Subgroups — C₅×#10649 #11933369

Order 1280.

350 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	C₅×#1947	C₂	1	2	yes	maximal
2	C₅×#1939	C₂	1	1	yes	maximal, characteristic
3	C₅×#1845	C₂	1	1	yes	maximal, characteristic
4	C₅×#1296	C₂	1	2	yes	maximal
5	C₅×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×#107	—	2	1	no	—
7	C₂×C₅×#107	—	2	1	no	—
8	C₂×C₅×#107	—	2	2	no	—
9	C₂×C₅×C₄⋊C₈	—	2	1	no	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×#384	C₂²	1	2	yes	—
14	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₅×#340	—	2	2	no	—
16	C₅×#336	—	2	1	no	—
17	#10649	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₅×C₈	—	2	2	no	—
19	C₂²×C₅×C₈	—	2	1	no	—
20	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
22	C₂×C₅×C₄⋊C₄	—	2	2	no	—
23	C₂×C₅×C₄⋊C₄	—	2	2	no	—
24	C₂×C₅×C₄⋊C₄	—	2	2	no	—
25	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
26	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	—	2	1	no	—
28	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₅×#107	—	4	2	no	—
30	C₅×#107	—	4	2	no	—
31	C₅×#107	—	4	4	no	—
32	C₅×C₄⋊C₈	—	2	2	no	—
33	C₅×C₄⋊C₈	—	2	2	no	—
34	C₅×C₄⋊C₈	—	4	2	no	—
35	C₅×C₄⋊C₈	Q₈	1	4	yes	—
36	C₄×C₅×C₈	—	2	2	no	—
37	C₅×#95	—	2	1	no	—
38	#1947	C₂×C₅	1	2	yes	—
39	#1939	C₂×C₅	1	1	yes	characteristic
40	#1845	C₂×C₅	1	1	yes	characteristic
41	#1296	C₂×C₅	1	2	yes	—
42	#1277	C₂×C₅	1	1	yes	characteristic
43	C₂²×C₄×C₅	—	2	1	no	—
44	C₂²×C₄×C₅	—	2	1	no	—
45	C₂²×C₄×C₅	—	4	2	no	—
46	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	#41	1	1	yes	characteristic
49	C₂×C₅×C₈	—	2	1	no	—
50	C₂×C₅×C₈	—	2	1	no	—
51	C₂×C₅×C₈	—	2	1	no	—
52	C₂×C₅×C₈	—	2	1	no	—
53	C₂×C₅×C₈	—	2	4	no	—
54	C₂×C₅×C₈	—	4	2	no	—
55	C₂×C₅×C₈	—	4	2	no	—
56	C₂×C₅×C₈	—	4	1	no	—
57	C₂×C₅×C₈	—	4	1	no	—
58	C₂×C₅×C₈	—	4	1	no	—
59	C₂×C₅×C₈	D₈	1	2	yes	—
60	C₂×C₅×C₈	Q₁₆	1	2	yes	—
61	C₅×C₄⋊C₄	—	2	1	no	—
62	C₅×C₄⋊C₄	—	2	2	no	—
63	C₅×C₄⋊C₄	—	4	2	no	—
64	C₅×C₄⋊C₄	—	4	2	no	—
65	C₅×C₄⋊C₄	—	4	2	no	—
66	C₅×C₄⋊C₄	—	4	2	no	—
67	C₅×C₄⋊C₄	—	8	2	no	—
68	C₅×C₄⋊C₄	D₈	1	2	yes	—
69	C₄²×C₅	—	4	1	no	—
70	C₄²×C₅	—	4	1	no	—
71	C₄²×C₅	C₂×Q₈	1	2	yes	—
72	C₄²×C₅	#41	1	2	yes	—
73	C₂×#107	—	2	1	no	—
74	C₂×#107	—	2	1	no	—
75	C₂×#107	—	2	2	no	—
76	C₂×C₄⋊C₈	—	2	1	no	—
77	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
78	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
79	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
80	#384	C₂²×C₅	1	2	yes	—
81	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
82	#340	—	2	2	no	—
83	#336	—	2	1	no	—
84	C₂³×C₅	#121	1	1	yes	characteristic
85	C₂×C₄×C₅	—	2	1	no	—
86	C₂×C₄×C₅	—	2	1	no	—
87	C₂×C₄×C₅	—	2	2	no	—
88	C₂×C₄×C₅	—	4	1	no	—
89	C₂×C₄×C₅	—	4	1	no	—
90	C₂×C₄×C₅	—	4	1	no	—
91	C₂×C₄×C₅	—	4	1	no	—
92	C₂×C₄×C₅	—	4	1	no	—
93	C₂×C₄×C₅	—	8	2	no	—
94	C₂×C₄×C₅	—	8	2	no	—
95	C₂×C₄×C₅	—	8	2	no	—
96	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
97	C₂×C₄×C₅	#123	1	1	yes	characteristic
98	C₂×C₄×C₅	#125	1	2	yes	—
99	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄×C₅	#135	1	1	yes	characteristic
104	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂×C₄×C₅	#137	1	2	yes	—
106	C₂×C₄×C₅	#137	1	1	yes	characteristic
107	C₅×C₈	—	2	2	no	—
108	C₅×C₈	—	2	2	no	—
109	C₅×C₈	—	4	4	no	—
110	C₅×C₈	—	8	1	no	—
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	2	1	no	—
113	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
114	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
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	C₅×D₄	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
120	C₂×C₄²	—	2	1	no	—
