Subgroups — C₂×C₅×#3796 #13089309

Order 1440.

710 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#626	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×D₃×Dic₃	C₂	1	2	yes	maximal, complemented
6	C₅×#3796	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₅×#239	—	3	1	no	maximal
8	C₂×C₅×#237	—	3	1	no	maximal
9	C₂²×C₅×C₃⋊D₃	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₅×D₃	C₂²	1	2	yes	—
11	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
13	C₄×C₅×C₃⋊D₃	C₂²	1	4	yes	complemented
14	C₂×C₃×C₅×Dic₃	C₂²	1	2	yes	—
15	C₃×C₅×D₁₂	C₂²	1	4	yes	complemented
16	C₃×C₅×Dic₆	C₂²	1	4	yes	complemented
17	C₅×#626	C₂²	1	8	yes	complemented
18	C₅×D₃×Dic₃	C₂²	1	8	yes	complemented
19	C₂×#3796	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂×C₃×C₅×Q₈	—	3	1	no	—
21	C₂×C₃×C₅×D₄	—	3	1	no	—
22	C₂×C₅×#67	—	3	2	no	—
23	C₂²×C₅×Dic₃	—	3	2	no	—
24	C₅×#239	—	3	8	no	—
25	C₅×#237	—	3	8	no	—
26	C₂×C₅×D₁₂	—	3	2	no	—
27	C₂×C₅×D₁₂	D₃	1	1	yes	characteristic, complemented
28	C₂×C₄×C₅×D₃	—	3	2	no	—
29	C₂×C₄×C₅×D₃	—	3	1	no	—
30	C₂×C₄×C₅×D₃	—	3	1	no	—
31	C₂×C₄×C₅×D₃	—	6	1	no	—
32	C₂×C₅×Dic₆	D₃	1	1	yes	characteristic, complemented
33	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
34	C₂×C₅×C₃⋊D₃	—	2	2	no	—
35	C₂×C₅×C₃⋊D₃	C₂³	1	2	yes	—
36	C₂×C₃×C₅×D₃	—	2	4	no	—
37	C₂×C₃×C₅×D₃	C₂³	1	4	yes	—
38	C₃²×C₄×C₅	C₂³	1	2	yes	—
39	C₅×C₃⋊Dic₃	C₂³	1	2	yes	—
40	C₃×C₅×Dic₃	C₂³	1	4	yes	—
41	C₂×C₅×#41	—	9	1	no	Hall
42	C₂×C₄×C₃⋊D₃	C₂×C₅	1	1	yes	characteristic, complemented
43	C₂×C₃×D₁₂	C₂×C₅	1	1	yes	characteristic, complemented
44	C₂×C₃×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
45	C₂×#626	C₂×C₅	1	2	yes	complemented
46	C₂×D₃×Dic₃	C₂×C₅	1	2	yes	complemented
47	#3796	C₂×C₅	1	8	yes	direct product factor
48	C₂³×C₃×C₅	—	3	2	no	—
49	C₂²×C₅×D₃	—	3	1	no	—
50	C₂²×C₅×D₃	—	3	1	no	—
51	C₂²×C₅×D₃	—	6	1	no	—
52	C₂²×C₅×D₃	C₂×D₃	1	2	yes	—
53	C₃×C₅×Q₈	—	3	4	no	—
54	C₃×C₅×D₄	—	3	4	no	—
55	C₂×C₃×C₄×C₅	—	2	1	no	—
56	C₂×C₃×C₄×C₅	—	3	2	no	—
57	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
59	C₅×#67	—	3	8	no	—
60	C₂×C₅×Dic₃	—	3	1	no	—
61	C₂×C₅×Dic₃	—	3	1	no	—
62	C₂×C₅×Dic₃	—	3	8	no	—
63	C₂×C₅×Dic₃	—	6	1	no	—
64	C₂×C₅×Dic₃	C₂×D₃	1	2	yes	—
65	C₅×D₁₂	—	3	8	no	—
66	C₅×D₁₂	C₂×D₃	1	4	yes	complemented
67	C₄×C₅×D₃	—	3	8	no	—
68	C₄×C₅×D₃	—	3	4	no	—
69	C₄×C₅×D₃	—	3	4	no	—
70	C₄×C₅×D₃	—	6	4	no	—
71	C₅×Dic₆	C₂×D₃	1	4	yes	complemented
72	C₂×#239	—	3	1	no	—
73	C₂×#237	—	3	1	no	—
74	C₂×C₃²×C₅	#41	1	2	yes	complemented
