Subgroups — C₅×#8587 #11931307

Order 1280.

516 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#1842	C₂	1	2	yes	maximal
3	C₂×C₅×#333	C₂	1	2	yes	maximal
4	C₅×#1296	C₂	1	2	yes	maximal
5	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₅×Q₈	C₄	1	2	yes	—
7	C₂×C₅×C₄⋊C₈	—	2	2	no	—
8	C₂×C₅×C₄⋊C₈	—	2	2	no	—
9	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
10	C₅×#391	C₄	1	2	yes	—
11	C₅×#389	—	2	2	no	—
12	C₅×#384	C₂²	1	1	yes	characteristic
13	C₄×C₅×C₄⋊C₄	—	2	2	no	—
14	C₄³×C₅	C₂²	1	1	yes	characteristic
15	C₅×#333	—	2	4	no	—
16	#8587	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
18	C₂²×C₅×C₈	—	2	4	no	—
19	C₅×C₄⋊Q₈	—	2	2	no	—
20	C₅×C₄⋊Q₈	D₄	1	2	yes	—
21	C₅×C₄⋊Q₈	Q₈	1	2	yes	—
22	C₄×C₅×Q₈	—	4	4	no	—
23	C₂×C₅×C₄⋊C₄	—	2	2	no	—
24	C₂×C₅×C₄⋊C₄	—	2	2	no	—
25	C₂×C₅×C₄⋊C₄	—	2	2	no	—
26	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅	—	2	2	no	—
28	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅	D₄	1	2	yes	—
31	C₂×C₄²×C₅	D₄	1	2	yes	—
32	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
33	C₅×C₄⋊C₈	—	2	4	no	—
34	C₅×C₄⋊C₈	—	2	4	no	—
35	C₅×C₄⋊C₈	—	4	4	no	—
36	C₅×C₄⋊C₈	—	4	4	no	—
37	C₅×#95	—	2	4	no	—
38	C₄×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
39	#1842	C₂×C₅	1	2	yes	—
40	C₂×#333	C₂×C₅	1	2	yes	—
41	#1296	C₂×C₅	1	2	yes	—
42	C₂×C₅×Q₈	—	2	2	no	—
43	C₂×C₅×Q₈	—	4	2	no	—
44	C₂×C₅×Q₈	—	4	2	no	—
45	C₂×C₅×Q₈	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₅	—	2	2	no	—
47	C₂²×C₄×C₅	—	2	2	no	—
48	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
53	C₂×C₅×C₈	—	4	4	no	—
54	C₂×C₅×C₈	—	4	4	no	—
55	C₂×C₅×C₈	—	4	4	no	—
56	C₅×C₄⋊C₄	—	2	4	no	—
57	C₅×C₄⋊C₄	—	2	4	no	—
58	C₅×C₄⋊C₄	—	4	2	no	—
59	C₅×C₄⋊C₄	—	4	4	no	—
60	C₅×C₄⋊C₄	—	4	2	no	—
61	C₅×C₄⋊C₄	—	4	2	no	—
62	C₄²×C₅	—	2	2	no	—
63	C₄²×C₅	—	2	2	no	—
64	C₄²×C₅	—	2	2	no	—
65	C₄²×C₅	—	4	1	no	—
66	C₄²×C₅	—	4	4	no	—
67	C₄²×C₅	SD₁₆	1	4	yes	—
68	C₄²×C₅	Q₁₆	1	4	yes	—
69	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₅	#41	1	2	yes	—
72	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
73	C₂×C₄×Q₈	C₄×C₅	1	2	yes	—
74	C₂×C₄⋊C₈	—	2	2	no	—
75	C₂×C₄⋊C₈	—	2	2	no	—
76	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
77	#391	C₄×C₅	1	2	yes	—
78	#389	—	2	2	no	—
79	#384	C₂²×C₅	1	1	yes	characteristic
80	C₄×C₄⋊C₄	—	2	2	no	—
81	C₄³	C₂²×C₅	1	1	yes	characteristic
82	#333	—	2	4	no	—
83	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₅×Q₈	—	4	4	no	—
85	C₅×Q₈	—	4	4	no	—
86	C₂×C₄×C₅	—	2	1	no	—
87	C₂×C₄×C₅	—	2	2	no	—
88	C₂×C₄×C₅	—	2	1	no	—
89	C₂×C₄×C₅	—	2	2	no	—
90	C₂×C₄×C₅	—	2	4	no	—
91	C₂×C₄×C₅	—	2	1	no	—
92	C₂×C₄×C₅	—	2	2	no	—
93	C₂×C₄×C₅	—	4	2	no	—
94	C₂×C₄×C₅	—	4	2	no	—
95	C₂×C₄×C₅	—	4	2	no	—
96	C₂×C₄×C₅	—	4	4	no	—
97	C₂×C₄×C₅	#100	1	2	yes	—
98	C₂×C₄×C₅	#103	1	4	yes	—
99	C₂×C₄×C₅	#103	1	4	yes	—
100	C₂×C₄×C₅	C₂×C₄⋊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×SD₁₆	1	2	yes	—
105	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
106	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
107	C₂×C₄×C₅	#137	1	2	yes	—
108	C₅×C₈	—	8	4	no	—
109	C₂²×Q₈	C₂×C₄×C₅	1	2	yes	—
110	C₂²×C₈	—	2	4	no	—
111	C₄⋊Q₈	—	2	2	no	—
112	C₄⋊Q₈	C₅×D₄	1	2	yes	—
113	C₄⋊Q₈	C₅×Q₈	1	2	yes	—
114	C₄×Q₈	—	4	4	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
119	C₂×C₄²	—	2	2	no	—
