Subgroups — C₂×C₅×#1385 #11935285

Order 1280.

554 conjugacy classes of subgroups.

The subgroup lattice has 270 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	C₂²×C₅×#108	C₂	1	2	yes	maximal
3	C₅×#1385	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₅×M₁₆	—	2	2	no	—
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₅×M₃₂	—	2	2	no	—
7	C₂×C₅×M₃₂	—	2	2	no	—
8	C₂×C₅×M₃₂	C₄	1	2	yes	—
9	C₂×C₅×M₃₂	C₂²	1	2	yes	—
10	C₂²×C₅×C₁₆	C₄	1	2	yes	—
11	C₂×C₅×#108	—	2	2	no	—
12	C₂×C₅×#108	—	2	2	no	—
13	C₂×C₅×#108	—	2	2	no	—
14	C₂×C₅×#108	C₄	1	4	yes	—
15	C₂×C₅×#108	C₄	1	4	yes	—
16	C₂×C₅×#108	C₂²	1	4	yes	—
17	C₂×#1385	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
19	C₂×C₅×M₁₆	—	2	4	no	—
20	C₂×C₅×M₁₆	—	2	4	no	—
21	C₂×C₅×M₁₆	—	2	4	no	—
22	C₂×C₅×M₁₆	—	4	2	no	—
23	C₂×C₅×M₁₆	—	4	2	no	—
24	C₂×C₅×M₁₆	—	4	2	no	—
25	C₂²×C₅×C₈	—	4	2	no	—
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	D₄	1	2	yes	—
32	C₂²×C₅×C₈	D₄	1	2	yes	—
33	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
34	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
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	—
41	C₅×#108	—	2	4	no	—
42	C₅×#108	—	2	4	no	—
43	C₅×#108	—	2	4	no	—
44	C₅×#108	C₂×C₄	1	8	yes	—
45	C₂²×M₃₂	C₂×C₅	1	1	yes	characteristic
46	C₂²×#108	C₂×C₅	1	2	yes	—
47	#1385	C₂×C₅	1	4	yes	direct product factor
48	C₂⁴×C₅	Q₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	D₈	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	C₂×D₄	1	1	yes	characteristic
59	C₅×M₁₆	—	2	8	no	—
60	C₅×M₁₆	—	4	4	no	—
61	C₅×M₁₆	—	4	4	no	—
62	C₅×M₁₆	—	4	4	no	—
63	C₂×C₅×C₈	—	4	4	no	—
64	C₂×C₅×C₈	—	4	4	no	—
65	C₂×C₅×C₈	—	4	4	no	—
66	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
67	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
68	C₂×C₅×C₈	C₄²	1	2	yes	—
69	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
70	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
71	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
74	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
77	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
78	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
79	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
80	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
82	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
83	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
84	C₅×C₁₆	—	2	4	no	—
85	C₂²×M₁₆	—	2	2	no	—
86	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
87	C₂×M₃₂	—	2	2	no	—
88	C₂×M₃₂	—	2	2	no	—
89	C₂×M₃₂	C₄×C₅	1	2	yes	—
90	C₂×M₃₂	C₂²×C₅	1	2	yes	—
91	C₂²×C₁₆	C₄×C₅	1	2	yes	—
92	C₂×#108	—	2	2	no	—
93	C₂×#108	—	2	2	no	—
94	C₂×#108	—	2	2	no	—
95	C₂×#108	C₄×C₅	1	4	yes	—
96	C₂×#108	C₄×C₅	1	4	yes	—
97	C₂×#108	C₂²×C₅	1	4	yes	—
98	C₂³×C₅	—	4	2	no	—
99	C₂³×C₅	#103	1	2	yes	—
100	C₂³×C₅	#103	1	2	yes	—
101	C₂³×C₅	#107	1	1	yes	characteristic
102	C₂³×C₅	#107	1	1	yes	characteristic
103	C₂³×C₅	C₂×Q₁₆	1	1	yes	characteristic
104	C₂×C₄×C₅	#95	1	2	yes	—
105	C₂×C₄×C₅	#95	1	1	yes	characteristic
106	C₂×C₄×C₅	#95	1	1	yes	characteristic
107	C₂×C₄×C₅	#102	1	2	yes	—
108	C₂×C₄×C₅	#102	1	2	yes	—
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	#106	1	2	yes	—
114	C₂×C₄×C₅	#106	1	1	yes	characteristic
115	C₂×C₄×C₅	#106	1	1	yes	characteristic
116	C₂×C₄×C₅	#107	1	1	yes	characteristic
117	C₂×C₄×C₅	#107	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
119	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
122	C₅×C₈	—	4	8	no	—
123	C₅×C₈	#95	1	1	yes	characteristic
124	C₅×C₈	#95	1	1	yes	characteristic
125	C₅×C₈	#95	1	2	yes	—
126	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
127	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
129	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
130	C₂×M₁₆	—	2	4	no	—
131	C₂×M₁₆	—	2	4	no	—
132	C₂×M₁₆	—	2	4	no	—
133	C₂×M₁₆	—	4	2	no	—
134	C₂×M₁₆	—	4	2	no	—
135	C₂×M₁₆	—	4	2	no	—
136	C₂²×C₈	—	4	2	no	—
137	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
138	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
139	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
140	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
141	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
142	C₂²×C₈	C₅×D₄	1	2	yes	—
143	C₂²×C₈	C₅×D₄	1	2	yes	—
144	C₂²×C₈	C₅×Q₈	1	1	yes	characteristic
145	C₂²×C₈	C₅×Q₈	1	1	yes	characteristic
146	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
147	M₃₂	—	2	4	no	—
148	M₃₂	—	2	4	no	—
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	—
152	#108	—	2	4	no	—
153	#108	—	2	4	no	—
154	#108	—	2	4	no	—
155	#108	C₂×C₄×C₅	1	8	yes	—
156	C₂²×C₅	—	2	4	no	—
157	C₂²×C₅	—	2	2	no	—
158	C₂²×C₅	—	4	2	no	—
159	C₂²×C₅	—	4	1	no	—
160	C₂²×C₅	—	4	1	no	—
