Subgroups — C₅×#8616 #11931336

Order 1280.

486 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₅×#1800	C₂	1	2	yes	maximal
3	C₅×#1567	C₂	1	2	yes	maximal, complemented
4	C₅×#1292	C₂	1	2	yes	maximal, complemented
5	C₂³×C₅×C₈	C₄	1	2	yes	—
6	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
7	C₅×C₈×D₄	—	2	2	no	—
8	C₅×C₈×D₄	—	2	2	no	—
9	C₅×#423	C₂²	1	4	yes	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×#100	—	2	2	no	—
12	C₂×C₅×C₂²⋊C₈	C₄	1	2	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₅×#343	—	2	2	no	—
15	C₅×#335	—	2	2	no	—
16	C₅×#335	—	2	2	no	—
17	#8616	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₅×D₄	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
20	C₂×C₅×M₁₆	—	2	4	no	—
21	C₂²×C₅×C₈	—	2	2	no	—
22	C₂²×C₅×C₈	—	2	2	no	—
23	C₂²×C₅×C₈	—	4	4	no	—
24	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₅×D₄	—	2	2	no	—
28	C₄×C₅×D₄	—	2	2	no	—
29	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
30	C₂×C₅×C₂²⋊C₄	Q₈	1	2	yes	—
31	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
32	C₅×C₄⋊C₈	—	2	2	no	—
33	C₅×C₄⋊C₈	—	2	4	no	—
34	C₅×C₄⋊C₈	—	2	4	no	—
35	C₅×#100	—	4	4	no	—
36	C₅×C₂²⋊C₈	—	2	4	no	—
37	C₄×C₅×C₈	D₄	1	2	yes	—
38	C₄×C₅×C₈	D₄	1	2	yes	—
39	C₂×C₈×D₄	C₂×C₅	1	1	yes	characteristic
40	#1800	C₂×C₅	1	2	yes	—
41	#1567	C₂×C₅	1	2	yes	complemented
42	#1292	C₂×C₅	1	2	yes	complemented
43	C₂⁴×C₅	C₄⋊C₄	1	2	yes	—
44	C₂×C₅×D₄	—	2	2	no	—
45	C₂×C₅×D₄	—	2	2	no	—
46	C₂×C₅×D₄	D₈	1	2	yes	—
47	C₂×C₅×D₄	Q₁₆	1	2	yes	—
48	C₂²×C₄×C₅	—	2	2	no	—
49	C₂²×C₄×C₅	—	2	2	no	—
50	C₂²×C₄×C₅	—	4	2	no	—
51	C₂²×C₄×C₅	—	4	2	no	—
52	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
56	C₅×M₁₆	—	4	4	no	—
57	C₅×M₁₆	—	4	4	no	—
58	C₂×C₅×C₈	—	2	2	no	—
59	C₂×C₅×C₈	—	2	2	no	—
60	C₂×C₅×C₈	—	4	1	no	—
61	C₂×C₅×C₈	—	4	4	no	—
62	C₂×C₅×C₈	—	4	4	no	—
63	C₂×C₅×C₈	—	4	2	no	—
64	C₂×C₅×C₈	—	4	4	no	—
65	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
67	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₅×C₄⋊C₄	—	2	2	no	—
69	C₅×C₂²⋊C₄	—	2	4	no	—
70	C₄²×C₅	C₂×D₄	1	2	yes	—
71	C₄²×C₅	#41	1	2	yes	—
72	C₂³×C₈	C₄×C₅	1	2	yes	—
73	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
74	C₈×D₄	—	2	2	no	—
75	C₈×D₄	—	2	2	no	—
76	#423	C₂²×C₅	1	4	yes	—
77	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
78	C₂×#100	—	2	2	no	—
79	C₂×C₂²⋊C₈	C₄×C₅	1	2	yes	—
80	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
81	#343	—	2	2	no	—
82	#335	—	2	2	no	—
83	#335	—	2	2	no	—
84	C₂³×C₅	—	2	2	no	—
85	C₂³×C₅	—	2	2	no	—
86	C₂³×C₅	—	4	2	no	—
87	C₂³×C₅	—	4	2	no	—
88	C₂³×C₅	#107	1	4	yes	—
89	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₅×D₄	—	2	4	no	—
91	C₅×D₄	—	2	4	no	—
92	C₂×C₄×C₅	—	2	1	no	—
93	C₂×C₄×C₅	—	2	1	no	—
94	C₂×C₄×C₅	—	2	2	no	—
95	C₂×C₄×C₅	—	4	1	no	—
96	C₂×C₄×C₅	—	4	4	no	—
97	C₂×C₄×C₅	—	4	2	no	—
98	C₂×C₄×C₅	—	4	2	no	—
99	C₂×C₄×C₅	—	4	2	no	—
100	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅	#117	1	1	yes	characteristic
102	C₂×C₄×C₅	#121	1	2	yes	—
103	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
106	C₂×C₄×C₅	#135	1	2	yes	—
107	C₅×C₈	—	2	1	no	—
108	C₅×C₈	—	2	2	no	—
109	C₅×C₈	—	4	2	no	—
110	C₅×C₈	—	8	4	no	—
111	C₅×C₈	#100	1	2	yes	complemented
112	C₂²×D₄	C₅×D₄	1	1	yes	characteristic
113	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
114	C₂×M₁₆	—	2	4	no	—
115	C₂²×C₈	—	2	2	no	—
116	C₂²×C₈	—	2	2	no	—
117	C₂²×C₈	—	4	4	no	—
118	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
119	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
120	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
121	C₄×D₄	—	2	2	no	—
122	C₄×D₄	—	2	2	no	—
123	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
