Subgroups — C₂×C₃×C₅×#498 #14276191

Order 1920.

848 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×C₄⋊Q₈	C₂	1	2	yes	maximal
2	C₃×C₅×#498	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₃×C₅×#106	C₂	1	4	yes	maximal
4	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×#498	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	—	2	2	no	—
7	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
8	C₃×C₅×C₄⋊Q₈	—	2	4	no	—
9	C₃×C₅×C₄⋊Q₈	C₂²	1	8	yes	—
10	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₃×C₅×#106	C₂²	1	16	yes	—
14	C₃×C₄×C₅×C₈	C₂²	1	4	yes	—
15	C₂×C₃×#498	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	2	yes	—
17	C₅×#498	C₂×C₃	1	8	yes	direct product factor
18	C₂×C₅×#106	C₂×C₃	1	4	yes	—
19	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×C₅×Q₈	—	2	8	no	—
21	C₂×C₃×C₅×Q₈	—	4	8	no	—
22	C₂²×C₃×C₄×C₅	—	2	4	no	—
23	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	Q₈	1	8	yes	complemented
26	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
27	C₃×C₅×C₄⋊C₄	—	2	8	no	—
28	C₃×C₅×C₄⋊C₄	—	2	4	no	—
29	C₃×C₅×C₄⋊C₄	C₂³	1	8	yes	—
30	C₃×C₄²×C₅	D₄	1	2	yes	—
31	C₃×C₄²×C₅	C₂³	1	2	yes	—
32	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	2	yes	—
33	C₃×#498	C₂×C₅	1	8	yes	direct product factor
34	C₂×C₃×#106	C₂×C₅	1	4	yes	—
35	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
36	C₂²×C₅×Q₈	—	2	2	no	—
37	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
38	C₅×C₄⋊Q₈	—	2	4	no	—
39	C₅×C₄⋊Q₈	C₂²×C₃	1	8	yes	—
40	C₂×C₅×C₄⋊C₄	—	2	2	no	—
41	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
42	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
43	C₅×#106	C₂²×C₃	1	16	yes	—
44	C₄×C₅×C₈	C₂²×C₃	1	4	yes	—
45	C₂×#498	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
46	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
47	C₃×C₅×Q₈	—	4	16	no	—
48	C₂×C₃×C₄×C₅	—	2	8	no	—
49	C₂×C₃×C₄×C₅	—	4	8	no	—
50	C₂×C₃×C₄×C₅	SD₁₆	1	8	yes	—
51	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
53	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
54	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
55	C₃×C₅×C₈	C₂×Q₈	1	8	yes	complemented
56	C₂²×C₃×Q₈	—	2	2	no	—
57	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
58	C₃×C₄⋊Q₈	—	2	4	no	—
59	C₃×C₄⋊Q₈	C₂²×C₅	1	8	yes	—
60	C₂×C₃×C₄⋊C₄	—	2	2	no	—
61	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
62	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
63	C₃×#106	C₂²×C₅	1	16	yes	—
64	C₃×C₄×C₈	C₂²×C₅	1	4	yes	—
65	C₂×C₅×Q₈	—	2	8	no	—
66	C₂×C₅×Q₈	—	4	8	no	—
67	C₂²×C₄×C₅	—	2	4	no	—
68	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
70	C₂×C₅×C₈	C₃×Q₈	1	8	yes	complemented
71	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
72	C₅×C₄⋊C₄	—	2	8	no	—
73	C₅×C₄⋊C₄	—	2	4	no	—
74	C₅×C₄⋊C₄	C₂³×C₃	1	8	yes	—
75	C₄²×C₅	C₃×D₄	1	2	yes	—
76	C₄²×C₅	C₂³×C₃	1	2	yes	—
77	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	2	yes	—
78	#498	C₂×C₃×C₅	1	8	yes	direct product factor
79	C₂×#106	C₂×C₃×C₅	1	4	yes	—
80	C₂×C₄×C₈	C₂×C₃×C₅	1	1	yes	characteristic
81	C₂²×C₃×C₅	C₄⋊Q₈	1	2	yes	—
82	C₂²×C₃×C₅	C₂×SD₁₆	1	4	yes	—
83	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
84	C₃×C₄×C₅	—	4	8	no	—
85	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
86	C₃×C₄×C₅	C₂×SD₁₆	1	8	yes	—
87	C₃×C₄×C₅	C₂²×Q₈	1	2	yes	—
88	C₂×C₃×Q₈	—	2	8	no	—
89	C₂×C₃×Q₈	—	4	8	no	—
90	C₂²×C₃×C₄	—	2	4	no	—
91	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
92	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
93	C₂×C₃×C₈	C₅×Q₈	1	8	yes	complemented
94	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
95	C₃×C₄⋊C₄	—	2	8	no	—
96	C₃×C₄⋊C₄	—	2	4	no	—
97	C₃×C₄⋊C₄	C₂³×C₅	1	8	yes	—
98	C₃×C₄²	C₅×D₄	1	2	yes	—
99	C₃×C₄²	C₂³×C₅	1	2	yes	—
100	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
101	C₅×Q₈	—	4	16	no	—
102	C₂×C₄×C₅	—	2	8	no	—
103	C₂×C₄×C₅	—	4	8	no	—
104	C₂×C₄×C₅	C₃×SD₁₆	1	8	yes	—
105	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
107	C₂×C₄×C₅	C₂×C₃×Q₈	1	4	yes	—
108	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
