Subgroups — C₂×C₅×#1332 #11934310

Order 1280.

406 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#335	C₂	1	1	yes	maximal, characteristic
4	C₅×#1332	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₅×Q₁₆	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×Q₈	—	2	1	no	—
7	C₂²×C₅×C₁₆	—	2	1	no	—
8	C₄×C₅×Q₁₆	—	2	1	no	—
9	C₄×C₅×Q₁₆	—	2	1	no	—
10	C₄×C₅×Q₁₆	C₄	1	2	yes	—
11	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
12	C₂×C₅×#107	C₄	1	1	yes	characteristic
13	C₂×C₅×C₄⋊C₈	—	2	1	no	—
14	C₂×C₅×#103	—	2	1	no	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₄×C₅×C₁₆	—	2	1	no	—
17	C₄×C₅×C₁₆	C₂²	1	2	yes	—
18	C₅×#335	—	2	1	no	—
19	C₅×#335	C₂²	1	2	yes	—
20	C₂×#1332	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×Q₈	—	2	1	no	—
22	C₂×C₅×Q₁₆	—	2	1	no	—
23	C₂×C₅×Q₁₆	—	2	2	no	—
24	C₂×C₅×Q₁₆	C₈	1	4	yes	complemented
25	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
26	C₂²×C₅×C₈	—	4	1	no	—
27	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
29	C₄×C₅×Q₈	—	2	2	no	—
30	C₄×C₅×Q₈	—	2	2	no	—
31	C₄×C₅×Q₈	—	4	1	no	—
32	C₄×C₅×Q₈	—	4	1	no	—
33	C₂×C₅×C₄⋊C₄	—	2	1	no	—
34	C₂×C₅×C₄⋊C₄	—	4	1	no	—
35	C₂×C₄²×C₅	—	4	1	no	—
36	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₂×C₅×C₁₆	—	2	4	no	—
38	C₂×C₅×C₁₆	—	2	2	no	—
39	C₅×#107	—	2	1	no	—
40	C₅×#107	C₂×C₄	1	2	yes	—
41	C₅×C₄⋊C₈	—	2	2	no	—
42	C₅×C₄⋊C₈	—	4	1	no	—
43	C₅×#103	—	2	2	no	—
44	C₅×#103	—	2	2	no	—
45	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₅×C₈	D₄	1	1	yes	characteristic
47	C₄×C₅×C₈	D₄	1	1	yes	characteristic
48	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
49	C₂×C₄×Q₁₆	C₂×C₅	1	1	yes	characteristic
50	C₂×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic
51	C₂×#335	C₂×C₅	1	1	yes	characteristic
52	#1332	C₂×C₅	1	4	yes	direct product factor
53	C₂×C₅×Q₈	—	2	4	no	—
54	C₂×C₅×Q₈	—	2	2	no	—
55	C₂×C₅×Q₈	—	4	1	no	—
56	C₂×C₅×Q₈	—	4	2	no	—
57	C₂²×C₄×C₅	—	4	1	no	—
58	C₂²×C₄×C₅	—	4	1	no	—
59	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
62	C₅×Q₁₆	—	2	4	no	—
63	C₅×Q₁₆	—	2	2	no	—
64	C₅×Q₁₆	C₂×C₈	1	4	yes	complemented
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	—	4	2	no	—
67	C₂×C₅×C₈	—	8	2	no	—
68	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
71	C₂×C₅×C₈	M₁₆	1	2	yes	—
72	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
74	C₅×C₄⋊C₄	—	2	2	no	—
75	C₅×C₄⋊C₄	—	4	1	no	—
76	C₅×C₄⋊C₄	—	4	2	no	—
77	C₅×C₄⋊C₄	—	4	2	no	—
78	C₄²×C₅	—	4	2	no	—
79	C₄²×C₅	—	4	2	no	—
80	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
81	C₄²×C₅	D₈	1	1	yes	characteristic
82	C₄²×C₅	SD₁₆	1	1	yes	characteristic
83	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
84	C₅×C₁₆	—	2	4	no	—
85	C₂²×Q₁₆	C₄×C₅	1	1	yes	characteristic
86	C₂×C₄×Q₈	—	2	1	no	—
87	C₂²×C₁₆	—	2	1	no	—
88	C₄×Q₁₆	—	2	1	no	—
89	C₄×Q₁₆	—	2	1	no	—
90	C₄×Q₁₆	C₄×C₅	1	2	yes	—
91	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