124	C₂×C₂²⋊C₄	C₅×Q₈	1	2	yes	—
125	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	2	4	no	—
128	C₄⋊C₈	—	2	4	no	—
129	#100	—	4	4	no	—
130	C₂²⋊C₈	—	2	4	no	—
131	C₄×C₈	C₅×D₄	1	2	yes	—
132	C₄×C₈	C₅×D₄	1	2	yes	—
133	C₂²×C₅	—	2	1	no	—
134	C₂²×C₅	—	2	1	no	—
135	C₂²×C₅	—	2	4	no	—
136	C₂²×C₅	—	4	4	no	—
137	C₂²×C₅	—	4	4	no	—
138	C₂²×C₅	—	4	2	no	—
139	C₂²×C₅	—	4	2	no	—
140	C₂²×C₅	#380	1	1	yes	characteristic
141	C₂²×C₅	C₂×#107	1	1	yes	characteristic
142	C₂²×C₅	#427	1	1	yes	characteristic
143	C₄×C₅	—	2	1	no	—
144	C₄×C₅	—	2	2	no	—
145	C₄×C₅	—	4	2	no	—
146	C₄×C₅	C₂×#100	1	1	yes	characteristic
147	C₄×C₅	#413	1	1	yes	characteristic
148	C₄×C₅	#466	1	2	yes	—
149	C₄×C₅	#467	1	2	yes	—
150	C₂⁴	C₅×C₄⋊C₄	1	2	yes	—
151	C₂×D₄	—	2	2	no	—
152	C₂×D₄	—	2	2	no	—
153	C₂×D₄	C₅×D₈	1	2	yes	—
154	C₂×D₄	C₅×Q₁₆	1	2	yes	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	—	4	2	no	—
159	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
160	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
161	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
162	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
163	M₁₆	—	4	4	no	—
164	M₁₆	—	4	4	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	4	no	—
169	C₂×C₈	—	4	4	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	—	4	4	no	—
172	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
173	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
174	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
175	C₄⋊C₄	—	2	2	no	—
176	C₂²⋊C₄	—	2	4	no	—
177	C₄²	C₂×C₅×D₄	1	2	yes	—
178	C₄²	C₅×#41	1	2	yes	—
179	C₂×C₅	—	2	1	no	—
180	C₂×C₅	—	2	1	no	—
181	C₂×C₅	—	4	4	no	—
182	C₂×C₅	#1800	1	1	yes	characteristic
183	C₂×C₅	#1808	1	1	yes	characteristic
184	C₂×C₅	#1813	1	1	yes	characteristic
185	C₂³	—	2	2	no	—
186	C₂³	—	2	2	no	—
187	C₂³	—	4	2	no	—
188	C₂³	—	4	2	no	—
189	C₂³	C₅×#107	1	4	yes	—
190	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
191	D₄	—	2	4	no	—
192	D₄	—	2	4	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
202	C₂×C₄	C₅×#117	1	1	yes	characteristic
203	C₂×C₄	C₅×#121	1	2	yes	—
204	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
205	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
206	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
207	C₂×C₄	C₅×#135	1	2	yes	—
208	C₈	—	2	1	no	—
209	C₈	—	2	2	no	—
210	C₈	—	4	2	no	—
211	C₈	—	8	4	no	—
212	C₈	C₅×#100	1	2	yes	complemented
213	C₅	#8616	1	1	yes	characteristic, direct product factor, Sylow
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	4	4	no	—
218	C₂²	—	4	4	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	C₅×#380	1	1	yes	characteristic
222	C₂²	C₂×C₅×#107	1	1	yes	characteristic
223	C₂²	C₅×#427	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	2	2	no	—
226	C₄	—	4	2	no	—
227	C₄	C₂×C₅×#100	1	1	yes	characteristic
228	C₄	C₅×#413	1	1	yes	characteristic
229	C₄	C₅×#466	1	2	yes	—
230	C₄	C₅×#467	1	2	yes	—
231	C₂	—	2	1	no	—
232	C₂	—	2	1	no	—
233	C₂	—	4	4	no	—
234	C₂	C₅×#1800	1	1	yes	characteristic
235	C₂	C₅×#1808	1	1	yes	characteristic
236	C₂	C₅×#1813	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#380
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂²×C₅×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1567	C₂	semidirect
4. C₅×#1292	C₂	semidirect
17. #8616	C₅	direct (N × H)
41. #1567	C₂×C₅	semidirect
42. #1292	C₂×C₅	semidirect
111. C₅×C₈	#100	semidirect
212. C₈	C₅×#100	semidirect
213. C₅	#8616	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×C₅×C₈×D₄	2
C₅×#1800	2
C₅×#1567	2
C₅×#1292	2
#8616	5

Subgroups — C5×#8616 #11931336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8616 #11931336