Subgroups — C₂×C₃×C₅×#334 #14274596

Order 1920.

400 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	C₂×C₃×C₅×C₄⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#334	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅×#334	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₅×C₈	—	2	2	no	—
6	C₂²×C₃×C₅×C₈	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	C₄	1	1	yes	characteristic
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₃×C₅×C₄⋊C₈	—	2	2	no	—
10	C₃×C₅×C₄⋊C₈	C₂²	1	4	yes	—
11	C₃×C₄×C₅×C₈	C₄	1	2	yes	—
12	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
13	C₂×C₃×#334	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₅×C₄⋊C₈	C₂×C₃	1	2	yes	—
15	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
16	C₅×#334	C₂×C₃	1	4	yes	direct product factor
17	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
19	C₂²×C₃×C₄×C₅	Q₈	1	1	yes	characteristic
20	C₂×C₃×C₅×C₈	—	2	2	no	—
21	C₂×C₃×C₅×C₈	—	2	4	no	—
22	C₂×C₃×C₅×C₈	—	4	4	no	—
23	C₂×C₃×C₅×C₈	C₈	1	4	yes	complemented
24	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
26	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
28	C₃×C₄²×C₅	Q₈	1	1	yes	characteristic
29	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
30	C₂×C₃×C₄⋊C₈	C₂×C₅	1	2	yes	—
31	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
32	C₃×#334	C₂×C₅	1	4	yes	direct product factor
33	C₂²×C₅×C₈	—	2	2	no	—
34	C₂²×C₅×C₈	C₃×C₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₃×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
37	C₅×C₄⋊C₈	—	2	2	no	—
38	C₅×C₄⋊C₈	C₂²×C₃	1	4	yes	—
39	C₄×C₅×C₈	C₃×C₄	1	2	yes	—
40	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
41	C₂×#334	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
42	C₂³×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₃×C₄×C₅	—	2	2	no	—
44	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
48	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
49	C₂×C₃×C₄×C₅	SD₁₆	1	4	yes	—
50	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
53	C₃×C₅×C₈	—	2	2	no	—
54	C₃×C₅×C₈	—	4	4	no	—
55	C₃×C₅×C₈	C₂×C₈	1	4	yes	complemented
56	C₂²×C₃×C₈	—	2	2	no	—
57	C₂²×C₃×C₈	C₄×C₅	1	1	yes	characteristic
58	C₂²×C₃×C₈	C₄×C₅	1	1	yes	characteristic
59	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
60	C₃×C₄⋊C₈	—	2	2	no	—
61	C₃×C₄⋊C₈	C₂²×C₅	1	4	yes	—
62	C₃×C₄×C₈	C₄×C₅	1	2	yes	—
63	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
64	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₃×Q₈	1	1	yes	characteristic
67	C₂×C₅×C₈	—	2	2	no	—
68	C₂×C₅×C₈	—	2	4	no	—
69	C₂×C₅×C₈	—	4	4	no	—
70	C₂×C₅×C₈	C₃×C₈	1	4	yes	complemented
71	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
72	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
73	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
74	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
75	C₄²×C₅	C₃×Q₈	1	1	yes	characteristic
76	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
77	C₂×C₄⋊C₈	C₂×C₃×C₅	1	2	yes	—
78	C₂×C₄×C₈	C₂×C₃×C₅	1	1	yes	characteristic
79	#334	C₂×C₃×C₅	1	4	yes	direct product factor
80	C₂²×C₃×C₅	C₄⋊C₈	1	2	yes	—
