Subgroups — C₂×C₃×C₅×#335 #14274597

Order 1920.

400 conjugacy classes of subgroups.

The subgroup lattice has 228 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×#335	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅×#335	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×#335	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×#335	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×#335	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×#335	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	D₈	1	2	yes	—
50	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
54	C₃×C₅×C₈	—	2	2	no	—
55	C₃×C₅×C₈	—	4	4	no	—
56	C₃×C₅×C₈	C₂×C₈	1	4	yes	complemented
57	C₂²×C₃×C₈	—	2	2	no	—
58	C₂²×C₃×C₈	C₄×C₅	1	1	yes	characteristic
59	C₂²×C₃×C₈	C₄×C₅	1	1	yes	characteristic
60	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
61	C₃×C₄⋊C₈	—	2	2	no	—
62	C₃×C₄⋊C₈	C₂²×C₅	1	4	yes	—
63	C₃×C₄×C₈	C₄×C₅	1	2	yes	—
64	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
65	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₃×Q₈	1	1	yes	characteristic
68	C₂×C₅×C₈	—	2	2	no	—
69	C₂×C₅×C₈	—	2	4	no	—
70	C₂×C₅×C₈	—	4	4	no	—
71	C₂×C₅×C₈	C₃×C₈	1	4	yes	complemented
72	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
73	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
74	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
75	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
76	C₄²×C₅	C₃×Q₈	1	1	yes	characteristic
77	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
78	C₂×C₄⋊C₈	C₂×C₃×C₅	1	2	yes	—
79	C₂×C₄×C₈	C₂×C₃×C₅	1	1	yes	characteristic
80	#335	C₂×C₃×C₅	1	4	yes	direct product factor
81	C₂²×C₃×C₅	C₄⋊C₈	1	2	yes	—
82	C₂²×C₃×C₅	#107	1	2	yes	—
83	C₂²×C₃×C₅	#108	1	2	yes	—
84	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₃×C₄×C₅	—	2	1	no	—
86	C₃×C₄×C₅	—	2	1	no	—
87	C₃×C₄×C₅	C₄⋊C₈	1	2	yes	—
88	C₃×C₄×C₅	#107	1	2	yes	—
89	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
90	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
91	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
92	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₅×Q₈	1	1	yes	characteristic
96	C₂×C₃×C₈	—	2	2	no	—
97	C₂×C₃×C₈	—	2	4	no	—
98	C₂×C₃×C₈	—	4	4	no	—
99	C₂×C₃×C₈	C₅×C₈	1	4	yes	complemented
100	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
101	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
102	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
103	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
104	C₃×C₄²	C₅×Q₈	1	1	yes	characteristic
105	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
106	C₂³×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
112	C₂×C₄×C₅	C₃×M₁₆	1	2	yes	—
113	C₂×C₄×C₅	C₃×D₈	1	2	yes	—
114	C₂×C₄×C₅	C₃×Q₁₆	1	2	yes	—
115	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
118	C₅×C₈	—	2	2	no	—
119	C₅×C₈	—	4	4	no	—
120	C₅×C₈	C₂×C₃×C₈	1	4	yes	complemented
121	C₂²×C₈	—	2	2	no	—
122	C₂²×C₈	C₃×C₄×C₅	1	1	yes	characteristic
123	C₂²×C₈	C₃×C₄×C₅	1	1	yes	characteristic
124	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	C₂²×C₃×C₅	1	4	yes	—
127	C₄×C₈	C₃×C₄×C₅	1	2	yes	—
128	C₄×C₈	C₂²×C₃×C₅	1	2	yes	—
129	C₂×C₃×C₅	#335	1	4	yes	direct product factor
130	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
131	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
132	C₂×C₃×C₅	C₂×#108	1	1	yes	characteristic
133	C₂³×C₃	C₅×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×C₅×C₈	1	2	yes	—
139	C₂×C₃×C₄	C₅×M₁₆	1	2	yes	—
140	C₂×C₃×C₄	C₅×D₈	1	2	yes	—
141	C₂×C₃×C₄	C₅×Q₁₆	1	2	yes	—
142	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
145	C₃×C₈	—	2	2	no	—
146	C₃×C₈	—	4	4	no	—
147	C₃×C₈	C₂×C₅×C₈	1	4	yes	complemented
148	C₂²×C₅	C₃×C₄⋊C₈	1	2	yes	—
149	C₂²×C₅	C₃×#107	1	2	yes	—
150	C₂²×C₅	C₃×#108	1	2	yes	—
151	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₅	—	2	1	no	—
153	C₄×C₅	—	2	1	no	—
154	C₄×C₅	C₃×C₄⋊C₈	1	2	yes	—
155	C₄×C₅	C₃×#107	1	2	yes	—
