Subgroups — C₂×C₃×C₅×#494 #14276179

Order 1920.

1104 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×Q₁₆	C₂	1	4	yes	maximal
2	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	2	yes	maximal
3	C₃×C₅×#494	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×#494	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	C₂²	1	4	yes	—
7	C₂×C₃×C₅×Q₁₆	—	2	16	no	—
8	C₂×C₃×C₅×Q₁₆	C₂²	1	16	yes	—
9	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
10	C₃×C₅×C₄⋊Q₈	—	2	4	no	—
11	C₃×C₅×C₄⋊Q₈	C₂²	1	8	yes	—
12	C₂×C₃×C₅×C₄⋊C₄	—	2	4	no	—
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₅×C₈	C₂²	1	4	yes	—
15	C₂×C₃×#494	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂²×C₅×Q₁₆	C₂×C₃	1	4	yes	—
17	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	2	yes	—
18	C₅×#494	C₂×C₃	1	8	yes	direct product factor
19	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×C₅×Q₈	—	2	4	no	—
21	C₂×C₃×C₅×Q₈	—	2	16	no	—
22	C₂×C₃×C₅×Q₈	C₂³	1	8	yes	—
23	C₂²×C₃×C₄×C₅	—	2	4	no	—
24	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₃×C₅×Q₁₆	—	2	32	no	—
27	C₂×C₃×C₅×C₈	D₄	1	8	yes	—
28	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
29	C₃×C₅×C₄⋊C₄	—	2	16	no	—
30	C₃×C₄²×C₅	D₄	1	2	yes	—
31	C₃×C₄²×C₅	C₂³	1	2	yes	—
32	C₂²×C₃×Q₁₆	C₂×C₅	1	4	yes	—
33	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	2	yes	—
34	C₃×#494	C₂×C₅	1	8	yes	direct product factor
35	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
36	C₂²×C₅×Q₈	C₂²×C₃	1	4	yes	—
37	C₂×C₅×Q₁₆	—	2	16	no	—
38	C₂×C₅×Q₁₆	C₂²×C₃	1	16	yes	—
39	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
40	C₅×C₄⋊Q₈	—	2	4	no	—
41	C₅×C₄⋊Q₈	C₂²×C₃	1	8	yes	—
42	C₂×C₅×C₄⋊C₄	—	2	4	no	—
43	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
44	C₄×C₅×C₈	C₂²×C₃	1	4	yes	—
45	C₂×#494	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
46	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
47	C₃×C₅×Q₈	—	2	16	no	—
48	C₃×C₅×Q₈	—	4	8	no	—
49	C₂×C₃×C₄×C₅	—	2	8	no	—
50	C₂×C₃×C₄×C₅	—	4	8	no	—
51	C₂×C₃×C₄×C₅	Q₁₆	1	8	yes	complemented
52	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
54	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
55	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
56	C₃×C₅×C₈	C₂×D₄	1	8	yes	—
57	C₂²×C₃×Q₈	C₂²×C₅	1	4	yes	—
58	C₂×C₃×Q₁₆	—	2	16	no	—
59	C₂×C₃×Q₁₆	C₂²×C₅	1	16	yes	—
60	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
61	C₃×C₄⋊Q₈	—	2	4	no	—
62	C₃×C₄⋊Q₈	C₂²×C₅	1	8	yes	—
63	C₂×C₃×C₄⋊C₄	—	2	4	no	—
64	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
65	C₃×C₄×C₈	C₂²×C₅	1	4	yes	—
66	C₂×C₅×Q₈	—	2	4	no	—
67	C₂×C₅×Q₈	—	2	16	no	—
68	C₂×C₅×Q₈	C₂³×C₃	1	8	yes	—
69	C₂²×C₄×C₅	—	2	4	no	—
70	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
71	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
