Subgroups — C₄×C₅×#498 #11944439

Order 1280.

878 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×C₅×#498	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×C₄⋊Q₈	C₂	1	2	yes	maximal
3	C₅×#1990	C₂	1	2	yes	maximal
4	C₅×#1941	C₂	1	4	yes	maximal
5	C₅×#1844	C₂	1	1	yes	maximal, characteristic
6	C₄×C₅×#106	C₂	1	4	yes	maximal
7	C₄²×C₅×C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₄×C₅×Q₈	—	2	2	no	—
10	C₅×#498	C₄	1	8	yes	direct product factor
11	C₂×C₅×#106	C₂²	1	4	yes	—
12	C₂×C₅×#106	C₂²	1	4	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
17	C₅×#391	—	2	2	no	—
18	C₅×#389	—	2	2	no	—
19	C₅×#389	C₂²	1	4	yes	—
20	C₅×#384	C₂²	1	2	yes	—
21	C₄×C₅×C₄⋊C₄	—	2	2	no	—
22	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
23	C₄³×C₅	C₂²	1	1	yes	characteristic
24	C₅×#340	C₂²	1	8	yes	—
25	C₄×#498	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×C₅×Q₈	—	2	2	no	—
27	C₂²×C₅×C₈	C₂³	1	2	yes	—
28	C₂²×C₅×C₈	C₂³	1	2	yes	—
29	C₅×C₄⋊Q₈	—	2	4	no	—
30	C₅×C₄⋊Q₈	C₂×C₄	1	8	yes	—
31	C₄×C₅×Q₈	—	4	8	no	—
32	C₂×C₅×C₄⋊C₄	—	2	2	no	—
33	C₂×C₅×C₄⋊C₄	—	2	2	no	—
34	C₂×C₅×C₄⋊C₄	—	2	2	no	—
35	C₂×C₅×C₄⋊C₄	—	2	4	no	—
36	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
37	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
38	C₂×C₄²×C₅	—	2	4	no	—
39	C₂×C₄²×C₅	D₄	1	2	yes	—
40	C₂×C₄²×C₅	D₄	1	2	yes	—
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₅×#106	—	2	8	no	—
45	C₅×#106	C₂×C₄	1	16	yes	—
46	C₄×C₅×C₈	—	2	4	no	—
47	C₄×C₅×C₈	—	2	2	no	—
48	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
49	C₄×C₅×C₈	Q₈	1	8	yes	complemented
50	C₅×#95	—	2	4	no	—
51	C₅×#95	—	2	4	no	—
52	C₂×#498	C₂×C₅	1	1	yes	characteristic
53	C₄×C₄⋊Q₈	C₂×C₅	1	2	yes	—
54	#1990	C₂×C₅	1	2	yes	—
55	#1941	C₂×C₅	1	4	yes	—
56	#1844	C₂×C₅	1	1	yes	characteristic
57	C₄×#106	C₂×C₅	1	4	yes	—
58	C₄²×C₈	C₂×C₅	1	1	yes	characteristic
59	C₂×C₅×Q₈	—	2	8	no	—
60	C₂×C₅×Q₈	—	4	4	no	—
61	C₂×C₅×Q₈	—	4	4	no	—
62	C₂²×C₄×C₅	—	2	4	no	—
63	C₂²×C₄×C₅	—	2	4	no	—
64	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
69	C₂×C₅×C₈	—	2	2	no	—
70	C₂×C₅×C₈	—	2	2	no	—
71	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
72	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
73	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
74	C₂×C₅×C₈	#41	1	4	yes	—
75	C₅×C₄⋊C₄	—	2	8	no	—
76	C₅×C₄⋊C₄	—	2	4	no	—
77	C₅×C₄⋊C₄	—	2	8	no	—
78	C₅×C₄⋊C₄	—	4	8	no	—
79	C₅×C₄⋊C₄	—	4	4	no	—
80	C₅×C₄⋊C₄	—	4	4	no	—
81	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
82	C₄²×C₅	—	2	4	no	—
83	C₄²×C₅	—	2	2	no	—
84	C₄²×C₅	—	4	8	no	—
85	C₄²×C₅	SD₁₆	1	8	yes	—
86	C₄²×C₅	C₂²×C₄	1	2	yes	—
87	C₄²×C₅	C₂×Q₈	1	4	yes	—
88	C₄²×C₅	#41	1	2	yes	—
89	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
90	C₂×C₄×Q₈	—	2	2	no	—
91	#498	C₄×C₅	1	8	yes	direct product factor
