Subgroups — C₂×C₅×#1376 #11935268

Order 1280.

670 conjugacy classes of subgroups.

The subgroup lattice has 264 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	2	yes	maximal
2	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#1376	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₅	C₄	1	1	yes	characteristic
7	C₂×C₅×M₃₂	—	2	2	no	—
8	C₂×C₅×M₃₂	—	2	4	no	—
9	C₂×C₅×M₃₂	C₄	1	8	yes	complemented
10	C₂×C₅×M₃₂	C₂²	1	4	yes	—
11	C₂²×C₅×C₁₆	—	2	2	no	—
12	C₂×C₄×C₅×C₈	—	2	4	no	—
13	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
15	C₂×#1376	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₄×C₅	—	2	1	no	—
17	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	—	2	2	no	—
19	C₂²×C₅×C₈	—	2	2	no	—
20	C₂²×C₅×C₈	C₈	1	2	yes	—
21	C₂²×C₅×C₈	C₈	1	2	yes	—
22	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	D₄	1	4	yes	—
27	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
29	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₅	—	2	2	no	—
32	C₂×C₄²×C₅	—	2	1	no	—
33	C₂×C₄²×C₅	C₈	1	4	yes	—
34	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
35	C₅×M₃₂	—	2	8	no	—
36	C₅×M₃₂	—	2	4	no	—
37	C₅×M₃₂	C₂×C₄	1	8	yes	complemented
38	C₂×C₅×C₁₆	—	2	8	no	—
39	C₂×C₅×C₁₆	—	2	4	no	—
40	C₄×C₅×C₈	—	2	4	no	—
41	C₄×C₅×C₈	—	2	4	no	—
42	C₂²×M₃₂	C₂×C₅	1	2	yes	—
43	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
44	#1376	C₂×C₅	1	4	yes	direct product factor
45	C₂⁴×C₅	M₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₅	—	2	8	no	—
47	C₂²×C₄×C₅	—	2	4	no	—
48	C₂²×C₄×C₅	—	2	2	no	—
49	C₂²×C₄×C₅	C₄²	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	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₅	M₁₆	1	2	yes	—
58	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₅×C₈	—	2	4	no	—
60	C₂×C₅×C₈	—	2	4	no	—
61	C₂×C₅×C₈	—	2	2	no	—
62	C₂×C₅×C₈	—	2	2	no	—
63	C₂×C₅×C₈	C₄²	1	2	yes	—
64	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
65	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
66	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
67	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
70	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
71	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
72	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
73	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
75	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
76	C₄²×C₅	—	2	4	no	—
77	C₄²×C₅	—	2	2	no	—
78	C₄²×C₅	C₂×C₈	1	4	yes	—
79	C₅×C₁₆	—	2	8	no	—
80	C₂³×C₈	C₄×C₅	1	1	yes	characteristic
81	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
82	C₂²×C₄²	C₄×C₅	1	1	yes	characteristic
83	C₂×M₃₂	—	2	4	no	—
84	C₂×M₃₂	—	2	2	no	—
85	C₂×M₃₂	C₄×C₅	1	8	yes	complemented
86	C₂×M₃₂	C₂²×C₅	1	4	yes	—
87	C₂²×C₁₆	—	2	2	no	—
88	C₂×C₄×C₈	—	2	4	no	—
89	C₂×C₄×C₈	C₄×C₅	1	2	yes	—
90	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
91	C₂³×C₅	—	2	4	no	—
92	C₂³×C₅	#97	1	2	yes	—
93	C₂³×C₅	C₂²⋊C₈	1	2	yes	—
94	C₂³×C₅	C₄⋊C₈	1	1	yes	characteristic
95	C₂³×C₅	C₄⋊C₈	1	1	yes	characteristic
96	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₅	—	2	16	no	—
98	C₂×C₄×C₅	—	2	8	no	—
99	C₂×C₄×C₅	—	2	4	no	—
100	C₂×C₄×C₅	#95	1	2	yes	—
101	C₂×C₄×C₅	#95	1	2	yes	—
102	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
103	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
104	C₂×C₄×C₅	#97	1	2	yes	—
105	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
106	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
107	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
108	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
109	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
110	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
111	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
112	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
118	C₅×C₈	—	2	4	no	—
119	C₅×C₈	#95	1	2	yes	—
120	C₅×C₈	#95	1	2	yes	—
121	C₅×C₈	C₂×C₂²⋊C₄	1	2	yes	—
122	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
123	C₂³×C₄	—	2	1	no	—
124	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	C₅×C₈	1	2	yes	—
128	C₂²×C₈	C₅×C₈	1	2	yes	—
129	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
130	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
131	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
132	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
133	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
134	C₂²×C₈	C₅×D₄	1	4	yes	—
135	C₂²×C₈	C₅×Q₈	1	1	yes	characteristic
136	C₂²×C₈	C₅×Q₈	1	1	yes	characteristic
137	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
138	C₂×C₄²	—	2	2	no	—
139	C₂×C₄²	—	2	1	no	—
140	C₂×C₄²	C₅×C₈	1	4	yes	—
141	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
142	M₃₂	—	2	8	no	—
143	M₃₂	—	2	4	no	—
144	M₃₂	C₂×C₄×C₅	1	8	yes	complemented
145	C₂×C₁₆	—	2	8	no	—
