Subgroups — C₂×C₅×#1384 #11935284

Order 1280.

586 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#427	C₂	1	2	yes	maximal
3	C₅×#1384	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
5	C₂×C₅×#117	—	2	2	no	—
6	C₂×C₅×M₃₂	—	2	2	no	—
7	C₂×C₅×M₃₂	—	2	2	no	—
8	C₂×C₅×M₃₂	C₄	1	2	yes	complemented
9	C₂×C₅×M₃₂	C₂²	1	2	yes	—
10	C₂²×C₅×C₁₆	C₄	1	2	yes	complemented
11	C₅×#427	—	2	2	no	—
12	C₅×#427	—	2	2	no	—
13	C₅×#427	C₄	1	4	yes	complemented
14	C₅×#427	C₂²	1	4	yes	—
15	C₂×C₅×#107	—	2	2	no	—
16	C₂×C₅×#106	C₄	1	4	yes	complemented
17	C₂×#1384	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	D₄	1	2	yes	—
25	C₂²×C₅×C₈	D₄	1	2	yes	—
26	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
27	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
29	C₅×#117	—	2	4	no	—
30	C₅×#117	—	2	4	no	—
31	C₅×#117	—	4	2	no	—
32	C₅×#117	—	4	2	no	—
33	C₂×C₅×C₄⋊C₄	—	2	4	no	—
34	C₂×C₅×C₂²⋊C₄	—	4	2	no	—
35	C₂×C₄²×C₅	—	4	2	no	—
36	C₅×M₃₂	—	2	4	no	—
37	C₅×M₃₂	—	2	4	no	—
38	C₂×C₅×C₁₆	—	2	2	no	—
39	C₂×C₅×C₁₆	—	2	2	no	—
40	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
41	C₅×#107	—	2	4	no	—
42	C₅×#107	—	2	4	no	—
43	C₅×#106	—	2	4	no	—
44	C₅×#106	C₂×C₄	1	8	yes	complemented
45	C₂²×M₃₂	C₂×C₅	1	1	yes	characteristic
46	C₂×#427	C₂×C₅	1	2	yes	—
47	#1384	C₂×C₅	1	4	yes	direct product factor
48	C₂⁴×C₅	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₅	—	4	4	no	—
50	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
56	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
57	C₂²×C₄×C₅	Q₁₆	1	1	yes	characteristic
58	C₂²×C₄×C₅	Q₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
61	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
62	C₂×C₅×C₈	C₄²	1	2	yes	—
63	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
64	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
65	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
68	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
71	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
72	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
73	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
74	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
76	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
77	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
78	C₅×C₄⋊C₄	—	2	8	no	—
79	C₅×C₄⋊C₄	—	4	4	no	—
80	C₅×C₂²⋊C₄	—	4	4	no	—
81	C₅×C₂²⋊C₄	—	4	4	no	—
82	C₄²×C₅	—	4	4	no	—
83	C₄²×C₅	—	4	4	no	—
84	C₅×C₁₆	—	2	4	no	—
85	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
86	C₂×#117	—	2	2	no	—
87	C₂×M₃₂	—	2	2	no	—
88	C₂×M₃₂	—	2	2	no	—
89	C₂×M₃₂	C₄×C₅	1	2	yes	complemented
90	C₂×M₃₂	C₂²×C₅	1	2	yes	—
91	C₂²×C₁₆	C₄×C₅	1	2	yes	complemented
92	#427	—	2	2	no	—
93	#427	—	2	2	no	—
94	#427	C₄×C₅	1	4	yes	complemented
95	#427	C₂²×C₅	1	4	yes	—
96	C₂×#107	—	2	2	no	—
97	C₂×#106	C₄×C₅	1	4	yes	complemented
98	C₂³×C₅	—	4	2	no	—
99	C₂³×C₅	#102	1	2	yes	—
100	C₂³×C₅	#102	1	2	yes	—
101	C₂³×C₅	#107	1	1	yes	characteristic
102	C₂³×C₅	#107	1	1	yes	characteristic
103	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₅	—	4	8	no	—
105	C₂×C₄×C₅	—	8	8	no	—
106	C₂×C₄×C₅	#95	1	2	yes	—
107	C₂×C₄×C₅	#95	1	1	yes	characteristic
108	C₂×C₄×C₅	#95	1	1	yes	characteristic
109	C₂×C₄×C₅	#102	1	2	yes	—
110	C₂×C₄×C₅	#102	1	2	yes	—
111	C₂×C₄×C₅	#103	1	2	yes	—
112	C₂×C₄×C₅	#103	1	2	yes	—
113	C₂×C₄×C₅	#103	1	2	yes	—
114	C₂×C₄×C₅	#103	1	2	yes	—
115	C₂×C₄×C₅	#106	1	2	yes	—
116	C₂×C₄×C₅	#106	1	1	yes	characteristic
