Subgroups — C₅×#9820 #11932540

Order 1280.

642 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#492	C₂	1	1	yes	maximal, characteristic
2	C₅×#1990	C₂	1	2	yes	maximal, complemented
3	C₅×#1957	C₂	1	2	yes	maximal
4	C₅×#1844	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₅×SD₁₆	—	2	2	no	—
7	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×#127	C₂²	1	1	yes	characteristic
9	C₅×#492	—	2	2	no	—
10	C₅×#492	C₄	1	4	yes	complemented
11	C₂×C₅×#106	—	2	2	no	—
12	C₂×C₅×#102	—	2	4	no	—
13	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×#384	C₂²	1	2	yes	—
16	C₅×#340	—	2	4	no	—
17	C₅×C₈²	—	2	2	no	—
18	#9820	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂²×C₅×Q₈	—	2	1	no	—
20	C₂²×C₅×D₄	—	2	2	no	—
21	C₂²×C₅×D₄	—	2	1	no	—
22	C₂×C₅×SD₁₆	—	2	8	no	—
23	C₂×C₅×SD₁₆	—	4	4	no	—
24	C₂×C₅×SD₁₆	—	4	4	no	—
25	C₂²×C₅×C₈	D₄	1	4	yes	—
26	C₂²×C₅×C₈	D₄	1	2	yes	—
27	C₅×C₄⋊Q₈	—	2	1	no	—
28	C₅×C₄⋊Q₈	—	2	2	no	—
29	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
30	C₅×#127	—	2	1	no	—
31	C₅×#127	—	2	2	no	—
32	C₅×#127	C₂×C₄	1	2	yes	—
33	C₂×C₅×C₄⋊C₄	—	2	2	no	—
34	C₂×C₅×C₄⋊C₄	—	2	2	no	—
35	C₂×C₅×C₄⋊C₄	—	2	2	no	—
36	C₂×C₅×C₄⋊C₄	—	2	2	no	—
37	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
38	C₅×#106	—	4	4	no	—
39	C₅×#102	—	4	8	no	—
40	C₄×C₅×C₈	—	2	2	no	—
41	C₄×C₅×C₈	—	2	2	no	—
42	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
43	C₄×C₅×C₈	D₄	1	2	yes	—
44	C₂×#492	C₂×C₅	1	1	yes	characteristic
45	#1990	C₂×C₅	1	2	yes	complemented
46	#1957	C₂×C₅	1	2	yes	—
47	#1844	C₂×C₅	1	1	yes	characteristic, complemented
48	C₂×C₈²	C₂×C₅	1	1	yes	characteristic, complemented
49	C₂⁴×C₅	—	4	1	no	—
50	C₂×C₅×Q₈	—	2	2	no	—
51	C₂×C₅×Q₈	—	4	1	no	—
52	C₂×C₅×Q₈	—	4	2	no	—
53	C₂×C₅×Q₈	—	4	2	no	—
54	C₂×C₅×D₄	—	2	2	no	—
55	C₂×C₅×D₄	—	2	4	no	—
56	C₂×C₅×D₄	—	4	2	no	—
57	C₂×C₅×D₄	—	4	1	no	—
58	C₂×C₅×D₄	—	4	2	no	—
59	C₂×C₅×D₄	—	4	2	no	—
60	C₂×C₅×D₄	—	4	4	no	—
61	C₂×C₅×D₄	—	4	4	no	—
62	C₂²×C₄×C₅	—	4	1	no	—
63	C₂²×C₄×C₅	—	4	2	no	—
64	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
66	C₅×SD₁₆	—	4	16	no	—
67	C₂×C₅×C₈	—	2	4	no	—
68	C₂×C₅×C₈	—	2	4	no	—
69	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
70	C₂×C₅×C₈	D₈	1	4	yes	—
71	C₂×C₅×C₈	SD₁₆	1	4	yes	—
72	C₂×C₅×C₈	SD₁₆	1	8	yes	complemented
73	C₅×C₄⋊C₄	—	2	4	no	—
74	C₅×C₄⋊C₄	—	4	2	no	—
75	C₅×C₄⋊C₄	—	4	4	no	—
76	C₅×C₄⋊C₄	—	4	4	no	—
77	C₅×C₄⋊C₄	—	4	4	no	—
78	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₅	#41	1	2	yes	—
81	C₂²×SD₁₆	—	2	2	no	—
82	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
