Subgroups — C₂×C₃×C₅×#399 #14275427

Order 1920.

1816 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
2	C₂²×C₃×C₅×C₂²⋊C₄	C₂	1	3	yes	maximal
3	C₃×C₅×#399	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×C₅×#95	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#399	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂⁵×C₃×C₅	C₂²	1	1	yes	characteristic
7	C₂³×C₃×C₄×C₅	C₂²	1	3	yes	—
8	C₂³×C₃×C₄×C₅	C₂²	1	3	yes	—
9	C₂×C₃×C₅×C₄⋊C₄	—	2	12	no	—
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	12	yes	complemented
11	C₂×C₃×C₅×C₂²⋊C₄	—	2	12	no	—
12	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	12	yes	—
13	C₃×C₅×#95	C₂²	1	4	yes	complemented
14	C₂×C₃×#399	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂²×C₅×C₄⋊C₄	C₂×C₃	1	3	yes	complemented
16	C₂²×C₅×C₂²⋊C₄	C₂×C₃	1	3	yes	—
17	C₅×#399	C₂×C₃	1	8	yes	direct product factor
18	C₂×C₅×#95	C₂×C₃	1	1	yes	characteristic, complemented
19	C₂⁴×C₃×C₅	—	2	6	no	—
20	C₂⁴×C₃×C₅	—	2	6	no	—
21	C₂⁴×C₃×C₅	Q₈	1	4	yes	—
22	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
23	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	—	2	12	no	—
25	C₂²×C₃×C₄×C₅	—	2	6	no	—
26	C₂²×C₃×C₄×C₅	—	2	12	no	—
27	C₂²×C₃×C₄×C₅	D₄	1	12	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
29	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
30	C₃×C₅×C₄⋊C₄	—	2	24	no	—
31	C₃×C₅×C₂²⋊C₄	—	2	24	no	—
32	C₂²×C₃×C₄⋊C₄	C₂×C₅	1	3	yes	complemented
33	C₂²×C₃×C₂²⋊C₄	C₂×C₅	1	3	yes	—
34	C₃×#399	C₂×C₅	1	8	yes	direct product factor
35	C₂×C₃×#95	C₂×C₅	1	1	yes	characteristic, complemented
36	C₂⁵×C₅	C₂²×C₃	1	1	yes	characteristic
37	C₂³×C₄×C₅	C₂²×C₃	1	3	yes	—
38	C₂³×C₄×C₅	C₂²×C₃	1	3	yes	—
39	C₂×C₅×C₄⋊C₄	—	2	12	no	—
40	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	12	yes	complemented
41	C₂×C₅×C₂²⋊C₄	—	2	12	no	—
42	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
43	C₅×#95	C₂²×C₃	1	4	yes	complemented
44	C₂×#399	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
45	C₂³×C₃×C₅	—	2	24	no	—
46	C₂³×C₃×C₅	—	2	6	no	—
47	C₂³×C₃×C₅	—	2	6	no	—
48	C₂³×C₃×C₅	—	4	12	no	—
49	C₂³×C₃×C₅	—	4	4	no	—
50	C₂³×C₃×C₅	C₂×D₄	1	6	yes	—
51	C₂³×C₃×C₅	C₂×Q₈	1	4	yes	—
52	C₂³×C₃×C₅	C₂×Q₈	1	2	yes	—
53	C₂³×C₃×C₅	#41	1	6	yes	—
54	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	—	2	24	no	—
56	C₂×C₃×C₄×C₅	—	2	12	no	—
57	C₂×C₃×C₄×C₅	—	2	6	no	—
58	C₂×C₃×C₄×C₅	—	2	12	no	—
59	C₂×C₃×C₄×C₅	—	4	12	no	—
60	C₂×C₃×C₄×C₅	C₂×D₄	1	12	yes	—
61	C₂⁵×C₃	C₂²×C₅	1	1	yes	characteristic
