Subgroups — C₂×C₃×C₅×#393 #14275419

Order 1920.

2072 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×C₅×C₂²⋊C₄	C₂	1	3	yes	maximal
3	C₃×C₅×#393	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×C₅×#95	C₂	1	3	yes	maximal, complemented
5	C₂×C₅×#393	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂⁵×C₃×C₅	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₅×Q₈	—	2	12	no	—
8	C₂²×C₃×C₅×Q₈	C₂²	1	4	yes	complemented
9	C₂³×C₃×C₄×C₅	C₂²	1	3	yes	—
10	C₂³×C₃×C₄×C₅	C₂²	1	3	yes	—
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	12	yes	complemented
14	C₂×C₃×#393	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂³×C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
16	C₂²×C₅×C₂²⋊C₄	C₂×C₃	1	3	yes	—
17	C₅×#393	C₂×C₃	1	8	yes	direct product factor
18	C₂×C₅×#95	C₂×C₃	1	3	yes	complemented
19	C₂⁴×C₃×C₅	—	2	6	no	—
20	C₂⁴×C₃×C₅	—	2	6	no	—
21	C₂⁴×C₃×C₅	Q₈	1	4	yes	complemented
22	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
23	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
24	C₂×C₃×C₅×Q₈	—	2	32	no	—
25	C₂×C₃×C₅×Q₈	—	2	24	no	—
26	C₂²×C₃×C₄×C₅	—	2	12	no	—
27	C₂²×C₃×C₄×C₅	—	2	6	no	—
28	C₂²×C₃×C₄×C₅	—	2	12	no	—
29	C₂²×C₃×C₄×C₅	D₄	1	12	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
32	C₃×C₅×C₂²⋊C₄	—	2	24	no	—
33	C₂³×C₃×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
34	C₂²×C₃×C₂²⋊C₄	C₂×C₅	1	3	yes	—
35	C₃×#393	C₂×C₅	1	8	yes	direct product factor
36	C₂×C₃×#95	C₂×C₅	1	3	yes	complemented
37	C₂⁵×C₅	C₂²×C₃	1	1	yes	characteristic
38	C₂²×C₅×Q₈	—	2	12	no	—
39	C₂²×C₅×Q₈	C₂²×C₃	1	4	yes	complemented
40	C₂³×C₄×C₅	C₂²×C₃	1	3	yes	—
41	C₂³×C₄×C₅	C₂²×C₃	1	3	yes	—
42	C₂×C₅×C₂²⋊C₄	—	2	12	no	—
43	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
44	C₅×#95	C₂²×C₃	1	12	yes	complemented
45	C₂×#393	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
46	C₂³×C₃×C₅	—	2	24	no	—
47	C₂³×C₃×C₅	—	2	6	no	—
48	C₂³×C₃×C₅	—	2	6	no	—
49	C₂³×C₃×C₅	—	4	4	no	—
50	C₂³×C₃×C₅	—	4	12	no	—
51	C₂³×C₃×C₅	C₂×D₄	1	6	yes	—
52	C₂³×C₃×C₅	C₂×Q₈	1	2	yes	—
53	C₂³×C₃×C₅	C₂×Q₈	1	4	yes	complemented
54	C₂³×C₃×C₅	#41	1	6	yes	—
55	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
56	C₃×C₅×Q₈	—	2	32	no	—
57	C₂×C₃×C₄×C₅	—	2	24	no	—
58	C₂×C₃×C₄×C₅	—	2	12	no	—
59	C₂×C₃×C₄×C₅	—	2	6	no	—
60	C₂×C₃×C₄×C₅	—	2	12	no	—
61	C₂×C₃×C₄×C₅	—	4	12	no	—
62	C₂×C₃×C₄×C₅	C₂×D₄	1	12	yes	—
63	C₂⁵×C₃	C₂²×C₅	1	1	yes	characteristic
64	C₂²×C₃×Q₈	—	2	12	no	—
65	C₂²×C₃×Q₈	C₂²×C₅	1	4	yes	complemented
66	C₂³×C₃×C₄	C₂²×C₅	1	3	yes	—