161	C₂²×C₅	#340	1	2	yes	—
162	C₂²×C₅	#340	1	1	yes	characteristic
163	C₂²×C₅	#340	1	1	yes	characteristic
164	C₂²×C₅	C₂×#103	1	2	yes	—
165	C₂²×C₅	C₂×#107	1	1	yes	characteristic
166	C₄×C₅	#340	1	2	yes	—
167	C₄×C₅	#340	1	1	yes	characteristic
168	C₄×C₅	#340	1	1	yes	characteristic
169	C₄×C₅	C₂×#95	1	1	yes	characteristic
170	C₄×C₅	C₂×#102	1	2	yes	—
171	C₄×C₅	C₂×#106	1	1	yes	characteristic
172	C₂⁴	C₅×Q₁₆	1	1	yes	characteristic
173	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
174	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
175	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
176	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
177	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
178	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
179	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
180	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
181	C₂²×C₄	C₅×Q₁₆	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
183	M₁₆	—	2	8	no	—
184	M₁₆	—	4	4	no	—
185	M₁₆	—	4	4	no	—
186	M₁₆	—	4	4	no	—
187	C₂×C₈	—	4	4	no	—
188	C₂×C₈	—	4	4	no	—
189	C₂×C₈	—	4	4	no	—
190	C₂×C₈	C₄²×C₅	1	1	yes	characteristic
191	C₂×C₈	C₄²×C₅	1	1	yes	characteristic
192	C₂×C₈	C₄²×C₅	1	2	yes	—
193	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
194	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
195	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
196	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
197	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
198	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
199	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
200	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
201	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
202	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
203	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
204	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
205	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
206	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
207	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
208	C₁₆	—	2	4	no	—
209	C₂×C₅	—	2	2	no	—
210	C₂×C₅	—	2	1	no	—
211	C₂×C₅	—	2	2	no	—
212	C₂×C₅	—	2	1	no	—
213	C₂×C₅	#1385	1	2	yes	direct product factor
214	C₂×C₅	C₂×#340	1	1	yes	characteristic
215	C₂³	—	4	2	no	—
216	C₂³	C₅×#103	1	2	yes	—
217	C₂³	C₅×#103	1	2	yes	—
218	C₂³	C₅×#107	1	1	yes	characteristic
219	C₂³	C₅×#107	1	1	yes	characteristic
220	C₂³	C₂×C₅×Q₁₆	1	1	yes	characteristic
221	C₂×C₄	C₅×#95	1	1	yes	characteristic
222	C₂×C₄	C₅×#95	1	2	yes	—
223	C₂×C₄	C₅×#95	1	1	yes	characteristic
224	C₂×C₄	C₅×#102	1	2	yes	—
225	C₂×C₄	C₅×#102	1	2	yes	—
226	C₂×C₄	C₅×#102	1	2	yes	—
227	C₂×C₄	C₅×#102	1	2	yes	—
228	C₂×C₄	C₅×#103	1	2	yes	—
229	C₂×C₄	C₅×#103	1	2	yes	—
230	C₂×C₄	C₅×#106	1	2	yes	—
231	C₂×C₄	C₅×#106	1	1	yes	characteristic
232	C₂×C₄	C₅×#106	1	1	yes	characteristic
233	C₂×C₄	C₅×#107	1	1	yes	characteristic
234	C₂×C₄	C₅×#107	1	1	yes	characteristic
235	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
236	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
237	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
238	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
239	C₈	—	4	8	no	—
240	C₈	C₅×#95	1	1	yes	characteristic
241	C₈	C₅×#95	1	1	yes	characteristic
242	C₈	C₅×#95	1	2	yes	—
243	C₈	C₂×C₄²×C₅	1	1	yes	characteristic
244	C₈	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
245	C₈	C₂×C₅×C₄⋊C₄	1	2	yes	—
246	C₅	C₂×#1385	1	1	yes	characteristic, direct product factor, Sylow
247	C₂²	—	2	4	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	4	1	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	1	no	—
252	C₂²	C₅×#340	1	2	yes	—
253	C₂²	C₅×#340	1	1	yes	characteristic
254	C₂²	C₅×#340	1	1	yes	characteristic
255	C₂²	C₂×C₅×#103	1	2	yes	—
256	C₂²	C₂×C₅×#107	1	1	yes	characteristic
257	C₄	C₅×#340	1	1	yes	characteristic
258	C₄	C₅×#340	1	2	yes	—
259	C₄	C₅×#340	1	1	yes	characteristic
260	C₄	C₂×C₅×#95	1	1	yes	characteristic
261	C₄	C₂×C₅×#102	1	2	yes	—
262	C₄	C₂×C₅×#106	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	2	1	no	—
265	C₂	—	2	2	no	—
266	C₂	—	2	1	no	—
267	C₂	C₅×#1385	1	2	yes	direct product factor
268	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	Q₁₆
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×#1385	C₂	direct (N × H)
17. C₂×#1385	C₅	direct (N × H)
47. #1385	C₂×C₅	direct (N × H)
213. C₂×C₅	#1385	direct (N × H)
246. C₅	C₂×#1385	direct (N × H)
267. C₂	C₅×#1385	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
C₂²×C₅×#108	2
C₅×#1385	2
C₂×#1385	5

Subgroups — C2×C5×#1385 #11935285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1385 #11935285