62	C₂³×C₃×C₄	C₂²×C₅	1	3	yes	—
63	C₂³×C₃×C₄	C₂²×C₅	1	3	yes	—
64	C₂×C₃×C₄⋊C₄	—	2	12	no	—
65	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	12	yes	complemented
66	C₂×C₃×C₂²⋊C₄	—	2	12	no	—
67	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	12	yes	—
68	C₃×#95	C₂²×C₅	1	4	yes	complemented
69	C₂⁴×C₅	—	2	6	no	—
70	C₂⁴×C₅	—	2	6	no	—
71	C₂⁴×C₅	C₃×Q₈	1	4	yes	—
72	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
73	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
74	C₂²×C₄×C₅	—	2	12	no	—
75	C₂²×C₄×C₅	—	2	6	no	—
76	C₂²×C₄×C₅	—	2	12	no	—
77	C₂²×C₄×C₅	C₃×D₄	1	12	yes	—
78	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
79	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
80	C₅×C₄⋊C₄	—	2	24	no	—
81	C₅×C₂²⋊C₄	—	2	24	no	—
82	C₂²×C₄⋊C₄	C₂×C₃×C₅	1	3	yes	complemented
83	C₂²×C₂²⋊C₄	C₂×C₃×C₅	1	3	yes	—
84	#399	C₂×C₃×C₅	1	8	yes	direct product factor
85	C₂×#95	C₂×C₃×C₅	1	1	yes	characteristic, complemented
86	C₂²×C₃×C₅	—	2	12	no	—
87	C₂²×C₃×C₅	—	4	6	no	—
88	C₂²×C₃×C₅	—	4	2	no	—
89	C₂²×C₃×C₅	—	4	16	no	—
90	C₂²×C₃×C₅	C₂²⋊Q₈	1	12	yes	—
91	C₂²×C₃×C₅	#123	1	12	yes	—
92	C₂²×C₃×C₅	#127	1	4	yes	—
93	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
94	C₂²×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
95	C₂²×C₃×C₅	C₂×#41	1	3	yes	—
96	C₃×C₄×C₅	—	2	12	no	—
97	C₃×C₄×C₅	—	4	6	no	—
98	C₂⁴×C₃	—	2	6	no	—
99	C₂⁴×C₃	—	2	6	no	—
100	C₂⁴×C₃	C₅×Q₈	1	4	yes	—
101	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
102	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
103	C₂²×C₃×C₄	—	2	12	no	—
104	C₂²×C₃×C₄	—	2	6	no	—
105	C₂²×C₃×C₄	—	2	12	no	—
106	C₂²×C₃×C₄	C₅×D₄	1	12	yes	—
107	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
108	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
109	C₃×C₄⋊C₄	—	2	24	no	—
110	C₃×C₂²⋊C₄	—	2	24	no	—
111	C₂³×C₅	—	2	24	no	—
112	C₂³×C₅	—	2	6	no	—
113	C₂³×C₅	—	2	6	no	—
114	C₂³×C₅	—	4	12	no	—
115	C₂³×C₅	—	4	4	no	—
116	C₂³×C₅	C₂×C₃×D₄	1	6	yes	—
117	C₂³×C₅	C₂×C₃×Q₈	1	4	yes	—
118	C₂³×C₅	C₂×C₃×Q₈	1	2	yes	—
119	C₂³×C₅	C₃×#41	1	6	yes	—
120	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
121	C₂×C₄×C₅	—	2	24	no	—
122	C₂×C₄×C₅	—	2	12	no	—
123	C₂×C₄×C₅	—	2	6	no	—
124	C₂×C₄×C₅	—	2	12	no	—
125	C₂×C₄×C₅	—	4	12	no	—
126	C₂×C₄×C₅	C₂×C₃×D₄	1	12	yes	—
127	C₂⁵	C₂²×C₃×C₅	1	1	yes	characteristic
128	C₂³×C₄	C₂²×C₃×C₅	1	3	yes	—
129	C₂³×C₄	C₂²×C₃×C₅	1	3	yes	—
130	C₂×C₄⋊C₄	—	2	12	no	—