117	C₂×C₄×C₅	#106	1	1	yes	characteristic
118	C₂×C₄×C₅	#107	1	1	yes	characteristic
119	C₂×C₄×C₅	#107	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
121	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
123	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
124	C₅×C₈	#95	1	1	yes	characteristic
125	C₅×C₈	#95	1	2	yes	—
126	C₅×C₈	#95	1	1	yes	characteristic
127	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
128	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
130	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
131	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
132	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
133	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
134	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
135	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
136	C₂²×C₈	C₅×D₄	1	2	yes	—
137	C₂²×C₈	C₅×D₄	1	2	yes	—
138	C₂²×C₈	C₅×Q₈	1	1	yes	characteristic
139	C₂²×C₈	C₅×Q₈	1	1	yes	characteristic
140	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
141	#117	—	2	4	no	—
142	#117	—	2	4	no	—
143	#117	—	4	2	no	—
144	#117	—	4	2	no	—
145	C₂×C₄⋊C₄	—	2	4	no	—
146	C₂×C₂²⋊C₄	—	4	2	no	—
147	C₂×C₄²	—	4	2	no	—
148	M₃₂	—	2	4	no	—
149	M₃₂	—	2	4	no	—
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	—	2	2	no	—
152	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	complemented
153	#107	—	2	4	no	—
154	#107	—	2	4	no	—
155	#106	—	2	4	no	—
156	#106	C₂×C₄×C₅	1	8	yes	complemented
157	C₂²×C₅	—	2	4	no	—
158	C₂²×C₅	—	2	2	no	—
159	C₂²×C₅	—	4	2	no	—
160	C₂²×C₅	—	4	1	no	—
161	C₂²×C₅	—	4	1	no	—
162	C₂²×C₅	#340	1	2	yes	—
163	C₂²×C₅	#340	1	1	yes	characteristic
164	C₂²×C₅	#340	1	1	yes	characteristic
165	C₂²×C₅	C₂×#102	1	2	yes	—
166	C₂²×C₅	C₂×#107	1	1	yes	characteristic
167	C₄×C₅	—	8	8	no	—
168	C₄×C₅	#340	1	2	yes	—
169	C₄×C₅	#340	1	1	yes	characteristic
170	C₄×C₅	#340	1	1	yes	characteristic
171	C₄×C₅	C₂×#95	1	1	yes	characteristic
172	C₄×C₅	C₂×#103	1	2	yes	—
173	C₄×C₅	C₂×#106	1	1	yes	characteristic
174	C₂⁴	C₅×D₈	1	1	yes	characteristic
175	C₂²×C₄	—	4	4	no	—
176	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
177	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
178	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
179	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
180	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
181	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
182	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
183	C₂²×C₄	C₅×Q₁₆	1	1	yes	characteristic
184	C₂²×C₄	C₅×Q₁₆	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
186	C₂×C₈	C₄²×C₅	1	1	yes	characteristic
187	C₂×C₈	C₄²×C₅	1	1	yes	characteristic
188	C₂×C₈	C₄²×C₅	1	2	yes	—
189	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
190	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
191	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
192	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
193	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
194	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
195	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
196	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
197	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
198	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
199	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
200	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