67	C₂³×C₃×C₄	C₂²×C₅	1	3	yes	—
68	C₂×C₃×C₂²⋊C₄	—	2	12	no	—
69	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	12	yes	—
70	C₃×#95	C₂²×C₅	1	12	yes	complemented
71	C₂⁴×C₅	—	2	6	no	—
72	C₂⁴×C₅	—	2	6	no	—
73	C₂⁴×C₅	C₃×Q₈	1	4	yes	complemented
74	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
75	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
76	C₂×C₅×Q₈	—	2	32	no	—
77	C₂×C₅×Q₈	—	2	24	no	—
78	C₂²×C₄×C₅	—	2	12	no	—
79	C₂²×C₄×C₅	—	2	6	no	—
80	C₂²×C₄×C₅	—	2	12	no	—
81	C₂²×C₄×C₅	C₃×D₄	1	12	yes	—
82	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
83	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
84	C₅×C₂²⋊C₄	—	2	24	no	—
85	C₂³×Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
86	C₂²×C₂²⋊C₄	C₂×C₃×C₅	1	3	yes	—
87	#393	C₂×C₃×C₅	1	8	yes	direct product factor
88	C₂×#95	C₂×C₃×C₅	1	3	yes	complemented
89	C₂²×C₃×C₅	—	2	12	no	—
90	C₂²×C₃×C₅	—	4	2	no	—
91	C₂²×C₃×C₅	—	4	6	no	—
92	C₂²×C₃×C₅	—	4	16	no	—
93	C₂²×C₃×C₅	#120	1	4	yes	—
94	C₂²×C₃×C₅	C₂²⋊Q₈	1	12	yes	—
95	C₂²×C₃×C₅	#124	1	12	yes	—
96	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
97	C₂²×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
98	C₂²×C₃×C₅	C₂×#41	1	3	yes	—
99	C₃×C₄×C₅	—	2	12	no	—
100	C₃×C₄×C₅	—	4	6	no	—
101	C₂⁴×C₃	—	2	6	no	—
102	C₂⁴×C₃	—	2	6	no	—
103	C₂⁴×C₃	C₅×Q₈	1	4	yes	complemented
104	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
105	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
106	C₂×C₃×Q₈	—	2	32	no	—
107	C₂×C₃×Q₈	—	2	24	no	—
108	C₂²×C₃×C₄	—	2	12	no	—
109	C₂²×C₃×C₄	—	2	6	no	—
110	C₂²×C₃×C₄	—	2	12	no	—
111	C₂²×C₃×C₄	C₅×D₄	1	12	yes	—
112	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
113	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
114	C₃×C₂²⋊C₄	—	2	24	no	—
115	C₂³×C₅	—	2	24	no	—
116	C₂³×C₅	—	2	6	no	—
117	C₂³×C₅	—	2	6	no	—
118	C₂³×C₅	—	4	4	no	—
119	C₂³×C₅	—	4	12	no	—
120	C₂³×C₅	C₂×C₃×D₄	1	6	yes	—
121	C₂³×C₅	C₂×C₃×Q₈	1	2	yes	—
122	C₂³×C₅	C₂×C₃×Q₈	1	4	yes	complemented
123	C₂³×C₅	C₃×#41	1	6	yes	—
124	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
125	C₅×Q₈	—	2	32	no	—
126	C₂×C₄×C₅	—	2	24	no	—
127	C₂×C₄×C₅	—	2	12	no	—
128	C₂×C₄×C₅	—	2	6	no	—
129	C₂×C₄×C₅	—	2	12	no	—
130	C₂×C₄×C₅	—	4	12	no	—
131	C₂×C₄×C₅	C₂×C₃×D₄	1	12	yes	—
132	C₂⁵	C₂²×C₃×C₅	1	1	yes	characteristic
133	C₂²×Q₈	—	2	12	no	—
134	C₂²×Q₈	C₂²×C₃×C₅	1	4	yes	complemented
135	C₂³×C₄	C₂²×C₃×C₅	1	3	yes	—
136	C₂³×C₄	C₂²×C₃×C₅	1	3	yes	—
137	C₂×C₂²⋊C₄	—	2	12	no	—
