Subgroups — C₅×#8614 #11931334

Order 1280.

486 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	C₂×C₅×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₅×#1800	C₂	1	2	yes	maximal
3	C₅×#1566	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×#334	—	2	2	no	—
16	C₅×#334	—	2	2	no	—
17	#8614	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	#1566	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	SD₁₆	1	4	yes	—
47	C₂²×C₄×C₅	—	2	2	no	—
48	C₂²×C₄×C₅	—	2	2	no	—
49	C₂²×C₄×C₅	—	4	2	no	—
50	C₂²×C₄×C₅	—	4	2	no	—
51	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
55	C₅×M₁₆	—	4	4	no	—
56	C₅×M₁₆	—	4	4	no	—
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	—	2	2	no	—
59	C₂×C₅×C₈	—	4	1	no	—
60	C₂×C₅×C₈	—	4	4	no	—
61	C₂×C₅×C₈	—	4	4	no	—
62	C₂×C₅×C₈	—	4	2	no	—
63	C₂×C₅×C₈	—	4	4	no	—
64	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
66	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₅×C₄⋊C₄	—	2	2	no	—
68	C₅×C₂²⋊C₄	—	2	4	no	—
69	C₄²×C₅	C₂×D₄	1	2	yes	—
70	C₄²×C₅	#41	1	2	yes	—
71	C₂³×C₈	C₄×C₅	1	2	yes	—
72	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
73	C₈×D₄	—	2	2	no	—
74	C₈×D₄	—	2	2	no	—
75	#423	C₂²×C₅	1	4	yes	—
76	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
77	C₂×#100	—	2	2	no	—
78	C₂×C₂²⋊C₈	C₄×C₅	1	2	yes	—
79	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
80	#343	—	2	2	no	—
81	#334	—	2	2	no	—
82	#334	—	2	2	no	—
83	C₂³×C₅	—	2	2	no	—
84	C₂³×C₅	—	2	2	no	—
85	C₂³×C₅	—	4	2	no	—
86	C₂³×C₅	—	4	2	no	—
87	C₂³×C₅	#106	1	4	yes	—
88	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₅×D₄	—	2	8	no	—
90	C₂×C₄×C₅	—	2	1	no	—
91	C₂×C₄×C₅	—	2	1	no	—
92	C₂×C₄×C₅	—	2	2	no	—
93	C₂×C₄×C₅	—	4	1	no	—
94	C₂×C₄×C₅	—	4	4	no	—
95	C₂×C₄×C₅	—	4	2	no	—
96	C₂×C₄×C₅	—	4	2	no	—
97	C₂×C₄×C₅	—	4	2	no	—
98	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅	#117	1	1	yes	characteristic
100	C₂×C₄×C₅	#121	1	2	yes	—
101	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
103	C₂×C₄×C₅	#135	1	2	yes	—
104	C₅×C₈	—	2	1	no	—
105	C₅×C₈	—	2	2	no	—
106	C₅×C₈	—	4	2	no	—
107	C₅×C₈	—	8	4	no	—
108	C₅×C₈	#100	1	2	yes	complemented
109	C₂²×D₄	C₅×D₄	1	1	yes	characteristic
110	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
111	C₂×M₁₆	—	2	4	no	—
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	—	4	4	no	—
115	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
116	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
117	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
118	C₄×D₄	—	2	2	no	—
119	C₄×D₄	—	2	2	no	—
120	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
121	C₂×C₂²⋊C₄	C₅×Q₈	1	2	yes	—
122	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	4	no	—
125	C₄⋊C₈	—	2	4	no	—
126	#100	—	4	4	no	—
127	C₂²⋊C₈	—	2	4	no	—
128	C₄×C₈	C₅×D₄	1	2	yes	—
129	C₄×C₈	C₅×D₄	1	2	yes	—
130	C₂²×C₅	—	2	1	no	—
131	C₂²×C₅	—	2	1	no	—
132	C₂²×C₅	—	2	4	no	—
133	C₂²×C₅	—	4	4	no	—
134	C₂²×C₅	—	4	4	no	—
135	C₂²×C₅	—	4	2	no	—
136	C₂²×C₅	—	4	2	no	—
137	C₂²×C₅	#380	1	1	yes	characteristic
138	C₂²×C₅	C₂×#106	1	1	yes	characteristic
139	C₂²×C₅	#427	1	1	yes	characteristic
140	C₄×C₅	—	2	1	no	—
141	C₄×C₅	—	2	2	no	—
142	C₄×C₅	—	4	2	no	—
143	C₄×C₅	C₂×#100	1	1	yes	characteristic
144	C₄×C₅	#413	1	1	yes	characteristic
145	C₄×C₅	#465	1	4	yes	—
146	C₂⁴	C₅×C₄⋊C₄	1	2	yes	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	—	2	2	no	—
149	C₂×D₄	C₅×SD₁₆	1	4	yes	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	—	4	2	no	—
154	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
155	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
158	M₁₆	—	4	4	no	—
159	M₁₆	—	4	4	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	4	1	no	—
163	C₂×C₈	—	4	4	no	—
164	C₂×C₈	—	4	4	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	4	4	no	—
167	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
168	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
170	C₄⋊C₄	—	2	2	no	—
171	C₂²⋊C₄	—	2	4	no	—
172	C₄²	C₂×C₅×D₄	1	2	yes	—
173	C₄²	C₅×#41	1	2	yes	—
174	C₂×C₅	—	2	1	no	—
175	C₂×C₅	—	2	1	no	—
176	C₂×C₅	—	4	4	no	—
177	C₂×C₅	#1800	1	1	yes	characteristic
178	C₂×C₅	#1807	1	1	yes	characteristic
179	C₂×C₅	#1813	1	1	yes	characteristic
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	4	2	no	—
183	C₂³	—	4	2	no	—
184	C₂³	C₅×#106	1	4	yes	—
185	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
186	D₄	—	2	8	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
196	C₂×C₄	C₅×#117	1	1	yes	characteristic
197	C₂×C₄	C₅×#121	1	2	yes	—
198	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
199	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
200	C₂×C₄	C₅×#135	1	2	yes	—
201	C₈	—	2	1	no	—
202	C₈	—	2	2	no	—
203	C₈	—	4	2	no	—
204	C₈	—	8	4	no	—
205	C₈	C₅×#100	1	2	yes	complemented
206	C₅	#8614	1	1	yes	characteristic, direct product factor, Sylow
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	4	4	no	—
211	C₂²	—	4	4	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	C₅×#380	1	1	yes	characteristic
215	C₂²	C₂×C₅×#106	1	1	yes	characteristic
216	C₂²	C₅×#427	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	2	no	—
219	C₄	—	4	2	no	—
220	C₄	C₂×C₅×#100	1	1	yes	characteristic
221	C₄	C₅×#413	1	1	yes	characteristic
222	C₄	C₅×#465	1	4	yes	—
223	C₂	—	2	1	no	—
224	C₂	—	2	1	no	—
225	C₂	—	4	4	no	—
226	C₂	C₅×#1800	1	1	yes	characteristic
227	C₂	C₅×#1807	1	1	yes	characteristic
228	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×#1566	C₂	semidirect
4. C₅×#1292	C₂	semidirect
17. #8614	C₅	direct (N × H)
41. #1566	C₂×C₅	semidirect
42. #1292	C₂×C₅	semidirect
108. C₅×C₈	#100	semidirect
205. C₈	C₅×#100	semidirect
206. C₅	#8614	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×#1566	2
C₅×#1292	2
#8614	5

Subgroups — C5×#8614 #11931334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8614 #11931334