83	C₂×#127	C₂²×C₅	1	1	yes	characteristic
84	#492	—	2	2	no	—
85	#492	C₄×C₅	1	4	yes	complemented
86	C₂×#106	—	2	2	no	—
87	C₂×#102	—	2	4	no	—
88	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
89	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
90	#384	C₂²×C₅	1	2	yes	—
91	#340	—	2	4	no	—
92	C₈²	—	2	2	no	—
93	C₂³×C₅	—	4	2	no	—
94	C₂³×C₅	—	8	1	no	—
95	C₂³×C₅	—	8	2	no	—
96	C₂³×C₅	—	8	1	no	—
97	C₂³×C₅	—	8	2	no	—
98	C₂³×C₅	#127	1	1	yes	characteristic
99	C₅×Q₈	—	4	4	no	—
100	C₅×Q₈	—	8	2	no	—
101	C₅×D₄	—	4	4	no	—
102	C₅×D₄	—	4	8	no	—
103	C₅×D₄	—	8	4	no	—
104	C₅×D₄	—	8	2	no	—
105	C₂×C₄×C₅	—	4	2	no	—
106	C₂×C₄×C₅	—	8	1	no	—
107	C₂×C₄×C₅	—	8	1	no	—
108	C₂×C₄×C₅	—	8	2	no	—
109	C₂×C₄×C₅	—	8	2	no	—
110	C₂×C₄×C₅	—	8	2	no	—
111	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₅	#121	1	2	yes	—
114	C₂×C₄×C₅	#124	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
117	C₂×C₄×C₅	C₂×SD₁₆	1	4	yes	—
118	C₂×C₄×C₅	#135	1	4	yes	—
119	C₅×C₈	—	2	8	no	—
120	C₅×C₈	#102	1	8	yes	complemented
121	C₂²×Q₈	—	2	1	no	—
122	C₂²×D₄	—	2	1	no	—
123	C₂²×D₄	—	2	2	no	—
124	C₂×SD₁₆	—	2	8	no	—
125	C₂×SD₁₆	—	4	4	no	—
126	C₂×SD₁₆	—	4	4	no	—
127	C₂²×C₈	C₅×D₄	1	4	yes	—
128	C₂²×C₈	C₅×D₄	1	2	yes	—
129	C₄⋊Q₈	—	2	1	no	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
132	#127	—	2	1	no	—
133	#127	—	2	2	no	—
134	#127	C₂×C₄×C₅	1	2	yes	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
140	#106	—	4	4	no	—
141	#102	—	4	8	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
145	C₄×C₈	C₅×D₄	1	2	yes	—
146	C₂²×C₅	—	8	2	no	—
147	C₂²×C₅	—	8	4	no	—
148	C₂²×C₅	—	16	1	no	—
149	C₂²×C₅	—	16	2	no	—
150	C₂²×C₅	—	16	1	no	—
151	C₂²×C₅	#390	1	1	yes	characteristic
152	C₂²×C₅	#492	1	1	yes	characteristic
153	C₂²×C₅	#492	1	2	yes	—
154	C₂²×C₅	#493	1	1	yes	characteristic
155	C₂²×C₅	#495	1	2	yes	—
156	C₄×C₅	—	8	2	no	—
157	C₄×C₅	—	16	2	no	—
158	C₄×C₅	C₂×#102	1	2	yes	—
159	C₄×C₅	C₄×SD₁₆	1	4	yes	—
160	C₄×C₅	#465	1	4	yes	—
161	C₄×C₅	#486	1	2	yes	—
162	C₂⁴	—	4	1	no	—
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂×D₄	—	2	2	no	—
168	C₂×D₄	—	2	4	no	—
169	C₂×D₄	—	4	2	no	—
170	C₂×D₄	—	4	1	no	—
171	C₂×D₄	—	4	2	no	—
172	C₂×D₄	—	4	4	no	—
173	C₂×D₄	—	4	2	no	—
174	C₂×D₄	—	4	4	no	—
175	C₂²×C₄	—	4	1	no	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
179	SD₁₆	—	4	16	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
183	C₂×C₈	C₅×D₈	1	4	yes	—
184	C₂×C₈	C₅×SD₁₆	1	4	yes	—
185	C₂×C₈	C₅×SD₁₆	1	8	yes	complemented
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	—	4	4	no	—