146	C₂×C₁₆	—	2	4	no	—
147	C₄×C₈	—	2	4	no	—
148	C₄×C₈	—	2	4	no	—
149	C₂²×C₅	—	2	4	no	—
150	C₂²×C₅	—	2	4	no	—
151	C₂²×C₅	—	2	2	no	—
152	C₂²×C₅	—	2	2	no	—
153	C₂²×C₅	#336	1	2	yes	—
154	C₂²×C₅	#336	1	2	yes	—
155	C₂²×C₅	#364	1	4	yes	—
156	C₂²×C₅	C₂×#97	1	1	yes	characteristic
157	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
158	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
159	C₄×C₅	—	2	8	no	—
160	C₄×C₅	#336	1	2	yes	—
161	C₄×C₅	#336	1	2	yes	—
162	C₄×C₅	C₂×#95	1	1	yes	characteristic
163	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
164	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
165	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
166	C₂⁴	C₅×M₁₆	1	1	yes	characteristic
167	C₂²×C₄	—	2	8	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	C₄²×C₅	1	2	yes	—
171	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
172	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
173	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
177	C₂²×C₄	C₅×M₁₆	1	2	yes	—
178	C₂²×C₄	C₅×M₁₆	1	1	yes	characteristic
179	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	C₄²×C₅	1	2	yes	—
185	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
186	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
187	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
188	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
189	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
190	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
191	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
192	C₂×C₈	C₂×C₅×C₈	1	4	yes	—
193	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
194	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
195	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
196	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
197	C₄²	—	2	2	no	—
198	C₄²	—	2	4	no	—
199	C₄²	C₂×C₅×C₈	1	4	yes	—
200	C₁₆	—	2	8	no	—
201	C₂×C₅	—	2	1	no	—
202	C₂×C₅	—	2	2	no	—
203	C₂×C₅	—	2	1	no	—
204	C₂×C₅	#1376	1	4	yes	direct product factor
205	C₂×C₅	C₂×#336	1	1	yes	characteristic
206	C₂×C₅	C₂×#364	1	2	yes	—
207	C₂³	—	2	4	no	—
208	C₂³	C₅×#97	1	2	yes	—
209	C₂³	C₅×C₂²⋊C₈	1	2	yes	—
210	C₂³	C₅×C₄⋊C₈	1	1	yes	characteristic
211	C₂³	C₅×C₄⋊C₈	1	1	yes	characteristic
212	C₂³	C₂×C₅×M₁₆	1	1	yes	characteristic
213	C₂×C₄	—	2	16	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	C₅×#95	1	2	yes	—
217	C₂×C₄	C₅×#95	1	2	yes	—
218	C₂×C₄	C₄×C₅×C₈	1	2	yes	—
219	C₂×C₄	C₄×C₅×C₈	1	2	yes	—
220	C₂×C₄	C₅×#97	1	2	yes	—
221	C₂×C₄	C₅×C₂²⋊C₈	1	2	yes	—
222	C₂×C₄	C₅×C₂²⋊C₈	1	2	yes	—
223	C₂×C₄	C₅×C₂²⋊C₈	1	2	yes	—
224	C₂×C₄	C₅×C₄⋊C₈	1	1	yes	characteristic
225	C₂×C₄	C₅×C₄⋊C₈	1	1	yes	characteristic
226	C₂×C₄	C₅×C₄⋊C₈	1	2	yes	—
227	C₂×C₄	C₅×C₄⋊C₈	1	2	yes	—
228	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
229	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
230	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
231	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
232	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
233	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
234	C₈	—	2	4	no	—
235	C₈	C₅×#95	1	2	yes	—
236	C₈	C₅×#95	1	2	yes	—
237	C₈	C₂×C₅×C₂²⋊C₄	1	2	yes	—
238	C₈	C₂×C₅×C₄⋊C₄	1	2	yes	—
239	C₅	C₂×#1376	1	1	yes	characteristic, direct product factor, Sylow
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	C₅×#336	1	2	yes	—
245	C₂²	C₅×#336	1	2	yes	—
246	C₂²	C₅×#364	1	4	yes	—
247	C₂²	C₂×C₅×#97	1	1	yes	characteristic
248	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
249	C₂²	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
250	C₄	—	2	8	no	—
251	C₄	C₅×#336	1	2	yes	—
252	C₄	C₅×#336	1	2	yes	—
253	C₄	C₂×C₅×#95	1	1	yes	characteristic
254	C₄	C₂×C₄×C₅×C₈	1	1	yes	characteristic
255	C₄	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
256	C₄	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
257	C₂	—	2	1	no	—
258	C₂	—	2	1	no	—
259	C₂	—	2	2	no	—
260	C₂	C₅×#1376	1	4	yes	direct product factor
261	C₂	C₂×C₅×#336	1	1	yes	characteristic
262	C₂	C₂×C₅×#364	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅×C₈	D₄
commutator	C₄	C₂×C₄×C₅×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂³×C₅	C₄⋊C₈
prime radical	C₂⁴×C₅	M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅×C₈ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1376	C₂	direct (N × H)
9. C₂×C₅×M₃₂	C₄	semidirect
15. C₂×#1376	C₅	direct (N × H)
37. C₅×M₃₂	C₂×C₄	semidirect
44. #1376	C₂×C₅	direct (N × H)
85. C₂×M₃₂	C₄×C₅	semidirect
144. M₃₂	C₂×C₄×C₅	semidirect
204. C₂×C₅	#1376	direct (N × H)
239. C₅	C₂×#1376	direct (N × H)
260. C₂	C₅×#1376	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×C₄×C₅×C₈	2
C₅×#1376	2
C₂×#1376	5

Subgroups — C2×C5×#1376 #11935268

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1376 #11935268