201	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
202	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
203	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
204	C₄⋊C₄	—	2	8	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₂²⋊C₄	—	4	4	no	—
207	C₂²⋊C₄	—	4	4	no	—
208	C₄²	—	4	4	no	—
209	C₄²	—	4	4	no	—
210	C₁₆	—	2	4	no	—
211	C₂×C₅	—	2	2	no	—
212	C₂×C₅	—	2	2	no	—
213	C₂×C₅	—	2	1	no	—
214	C₂×C₅	—	2	1	no	—
215	C₂×C₅	#1384	1	2	yes	direct product factor
216	C₂×C₅	C₂×#340	1	1	yes	characteristic
217	C₂³	—	4	2	no	—
218	C₂³	C₅×#102	1	2	yes	—
219	C₂³	C₅×#102	1	2	yes	—
220	C₂³	C₅×#107	1	1	yes	characteristic
221	C₂³	C₅×#107	1	1	yes	characteristic
222	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
223	C₂×C₄	—	4	8	no	—
224	C₂×C₄	—	8	8	no	—
225	C₂×C₄	C₅×#95	1	1	yes	characteristic
226	C₂×C₄	C₅×#95	1	2	yes	—
227	C₂×C₄	C₅×#95	1	1	yes	characteristic
228	C₂×C₄	C₅×#102	1	2	yes	—
229	C₂×C₄	C₅×#102	1	2	yes	—
230	C₂×C₄	C₅×#103	1	2	yes	—
231	C₂×C₄	C₅×#103	1	2	yes	—
232	C₂×C₄	C₅×#103	1	2	yes	—
233	C₂×C₄	C₅×#103	1	2	yes	—
234	C₂×C₄	C₅×#106	1	2	yes	—
235	C₂×C₄	C₅×#106	1	1	yes	characteristic
236	C₂×C₄	C₅×#106	1	1	yes	characteristic
237	C₂×C₄	C₅×#107	1	1	yes	characteristic
238	C₂×C₄	C₅×#107	1	1	yes	characteristic
239	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
240	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
241	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
242	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
243	C₈	C₅×#95	1	1	yes	characteristic
244	C₈	C₅×#95	1	1	yes	characteristic
245	C₈	C₅×#95	1	2	yes	—
246	C₈	C₂×C₄²×C₅	1	1	yes	characteristic
247	C₈	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
248	C₈	C₂×C₅×C₄⋊C₄	1	2	yes	—
249	C₅	C₂×#1384	1	1	yes	characteristic, direct product factor, Sylow
250	C₂²	—	2	4	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	4	1	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	1	no	—
255	C₂²	C₅×#340	1	2	yes	—
256	C₂²	C₅×#340	1	1	yes	characteristic
257	C₂²	C₅×#340	1	1	yes	characteristic
258	C₂²	C₂×C₅×#102	1	2	yes	—
259	C₂²	C₂×C₅×#107	1	1	yes	characteristic
260	C₄	—	8	8	no	—
261	C₄	C₅×#340	1	1	yes	characteristic
262	C₄	C₅×#340	1	2	yes	—
263	C₄	C₅×#340	1	1	yes	characteristic
264	C₄	C₂×C₅×#95	1	1	yes	characteristic
265	C₄	C₂×C₅×#103	1	2	yes	—
266	C₄	C₂×C₅×#106	1	1	yes	characteristic
267	C₂	—	2	2	no	—
268	C₂	—	2	2	no	—
269	C₂	—	2	1	no	—
270	C₂	—	2	1	no	—
271	C₂	C₅×#1384	1	2	yes	direct product factor
272	C₂	C₂×C₅×#340	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#106
commutator	C₈	C₂×C₄²×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	#340
prime radical	C₂⁴×C₅	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1384	C₂	direct (N × H)
8. C₂×C₅×M₃₂	C₄	semidirect
10. C₂²×C₅×C₁₆	C₄	semidirect
13. C₅×#427	C₄	semidirect
16. C₂×C₅×#106	C₄	semidirect
17. C₂×#1384	C₅	direct (N × H)
40. C₂×C₅×C₁₆	C₂×C₄	semidirect
44. C₅×#106	C₂×C₄	semidirect
47. #1384	C₂×C₅	direct (N × H)
89. C₂×M₃₂	C₄×C₅	semidirect
91. C₂²×C₁₆	C₄×C₅	semidirect
94. #427	C₄×C₅	semidirect
97. C₂×#106	C₄×C₅	semidirect
152. C₂×C₁₆	C₂×C₄×C₅	semidirect
156. #106	C₂×C₄×C₅	semidirect
215. C₂×C₅	#1384	direct (N × H)
249. C₅	C₂×#1384	direct (N × H)
271. C₂	C₅×#1384	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×C₅×M₃₂	2
C₂×C₅×#427	2
C₅×#1384	2
C₂×#1384	5

Subgroups — C2×C5×#1384 #11935284

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1384 #11935284