72	C₅×Q₁₆	—	2	32	no	—
73	C₂×C₅×C₈	C₃×D₄	1	8	yes	—
74	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
75	C₅×C₄⋊C₄	—	2	16	no	—
76	C₄²×C₅	C₃×D₄	1	2	yes	—
77	C₄²×C₅	C₂³×C₃	1	2	yes	—
78	C₂²×Q₁₆	C₂×C₃×C₅	1	4	yes	—
79	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	2	yes	—
80	#494	C₂×C₃×C₅	1	8	yes	direct product factor
81	C₂×C₄×C₈	C₂×C₃×C₅	1	1	yes	characteristic
82	C₂²×C₃×C₅	#127	1	2	yes	—
83	C₂²×C₃×C₅	C₂×Q₁₆	1	4	yes	—
84	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
85	C₃×C₄×C₅	—	4	8	no	—
86	C₃×C₄×C₅	#127	1	2	yes	—
87	C₃×C₄×C₅	C₂×Q₁₆	1	8	yes	complemented
88	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
89	C₂×C₃×Q₈	—	2	4	no	—
90	C₂×C₃×Q₈	—	2	16	no	—
91	C₂×C₃×Q₈	C₂³×C₅	1	8	yes	—
92	C₂²×C₃×C₄	—	2	4	no	—
93	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
94	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
95	C₃×Q₁₆	—	2	32	no	—
96	C₂×C₃×C₈	C₅×D₄	1	8	yes	—
97	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
98	C₃×C₄⋊C₄	—	2	16	no	—
99	C₃×C₄²	C₅×D₄	1	2	yes	—
100	C₃×C₄²	C₂³×C₅	1	2	yes	—
101	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
102	C₅×Q₈	—	2	16	no	—
103	C₅×Q₈	—	4	8	no	—
104	C₂×C₄×C₅	—	2	8	no	—
105	C₂×C₄×C₅	—	4	8	no	—
106	C₂×C₄×C₅	C₃×Q₁₆	1	8	yes	complemented
107	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
109	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
110	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
111	C₅×C₈	C₂×C₃×D₄	1	8	yes	—
112	C₂²×Q₈	C₂²×C₃×C₅	1	4	yes	—
113	C₂×Q₁₆	—	2	16	no	—
114	C₂×Q₁₆	C₂²×C₃×C₅	1	16	yes	—
115	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
116	C₄⋊Q₈	—	2	4	no	—
117	C₄⋊Q₈	C₂²×C₃×C₅	1	8	yes	—
118	C₂×C₄⋊C₄	—	2	4	no	—
119	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
120	C₄×C₈	C₂²×C₃×C₅	1	4	yes	—
121	C₂×C₃×C₅	#494	1	4	yes	direct product factor
122	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
123	C₂×C₃×C₅	C₂²×Q₁₆	1	2	yes	—
124	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
125	C₃×Q₈	—	2	16	no	—
126	C₃×Q₈	—	4	8	no	—
127	C₂×C₃×C₄	—	2	8	no	—
128	C₂×C₃×C₄	—	4	8	no	—
129	C₂×C₃×C₄	C₅×Q₁₆	1	8	yes	complemented
130	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
132	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
133	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
134	C₃×C₈	C₂×C₅×D₄	1	8	yes	—
135	C₂²×C₅	C₃×#127	1	2	yes	—
136	C₂²×C₅	C₂×C₃×Q₁₆	1	4	yes	—
137	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
138	C₄×C₅	—	4	8	no	—
139	C₄×C₅	C₃×#127	1	2	yes	—
140	C₄×C₅	C₂×C₃×Q₁₆	1	8	yes	complemented
141	C₄×C₅	C₂²×C₃×D₄	1	2	yes	—
142	C₂×Q₈	—	2	4	no	—
143	C₂×Q₈	—	2	16	no	—
144	C₂×Q₈	C₂³×C₃×C₅	1	8	yes	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
147	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
148	Q₁₆	—	2	32	no	—
