²×C₃×C₅	—	4	1	no	—
92	C₂²×C₃×C₅	—	4	6	no	—
93	C₂²×C₃×C₅	#120	1	4	yes	—
94	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
95	C₂²×C₃×C₅	2₋¹⁺⁴	1	4	yes	—
96	C₃×C₄×C₅	—	2	12	no	—
97	C₃×C₄×C₅	—	4	1	no	—
98	C₃×C₄×C₅	—	4	6	no	—
99	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
100	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
101	C₂×C₃×Q₈	—	2	32	no	—
102	C₂×C₃×Q₈	—	2	24	no	—
103	C₂×C₃×D₄	—	2	6	no	—
104	C₂²×C₃×C₄	—	2	12	no	—
105	C₂²×C₃×C₄	C₅×D₄	1	12	yes	—
106	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
107	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
109	C₃×C₄⋊C₄	—	2	12	no	—
110	C₃×C₂²⋊C₄	—	2	6	no	—
111	C₃×C₂²⋊C₄	—	2	12	no	—
112	C₂³×C₅	—	2	4	no	—
113	C₂³×C₅	—	2	3	no	—
114	C₂³×C₅	—	2	4	no	—
115	C₂³×C₅	C₂×C₃×D₄	1	6	yes	—
116	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
117	C₅×Q₈	—	2	32	no	—
118	C₅×D₄	—	4	4	no	—
119	C₂×C₄×C₅	—	2	24	no	—
120	C₂×C₄×C₅	—	2	12	no	—
121	C₂×C₄×C₅	—	2	6	no	—
122	C₂×C₄×C₅	—	2	3	no	—
123	C₂×C₄×C₅	—	2	12	no	—
124	C₂×C₄×C₅	C₂×C₃×D₄	1	12	yes	—
125	C₂²×Q₈	—	2	12	no	—
126	C₂²×Q₈	C₂²×C₃×C₅	1	4	yes	—
127	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
128	C₂³×C₄	C₂²×C₃×C₅	1	3	yes	—
129	#123	—	2	12	no	—
130	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	6	yes	—
131	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	3	yes	—
132	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	6	yes	—
133	#95	C₂²×C₃×C₅	1	12	yes	—
134	C₂×C₃×C₅	—	2	4	no	—
135	C₂×C₃×C₅	—	8	1	no	—
136	C₂×C₃×C₅	C₂×#120	1	1	yes	characteristic
137	C₂×C₃×C₅	#536	1	6	yes	—
138	C₂³×C₃	—	2	4	no	—
139	C₂³×C₃	—	2	3	no	—
140	C₂³×C₃	—	2	4	no	—
141	C₂³×C₃	C₂×C₅×D₄	1	6	yes	—
142	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
143	C₃×Q₈	—	2	32	no	—
144	C₃×D₄	—	4	4	no	—
145	C₂×C₃×C₄	—	2	24	no	—
146	C₂×C₃×C₄	—	2	12	no	—
147	C₂×C₃×C₄	—	2	6	no	—
148	C₂×C₃×C₄	—	2	3	no	—
149	C₂×C₃×C₄	—	2	12	no	—
150	C₂×C₃×C₄	C₂×C₅×D₄	1	12	yes	—
151	C₂²×C₅	—	2	12	no	—
152	C₂²×C₅	—	4	1	no	—
153	C₂²×C₅	—	4	6	no	—
154	C₂²×C₅	C₃×#120	1	4	yes	—
155	C₂²×C₅	C₂²×C₃×D₄	1	3	yes	—
156	C₂²×C₅	C₃×2₋¹⁺⁴	1	4	yes	—
157	C₄×C₅	—	2	12	no	—
158	C₄×C₅	—	4	1	no	—
159	C₄×C₅	—	4	6	no	—
160	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
161	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
162	C₂×Q₈	—	2	32	no	—
163	C₂×Q₈	—	2	24	no	—
164	C₂×D₄	—	2	6	no	—
165	C₂²×C₄	—	2	12	no	—
166	C₂²×C₄	C₃×C₅×D₄	1	12	yes	—