75	C₂×C₃²×C₅	C₂⁴	1	1	yes	characteristic
76	C₅×C₃⋊D₃	—	2	2	no	—
77	C₃×C₅×D₃	—	2	4	no	—
78	C₅×#41	—	9	8	no	—
79	C₂×C₅×Q₈	—	3	1	no	—
80	C₂×C₅×D₄	—	3	1	no	—
81	C₂×C₅×D₄	—	9	2	no	—
82	C₂²×C₄×C₅	—	9	2	no	—
83	C₂²×C₄×C₅	—	9	1	no	—
84	C₂²×C₃⋊D₃	C₂²×C₅	1	1	yes	characteristic
85	C₂²×C₃×D₃	C₂²×C₅	1	2	yes	—
86	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
87	C₂×C₃⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
88	C₄×C₃⋊D₃	C₂²×C₅	1	4	yes	complemented
89	C₂×C₃×Dic₃	C₂²×C₅	1	2	yes	—
90	C₃×D₁₂	C₂²×C₅	1	4	yes	complemented
91	C₃×Dic₆	C₂²×C₅	1	4	yes	complemented
92	#626	C₂²×C₅	1	8	yes	complemented
93	D₃×Dic₃	C₂²×C₅	1	8	yes	complemented
94	C₂²×C₃×C₅	—	2	1	no	—
95	C₂²×C₃×C₅	—	3	4	no	—
96	C₂²×C₃×C₅	—	6	4	no	—
97	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
98	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
99	C₂×C₅×D₃	—	2	4	no	—
100	C₂×C₅×D₃	—	3	2	no	—
101	C₂×C₅×D₃	—	3	2	no	—
102	C₂×C₅×D₃	—	6	2	no	—
103	C₂×C₅×D₃	—	6	4	no	—
104	C₂×C₅×D₃	—	6	2	no	—
105	C₂×C₅×D₃	—	6	2	no	—
106	C₂×C₅×D₃	C₂²×D₃	1	4	yes	—
107	C₃×C₄×C₅	—	2	2	no	—
108	C₃×C₄×C₅	—	3	4	no	—
109	C₃×C₄×C₅	C₂²×D₃	1	2	yes	—
110	C₃×C₄×C₅	C₂²×D₃	1	2	yes	—
111	C₅×Dic₃	—	3	2	no	—
112	C₅×Dic₃	—	3	2	no	—
113	C₅×Dic₃	—	6	2	no	—
114	C₅×Dic₃	C₂²×D₃	1	4	yes	—
115	C₂×C₃×Q₈	—	3	1	no	—
116	C₂×C₃×D₄	—	3	1	no	—
117	C₂×#67	—	3	2	no	—
118	C₂²×Dic₃	—	3	2	no	—
119	#239	—	3	8	no	—
120	#237	—	3	8	no	—
121	C₂×D₁₂	—	3	2	no	—
122	C₂×D₁₂	C₅×D₃	1	1	yes	characteristic, complemented
123	C₂×C₄×D₃	—	3	2	no	—
124	C₂×C₄×D₃	—	3	1	no	—
125	C₂×C₄×D₃	—	3	1	no	—
126	C₂×C₄×D₃	—	6	1	no	—
127	C₂×Dic₆	C₅×D₃	1	1	yes	characteristic, complemented
128	C₃²×C₅	C₂×#41	1	1	yes	characteristic, complemented, Hall
129	C₂³×C₅	—	3	2	no	—
130	C₂³×C₅	—	9	1	no	—
131	C₅×Q₈	—	3	4	no	—
132	C₅×D₄	—	3	4	no	—
133	C₅×D₄	—	9	8	no	—
134	C₂×C₄×C₅	—	3	2	no	—
135	C₂×C₄×C₅	—	9	8	no	—
136	C₂×C₄×C₅	—	9	1	no	—
137	C₂×C₄×C₅	—	9	4	no	—
138	C₂×C₄×C₅	D₃²	1	1	yes	characteristic
139	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
140	C₂×C₃⋊D₃	—	2	2	no	—
141	C₂×C₃⋊D₃	C₂³×C₅	1	2	yes	—
142	C₂×C₃×D₃	—	2	4	no	—
143	C₂×C₃×D₃	C₂³×C₅	1	4	yes	—
144	C₃²×C₄	C₂³×C₅	1	2	yes	—
145	C₃⋊Dic₃	C₂³×C₅	1	2	yes	—
146	C₃×Dic₃	C₂³×C₅	1	4	yes	—
147	C₂×#41	—	9	1	no	Sylow
148	C₂×C₃×C₅	—	2	1	no	—
149	C₂×C₃×C₅	—	2	2	no	—
150	C₂×C₃×C₅	—	6	4	no	—
151	C₂×C₃×C₅	#237	1	2	yes	complemented
152	C₂×C₃×C₅	#239	1	2	yes	complemented
153	C₂×C₃×C₅	C₂³×D₃	1	1	yes	characteristic
154	C₂×C₃×C₅	C₂³×D₃	1	1	yes	characteristic
