Subgroups — C₂×C₃×C₅×#331 #14274572

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#331	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅×#331	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₅×D₄	—	2	2	no	—
6	C₂²×C₃×C₅×D₄	C₄	1	2	yes	—
7	C₂²×C₃×C₅×C₈	—	2	2	no	—
8	C₃×C₅×#127	—	2	2	no	—
9	C₃×C₅×#127	C₄	1	2	yes	—
10	C₃×C₅×#127	C₂²	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊C₈	—	2	2	no	—
13	C₃×C₅×C₄⋊C₈	C₂²	1	4	yes	complemented
14	C₂×C₃×#331	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂×C₅×#127	C₂×C₃	1	1	yes	characteristic
16	C₂×C₅×C₄⋊C₈	C₂×C₃	1	2	yes	complemented
17	C₅×#331	C₂×C₃	1	4	yes	direct product factor
18	C₂⁴×C₃×C₅	—	2	2	no	—
19	C₂×C₃×C₅×D₄	—	2	4	no	—
20	C₂×C₃×C₅×D₄	—	2	2	no	—
21	C₂×C₃×C₅×D₄	—	2	4	no	—
22	C₂×C₃×C₅×D₄	—	4	4	no	—
23	C₂×C₃×C₅×D₄	—	4	4	no	—
24	C₂×C₃×C₅×D₄	—	4	4	no	—
25	C₂×C₃×C₅×D₄	C₂×C₄	1	4	yes	—
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
28	C₂×C₃×C₅×C₈	—	2	4	no	—
29	C₂×C₃×C₅×C₈	—	4	4	no	—
30	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
31	C₃×C₄²×C₅	D₄	1	2	yes	—
32	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₃×#127	C₂×C₅	1	1	yes	characteristic
34	C₂×C₃×C₄⋊C₈	C₂×C₅	1	2	yes	complemented
35	C₃×#331	C₂×C₅	1	4	yes	direct product factor
36	C₂²×C₅×D₄	—	2	2	no	—
37	C₂²×C₅×D₄	C₃×C₄	1	2	yes	—
38	C₂²×C₅×C₈	—	2	2	no	—
39	C₅×#127	—	2	2	no	—
40	C₅×#127	C₃×C₄	1	2	yes	—
41	C₅×#127	C₂²×C₃	1	2	yes	—
42	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
43	C₅×C₄⋊C₈	—	2	2	no	—
44	C₅×C₄⋊C₈	C₂²×C₃	1	4	yes	complemented
45	C₂×#331	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
46	C₂³×C₃×C₅	—	2	4	no	—
47	C₂³×C₃×C₅	—	4	4	no	—
48	C₂³×C₃×C₅	—	4	8	no	—
49	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
50	C₃×C₅×D₄	—	4	4	no	—
51	C₃×C₅×D₄	—	4	4	no	—
52	C₃×C₅×D₄	—	4	8	no	—
53	C₃×C₅×D₄	—	4	4	no	—
54	C₃×C₅×D₄	—	4	4	no	—
55	C₂×C₃×C₄×C₅	—	2	2	no	—
56	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
58	C₂×C₃×C₄×C₅	D₈	1	4	yes	—
59	C₂×C₃×C₄×C₅	SD₁₆	1	4	yes	—
60	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
62	C₃×C₅×C₈	—	4	4	no	—
63	C₂²×C₃×D₄	—	2	2	no	—
64	C₂²×C₃×D₄	C₄×C₅	1	2	yes	—
65	C₂²×C₃×C₈	—	2	2	no	—
66	C₃×#127	—	2	2	no	—
67	C₃×#127	C₄×C₅	1	2	yes	—
68	C₃×#127	C₂²×C₅	1	2	yes	—
69	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
70	C₃×C₄⋊C₈	—	2	2	no	—
71	C₃×C₄⋊C₈	C₂²×C₅	1	4	yes	complemented
72	C₂⁴×C₅	—	2	2	no	—
73	C₂×C₅×D₄	—	2	2	no	—
74	C₂×C₅×D₄	—	2	4	no	—
75	C₂×C₅×D₄	—	2	4	no	—
76	C₂×C₅×D₄	—	4	4	no	—
77	C₂×C₅×D₄	—	4	4	no	—
78	C₂×C₅×D₄	—	4	4	no	—
79	C₂×C₅×D₄	C₂×C₃×C₄	1	4	yes	—
80	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
82	C₂×C₅×C₈	—	2	4	no	—
83	C₂×C₅×C₈	—	4	4	no	—
84	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
85	C₄²×C₅	C₃×D₄	1	2	yes	—
86	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
87	C₂×#127	C₂×C₃×C₅	1	1	yes	characteristic
88	C₂×C₄⋊C₈	C₂×C₃×C₅	1	2	yes	complemented