138	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	12	yes	—
139	#95	C₂²×C₃×C₅	1	12	yes	complemented
140	C₂×C₃×C₅	—	4	4	no	—
141	C₂×C₃×C₅	#393	1	8	yes	direct product factor
142	C₂×C₃×C₅	C₂×#120	1	1	yes	characteristic
143	C₂×C₃×C₅	C₂×C₂²⋊Q₈	1	3	yes	—
144	C₂×C₃×C₅	C₂×#124	1	3	yes	—
145	C₂³×C₃	—	2	24	no	—
146	C₂³×C₃	—	2	6	no	—
147	C₂³×C₃	—	2	6	no	—
148	C₂³×C₃	—	4	4	no	—
149	C₂³×C₃	—	4	12	no	—
150	C₂³×C₃	C₂×C₅×D₄	1	6	yes	—
151	C₂³×C₃	C₂×C₅×Q₈	1	2	yes	—
152	C₂³×C₃	C₂×C₅×Q₈	1	4	yes	complemented
153	C₂³×C₃	C₅×#41	1	6	yes	—
154	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
155	C₃×Q₈	—	2	32	no	—
156	C₂×C₃×C₄	—	2	24	no	—
157	C₂×C₃×C₄	—	2	12	no	—
158	C₂×C₃×C₄	—	2	6	no	—
159	C₂×C₃×C₄	—	2	12	no	—
160	C₂×C₃×C₄	—	4	12	no	—
161	C₂×C₃×C₄	C₂×C₅×D₄	1	12	yes	—
162	C₂²×C₅	—	2	12	no	—
163	C₂²×C₅	—	4	16	no	—
164	C₂²×C₅	—	4	2	no	—
165	C₂²×C₅	—	4	6	no	—
166	C₂²×C₅	C₃×#120	1	4	yes	—
167	C₂²×C₅	C₃×C₂²⋊Q₈	1	12	yes	—
168	C₂²×C₅	C₃×#124	1	12	yes	—
169	C₂²×C₅	C₂²×C₃×D₄	1	3	yes	—
170	C₂²×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
171	C₂²×C₅	C₂×C₃×#41	1	3	yes	—
172	C₄×C₅	—	2	12	no	—
173	C₄×C₅	—	4	6	no	—
174	C₂⁴	—	2	6	no	—
175	C₂⁴	—	2	6	no	—
176	C₂⁴	C₃×C₅×Q₈	1	4	yes	complemented
177	C₂⁴	C₂³×C₃×C₅	1	2	yes	—
178	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
179	C₂×Q₈	—	2	32	no	—
180	C₂×Q₈	—	2	24	no	—
181	C₂²×C₄	—	2	12	no	—
182	C₂²×C₄	—	2	6	no	—
183	C₂²×C₄	—	2	12	no	—
184	C₂²×C₄	C₃×C₅×D₄	1	12	yes	—
185	C₂²×C₄	C₂³×C₃×C₅	1	6	yes	—
186	C₂²×C₄	C₂³×C₃×C₅	1	6	yes	—
187	C₂²⋊C₄	—	2	24	no	—
188	C₃×C₅	C₂×#393	1	1	yes	characteristic, direct product factor, Hall
189	C₂²×C₃	—	2	12	no	—
190	C₂²×C₃	—	4	16	no	—
191	C₂²×C₃	—	4	2	no	—
192	C₂²×C₃	—	4	6	no	—
193	C₂²×C₃	C₅×#120	1	4	yes	—
194	C₂²×C₃	C₅×C₂²⋊Q₈	1	12	yes	—
195	C₂²×C₃	C₅×#124	1	12	yes	—
196	C₂²×C₃	C₂²×C₅×D₄	1	3	yes	—
197	C₂²×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
198	C₂²×C₃	C₂×C₅×#41	1	3	yes	—
199	C₃×C₄	—	2	12	no	—
200	C₃×C₄	—	4	6	no	—
201	C₂×C₅	—	4	4	no	—
202	C₂×C₅	C₃×#393	1	8	yes	direct product factor
203	C₂×C₅	C₂×C₃×#120	1	1	yes	characteristic
204	C₂×C₅	C₂×C₃×C₂²⋊Q₈	1	3	yes	—
205	C₂×C₅	C₂×C₃×#124	1	3	yes	—
206	C₂³	—	2	6	no	—
207	C₂³	—	2	24	no	—
208	C₂³	—	2	6	no	—
209	C₂³	—	4	4	no	—
210	C₂³	—	4	12	no	—
211	C₂³	C₂×C₃×C₅×D₄	1	6	yes	—
212	C₂³	C₂×C₃×C₅×Q₈	1	2	yes	—
213	C₂³	C₂×C₃×C₅×Q₈	1	4	yes	complemented
214	C₂³	C₃×C₅×#41	1	6	yes	—
215	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
216	Q₈	—	2	32	no	—
217	C₂×C₄	—	2	24	no	—
