Subgroups — C₂×C₅×D₃×Dic₆ #13089307

Order 1440.

662 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×C₅×#634	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₄×C₅×D₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#627	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×D₃×Dic₃	C₂	1	2	yes	maximal
6	C₅×D₃×Dic₆	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₅×D₃×Q₈	—	3	1	no	maximal
8	C₂²×C₅×Dic₆	—	3	1	no	maximal
9	C₂²×C₃×C₅×D₃	C₂²	1	1	yes	characteristic
10	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₃⋊Dic₃	C₂²	1	2	yes	—
12	C₅×#634	C₂²	1	4	yes	complemented
13	C₂×C₃×C₅×Dic₃	C₂²	1	2	yes	—
14	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
15	C₃×C₄×C₅×D₃	C₂²	1	4	yes	—
16	C₃×C₅×Dic₆	C₂²	1	4	yes	complemented
17	C₅×#627	C₂²	1	8	yes	complemented
18	C₅×D₃×Dic₃	C₂²	1	8	yes	—
19	C₂×D₃×Dic₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂×C₃×C₅×Q₈	—	3	1	no	—
21	C₂²×C₃×C₄×C₅	—	3	1	no	—
22	C₂²×C₅×Dic₃	—	3	2	no	—
23	C₅×D₃×Q₈	—	3	8	no	—
24	C₂×C₄×C₅×D₃	—	3	2	no	—
25	C₂×C₄×C₅×D₃	D₃	1	1	yes	characteristic
26	C₂×C₅×Dic₆	—	3	1	no	—
27	C₂×C₅×Dic₆	—	3	1	no	—
28	C₂×C₅×Dic₆	—	3	2	no	—
29	C₂×C₅×Dic₆	—	3	2	no	—
30	C₂×C₅×Dic₆	—	3	8	no	—
31	C₂×C₅×Dic₆	—	6	1	no	—
32	C₂×C₅×Dic₆	D₃	1	1	yes	characteristic, direct product factor
33	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₅×D₃	Q₈	1	4	yes	complemented
35	C₂×C₃×C₅×D₃	C₂³	1	2	yes	—
36	C₃²×C₄×C₅	C₂³	1	2	yes	—
37	C₅×C₃⋊Dic₃	C₂³	1	4	yes	—
38	C₃×C₅×Dic₃	C₂³	1	4	yes	—
39	C₃×C₅×Dic₃	C₂³	1	2	yes	—
40	C₂²×C₅×Q₈	—	9	1	no	Hall
41	C₂×#634	C₂×C₅	1	1	yes	characteristic, complemented
42	C₂×C₃×C₄×D₃	C₂×C₅	1	1	yes	characteristic
43	C₂×C₃×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
44	C₂×#627	C₂×C₅	1	2	yes	complemented
45	C₂×D₃×Dic₃	C₂×C₅	1	2	yes	—
46	D₃×Dic₆	C₂×C₅	1	8	yes	direct product factor
47	C₂³×C₃×C₅	—	3	1	no	—
48	C₂²×C₅×D₃	C₂×D₃	1	1	yes	characteristic
49	C₃×C₅×Q₈	—	3	4	no	—
50	C₂×C₃×C₄×C₅	—	2	1	no	—
51	C₂×C₃×C₄×C₅	—	3	1	no	—
52	C₂×C₃×C₄×C₅	—	3	2	no	—
53	C₂×C₃×C₄×C₅	—	3	4	no	—
54	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
56	C₂×C₅×Dic₃	—	3	2	no	—
57	C₂×C₅×Dic₃	—	3	2	no	—
58	C₂×C₅×Dic₃	—	3	8	no	—
59	C₂×C₅×Dic₃	—	6	2	no	—
60	C₂×C₅×Dic₃	C₂×D₃	1	1	yes	characteristic
61	C₂×C₅×Dic₃	C₂×D₃	1	2	yes	—
62	C₄×C₅×D₃	—	3	8	no	—
63	C₄×C₅×D₃	C₂×D₃	1	4	yes	—
64	C₅×Dic₆	—	3	4	no	—
65	C₅×Dic₆	—	3	4	no	—
66	C₅×Dic₆	—	3	8	no	—
67	C₅×Dic₆	—	3	8	no	—
68	C₅×Dic₆	—	6	4	no	—