92	C₂×#106	C₂²×C₅	1	4	yes	—
93	C₂×#106	C₂²×C₅	1	4	yes	—
94	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
95	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
98	#391	—	2	2	no	—
99	#389	—	2	2	no	—
100	#389	C₂²×C₅	1	4	yes	—
101	#384	C₂²×C₅	1	2	yes	—
102	C₄×C₄⋊C₄	—	2	2	no	—
103	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
104	C₄³	C₂²×C₅	1	1	yes	characteristic
105	#340	C₂²×C₅	1	8	yes	—
106	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
107	C₅×Q₈	—	4	16	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	2	2	no	—
110	C₂×C₄×C₅	—	2	2	no	—
111	C₂×C₄×C₅	—	2	8	no	—
112	C₂×C₄×C₅	—	4	4	no	—
113	C₂×C₄×C₅	—	4	4	no	—
114	C₂×C₄×C₅	—	4	4	no	—
115	C₂×C₄×C₅	—	4	4	no	—
116	C₂×C₄×C₅	—	4	4	no	—
117	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
118	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
119	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
120	C₂×C₄×C₅	C₂×SD₁₆	1	4	yes	—
121	C₂×C₄×C₅	C₂×SD₁₆	1	4	yes	—
122	C₂×C₄×C₅	C₂×SD₁₆	1	4	yes	—
123	C₂×C₄×C₅	#135	1	4	yes	—
124	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
126	C₂×C₄×C₅	C₂×#41	1	2	yes	—
127	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
128	C₅×C₈	—	2	4	no	—
129	C₅×C₈	C₄×Q₈	1	8	yes	complemented
130	C₂²×Q₈	—	2	2	no	—
131	C₂²×C₈	C₂³×C₅	1	2	yes	—
132	C₂²×C₈	C₂³×C₅	1	2	yes	—
133	C₄⋊Q₈	—	2	4	no	—
134	C₄⋊Q₈	C₂×C₄×C₅	1	8	yes	—
135	C₄×Q₈	—	4	8	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	4	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
141	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
142	C₂×C₄²	—	2	4	no	—
143	C₂×C₄²	C₅×D₄	1	2	yes	—
144	C₂×C₄²	C₅×D₄	1	2	yes	—
145	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
146	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
147	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
148	#106	—	2	8	no	—
149	#106	C₂×C₄×C₅	1	16	yes	—
150	C₄×C₈	—	2	4	no	—
151	C₄×C₈	—	2	2	no	—
152	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
153	C₄×C₈	C₅×Q₈	1	8	yes	complemented
154	#95	—	2	4	no	—
155	#95	—	2	4	no	—
156	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
157	C₂²×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
158	C₂²×C₅	#532	1	1	yes	characteristic
159	C₂²×C₅	C₂²×SD₁₆	1	2	yes	—
160	C₂²×C₅	C₂×#135	1	2	yes	—
161	C₄×C₅	—	2	2	no	—
162	C₄×C₅	—	2	4	no	—
163	C₄×C₅	—	4	8	no	—
164	C₄×C₅	—	8	4	no	—
165	C₄×C₅	C₄×SD₁₆	1	8	yes	—
166	C₄×C₅	#498	1	4	yes	direct product factor
167	C₄×C₅	C₂×C₄×Q₈	1	2	yes	—
168	C₄×C₅	#533	1	2	yes	—
169	C₂×Q₈	—	2	8	no	—
170	C₂×Q₈	—	4	4	no	—
171	C₂×Q₈	—	4	4	no	—
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
176	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
177	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
178	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