218	C₂×C₄	—	2	12	no	—
219	C₂×C₄	—	2	6	no	—
220	C₂×C₄	—	2	12	no	—
221	C₂×C₄	—	4	12	no	—
222	C₂×C₄	C₂×C₃×C₅×D₄	1	12	yes	—
223	C₂×C₃	—	4	4	no	—
224	C₂×C₃	C₅×#393	1	8	yes	direct product factor
225	C₂×C₃	C₂×C₅×#120	1	1	yes	characteristic
226	C₂×C₃	C₂×C₅×C₂²⋊Q₈	1	3	yes	—
227	C₂×C₃	C₂×C₅×#124	1	3	yes	—
228	C₅	C₂×C₃×#393	1	1	yes	characteristic, direct product factor, Sylow
229	C₂²	—	2	12	no	—
230	C₂²	—	4	2	no	—
231	C₂²	—	4	6	no	—
232	C₂²	—	4	16	no	—
233	C₂²	C₃×C₅×#120	1	4	yes	—
234	C₂²	C₃×C₅×C₂²⋊Q₈	1	12	yes	—
235	C₂²	C₃×C₅×#124	1	12	yes	—
236	C₂²	C₂²×C₃×C₅×D₄	1	3	yes	—
237	C₂²	C₂²×C₃×C₅×Q₈	1	1	yes	characteristic
238	C₂²	C₂×C₃×C₅×#41	1	3	yes	—
239	C₄	—	2	12	no	—
240	C₄	—	4	6	no	—
241	C₃	C₂×C₅×#393	1	1	yes	characteristic, direct product factor, Sylow
242	C₂	—	4	4	no	—
243	C₂	C₃×C₅×#393	1	8	yes	direct product factor
244	C₂	C₂×C₃×C₅×#120	1	1	yes	characteristic
245	C₂	C₂×C₃×C₅×C₂²⋊Q₈	1	3	yes	—
246	C₂	C₂×C₃×C₅×#124	1	3	yes	—

	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. C₂³×C₃×C₅×Q₈	C₂	semidirect
3. C₃×C₅×#393	C₂	direct (N × H)
4. C₂×C₃×C₅×#95	C₂	semidirect
5. C₂×C₅×#393	C₃	direct (N × H)
8. C₂²×C₃×C₅×Q₈	C₂²	semidirect
13. C₃×C₅×#95	C₂²	semidirect
14. C₂×C₃×#393	C₅	direct (N × H)
15. C₂³×C₅×Q₈	C₂×C₃	semidirect
17. C₅×#393	C₂×C₃	direct (N × H)
18. C₂×C₅×#95	C₂×C₃	semidirect
21. C₂⁴×C₃×C₅	Q₈	semidirect
33. C₂³×C₃×Q₈	C₂×C₅	semidirect
35. C₃×#393	C₂×C₅	direct (N × H)
36. C₂×C₃×#95	C₂×C₅	semidirect
39. C₂²×C₅×Q₈	C₂²×C₃	semidirect
44. C₅×#95	C₂²×C₃	semidirect
45. C₂×#393	C₃×C₅	direct (N × H)
53. C₂³×C₃×C₅	C₂×Q₈	semidirect
65. C₂²×C₃×Q₈	C₂²×C₅	semidirect
70. C₃×#95	C₂²×C₅	semidirect
73. C₂⁴×C₅	C₃×Q₈	semidirect
85. C₂³×Q₈	C₂×C₃×C₅	semidirect
87. #393	C₂×C₃×C₅	direct (N × H)
88. C₂×#95	C₂×C₃×C₅	semidirect
103. C₂⁴×C₃	C₅×Q₈	semidirect
122. C₂³×C₅	C₂×C₃×Q₈	semidirect
134. C₂²×Q₈	C₂²×C₃×C₅	semidirect
139. #95	C₂²×C₃×C₅	semidirect
141. C₂×C₃×C₅	#393	direct (N × H)
152. C₂³×C₃	C₂×C₅×Q₈	semidirect
176. C₂⁴	C₃×C₅×Q₈	semidirect
188. C₃×C₅	C₂×#393	direct (N × H)
202. C₂×C₅	C₃×#393	direct (N × H)
213. C₂³	C₂×C₃×C₅×Q₈	semidirect
224. C₂×C₃	C₅×#393	direct (N × H)
228. C₅	C₂×C₃×#393	direct (N × H)
241. C₃	C₂×C₅×#393	direct (N × H)
243. C₂	C₃×C₅×#393	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₃×C₅×Q₈	2
C₂²×C₃×C₅×C₂²⋊C₄	2
C₃×C₅×#393	2
C₂×C₃×C₅×#95	2
C₂×C₅×#393	3
C₂×C₃×#393	5

Subgroups — C2×C3×C5×#393 #14275419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#393 #14275419