69	C₅×Dic₆	C₂×D₃	1	4	yes	direct product factor
70	C₂×D₃×Q₈	—	3	1	no	—
71	C₂²×Dic₆	—	3	1	no	—
72	C₂×C₃²×C₅	C₂×Q₈	1	2	yes	complemented
73	C₂×C₃²×C₅	C₂⁴	1	1	yes	characteristic
74	C₃×C₅×D₃	C₂×Q₈	1	4	yes	complemented
75	C₂×C₅×Q₈	—	3	1	no	—
76	C₂×C₅×Q₈	—	9	2	no	—
77	C₂×C₅×Q₈	—	9	8	no	—
78	C₂×C₅×Q₈	—	9	1	no	—
79	C₂²×C₄×C₅	—	3	1	no	—
80	C₂²×C₄×C₅	—	9	2	no	—
81	C₂²×C₃×D₃	C₂²×C₅	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
83	C₂×C₃⋊Dic₃	C₂²×C₅	1	2	yes	—
84	#634	C₂²×C₅	1	4	yes	complemented
85	C₂×C₃×Dic₃	C₂²×C₅	1	2	yes	—
86	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
87	C₃×C₄×D₃	C₂²×C₅	1	4	yes	—
88	C₃×Dic₆	C₂²×C₅	1	4	yes	complemented
89	#627	C₂²×C₅	1	8	yes	complemented
90	D₃×Dic₃	C₂²×C₅	1	8	yes	—
91	C₂²×C₃×C₅	—	2	1	no	—
92	C₂²×C₃×C₅	—	3	4	no	—
93	C₂²×C₃×C₅	—	3	2	no	—
94	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
95	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
96	C₂×C₅×D₃	Dic₆	1	4	yes	direct product factor
97	C₂×C₅×D₃	C₂²×D₃	1	2	yes	—
98	C₃×C₄×C₅	—	2	2	no	—
99	C₃×C₄×C₅	—	3	2	no	—
100	C₃×C₄×C₅	—	3	4	no	—
101	C₃×C₄×C₅	C₂²×D₃	1	2	yes	—
102	C₃×C₄×C₅	C₂²×D₃	1	2	yes	—
103	C₅×Dic₃	—	3	4	no	—
104	C₅×Dic₃	—	3	4	no	—
105	C₅×Dic₃	—	6	4	no	—
106	C₅×Dic₃	C₂²×D₃	1	2	yes	—
107	C₅×Dic₃	C₂²×D₃	1	4	yes	—
108	C₂×C₃×Q₈	—	3	1	no	—
109	C₂²×C₃×C₄	—	3	1	no	—
110	C₂²×Dic₃	—	3	2	no	—
111	D₃×Q₈	—	3	8	no	—
112	C₂×C₄×D₃	—	3	2	no	—
113	C₂×C₄×D₃	C₅×D₃	1	1	yes	characteristic
114	C₂×Dic₆	—	3	1	no	—
115	C₂×Dic₆	—	3	1	no	—
116	C₂×Dic₆	—	3	2	no	—
117	C₂×Dic₆	—	3	2	no	—
118	C₂×Dic₆	—	3	8	no	—
119	C₂×Dic₆	—	6	1	no	—
120	C₂×Dic₆	C₅×D₃	1	1	yes	characteristic, direct product factor
121	C₃²×C₅	C₂²×Q₈	1	1	yes	characteristic, complemented, Hall
122	C₂³×C₅	—	3	1	no	—
123	C₅×Q₈	—	3	4	no	—
124	C₅×Q₈	—	9	8	no	—
125	C₅×Q₈	—	9	4	no	—
126	C₂×C₄×C₅	—	3	1	no	—
127	C₂×C₄×C₅	—	3	2	no	—
128	C₂×C₄×C₅	—	3	4	no	—
129	C₂×C₄×C₅	—	9	8	no	—
130	C₂×C₄×C₅	—	9	2	no	—
131	C₂×C₄×C₅	D₃²	1	1	yes	characteristic
132	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
133	C₂×C₃×D₃	C₅×Q₈	1	4	yes	complemented
134	C₂×C₃×D₃	C₂³×C₅	1	2	yes	—
135	C₃²×C₄	C₂³×C₅	1	2	yes	—
136	C₃⋊Dic₃	C₂³×C₅	1	4	yes	—
137	C₃×Dic₃	C₂³×C₅	1	4	yes	—
138	C₃×Dic₃	C₂³×C₅	1	2	yes	—
139	C₂²×Q₈	—	9	1	no	Sylow
140	C₂×C₃×C₅	—	2	1	no	—
141	C₂×C₃×C₅	—	2	2	no	—
142	C₂×C₃×C₅	—	3	4	no	—
143	C₂×C₃×C₅	C₂×Dic₆	1	2	yes	complemented
144	C₂×C₃×C₅	D₃×Q₈	1	2	yes	complemented
145	C₂×C₃×C₅	C₂³×D₃	1	1	yes	characteristic