89	#331	C₂×C₃×C₅	1	4	yes	direct product factor
90	C₂²×C₃×C₅	—	4	4	no	—
91	C₂²×C₃×C₅	—	4	8	no	—
92	C₂²×C₃×C₅	—	8	8	no	—
93	C₂²×C₃×C₅	#100	1	2	yes	—
94	C₂²×C₃×C₅	#102	1	4	yes	—
95	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₃×C₄×C₅	—	2	1	no	—
97	C₃×C₄×C₅	—	2	1	no	—
98	C₃×C₄×C₅	#102	1	4	yes	—
99	C₃×C₄×C₅	C₂×D₈	1	2	yes	—
100	C₃×C₄×C₅	C₂×SD₁₆	1	2	yes	—
101	C₂⁴×C₃	—	2	2	no	—
102	C₂×C₃×D₄	—	2	2	no	—
103	C₂×C₃×D₄	—	2	4	no	—
104	C₂×C₃×D₄	—	2	4	no	—
105	C₂×C₃×D₄	—	4	4	no	—
106	C₂×C₃×D₄	—	4	4	no	—
107	C₂×C₃×D₄	—	4	4	no	—
108	C₂×C₃×D₄	C₂×C₄×C₅	1	4	yes	—
109	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
111	C₂×C₃×C₈	—	2	4	no	—
112	C₂×C₃×C₈	—	4	4	no	—
113	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
114	C₃×C₄²	C₅×D₄	1	2	yes	—
115	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
116	C₂³×C₅	—	2	4	no	—
117	C₂³×C₅	—	4	4	no	—
118	C₂³×C₅	—	4	8	no	—
119	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
120	C₅×D₄	—	4	4	no	—
121	C₅×D₄	—	4	4	no	—
122	C₅×D₄	—	4	8	no	—
123	C₅×D₄	—	4	4	no	—
124	C₅×D₄	—	4	4	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
128	C₂×C₄×C₅	C₃×D₈	1	4	yes	—
129	C₂×C₄×C₅	C₃×SD₁₆	1	4	yes	—
130	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
132	C₅×C₈	—	4	4	no	—
133	C₂²×D₄	—	2	2	no	—
134	C₂²×D₄	C₃×C₄×C₅	1	2	yes	—
135	C₂²×C₈	—	2	2	no	—
136	#127	—	2	2	no	—
137	#127	C₃×C₄×C₅	1	2	yes	—
138	#127	C₂²×C₃×C₅	1	2	yes	—
139	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	C₂²×C₃×C₅	1	4	yes	complemented
142	C₂×C₃×C₅	—	8	4	no	—
143	C₂×C₃×C₅	#331	1	4	yes	direct product factor
144	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
145	C₂×C₃×C₅	C₂×#102	1	2	yes	—
146	C₂³×C₃	—	2	4	no	—
147	C₂³×C₃	—	4	4	no	—
148	C₂³×C₃	—	4	8	no	—
149	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
150	C₃×D₄	—	4	4	no	—
151	C₃×D₄	—	4	4	no	—
152	C₃×D₄	—	4	4	no	—
153	C₃×D₄	—	4	4	no	—
154	C₃×D₄	—	4	8	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
158	C₂×C₃×C₄	C₅×D₈	1	4	yes	—
159	C₂×C₃×C₄	C₅×SD₁₆	1	4	yes	—
160	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
162	C₃×C₈	—	4	4	no	—
163	C₂²×C₅	—	4	4	no	—
164	C₂²×C₅	—	4	8	no	—
165	C₂²×C₅	—	8	8	no	—
166	C₂²×C₅	C₃×#100	1	2	yes	—
167	C₂²×C₅	C₃×#102	1	4	yes	—
168	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
169	C₄×C₅	—	2	1	no	—
170	C₄×C₅	—	2	1	no	—
171	C₄×C₅	C₃×#102	1	4	yes	—
172	C₄×C₅	C₂×C₃×D₈	1	2	yes	—
173	C₄×C₅	C₂×C₃×SD₁₆	1	2	yes	—
174	C₂⁴	—	2	2	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	2	4	no	—
177	C₂×D₄	—	2	4	no	—
178	C₂×D₄	—	4	4	no	—
179	C₂×D₄	—	4	4	no	—
180	C₂×D₄	—	4	4	no	—
181	C₂×D₄	C₂×C₃×C₄×C₅	1	4	yes	—
182	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
183	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	4	4	no	—
186	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
187	C₄²	C₃×C₅×D₄	1	2	yes	—
188	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