155	C₅×D₃	—	2	4	no	—
156	C₅×D₃	—	6	4	no	—
157	C₅×D₃	—	6	2	no	—
158	C₅×D₃	—	6	2	no	—
159	C₂³×C₃	—	3	2	no	—
160	C₂²×D₃	—	3	1	no	—
161	C₂²×D₃	—	3	1	no	—
162	C₂²×D₃	—	6	1	no	—
163	C₂²×D₃	C₂×C₅×D₃	1	2	yes	—
164	C₃×Q₈	—	3	4	no	—
165	C₃×D₄	—	3	4	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	3	2	no	—
168	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
170	#67	—	3	8	no	—
171	C₂×Dic₃	—	3	1	no	—
172	C₂×Dic₃	—	3	1	no	—
173	C₂×Dic₃	—	3	8	no	—
174	C₂×Dic₃	—	6	1	no	—
175	C₂×Dic₃	C₂×C₅×D₃	1	2	yes	—
176	D₁₂	—	3	8	no	—
177	D₁₂	C₂×C₅×D₃	1	4	yes	complemented
178	C₄×D₃	—	3	8	no	—
179	C₄×D₃	—	3	4	no	—
180	C₄×D₃	—	3	4	no	—
181	C₄×D₃	—	6	4	no	—
182	Dic₆	C₂×C₅×D₃	1	4	yes	complemented
183	C₂²×C₅	—	3	4	no	—
184	C₂²×C₅	—	6	4	no	—
185	C₂²×C₅	—	9	2	no	—
186	C₂²×C₅	—	18	2	no	—
187	C₂²×C₅	C₂×D₃²	1	1	yes	characteristic
188	C₄×C₅	—	3	4	no	—
189	C₄×C₅	—	9	2	no	—
190	C₄×C₅	C₂×D₃²	1	2	yes	—
191	C₂×C₃²	C₅×#41	1	2	yes	complemented
192	C₂×C₃²	C₂⁴×C₅	1	1	yes	characteristic
193	C₃⋊D₃	—	2	2	no	—
194	C₃×D₃	—	2	4	no	—
195	#41	—	9	8	no	—
196	C₂×Q₈	—	3	1	no	—
197	C₂×D₄	—	3	1	no	—
198	C₂×D₄	—	9	2	no	—
199	C₂²×C₄	—	9	2	no	—
200	C₂²×C₄	—	9	1	no	—
201	C₃×C₅	—	2	1	no	—
202	C₃×C₅	C₂×#237	1	1	yes	characteristic, complemented
203	C₃×C₅	C₂×#239	1	1	yes	characteristic, complemented
204	C₂²×C₃	—	2	1	no	—
205	C₂²×C₃	—	3	4	no	—
206	C₂²×C₃	—	6	4	no	—
207	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
208	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
209	C₂×D₃	—	2	4	no	—
210	C₂×D₃	—	3	2	no	—
211	C₂×D₃	—	3	2	no	—
212	C₂×D₃	—	6	2	no	—
213	C₂×D₃	—	6	4	no	—
214	C₂×D₃	—	6	2	no	—
215	C₂×D₃	—	6	2	no	—
216	C₂×D₃	C₂²×C₅×D₃	1	4	yes	—
217	C₃×C₄	—	2	2	no	—
218	C₃×C₄	—	3	4	no	—
219	C₃×C₄	C₂²×C₅×D₃	1	2	yes	—
220	C₃×C₄	C₂²×C₅×D₃	1	2	yes	—
221	Dic₃	—	3	2	no	—
222	Dic₃	—	3	2	no	—
223	Dic₃	—	6	2	no	—
224	Dic₃	C₂²×C₅×D₃	1	4	yes	—
225	C₂×C₅	—	6	4	no	—
226	C₂×C₅	—	18	2	no	—
227	C₂×C₅	#3796	1	2	yes	direct product factor
228	C₂×C₅	C₂²×D₃²	1	1	yes	characteristic
229	C₃²	C₂×C₅×#41	1	1	yes	characteristic, complemented, Sylow
230	C₂³	—	3	2	no	—
231	C₂³	—	9	1	no	—
232	Q₈	—	3	4	no	—
233	D₄	—	3	4	no	—
234	D₄	—	9	8	no	—
235	C₂×C₄	—	3	2	no	—
236	C₂×C₄	—	9	8	no	—
237	C₂×C₄	—	9	1	no	—
238	C₂×C₄	—	9	4	no	—
239	C₂×C₄	C₅×D₃²	1	1	yes	characteristic
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	—	2	2	no	—
242	C₂×C₃	—	6	4	no	—
243	C₂×C₃	C₅×#237	1	2	yes	complemented
244	C₂×C₃	C₅×#239	1	2	yes	complemented