149	C₂×C₈	C₃×C₅×D₄	1	8	yes	—
150	C₂×C₈	C₂³×C₃×C₅	1	4	yes	—
151	C₄⋊C₄	—	2	16	no	—
152	C₄²	C₃×C₅×D₄	1	2	yes	—
153	C₄²	C₂³×C₃×C₅	1	2	yes	—
154	C₃×C₅	C₂×#494	1	1	yes	characteristic, direct product factor, Hall
155	C₂²×C₃	C₅×#127	1	2	yes	—
156	C₂²×C₃	C₂×C₅×Q₁₆	1	4	yes	—
157	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
158	C₃×C₄	—	4	8	no	—
159	C₃×C₄	C₅×#127	1	2	yes	—
160	C₃×C₄	C₂×C₅×Q₁₆	1	8	yes	complemented
161	C₃×C₄	C₂²×C₅×D₄	1	2	yes	—
162	C₂×C₅	C₃×#494	1	4	yes	direct product factor
163	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
164	C₂×C₅	C₂²×C₃×Q₁₆	1	2	yes	—
165	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
166	Q₈	—	2	16	no	—
167	Q₈	—	4	8	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	4	8	no	—
170	C₂×C₄	C₃×C₅×Q₁₆	1	8	yes	complemented
171	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
172	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
173	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
174	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
175	C₈	C₂×C₃×C₅×D₄	1	8	yes	—
176	C₂×C₃	C₅×#494	1	4	yes	direct product factor
177	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
178	C₂×C₃	C₂²×C₅×Q₁₆	1	2	yes	—
179	C₅	C₂×C₃×#494	1	1	yes	characteristic, direct product factor, Sylow
180	C₂²	C₃×C₅×#127	1	2	yes	—
181	C₂²	C₂×C₃×C₅×Q₁₆	1	4	yes	—
182	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
183	C₄	—	4	8	no	—
184	C₄	C₃×C₅×#127	1	2	yes	—
185	C₄	C₂×C₃×C₅×Q₁₆	1	8	yes	complemented
186	C₄	C₂²×C₃×C₅×D₄	1	2	yes	—
187	C₃	C₂×C₅×#494	1	1	yes	characteristic, direct product factor, Sylow
188	C₂	C₃×C₅×#494	1	4	yes	direct product factor
189	C₂	C₂×C₃×C₅×#127	1	1	yes	characteristic
190	C₂	C₂²×C₃×C₅×Q₁₆	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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×#494	C₂	direct (N × H)
5. C₂×C₅×#494	C₃	direct (N × H)
15. C₂×C₃×#494	C₅	direct (N × H)
18. C₅×#494	C₂×C₃	direct (N × H)
34. C₃×#494	C₂×C₅	direct (N × H)
45. C₂×#494	C₃×C₅	direct (N × H)
51. C₂×C₃×C₄×C₅	Q₁₆	semidirect
80. #494	C₂×C₃×C₅	direct (N × H)
87. C₃×C₄×C₅	C₂×Q₁₆	semidirect
106. C₂×C₄×C₅	C₃×Q₁₆	semidirect
121. C₂×C₃×C₅	#494	direct (N × H)
129. C₂×C₃×C₄	C₅×Q₁₆	semidirect
140. C₄×C₅	C₂×C₃×Q₁₆	semidirect
154. C₃×C₅	C₂×#494	direct (N × H)
160. C₃×C₄	C₂×C₅×Q₁₆	semidirect
162. C₂×C₅	C₃×#494	direct (N × H)
170. C₂×C₄	C₃×C₅×Q₁₆	semidirect
176. C₂×C₃	C₅×#494	direct (N × H)
179. C₅	C₂×C₃×#494	direct (N × H)
185. C₄	C₂×C₃×C₅×Q₁₆	semidirect
187. C₃	C₂×C₅×#494	direct (N × H)
188. C₂	C₃×C₅×#494	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×Q₁₆	2
C₂×C₃×C₅×C₄⋊Q₈	2
C₃×C₅×#494	2
C₂×C₃×C₄×C₅×C₈	2
C₂×C₅×#494	3
C₂×C₃×#494	5

Subgroups — C2×C3×C5×#494 #14276179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#494 #14276179