146	C₂×C₃×C₅	C₂³×D₃	1	1	yes	characteristic
147	C₅×D₃	C₂×Dic₆	1	4	yes	direct product factor
148	C₂³×C₃	—	3	1	no	—
149	C₂²×D₃	C₂×C₅×D₃	1	1	yes	characteristic
150	C₃×Q₈	—	3	4	no	—
151	C₂×C₃×C₄	—	2	1	no	—
152	C₂×C₃×C₄	—	3	1	no	—
153	C₂×C₃×C₄	—	3	2	no	—
154	C₂×C₃×C₄	—	3	4	no	—
155	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
157	C₂×Dic₃	—	3	2	no	—
158	C₂×Dic₃	—	3	2	no	—
159	C₂×Dic₃	—	3	8	no	—
160	C₂×Dic₃	—	6	2	no	—
161	C₂×Dic₃	C₂×C₅×D₃	1	1	yes	characteristic
162	C₂×Dic₃	C₂×C₅×D₃	1	2	yes	—
163	C₄×D₃	—	3	8	no	—
164	C₄×D₃	C₂×C₅×D₃	1	4	yes	—
165	Dic₆	—	3	4	no	—
166	Dic₆	—	3	4	no	—
167	Dic₆	—	3	8	no	—
168	Dic₆	—	3	8	no	—
169	Dic₆	—	6	4	no	—
170	Dic₆	C₂×C₅×D₃	1	4	yes	direct product factor
171	C₂²×C₅	—	3	2	no	—
172	C₂²×C₅	—	3	4	no	—
173	C₂²×C₅	C₂×D₃²	1	1	yes	characteristic
174	C₄×C₅	—	3	2	no	—
175	C₄×C₅	—	3	4	no	—
176	C₄×C₅	—	9	4	no	—
177	C₄×C₅	C₂×D₃²	1	2	yes	—
178	C₂×C₃²	C₂×C₅×Q₈	1	2	yes	complemented
179	C₂×C₃²	C₂⁴×C₅	1	1	yes	characteristic
180	C₃×D₃	C₂×C₅×Q₈	1	4	yes	complemented
181	C₂×Q₈	—	3	1	no	—
182	C₂×Q₈	—	9	2	no	—
183	C₂×Q₈	—	9	8	no	—
184	C₂×Q₈	—	9	1	no	—
185	C₂²×C₄	—	3	1	no	—
186	C₂²×C₄	—	9	2	no	—
187	C₃×C₅	—	2	1	no	—
188	C₃×C₅	C₂²×Dic₆	1	1	yes	characteristic, complemented
189	C₃×C₅	C₂×D₃×Q₈	1	1	yes	characteristic, complemented
190	C₂²×C₃	—	2	1	no	—
191	C₂²×C₃	—	3	4	no	—
192	C₂²×C₃	—	3	2	no	—
193	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
194	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
195	C₂×D₃	C₅×Dic₆	1	4	yes	direct product factor
196	C₂×D₃	C₂²×C₅×D₃	1	2	yes	—
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	3	2	no	—
199	C₃×C₄	—	3	4	no	—
200	C₃×C₄	C₂²×C₅×D₃	1	2	yes	—
201	C₃×C₄	C₂²×C₅×D₃	1	2	yes	—
202	Dic₃	—	3	4	no	—
203	Dic₃	—	3	4	no	—
204	Dic₃	—	6	4	no	—
205	Dic₃	C₂²×C₅×D₃	1	2	yes	—
206	Dic₃	C₂²×C₅×D₃	1	4	yes	—
207	C₂×C₅	—	3	4	no	—
208	C₂×C₅	D₃×Dic₆	1	2	yes	direct product factor
209	C₂×C₅	C₂²×D₃²	1	1	yes	characteristic
210	C₃²	C₂²×C₅×Q₈	1	1	yes	characteristic, complemented, Sylow
211	C₂³	—	3	1	no	—
212	Q₈	—	3	4	no	—
213	Q₈	—	9	8	no	—
214	Q₈	—	9	4	no	—
215	C₂×C₄	—	3	1	no	—
216	C₂×C₄	—	3	2	no	—
217	C₂×C₄	—	3	4	no	—
218	C₂×C₄	—	9	8	no	—
219	C₂×C₄	—	9	2	no	—
220	C₂×C₄	C₅×D₃²	1	1	yes	characteristic
221	C₂×C₃	—	2	1	no	—
222	C₂×C₃	—	2	2	no	—
223	C₂×C₃	—	3	4	no	—
224	C₂×C₃	C₂×C₅×Dic₆	1	2	yes	complemented
225	C₂×C₃	C₅×D₃×Q₈	1	2	yes	complemented
226	C₂×C₃	C₂³×C₅×D₃	1	1	yes	characteristic
227	C₂×C₃	C₂³×C₅×D₃	1	1	yes	characteristic