245	C₂×C₃	C₂³×C₅×D₃	1	1	yes	characteristic
246	C₂×C₃	C₂³×C₅×D₃	1	1	yes	characteristic
247	D₃	—	2	4	no	—
248	D₃	—	6	4	no	—
249	D₃	—	6	2	no	—
250	D₃	—	6	2	no	—
251	C₅	C₂×#3796	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²	—	3	4	no	—
253	C₂²	—	6	4	no	—
254	C₂²	—	9	2	no	—
255	C₂²	—	18	2	no	—
256	C₂²	C₂×C₅×D₃²	1	1	yes	characteristic
257	C₄	—	3	4	no	—
258	C₄	—	9	2	no	—
259	C₄	C₂×C₅×D₃²	1	2	yes	—
260	C₃	—	2	1	no	—
261	C₃	C₂×C₅×#237	1	1	yes	characteristic, complemented
262	C₃	C₂×C₅×#239	1	1	yes	characteristic, complemented
263	C₂	—	6	4	no	—
264	C₂	—	18	2	no	—
265	C₂	C₅×#3796	1	2	yes	direct product factor
266	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
hypercenter	C₂×C₄×C₅	D₃²
hypocenter	C₃²	C₂×C₅×#41
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×C₄×C₅×C₃⋊D₃	C₂	semidirect
2. C₂×C₃×C₅×D₁₂	C₂	semidirect
3. C₂×C₃×C₅×Dic₆	C₂	semidirect
4. C₂×C₅×#626	C₂	semidirect
5. C₂×C₅×D₃×Dic₃	C₂	semidirect
6. C₅×#3796	C₂	direct (N × H)
13. C₄×C₅×C₃⋊D₃	C₂²	semidirect
15. C₃×C₅×D₁₂	C₂²	semidirect
16. C₃×C₅×Dic₆	C₂²	semidirect
17. C₅×#626	C₂²	semidirect
18. C₅×D₃×Dic₃	C₂²	semidirect
19. C₂×#3796	C₅	direct (N × H)
27. C₂×C₅×D₁₂	D₃	semidirect
32. C₂×C₅×Dic₆	D₃	semidirect
42. C₂×C₄×C₃⋊D₃	C₂×C₅	semidirect
43. C₂×C₃×D₁₂	C₂×C₅	semidirect
44. C₂×C₃×Dic₆	C₂×C₅	semidirect
45. C₂×#626	C₂×C₅	semidirect
46. C₂×D₃×Dic₃	C₂×C₅	semidirect
47. #3796	C₂×C₅	direct (N × H)
66. C₅×D₁₂	C₂×D₃	semidirect
71. C₅×Dic₆	C₂×D₃	semidirect
74. C₂×C₃²×C₅	#41	semidirect
88. C₄×C₃⋊D₃	C₂²×C₅	semidirect
90. C₃×D₁₂	C₂²×C₅	semidirect
91. C₃×Dic₆	C₂²×C₅	semidirect
92. #626	C₂²×C₅	semidirect
93. D₃×Dic₃	C₂²×C₅	semidirect
122. C₂×D₁₂	C₅×D₃	semidirect
127. C₂×Dic₆	C₅×D₃	semidirect
128. C₃²×C₅	C₂×#41	semidirect
151. C₂×C₃×C₅	#237	semidirect
152. C₂×C₃×C₅	#239	semidirect
177. D₁₂	C₂×C₅×D₃	semidirect
182. Dic₆	C₂×C₅×D₃	semidirect
191. C₂×C₃²	C₅×#41	semidirect
202. C₃×C₅	C₂×#237	semidirect
203. C₃×C₅	C₂×#239	semidirect
227. C₂×C₅	#3796	direct (N × H)
229. C₃²	C₂×C₅×#41	semidirect
243. C₂×C₃	C₅×#237	semidirect
244. C₂×C₃	C₅×#239	semidirect
251. C₅	C₂×#3796	direct (N × H)
261. C₃	C₂×C₅×#237	semidirect
262. C₃	C₂×C₅×#239	semidirect
265. C₂	C₅×#3796	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×C₄×C₅×C₃⋊D₃	2
C₂×C₃×C₅×D₁₂	2
C₂×C₃×C₅×Dic₆	2
C₂×C₅×#626	2
C₂×C₅×D₃×Dic₃	2
C₅×#3796	2
C₂×C₅×#239	3
C₂×C₅×#237	3
C₂×#3796	5

Subgroups — C2×C5×#3796 #13089309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#3796 #13089309