189	C₃×C₅	C₂×#331	1	1	yes	characteristic, direct product factor, Hall
190	C₂²×C₃	—	4	4	no	—
191	C₂²×C₃	—	4	8	no	—
192	C₂²×C₃	—	8	8	no	—
193	C₂²×C₃	C₅×#100	1	2	yes	—
194	C₂²×C₃	C₅×#102	1	4	yes	—
195	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
196	C₃×C₄	—	2	1	no	—
197	C₃×C₄	—	2	1	no	—
198	C₃×C₄	C₅×#102	1	4	yes	—
199	C₃×C₄	C₂×C₅×D₈	1	2	yes	—
200	C₃×C₄	C₂×C₅×SD₁₆	1	2	yes	—
201	C₂×C₅	—	8	4	no	—
202	C₂×C₅	C₃×#331	1	4	yes	direct product factor
203	C₂×C₅	C₂×C₃×#100	1	1	yes	characteristic
204	C₂×C₅	C₂×C₃×#102	1	2	yes	—
205	C₂³	—	2	4	no	—
206	C₂³	—	4	4	no	—
207	C₂³	—	4	8	no	—
208	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
209	D₄	—	4	4	no	—
210	D₄	—	4	4	no	—
211	D₄	—	4	4	no	—
212	D₄	—	4	4	no	—
213	D₄	—	4	8	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
216	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
217	C₂×C₄	C₃×C₅×D₈	1	4	yes	—
218	C₂×C₄	C₃×C₅×SD₁₆	1	4	yes	—
219	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
220	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
221	C₈	—	4	4	no	—
222	C₂×C₃	—	8	4	no	—
223	C₂×C₃	C₅×#331	1	4	yes	direct product factor
224	C₂×C₃	C₂×C₅×#100	1	1	yes	characteristic
225	C₂×C₃	C₂×C₅×#102	1	2	yes	—
226	C₅	C₂×C₃×#331	1	1	yes	characteristic, direct product factor, Sylow
227	C₂²	—	4	4	no	—
228	C₂²	—	4	8	no	—
229	C₂²	—	8	8	no	—
230	C₂²	C₃×C₅×#100	1	2	yes	—
231	C₂²	C₃×C₅×#102	1	4	yes	—
232	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	C₃×C₅×#102	1	4	yes	—
236	C₄	C₂×C₃×C₅×D₈	1	2	yes	—
237	C₄	C₂×C₃×C₅×SD₁₆	1	2	yes	—
238	C₃	C₂×C₅×#331	1	1	yes	characteristic, direct product factor, Sylow
239	C₂	—	8	4	no	—
240	C₂	C₃×C₅×#331	1	4	yes	direct product factor
241	C₂	C₂×C₃×C₅×#100	1	1	yes	characteristic
242	C₂	C₂×C₃×C₅×#102	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₄²	C₂³×C₃×C₅
prime radical	C₂×C₃×C₅×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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. C₂×C₃×C₅×C₄⋊C₈	C₂	semidirect
3. C₃×C₅×#331	C₂	direct (N × H)
4. C₂×C₅×#331	C₃	direct (N × H)
13. C₃×C₅×C₄⋊C₈	C₂²	semidirect
14. C₂×C₃×#331	C₅	direct (N × H)
16. C₂×C₅×C₄⋊C₈	C₂×C₃	semidirect
17. C₅×#331	C₂×C₃	direct (N × H)
34. C₂×C₃×C₄⋊C₈	C₂×C₅	semidirect
35. C₃×#331	C₂×C₅	direct (N × H)
44. C₅×C₄⋊C₈	C₂²×C₃	semidirect
45. C₂×#331	C₃×C₅	direct (N × H)
71. C₃×C₄⋊C₈	C₂²×C₅	semidirect
88. C₂×C₄⋊C₈	C₂×C₃×C₅	semidirect
89. #331	C₂×C₃×C₅	direct (N × H)
141. C₄⋊C₈	C₂²×C₃×C₅	semidirect
143. C₂×C₃×C₅	#331	direct (N × H)
189. C₃×C₅	C₂×#331	direct (N × H)
202. C₂×C₅	C₃×#331	direct (N × H)
223. C₂×C₃	C₅×#331	direct (N × H)
226. C₅	C₂×C₃×#331	direct (N × H)
238. C₃	C₂×C₅×#331	direct (N × H)
240. C₂	C₃×C₅×#331	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×#127	2
C₂×C₃×C₅×C₄⋊C₈	2
C₃×C₅×#331	2
C₂×C₅×#331	3
C₂×C₃×#331	5

Subgroups — C2×C3×C5×#331 #14274572

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#331 #14274572