121	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
122	#107	—	4	2	no	—
123	#107	—	4	2	no	—
124	#107	—	4	4	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	4	2	no	—
128	C₄⋊C₈	C₅×Q₈	1	4	yes	—
129	C₄×C₈	—	2	2	no	—
130	#95	—	2	1	no	—
131	C₂²×C₅	#398	1	1	yes	characteristic
132	C₂²×C₅	#459	1	1	yes	characteristic
133	C₂²×C₅	#462	1	1	yes	characteristic
134	C₂²×C₅	#466	1	1	yes	characteristic
135	C₂²×C₅	#470	1	1	yes	characteristic
136	C₂²×C₅	#472	1	1	yes	characteristic
137	C₂²×C₅	#473	1	1	yes	characteristic
138	C₄×C₅	—	2	2	no	—
139	C₄×C₅	—	4	2	no	—
140	C₄×C₅	—	8	1	no	—
141	C₄×C₅	—	16	2	no	—
142	C₄×C₅	#474	1	2	yes	—
143	C₄×C₅	#479	1	2	yes	—
144	C₄×C₅	#480	1	1	yes	characteristic
145	C₄×C₅	#484	1	1	yes	characteristic
146	C₄×C₅	#500	1	1	yes	characteristic
147	C₄×C₅	#501	1	1	yes	characteristic
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
153	C₂²×C₄	C₅×#41	1	1	yes	characteristic
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	4	2	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	4	1	no	—
162	C₂×C₈	—	4	1	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	C₅×D₈	1	2	yes	—
165	C₂×C₈	C₅×Q₁₆	1	2	yes	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	8	2	no	—
173	C₄⋊C₄	C₅×D₈	1	2	yes	—
174	C₄²	—	4	1	no	—
175	C₄²	—	4	1	no	—
176	C₄²	C₂×C₅×Q₈	1	2	yes	—
177	C₄²	C₅×#41	1	2	yes	—
178	C₂×C₅	#1667	1	1	yes	characteristic
179	C₂×C₅	#1674	1	1	yes	characteristic
180	C₂×C₅	#1689	1	1	yes	characteristic
181	C₂×C₅	#1696	1	1	yes	characteristic
182	C₂×C₅	#2083	1	1	yes	characteristic
183	C₂×C₅	#2086	1	1	yes	characteristic
184	C₂×C₅	#2089	1	1	yes	characteristic
185	C₂³	C₅×#121	1	1	yes	characteristic
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	8	2	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
198	C₂×C₄	C₅×#123	1	1	yes	characteristic
199	C₂×C₄	C₅×#125	1	2	yes	—
200	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
201	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
202	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
203	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
204	C₂×C₄	C₅×#135	1	1	yes	characteristic
205	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
206	C₂×C₄	C₅×#137	1	2	yes	—
207	C₂×C₄	C₅×#137	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₈	—	4	4	no	—
211	C₈	—	8	1	no	—
212	C₅	#10649	1	1	yes	characteristic, direct product factor, Sylow
213	C₂²	C₅×#398	1	1	yes	characteristic
214	C₂²	C₅×#459	1	1	yes	characteristic
215	C₂²	C₅×#462	1	1	yes	characteristic
216	C₂²	C₅×#466	1	1	yes	characteristic
217	C₂²	C₅×#470	1	1	yes	characteristic
218	C₂²	C₅×#472	1	1	yes	characteristic
219	C₂²	C₅×#473	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	8	1	no	—
223	C₄	—	16	2	no	—
224	C₄	C₅×#474	1	2	yes	—
225	C₄	C₅×#479	1	2	yes	—
226	C₄	C₅×#480	1	1	yes	characteristic
227	C₄	C₅×#484	1	1	yes	characteristic
228	C₄	C₅×#500	1	1	yes	characteristic
229	C₄	C₅×#501	1	1	yes	characteristic
230	C₂	C₅×#1667	1	1	yes	characteristic
231	C₂	C₅×#1674	1	1	yes	characteristic
232	C₂	C₅×#1689	1	1	yes	characteristic
233	C₂	C₅×#1696	1	1	yes	characteristic
234	C₂	C₅×#2083	1	1	yes	characteristic
235	C₂	C₅×#2086	1	1	yes	characteristic
236	C₂	C₅×#2089	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₂×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
17. #10649	C₅	direct (N × H)
212. C₅	#10649	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#1947	2
C₅×#1939	2
C₅×#1845	2
C₅×#1296	2
C₅×#1277	2
#10649	5

Subgroups — C5×#10649 #11933369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10649 #11933369