120	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
121	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
122	C₂×C₄²	C₅×D₄	1	2	yes	—
123	C₂×C₄²	C₅×D₄	1	2	yes	—
124	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
125	C₄⋊C₈	—	2	4	no	—
126	C₄⋊C₈	—	2	4	no	—
127	C₄⋊C₈	—	4	4	no	—
128	C₄⋊C₈	—	4	4	no	—
129	#95	—	2	4	no	—
130	C₂²×C₅	#380	1	1	yes	characteristic
131	C₂²×C₅	C₂×#100	1	1	yes	characteristic
132	C₂²×C₅	#413	1	1	yes	characteristic
133	C₂²×C₅	C₂×#103	1	2	yes	—
134	C₂²×C₅	#417	1	2	yes	—
135	C₄×C₅	—	2	1	no	—
136	C₄×C₅	—	2	1	no	—
137	C₄×C₅	—	2	4	no	—
138	C₄×C₅	—	4	4	no	—
139	C₄×C₅	—	4	1	no	—
140	C₄×C₅	—	8	2	no	—
141	C₄×C₅	#333	1	4	yes	complemented
142	C₄×C₅	#460	1	2	yes	—
143	C₄×C₅	#462	1	2	yes	—
144	C₄×C₅	#475	1	2	yes	—
145	C₄×C₅	#477	1	2	yes	—
146	C₂×Q₈	—	2	2	no	—
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	C₅×C₄⋊C₄	1	4	yes	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
154	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
157	C₂×C₈	—	4	4	no	—
158	C₂×C₈	—	4	4	no	—
159	C₂×C₈	—	4	4	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	4	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	2	2	no	—
168	C₄²	—	2	2	no	—
169	C₄²	—	4	1	no	—
170	C₄²	—	4	4	no	—
171	C₄²	C₅×SD₁₆	1	4	yes	—
172	C₄²	C₅×Q₁₆	1	4	yes	—
173	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
174	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
175	C₄²	C₅×#41	1	2	yes	—
176	C₂×C₅	C₂×#333	1	1	yes	characteristic
177	C₂×C₅	#1550	1	2	yes	—
178	C₂×C₅	#1560	1	1	yes	characteristic
179	C₂×C₅	#1800	1	1	yes	characteristic
180	C₂×C₅	#1803	1	2	yes	—
181	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
182	Q₈	—	4	4	no	—
183	Q₈	—	4	4	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	C₅×#100	1	2	yes	—
196	C₂×C₄	C₅×#103	1	4	yes	—
197	C₂×C₄	C₅×#103	1	4	yes	—
198	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
199	C₂×C₄	C₅×#117	1	1	yes	characteristic
200	C₂×C₄	C₅×#121	1	2	yes	—
201	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
202	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
203	C₂×C₄	C₂×C₅×Q₁₆	1	2	yes	—
204	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
205	C₂×C₄	C₅×#137	1	2	yes	—
206	C₈	—	8	4	no	—
207	C₅	#8587	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²	C₅×#380	1	1	yes	characteristic
209	C₂²	C₂×C₅×#100	1	1	yes	characteristic
210	C₂²	C₅×#413	1	1	yes	characteristic
211	C₂²	C₂×C₅×#103	1	2	yes	—
212	C₂²	C₅×#417	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	4	no	—
216	C₄	—	4	4	no	—
217	C₄	—	4	1	no	—
218	C₄	—	8	2	no	—
219	C₄	C₅×#333	1	4	yes	complemented
220	C₄	C₅×#460	1	2	yes	—
221	C₄	C₅×#462	1	2	yes	—
222	C₄	C₅×#475	1	2	yes	—
223	C₄	C₅×#477	1	2	yes	—
224	C₂	C₂×C₅×#333	1	1	yes	characteristic
225	C₂	C₅×#1550	1	2	yes	—
226	C₂	C₅×#1560	1	1	yes	characteristic
227	C₂	C₅×#1800	1	1	yes	characteristic
228	C₂	C₅×#1803	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
16. #8587	C₅	direct (N × H)
141. C₄×C₅	#333	semidirect
207. C₅	#8587	direct (N × H)
219. C₄	C₅×#333	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₅×C₄⋊Q₈	2
C₅×#1842	2
C₂×C₅×#333	2
C₅×#1296	2
#8587	5

Subgroups — C5×#8587 #11931307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8587 #11931307