81	C₂²×C₃×C₅	#106	1	2	yes	—
82	C₂²×C₃×C₅	#108	1	2	yes	—
83	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₃×C₄×C₅	—	2	1	no	—
85	C₃×C₄×C₅	—	2	1	no	—
86	C₃×C₄×C₅	C₄⋊C₈	1	2	yes	—
87	C₃×C₄×C₅	#106	1	2	yes	—
88	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
89	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
90	C₃×C₄×C₅	C₂×SD₁₆	1	2	yes	—
91	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
92	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₅×Q₈	1	1	yes	characteristic
94	C₂×C₃×C₈	—	2	2	no	—
95	C₂×C₃×C₈	—	2	4	no	—
96	C₂×C₃×C₈	—	4	4	no	—
97	C₂×C₃×C₈	C₅×C₈	1	4	yes	complemented
98	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
99	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
100	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
101	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
102	C₃×C₄²	C₅×Q₈	1	1	yes	characteristic
103	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
104	C₂³×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
110	C₂×C₄×C₅	C₃×M₁₆	1	2	yes	—
111	C₂×C₄×C₅	C₃×SD₁₆	1	4	yes	—
112	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
115	C₅×C₈	—	2	2	no	—
116	C₅×C₈	—	4	4	no	—
117	C₅×C₈	C₂×C₃×C₈	1	4	yes	complemented
118	C₂²×C₈	—	2	2	no	—
119	C₂²×C₈	C₃×C₄×C₅	1	1	yes	characteristic
120	C₂²×C₈	C₃×C₄×C₅	1	1	yes	characteristic
121	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	C₂²×C₃×C₅	1	4	yes	—
124	C₄×C₈	C₃×C₄×C₅	1	2	yes	—
125	C₄×C₈	C₂²×C₃×C₅	1	2	yes	—
126	C₂×C₃×C₅	#334	1	4	yes	direct product factor
127	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
128	C₂×C₃×C₅	C₂×#106	1	1	yes	characteristic
129	C₂×C₃×C₅	C₂×#108	1	1	yes	characteristic
130	C₂³×C₃	C₅×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂×C₅×C₈	1	2	yes	—
136	C₂×C₃×C₄	C₅×M₁₆	1	2	yes	—
137	C₂×C₃×C₄	C₅×SD₁₆	1	4	yes	—
138	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
141	C₃×C₈	—	2	2	no	—
142	C₃×C₈	—	4	4	no	—
143	C₃×C₈	C₂×C₅×C₈	1	4	yes	complemented
144	C₂²×C₅	C₃×C₄⋊C₈	1	2	yes	—
145	C₂²×C₅	C₃×#106	1	2	yes	—
146	C₂²×C₅	C₃×#108	1	2	yes	—
147	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
148	C₄×C₅	—	2	1	no	—
149	C₄×C₅	—	2	1	no	—
150	C₄×C₅	C₃×C₄⋊C₈	1	2	yes	—
151	C₄×C₅	C₃×#106	1	2	yes	—
152	C₄×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
153	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
154	C₄×C₅	C₂×C₃×SD₁₆	1	2	yes	—
155	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
156	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
157	C₂²×C₄	C₃×C₅×Q₈	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	4	4	no	—
161	C₂×C₈	C₃×C₅×C₈	1	4	yes	complemented
162	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
163	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
164	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
165	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
166	C₄²	C₃×C₅×Q₈	1	1	yes	characteristic
167	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
168	C₃×C₅	C₂×#334	1	1	yes	characteristic, direct product factor, Hall
169	C₂²×C₃	C₅×C₄⋊C₈	1	2	yes	—
170	C₂²×C₃	C₅×#106	1	2	yes	—