156	C₄×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
157	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
158	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
159	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
160	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
161	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
162	C₂²×C₄	C₃×C₅×Q₈	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	4	4	no	—
166	C₂×C₈	C₃×C₅×C₈	1	4	yes	complemented
167	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
168	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
169	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
170	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
171	C₄²	C₃×C₅×Q₈	1	1	yes	characteristic
172	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
173	C₃×C₅	C₂×#335	1	1	yes	characteristic, direct product factor, Hall
174	C₂²×C₃	C₅×C₄⋊C₈	1	2	yes	—
175	C₂²×C₃	C₅×#107	1	2	yes	—
176	C₂²×C₃	C₅×#108	1	2	yes	—
177	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
178	C₃×C₄	—	2	1	no	—
179	C₃×C₄	—	2	1	no	—
180	C₃×C₄	C₅×C₄⋊C₈	1	2	yes	—
181	C₃×C₄	C₅×#107	1	2	yes	—
182	C₃×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
183	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
184	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
185	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
186	C₂×C₅	C₃×#335	1	4	yes	direct product factor
187	C₂×C₅	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic
188	C₂×C₅	C₂×C₃×#107	1	1	yes	characteristic
189	C₂×C₅	C₂×C₃×#108	1	1	yes	characteristic
190	C₂³	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
193	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
194	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
195	C₂×C₄	C₂×C₃×C₅×C₈	1	2	yes	—
196	C₂×C₄	C₃×C₅×M₁₆	1	2	yes	—
197	C₂×C₄	C₃×C₅×D₈	1	2	yes	—
198	C₂×C₄	C₃×C₅×Q₁₆	1	2	yes	—
199	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
200	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
201	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	4	4	no	—
204	C₈	C₂×C₃×C₅×C₈	1	4	yes	complemented
205	C₂×C₃	C₅×#335	1	4	yes	direct product factor
206	C₂×C₃	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
207	C₂×C₃	C₂×C₅×#107	1	1	yes	characteristic
208	C₂×C₃	C₂×C₅×#108	1	1	yes	characteristic
209	C₅	C₂×C₃×#335	1	1	yes	characteristic, direct product factor, Sylow
210	C₂²	C₃×C₅×C₄⋊C₈	1	2	yes	—
211	C₂²	C₃×C₅×#107	1	2	yes	—
212	C₂²	C₃×C₅×#108	1	2	yes	—
213	C₂²	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	C₃×C₅×C₄⋊C₈	1	2	yes	—
217	C₄	C₃×C₅×#107	1	2	yes	—
218	C₄	C₂²×C₃×C₅×C₈	1	1	yes	characteristic
219	C₄	C₂×C₃×C₅×M₁₆	1	1	yes	characteristic
220	C₄	C₂×C₃×C₅×D₈	1	1	yes	characteristic
221	C₄	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
222	C₃	C₂×C₅×#335	1	1	yes	characteristic, direct product factor, Sylow
223	C₂	C₃×C₅×#335	1	4	yes	direct product factor
224	C₂	C₂×C₃×C₅×C₄⋊C₈	1	1	yes	characteristic
225	C₂	C₂×C₃×C₅×#107	1	1	yes	characteristic
226	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×#335	C₂	direct (N × H)
4. C₂×C₅×#335	C₃	direct (N × H)
13. C₂×C₃×#335	C₅	direct (N × H)
16. C₅×#335	C₂×C₃	direct (N × H)
23. C₂×C₃×C₅×C₈	C₈	semidirect
32. C₃×#335	C₂×C₅	direct (N × H)
41. C₂×#335	C₃×C₅	direct (N × H)
56. C₃×C₅×C₈	C₂×C₈	semidirect
71. C₂×C₅×C₈	C₃×C₈	semidirect
80. #335	C₂×C₃×C₅	direct (N × H)
99. C₂×C₃×C₈	C₅×C₈	semidirect
120. C₅×C₈	C₂×C₃×C₈	semidirect
129. C₂×C₃×C₅	#335	direct (N × H)
147. C₃×C₈	C₂×C₅×C₈	semidirect
166. C₂×C₈	C₃×C₅×C₈	semidirect
173. C₃×C₅	C₂×#335	direct (N × H)
186. C₂×C₅	C₃×#335	direct (N × H)
204. C₈	C₂×C₃×C₅×C₈	semidirect
205. C₂×C₃	C₅×#335	direct (N × H)
209. C₅	C₂×C₃×#335	direct (N × H)
222. C₃	C₂×C₅×#335	direct (N × H)
223. C₂	C₃×C₅×#335	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×#335	2
C₂×C₅×#335	3
C₂×C₃×#335	5

Subgroups — C2×C3×C5×#335 #14274597

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#335 #14274597