182	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
183	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
184	C₂×C₈	C₅×#41	1	4	yes	—
185	C₄⋊C₄	—	2	8	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	2	8	no	—
188	C₄⋊C₄	—	4	8	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	—	4	4	no	—
191	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
192	C₄²	—	2	4	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	4	8	no	—
195	C₄²	C₅×SD₁₆	1	8	yes	—
196	C₄²	C₂²×C₄×C₅	1	2	yes	—
197	C₄²	C₂×C₅×Q₈	1	4	yes	—
198	C₄²	C₅×#41	1	2	yes	—
199	C₂×C₅	C₄×C₄⋊Q₈	1	1	yes	characteristic
200	C₂×C₅	C₂×C₄×SD₁₆	1	2	yes	—
201	C₂×C₅	C₂×#498	1	1	yes	characteristic
202	C₂×C₅	#3160	1	1	yes	characteristic
203	C₂×C₅	#3167	1	2	yes	—
204	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
205	Q₈	—	4	16	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	8	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
216	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
217	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
218	C₂×C₄	C₂×C₅×SD₁₆	1	4	yes	—
219	C₂×C₄	C₂×C₅×SD₁₆	1	4	yes	—
220	C₂×C₄	C₂×C₅×SD₁₆	1	4	yes	—
221	C₂×C₄	C₅×#135	1	4	yes	—
222	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
223	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
224	C₂×C₄	C₂×C₅×#41	1	2	yes	—
225	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
226	C₈	—	2	4	no	—
227	C₈	C₄×C₅×Q₈	1	8	yes	complemented
228	C₅	C₄×#498	1	1	yes	characteristic, direct product factor, Sylow
229	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
230	C₂²	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
231	C₂²	C₅×#532	1	1	yes	characteristic
232	C₂²	C₂²×C₅×SD₁₆	1	2	yes	—
233	C₂²	C₂×C₅×#135	1	2	yes	—
234	C₄	—	2	2	no	—
235	C₄	—	2	4	no	—
236	C₄	—	4	8	no	—
237	C₄	—	8	4	no	—
238	C₄	C₄×C₅×SD₁₆	1	8	yes	—
239	C₄	C₅×#498	1	4	yes	direct product factor
240	C₄	C₂×C₄×C₅×Q₈	1	2	yes	—
241	C₄	C₅×#533	1	2	yes	—
242	C₂	C₄×C₅×C₄⋊Q₈	1	1	yes	characteristic
243	C₂	C₂×C₄×C₅×SD₁₆	1	2	yes	—
244	C₂	C₂×C₅×#498	1	1	yes	characteristic
245	C₂	C₅×#3160	1	1	yes	characteristic
246	C₂	C₅×#3167	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle, prime radical	C₂³×C₅	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₅×#498	C₄	direct (N × H)
25. C₄×#498	C₅	direct (N × H)
49. C₄×C₅×C₈	Q₈	semidirect
91. #498	C₄×C₅	direct (N × H)
129. C₅×C₈	C₄×Q₈	semidirect
153. C₄×C₈	C₅×Q₈	semidirect
166. C₄×C₅	#498	direct (N × H)
227. C₈	C₄×C₅×Q₈	semidirect
228. C₅	C₄×#498	direct (N × H)
239. C₄	C₅×#498	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×#498	2
C₄×C₅×C₄⋊Q₈	2
C₅×#1990	2
C₅×#1941	2
C₅×#1844	2
C₄×C₅×#106	2
C₄²×C₅×C₈	2
C₄×#498	5

Subgroups — C4×C5×#498 #11944439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×#498 #11944439