92	C₂×#107	C₄×C₅	1	1	yes	characteristic
93	C₂×C₄⋊C₈	—	2	1	no	—
94	C₂×#103	—	2	1	no	—
95	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
96	C₄×C₁₆	—	2	1	no	—
97	C₄×C₁₆	C₂²×C₅	1	2	yes	—
98	#335	—	2	1	no	—
99	#335	C₂²×C₅	1	2	yes	—
100	C₂³×C₅	#102	1	1	yes	characteristic
101	C₅×Q₈	—	2	4	no	—
102	C₅×Q₈	—	4	4	no	—
103	C₅×Q₈	—	4	2	no	—
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	4	4	no	—
106	C₂×C₄×C₅	—	4	2	no	—
107	C₂×C₄×C₅	—	4	2	no	—
108	C₂×C₄×C₅	—	8	2	no	—
109	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
110	C₂×C₄×C₅	#102	1	1	yes	characteristic
111	C₂×C₄×C₅	#102	1	1	yes	characteristic
112	C₂×C₄×C₅	#102	1	1	yes	characteristic
113	C₂×C₄×C₅	#104	1	2	yes	—
114	C₂×C₄×C₅	SD₃₂	1	2	yes	—
115	C₂×C₄×C₅	Q₃₂	1	2	yes	—
116	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
119	C₅×C₈	—	2	1	no	—
120	C₅×C₈	—	2	1	no	—
121	C₅×C₈	—	8	2	no	—
122	C₅×C₈	C₂²⋊C₈	1	2	yes	—
123	C₅×C₈	C₂²×C₈	1	1	yes	characteristic
124	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
125	C₂²×Q₈	—	2	1	no	—
126	C₂×Q₁₆	—	2	1	no	—
127	C₂×Q₁₆	—	2	2	no	—
128	C₂×Q₁₆	C₅×C₈	1	4	yes	complemented
129	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
130	C₂²×C₈	—	4	1	no	—
131	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
132	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
133	C₄×Q₈	—	2	2	no	—
134	C₄×Q₈	—	2	2	no	—
135	C₄×Q₈	—	4	1	no	—
136	C₄×Q₈	—	4	1	no	—
137	C₂×C₄⋊C₄	—	2	1	no	—
138	C₂×C₄⋊C₄	—	4	1	no	—
139	C₂×C₄²	—	4	1	no	—
140	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
141	C₂×C₁₆	—	2	4	no	—
142	C₂×C₁₆	—	2	2	no	—
143	#107	—	2	1	no	—
144	#107	C₂×C₄×C₅	1	2	yes	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	4	1	no	—
147	#103	—	2	2	no	—
148	#103	—	2	2	no	—
149	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
150	C₄×C₈	C₅×D₄	1	1	yes	characteristic
151	C₄×C₈	C₅×D₄	1	1	yes	characteristic
152	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
153	C₂²×C₅	D₄⋊C₈	1	2	yes	—
154	C₂²×C₅	#358	1	2	yes	—
155	C₂²×C₅	#359	1	2	yes	—
156	C₂²×C₅	C₂×#102	1	1	yes	characteristic
157	C₄×C₅	—	2	1	no	—
158	C₄×C₅	—	2	1	no	—
159	C₄×C₅	—	4	4	no	—
160	C₄×C₅	—	8	2	no	—
161	C₄×C₅	D₄⋊C₈	1	2	yes	—
162	C₄×C₅	#358	1	2	yes	—
163	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
164	C₄×C₅	C₂×#104	1	1	yes	characteristic
165	C₄×C₅	C₂×SD₃₂	1	1	yes	characteristic
166	C₄×C₅	C₂×Q₃₂	1	1	yes	characteristic
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	2	4	no	—
169	C₂×Q₈	—	4	1	no	—
170	C₂×Q₈	—	4	2	no	—
171	C₂²×C₄	—	4	1	no	—
172	C₂²×C₄	—	4	1	no	—
173	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
174	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
175	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
176	Q₁₆	—	2	2	no	—
177	Q₁₆	—	2	4	no	—
178	Q₁₆	C₂×C₅×C₈	1	4	yes	complemented
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	—	8	2	no	—