131	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	12	yes	complemented
132	C₂×C₂²⋊C₄	—	2	12	no	—
133	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	12	yes	—
134	#95	C₂²×C₃×C₅	1	4	yes	complemented
135	C₂×C₃×C₅	—	4	4	no	—
136	C₂×C₃×C₅	#399	1	8	yes	direct product factor
137	C₂×C₃×C₅	C₂×C₂²⋊Q₈	1	3	yes	—
138	C₂×C₃×C₅	C₂×#123	1	3	yes	—
139	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
140	C₂³×C₃	—	2	24	no	—
141	C₂³×C₃	—	2	6	no	—
142	C₂³×C₃	—	2	6	no	—
143	C₂³×C₃	—	4	12	no	—
144	C₂³×C₃	—	4	4	no	—
145	C₂³×C₃	C₂×C₅×D₄	1	6	yes	—
146	C₂³×C₃	C₂×C₅×Q₈	1	4	yes	—
147	C₂³×C₃	C₂×C₅×Q₈	1	2	yes	—
148	C₂³×C₃	C₅×#41	1	6	yes	—
149	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
150	C₂×C₃×C₄	—	2	24	no	—
151	C₂×C₃×C₄	—	2	12	no	—
152	C₂×C₃×C₄	—	2	6	no	—
153	C₂×C₃×C₄	—	2	12	no	—
154	C₂×C₃×C₄	—	4	12	no	—
155	C₂×C₃×C₄	C₂×C₅×D₄	1	12	yes	—
156	C₂²×C₅	—	2	12	no	—
157	C₂²×C₅	—	4	16	no	—
158	C₂²×C₅	—	4	6	no	—
159	C₂²×C₅	—	4	2	no	—
160	C₂²×C₅	C₃×C₂²⋊Q₈	1	12	yes	—
161	C₂²×C₅	C₃×#123	1	12	yes	—
162	C₂²×C₅	C₃×#127	1	4	yes	—
163	C₂²×C₅	C₂²×C₃×D₄	1	3	yes	—
164	C₂²×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
165	C₂²×C₅	C₂×C₃×#41	1	3	yes	—
166	C₄×C₅	—	2	12	no	—
167	C₄×C₅	—	4	6	no	—
168	C₂⁴	—	2	6	no	—
169	C₂⁴	—	2	6	no	—
170	C₂⁴	C₃×C₅×Q₈	1	4	yes	—
171	C₂⁴	C₂³×C₃×C₅	1	2	yes	—
172	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
173	C₂²×C₄	—	2	12	no	—
174	C₂²×C₄	—	2	6	no	—
175	C₂²×C₄	—	2	12	no	—
176	C₂²×C₄	C₃×C₅×D₄	1	12	yes	—
177	C₂²×C₄	C₂³×C₃×C₅	1	6	yes	—
178	C₂²×C₄	C₂³×C₃×C₅	1	6	yes	—
179	C₄⋊C₄	—	2	24	no	—
180	C₂²⋊C₄	—	2	24	no	—
181	C₃×C₅	C₂×#399	1	1	yes	characteristic, direct product factor, Hall
182	C₂²×C₃	—	2	12	no	—
183	C₂²×C₃	—	4	16	no	—
184	C₂²×C₃	—	4	6	no	—
185	C₂²×C₃	—	4	2	no	—
186	C₂²×C₃	C₅×C₂²⋊Q₈	1	12	yes	—
187	C₂²×C₃	C₅×#123	1	12	yes	—
188	C₂²×C₃	C₅×#127	1	4	yes	—
189	C₂²×C₃	C₂²×C₅×D₄	1	3	yes	—
190	C₂²×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
191	C₂²×C₃	C₂×C₅×#41	1	3	yes	—
192	C₃×C₄	—	2	12	no	—
193	C₃×C₄	—	4	6	no	—
194	C₂×C₅	—	4	4	no	—
195	C₂×C₅	C₃×#399	1	8	yes	direct product factor
196	C₂×C₅	C₂×C₃×C₂²⋊Q₈	1	3	yes	—
197	C₂×C₅	C₂×C₃×#123	1	3	yes	—
198	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
199	C₂³	—	2	6	no	—
200	C₂³	—	2	24	no	—
201	C₂³	—	2	6	no	—
202	C₂³	—	4	12	no	—
203	C₂³	—	4	4	no	—
204	C₂³	C₂×C₃×C₅×D₄	1	6	yes	—
205	C₂³	C₂×C₃×C₅×Q₈	1	4	yes	—
206	C₂³	C₂×C₃×C₅×Q₈	1	2	yes	—
207	C₂³	C₃×C₅×#41	1	6	yes	—
208	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