171	C₂²×C₃	C₅×#108	1	2	yes	—
172	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
173	C₃×C₄	—	2	1	no	—
174	C₃×C₄	—	2	1	no	—
175	C₃×C₄	C₅×C₄⋊C₈	1	2	yes	—
176	C₃×C₄	C₅×#106	1	2	yes	—
177	C₃×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
178	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
179	C₃×C₄	C₂×C₅×SD₁₆	1	2	yes	—
180	C₂×C₅	C₃×#334	1	4	yes	direct product factor
181	C₂×C₅	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic
182	C₂×C₅	C₂×C₃×#106	1	1	yes	characteristic
183	C₂×C₅	C₂×C₃×#108	1	1	yes	characteristic
184	C₂³	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
187	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
188	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
189	C₂×C₄	C₂×C₃×C₅×C₈	1	2	yes	—
190	C₂×C₄	C₃×C₅×M₁₆	1	2	yes	—
191	C₂×C₄	C₃×C₅×SD₁₆	1	4	yes	—
192	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
193	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
194	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
195	C₈	—	2	2	no	—
196	C₈	—	4	4	no	—
197	C₈	C₂×C₃×C₅×C₈	1	4	yes	complemented
198	C₂×C₃	C₅×#334	1	4	yes	direct product factor
199	C₂×C₃	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
200	C₂×C₃	C₂×C₅×#106	1	1	yes	characteristic
201	C₂×C₃	C₂×C₅×#108	1	1	yes	characteristic
202	C₅	C₂×C₃×#334	1	1	yes	characteristic, direct product factor, Sylow
203	C₂²	C₃×C₅×C₄⋊C₈	1	2	yes	—
204	C₂²	C₃×C₅×#106	1	2	yes	—
205	C₂²	C₃×C₅×#108	1	2	yes	—
206	C₂²	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	C₃×C₅×C₄⋊C₈	1	2	yes	—
210	C₄	C₃×C₅×#106	1	2	yes	—
211	C₄	C₂²×C₃×C₅×C₈	1	1	yes	characteristic
212	C₄	C₂×C₃×C₅×M₁₆	1	1	yes	characteristic
213	C₄	C₂×C₃×C₅×SD₁₆	1	2	yes	—
214	C₃	C₂×C₅×#334	1	1	yes	characteristic, direct product factor, Sylow
215	C₂	C₃×C₅×#334	1	4	yes	direct product factor
216	C₂	C₂×C₃×C₅×C₄⋊C₈	1	1	yes	characteristic
217	C₂	C₂×C₃×C₅×#106	1	1	yes	characteristic
218	C₂	C₂×C₃×C₅×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄×C₅	D₄
commutator	C₄	C₂²×C₃×C₅×C₈
Frattini, cosocle	C₄²	C₂³×C₃×C₅
socle, prime radical	C₂³×C₃×C₅	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄×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×#334	C₂	direct (N × H)
4. C₂×C₅×#334	C₃	direct (N × H)
13. C₂×C₃×#334	C₅	direct (N × H)
16. C₅×#334	C₂×C₃	direct (N × H)
23. C₂×C₃×C₅×C₈	C₈	semidirect
32. C₃×#334	C₂×C₅	direct (N × H)
41. C₂×#334	C₃×C₅	direct (N × H)
55. C₃×C₅×C₈	C₂×C₈	semidirect
70. C₂×C₅×C₈	C₃×C₈	semidirect
79. #334	C₂×C₃×C₅	direct (N × H)
97. C₂×C₃×C₈	C₅×C₈	semidirect
117. C₅×C₈	C₂×C₃×C₈	semidirect
126. C₂×C₃×C₅	#334	direct (N × H)
143. C₃×C₈	C₂×C₅×C₈	semidirect
161. C₂×C₈	C₃×C₅×C₈	semidirect
168. C₃×C₅	C₂×#334	direct (N × H)
180. C₂×C₅	C₃×#334	direct (N × H)
197. C₈	C₂×C₃×C₅×C₈	semidirect
198. C₂×C₃	C₅×#334	direct (N × H)
202. C₅	C₂×C₃×#334	direct (N × H)
214. C₃	C₂×C₅×#334	direct (N × H)
215. C₂	C₃×C₅×#334	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⋊C₈	2
C₂×C₃×C₄×C₅×C₈	2
C₃×C₅×#334	2
C₂×C₅×#334	3
C₂×C₃×#334	5

Subgroups — C2×C3×C5×#334 #14274596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#334 #14274596