228	D₃	C₂×C₅×Dic₆	1	4	yes	direct product factor
229	C₅	C₂×D₃×Dic₆	1	1	yes	characteristic, direct product factor, Sylow
230	C₂²	—	3	2	no	—
231	C₂²	—	3	4	no	—
232	C₂²	C₂×C₅×D₃²	1	1	yes	characteristic
233	C₄	—	3	2	no	—
234	C₄	—	3	4	no	—
235	C₄	—	9	4	no	—
236	C₄	C₂×C₅×D₃²	1	2	yes	—
237	C₃	—	2	1	no	—
238	C₃	C₂²×C₅×Dic₆	1	1	yes	characteristic, complemented
239	C₃	C₂×C₅×D₃×Q₈	1	1	yes	characteristic, complemented
240	C₂	—	3	4	no	—
241	C₂	C₅×D₃×Dic₆	1	2	yes	direct product factor
242	C₂	C₂²×C₅×D₃²	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₃²
commutator, cosocle	C₂×C₃²	C₂⁴×C₅
Frattini	C₂	C₂²×C₅×D₃²
Fitting	C₂×C₃²×C₄×C₅	C₂²
socle	C₂²×C₃²×C₅	C₂³
prime radical	C₂²×C₃×C₅×D₃	C₂²
hypercenter	C₂×C₄×C₅	D₃²
hypocenter	C₃²	C₂²×C₅×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#634	C₂	semidirect
3. C₂×C₃×C₅×Dic₆	C₂	semidirect
4. C₂×C₅×#627	C₂	semidirect
6. C₅×D₃×Dic₆	C₂	direct (N × H)
12. C₅×#634	C₂²	semidirect
16. C₃×C₅×Dic₆	C₂²	semidirect
17. C₅×#627	C₂²	semidirect
19. C₂×D₃×Dic₆	C₅	direct (N × H)
32. C₂×C₅×Dic₆	D₃	direct (N × H)
34. C₂×C₃×C₅×D₃	Q₈	semidirect
41. C₂×#634	C₂×C₅	semidirect
43. C₂×C₃×Dic₆	C₂×C₅	semidirect
44. C₂×#627	C₂×C₅	semidirect
46. D₃×Dic₆	C₂×C₅	direct (N × H)
69. C₅×Dic₆	C₂×D₃	direct (N × H)
72. C₂×C₃²×C₅	C₂×Q₈	semidirect
74. C₃×C₅×D₃	C₂×Q₈	semidirect
84. #634	C₂²×C₅	semidirect
88. C₃×Dic₆	C₂²×C₅	semidirect
89. #627	C₂²×C₅	semidirect
96. C₂×C₅×D₃	Dic₆	direct (N × H)
120. C₂×Dic₆	C₅×D₃	direct (N × H)
121. C₃²×C₅	C₂²×Q₈	semidirect
133. C₂×C₃×D₃	C₅×Q₈	semidirect
143. C₂×C₃×C₅	C₂×Dic₆	semidirect
144. C₂×C₃×C₅	D₃×Q₈	semidirect
147. C₅×D₃	C₂×Dic₆	direct (N × H)
170. Dic₆	C₂×C₅×D₃	direct (N × H)
178. C₂×C₃²	C₂×C₅×Q₈	semidirect
180. C₃×D₃	C₂×C₅×Q₈	semidirect
188. C₃×C₅	C₂²×Dic₆	semidirect
189. C₃×C₅	C₂×D₃×Q₈	semidirect
195. C₂×D₃	C₅×Dic₆	direct (N × H)
208. C₂×C₅	D₃×Dic₆	direct (N × H)
210. C₃²	C₂²×C₅×Q₈	semidirect
224. C₂×C₃	C₂×C₅×Dic₆	semidirect
225. C₂×C₃	C₅×D₃×Q₈	semidirect
228. D₃	C₂×C₅×Dic₆	direct (N × H)
229. C₅	C₂×D₃×Dic₆	direct (N × H)
238. C₃	C₂²×C₅×Dic₆	semidirect
239. C₃	C₂×C₅×D₃×Q₈	semidirect
241. C₂	C₅×D₃×Dic₆	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×C₅×#634	2
C₂×C₃×C₄×C₅×D₃	2
C₂×C₃×C₅×Dic₆	2
C₂×C₅×#627	2
C₂×C₅×D₃×Dic₃	2
C₅×D₃×Dic₆	2
C₂×C₅×D₃×Q₈	3
C₂²×C₅×Dic₆	3
C₂×D₃×Dic₆	5

Subgroups — C2×C5×D3×Dic6 #13089307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×D₃×Dic₆ #13089307