182	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
183	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
184	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
185	C₂×C₈	C₅×M₁₆	1	2	yes	—
186	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
187	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	1	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄²	—	4	2	no	—
193	C₄²	—	4	2	no	—
194	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
195	C₄²	C₅×D₈	1	1	yes	characteristic
196	C₄²	C₅×SD₁₆	1	1	yes	characteristic
197	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
198	C₁₆	—	2	4	no	—
199	C₂×C₅	#1332	1	4	yes	direct product factor
200	C₂×C₅	C₂×D₄⋊C₈	1	1	yes	characteristic
201	C₂×C₅	C₂×#358	1	1	yes	characteristic
202	C₂×C₅	C₂×#359	1	1	yes	characteristic
203	C₂³	C₅×#102	1	1	yes	characteristic
204	Q₈	—	2	4	no	—
205	Q₈	—	4	4	no	—
206	Q₈	—	4	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	8	2	no	—
212	C₂×C₄	C₅×C₂²⋊C₈	1	2	yes	—
213	C₂×C₄	C₅×#102	1	1	yes	characteristic
214	C₂×C₄	C₅×#102	1	1	yes	characteristic
215	C₂×C₄	C₅×#102	1	1	yes	characteristic
216	C₂×C₄	C₅×#104	1	2	yes	—
217	C₂×C₄	C₅×SD₃₂	1	2	yes	—
218	C₂×C₄	C₅×Q₃₂	1	2	yes	—
219	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
220	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
221	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	8	2	no	—
225	C₈	C₅×C₂²⋊C₈	1	2	yes	—
226	C₈	C₂²×C₅×C₈	1	1	yes	characteristic
227	C₈	C₂×C₅×M₁₆	1	1	yes	characteristic
228	C₅	C₂×#1332	1	1	yes	characteristic, direct product factor, Sylow
229	C₂²	C₅×D₄⋊C₈	1	2	yes	—
230	C₂²	C₅×#358	1	2	yes	—
231	C₂²	C₅×#359	1	2	yes	—
232	C₂²	C₂×C₅×#102	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	4	no	—
236	C₄	—	8	2	no	—
237	C₄	C₅×D₄⋊C₈	1	2	yes	—
238	C₄	C₅×#358	1	2	yes	—
239	C₄	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
240	C₄	C₂×C₅×#104	1	1	yes	characteristic
241	C₄	C₂×C₅×SD₃₂	1	1	yes	characteristic
242	C₄	C₂×C₅×Q₃₂	1	1	yes	characteristic
243	C₂	C₅×#1332	1	4	yes	direct product factor
244	C₂	C₂×C₅×D₄⋊C₈	1	1	yes	characteristic
245	C₂	C₂×C₅×#358	1	1	yes	characteristic
246	C₂	C₂×C₅×#359	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	D₈
commutator	C₈	C₂²×C₅×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle, prime radical	C₂³×C₅	#102
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1332	C₂	direct (N × H)
20. C₂×#1332	C₅	direct (N × H)
24. C₂×C₅×Q₁₆	C₈	semidirect
52. #1332	C₂×C₅	direct (N × H)
64. C₅×Q₁₆	C₂×C₈	semidirect
128. C₂×Q₁₆	C₅×C₈	semidirect
178. Q₁₆	C₂×C₅×C₈	semidirect
199. C₂×C₅	#1332	direct (N × H)
228. C₅	C₂×#1332	direct (N × H)
243. C₂	C₅×#1332	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×Q₁₆	2
C₂×C₄×C₅×C₁₆	2
C₂×C₅×#335	2
C₅×#1332	2
C₂×#1332	5

Subgroups — C2×C5×#1332 #11934310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1332 #11934310