109	C₅×C₈	C₂×C₃×Q₈	1	8	yes	complemented
110	C₂²×Q₈	—	2	2	no	—
111	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
112	C₄⋊Q₈	—	2	4	no	—
113	C₄⋊Q₈	C₂²×C₃×C₅	1	8	yes	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
116	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
117	#106	C₂²×C₃×C₅	1	16	yes	—
118	C₄×C₈	C₂²×C₃×C₅	1	4	yes	—
119	C₂×C₃×C₅	#498	1	4	yes	direct product factor
120	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
121	C₂×C₃×C₅	C₂²×SD₁₆	1	2	yes	—
122	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
123	C₃×Q₈	—	4	16	no	—
124	C₂×C₃×C₄	—	2	8	no	—
125	C₂×C₃×C₄	—	4	8	no	—
126	C₂×C₃×C₄	C₅×SD₁₆	1	8	yes	—
127	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
129	C₂×C₃×C₄	C₂×C₅×Q₈	1	4	yes	—
130	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
131	C₃×C₈	C₂×C₅×Q₈	1	8	yes	complemented
132	C₂²×C₅	C₃×C₄⋊Q₈	1	2	yes	—
133	C₂²×C₅	C₂×C₃×SD₁₆	1	4	yes	—
134	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
135	C₄×C₅	—	4	8	no	—
136	C₄×C₅	C₃×C₄⋊Q₈	1	2	yes	—
137	C₄×C₅	C₂×C₃×SD₁₆	1	8	yes	—
138	C₄×C₅	C₂²×C₃×Q₈	1	2	yes	—
139	C₂×Q₈	—	2	8	no	—
140	C₂×Q₈	—	4	8	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
143	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
144	C₂×C₈	C₃×C₅×Q₈	1	8	yes	complemented
145	C₂×C₈	C₂³×C₃×C₅	1	4	yes	—
146	C₄⋊C₄	—	2	8	no	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	C₂³×C₃×C₅	1	8	yes	—
149	C₄²	C₃×C₅×D₄	1	2	yes	—
150	C₄²	C₂³×C₃×C₅	1	2	yes	—
151	C₃×C₅	C₂×#498	1	1	yes	characteristic, direct product factor, Hall
152	C₂²×C₃	C₅×C₄⋊Q₈	1	2	yes	—
153	C₂²×C₃	C₂×C₅×SD₁₆	1	4	yes	—
154	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
155	C₃×C₄	—	4	8	no	—
156	C₃×C₄	C₅×C₄⋊Q₈	1	2	yes	—
157	C₃×C₄	C₂×C₅×SD₁₆	1	8	yes	—
158	C₃×C₄	C₂²×C₅×Q₈	1	2	yes	—
159	C₂×C₅	C₃×#498	1	4	yes	direct product factor
160	C₂×C₅	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
161	C₂×C₅	C₂²×C₃×SD₁₆	1	2	yes	—
162	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
163	Q₈	—	4	16	no	—
164	C₂×C₄	—	2	8	no	—
165	C₂×C₄	—	4	8	no	—
166	C₂×C₄	C₃×C₅×SD₁₆	1	8	yes	—
167	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
168	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
169	C₂×C₄	C₂×C₃×C₅×Q₈	1	4	yes	—
170	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
171	C₈	C₂×C₃×C₅×Q₈	1	8	yes	complemented
172	C₂×C₃	C₅×#498	1	4	yes	direct product factor
173	C₂×C₃	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
174	C₂×C₃	C₂²×C₅×SD₁₆	1	2	yes	—
175	C₅	C₂×C₃×#498	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²	C₃×C₅×C₄⋊Q₈	1	2	yes	—
177	C₂²	C₂×C₃×C₅×SD₁₆	1	4	yes	—
178	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
179	C₄	—	4	8	no	—
180	C₄	C₃×C₅×C₄⋊Q₈	1	2	yes	—
181	C₄	C₂×C₃×C₅×SD₁₆	1	8	yes	—
182	C₄	C₂²×C₃×C₅×Q₈	1	2	yes	—
183	C₃	C₂×C₅×#498	1	1	yes	characteristic, direct product factor, Sylow
184	C₂	C₃×C₅×#498	1	4	yes	direct product factor
185	C₂	C₂×C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
186	C₂	C₂²×C₃×C₅×SD₁₆	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
2. C₃×C₅×#498	C₂	direct (N × H)
5. C₂×C₅×#498	C₃	direct (N × H)
15. C₂×C₃×#498	C₅	direct (N × H)
17. C₅×#498	C₂×C₃	direct (N × H)
25. C₂×C₃×C₅×C₈	Q₈	semidirect
33. C₃×#498	C₂×C₅	direct (N × H)
45. C₂×#498	C₃×C₅	direct (N × H)
55. C₃×C₅×C₈	C₂×Q₈	semidirect
70. C₂×C₅×C₈	C₃×Q₈	semidirect
78. #498	C₂×C₃×C₅	direct (N × H)
93. C₂×C₃×C₈	C₅×Q₈	semidirect
109. C₅×C₈	C₂×C₃×Q₈	semidirect
119. C₂×C₃×C₅	#498	direct (N × H)
131. C₃×C₈	C₂×C₅×Q₈	semidirect
144. C₂×C₈	C₃×C₅×Q₈	semidirect
151. C₃×C₅	C₂×#498	direct (N × H)
159. C₂×C₅	C₃×#498	direct (N × H)
171. C₈	C₂×C₃×C₅×Q₈	semidirect
172. C₂×C₃	C₅×#498	direct (N × H)
175. C₅	C₂×C₃×#498	direct (N × H)
183. C₃	C₂×C₅×#498	direct (N × H)
184. C₂	C₃×C₅×#498	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
C₃×C₅×#498	2
C₂×C₃×C₅×#106	2
C₂×C₃×C₄×C₅×C₈	2
C₂×C₅×#498	3
C₂×C₃×#498	5

Subgroups — C2×C3×C5×#498 #14276191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#498 #14276191