191	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
192	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
193	C₄²	C₅×#41	1	2	yes	—
194	C₂×C₅	—	16	2	no	—
195	C₂×C₅	#1705	1	1	yes	characteristic
196	C₂×C₅	#1706	1	2	yes	—
197	C₂×C₅	#1708	1	1	yes	characteristic
198	C₂×C₅	#1957	1	1	yes	characteristic
199	C₂×C₅	#1968	1	2	yes	—
200	C₂³	—	4	2	no	—
201	C₂³	—	8	1	no	—
202	C₂³	—	8	2	no	—
203	C₂³	—	8	1	no	—
204	C₂³	—	8	2	no	—
205	C₂³	C₅×#127	1	1	yes	characteristic
206	Q₈	—	4	4	no	—
207	Q₈	—	8	2	no	—
208	D₄	—	4	4	no	—
209	D₄	—	4	8	no	—
210	D₄	—	8	4	no	—
211	D₄	—	8	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	8	1	no	—
214	C₂×C₄	—	8	1	no	—
215	C₂×C₄	—	8	2	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
219	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
220	C₂×C₄	C₅×#121	1	2	yes	—
221	C₂×C₄	C₅×#124	1	1	yes	characteristic
222	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
223	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
224	C₂×C₄	C₂×C₅×SD₁₆	1	4	yes	—
225	C₂×C₄	C₅×#135	1	4	yes	—
226	C₈	—	2	8	no	—
227	C₈	C₅×#102	1	8	yes	complemented
228	C₅	#9820	1	1	yes	characteristic, direct product factor, Sylow
229	C₂²	—	8	2	no	—
230	C₂²	—	8	4	no	—
231	C₂²	—	16	1	no	—
232	C₂²	—	16	2	no	—
233	C₂²	—	16	1	no	—
234	C₂²	C₅×#390	1	1	yes	characteristic
235	C₂²	C₅×#492	1	1	yes	characteristic
236	C₂²	C₅×#492	1	2	yes	—
237	C₂²	C₅×#493	1	1	yes	characteristic
238	C₂²	C₅×#495	1	2	yes	—
239	C₄	—	8	2	no	—
240	C₄	—	16	2	no	—
241	C₄	C₂×C₅×#102	1	2	yes	—
242	C₄	C₄×C₅×SD₁₆	1	4	yes	—
243	C₄	C₅×#465	1	4	yes	—
244	C₄	C₅×#486	1	2	yes	—
245	C₂	—	16	2	no	—
246	C₂	C₅×#1705	1	1	yes	characteristic
247	C₂	C₅×#1706	1	2	yes	—
248	C₂	C₅×#1708	1	1	yes	characteristic
249	C₂	C₅×#1957	1	1	yes	characteristic
250	C₂	C₅×#1968	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	#127
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂×C₅×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1990	C₂	semidirect
4. C₅×#1844	C₂	semidirect
5. C₂×C₅×C₈²	C₂	semidirect
10. C₅×#492	C₄	semidirect
18. #9820	C₅	direct (N × H)
45. #1990	C₂×C₅	semidirect
47. #1844	C₂×C₅	semidirect
48. C₂×C₈²	C₂×C₅	semidirect
72. C₂×C₅×C₈	SD₁₆	semidirect
85. #492	C₄×C₅	semidirect
120. C₅×C₈	#102	semidirect
185. C₂×C₈	C₅×SD₁₆	semidirect
227. C₈	C₅×#102	semidirect
228. C₅	#9820	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×#492	2
C₅×#1990	2
C₅×#1957	2
C₅×#1844	2
C₂×C₅×C₈²	2
#9820	5

Subgroups — C5×#9820 #11932540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9820 #11932540