209	C₂×C₄	—	2	12	no	—
210	C₂×C₄	—	2	24	no	—
211	C₂×C₄	—	2	6	no	—
212	C₂×C₄	—	2	12	no	—
213	C₂×C₄	—	4	12	no	—
214	C₂×C₄	C₂×C₃×C₅×D₄	1	12	yes	—
215	C₂×C₃	—	4	4	no	—
216	C₂×C₃	C₅×#399	1	8	yes	direct product factor
217	C₂×C₃	C₂×C₅×C₂²⋊Q₈	1	3	yes	—
218	C₂×C₃	C₂×C₅×#123	1	3	yes	—
219	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
220	C₅	C₂×C₃×#399	1	1	yes	characteristic, direct product factor, Sylow
221	C₂²	—	2	12	no	—
222	C₂²	—	4	6	no	—
223	C₂²	—	4	2	no	—
224	C₂²	—	4	16	no	—
225	C₂²	C₃×C₅×C₂²⋊Q₈	1	12	yes	—
226	C₂²	C₃×C₅×#123	1	12	yes	—
227	C₂²	C₃×C₅×#127	1	4	yes	—
228	C₂²	C₂²×C₃×C₅×D₄	1	3	yes	—
229	C₂²	C₂²×C₃×C₅×Q₈	1	1	yes	characteristic
230	C₂²	C₂×C₃×C₅×#41	1	3	yes	—
231	C₄	—	2	12	no	—
232	C₄	—	4	6	no	—
233	C₃	C₂×C₅×#399	1	1	yes	characteristic, direct product factor, Sylow
234	C₂	—	4	4	no	—
235	C₂	C₃×C₅×#399	1	8	yes	direct product factor
236	C₂	C₂×C₃×C₅×C₂²⋊Q₈	1	3	yes	—
237	C₂	C₂×C₃×C₅×#123	1	3	yes	—
238	C₂	C₂×C₃×C₅×#127	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃×C₅	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
prime radical	C₂⁵×C₃×C₅	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃×C₅ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×C₃×C₅×C₄⋊C₄	C₂	semidirect
3. C₃×C₅×#399	C₂	direct (N × H)
4. C₂×C₃×C₅×#95	C₂	semidirect
5. C₂×C₅×#399	C₃	direct (N × H)
10. C₂×C₃×C₅×C₄⋊C₄	C₂²	semidirect
13. C₃×C₅×#95	C₂²	semidirect
14. C₂×C₃×#399	C₅	direct (N × H)
15. C₂²×C₅×C₄⋊C₄	C₂×C₃	semidirect
17. C₅×#399	C₂×C₃	direct (N × H)
18. C₂×C₅×#95	C₂×C₃	semidirect
32. C₂²×C₃×C₄⋊C₄	C₂×C₅	semidirect
34. C₃×#399	C₂×C₅	direct (N × H)
35. C₂×C₃×#95	C₂×C₅	semidirect
40. C₂×C₅×C₄⋊C₄	C₂²×C₃	semidirect
43. C₅×#95	C₂²×C₃	semidirect
44. C₂×#399	C₃×C₅	direct (N × H)
65. C₂×C₃×C₄⋊C₄	C₂²×C₅	semidirect
68. C₃×#95	C₂²×C₅	semidirect
82. C₂²×C₄⋊C₄	C₂×C₃×C₅	semidirect
84. #399	C₂×C₃×C₅	direct (N × H)
85. C₂×#95	C₂×C₃×C₅	semidirect
131. C₂×C₄⋊C₄	C₂²×C₃×C₅	semidirect
134. #95	C₂²×C₃×C₅	semidirect
136. C₂×C₃×C₅	#399	direct (N × H)
181. C₃×C₅	C₂×#399	direct (N × H)
195. C₂×C₅	C₃×#399	direct (N × H)
216. C₂×C₃	C₅×#399	direct (N × H)
220. C₅	C₂×C₃×#399	direct (N × H)
233. C₃	C₂×C₅×#399	direct (N × H)
235. C₂	C₃×C₅×#399	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×C₄⋊C₄	2
C₂²×C₃×C₅×C₂²⋊C₄	2
C₃×C₅×#399	2
C₂×C₃×C₅×#95	2
C₂×C₅×#399	3
C₂×C₃×#399	5

Subgroups — C2×C3×C5×#399 #14275427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#399 #14275427