167	C₂²×C₄	C₂³×C₃×C₅	1	6	yes	—
168	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₃×C₅	1	6	yes	—
170	C₄⋊C₄	—	2	12	no	—
171	C₂²⋊C₄	—	2	6	no	—
172	C₂²⋊C₄	—	2	12	no	—
173	C₃×C₅	#2614	1	1	yes	characteristic, direct product factor, Hall
174	C₂²×C₃	—	2	12	no	—
175	C₂²×C₃	—	4	1	no	—
176	C₂²×C₃	—	4	6	no	—
177	C₂²×C₃	C₅×#120	1	4	yes	—
178	C₂²×C₃	C₂²×C₅×D₄	1	3	yes	—
179	C₂²×C₃	C₅×2₋¹⁺⁴	1	4	yes	—
180	C₃×C₄	—	2	12	no	—
181	C₃×C₄	—	4	1	no	—
182	C₃×C₄	—	4	6	no	—
183	C₂×C₅	—	2	4	no	—
184	C₂×C₅	—	8	1	no	—
185	C₂×C₅	C₂×C₃×#120	1	1	yes	characteristic
186	C₂×C₅	C₃×#536	1	6	yes	—
187	C₂³	—	2	4	no	—
188	C₂³	—	2	3	no	—
189	C₂³	—	2	4	no	—
190	C₂³	C₂×C₃×C₅×D₄	1	6	yes	—
191	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
192	Q₈	—	2	32	no	—
193	D₄	—	4	4	no	—
194	C₂×C₄	—	2	24	no	—
195	C₂×C₄	—	2	12	no	—
196	C₂×C₄	—	2	6	no	—
197	C₂×C₄	—	2	3	no	—
198	C₂×C₄	—	2	12	no	—
199	C₂×C₄	C₂×C₃×C₅×D₄	1	12	yes	—
200	C₂×C₃	—	2	4	no	—
201	C₂×C₃	—	8	1	no	—
202	C₂×C₃	C₂×C₅×#120	1	1	yes	characteristic
203	C₂×C₃	C₅×#536	1	6	yes	—
204	C₅	C₃×#2614	1	1	yes	characteristic, direct product factor, Sylow
205	C₂²	—	2	12	no	—
206	C₂²	—	4	1	no	—
207	C₂²	—	4	6	no	—
208	C₂²	C₃×C₅×#120	1	4	yes	—
209	C₂²	C₂²×C₃×C₅×D₄	1	3	yes	—
210	C₂²	C₃×C₅×2₋¹⁺⁴	1	4	yes	—
211	C₄	—	2	12	no	—
212	C₄	—	4	1	no	—
213	C₄	—	4	6	no	—
214	C₃	C₅×#2614	1	1	yes	characteristic, direct product factor, Sylow
215	C₂	—	2	4	no	—
216	C₂	—	8	1	no	—
217	C₂	C₂×C₃×C₅×#120	1	1	yes	characteristic
218	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. C₂³×C₃×C₅×Q₈	C₂	semidirect
3. C₃×C₅×#395	C₂	semidirect
4. C₃×C₅×#393	C₂	semidirect
5. C₃×C₅×#381	C₂	semidirect
6. C₅×#2614	C₃	direct (N × H)
16. C₃×#2614	C₅	direct (N × H)
17. C₂³×C₅×Q₈	C₂×C₃	semidirect
19. C₅×#395	C₂×C₃	semidirect
20. C₅×#393	C₂×C₃	semidirect
21. C₅×#381	C₂×C₃	semidirect
35. C₂³×C₃×Q₈	C₂×C₅	semidirect
37. C₃×#395	C₂×C₅	semidirect
38. C₃×#393	C₂×C₅	semidirect
39. C₃×#381	C₂×C₅	semidirect
49. #2614	C₃×C₅	direct (N × H)
85. C₂³×Q₈	C₂×C₃×C₅	semidirect
87. #395	C₂×C₃×C₅	semidirect
88. #393	C₂×C₃×C₅	semidirect
89. #381	C₂×C₃×C₅	semidirect
173. C₃×C₅	#2614	direct (N × H)
204. C₅	C₃×#2614	direct (N × H)
214. C₃	C₅×#2614	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₃×C₅×Q₈	2
C₂×C₃×C₅×#123	2
C₃×C₅×#395	2
C₃×C₅×#393	2
C₃×C₅×#381	2
C₅×#2614	3
C₃×#2614	5

Subgroups — C3×C5×#2614 #14275648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2614 #14275648