Subgroups — C₅×#11983 #11934703

Order 1280.

314 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×#3170	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅×#1567	C₂	1	1	yes	maximal, characteristic
4	C₅×#1365	C₂	1	2	yes	maximal, complemented
5	C₅×#1362	C₂	1	2	yes	maximal, complemented
6	C₅×#533	—	2	1	no	—
7	C₂²×C₅×C₁₆	—	2	1	no	—
8	C₅×#500	C₂²	1	1	yes	characteristic
9	C₅×#494	C₂²	1	1	yes	characteristic
10	C₅×#487	—	2	1	no	—
11	C₅×#467	—	2	1	no	—
12	C₅×#467	C₄	1	2	yes	—
13	C₄×C₅×Q₁₆	—	2	1	no	—
14	C₄×C₅×Q₁₆	C₄	1	2	yes	—
15	C₅×#423	—	2	1	no	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×C₄⋊C₁₆	—	2	1	no	—
18	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
19	C₅×#335	—	2	1	no	—
20	C₅×#335	C₂²	1	2	yes	—
21	#11983	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₅×Q₁₆	—	2	1	no	—
23	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₅×M₁₆	—	4	1	no	—
25	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
27	C₅×C₄⋊Q₈	—	2	1	no	—
28	C₅×C₄⋊Q₈	—	2	1	no	—
29	C₅×#125	—	4	1	no	—
30	C₅×C₂²⋊Q₈	—	2	2	no	—
31	C₅×C₂²⋊Q₈	—	4	1	no	—
32	C₄×C₅×Q₈	—	2	2	no	—
33	C₄×C₅×Q₈	—	4	1	no	—
34	C₅×#117	—	4	1	no	—
35	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
36	C₂×C₅×C₁₆	—	2	2	no	—
37	C₂×C₅×C₁₆	—	4	2	no	—
38	C₅×#107	—	2	1	no	—
39	C₅×#107	C₂×C₄	1	2	yes	—
40	C₅×C₄⋊C₈	—	2	2	no	—
41	C₅×C₄⋊C₈	—	4	1	no	—
42	C₅×#103	—	2	2	no	—
43	C₅×#103	—	2	2	no	—
44	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
45	C₄×C₅×C₈	D₄	1	1	yes	characteristic
46	C₄×C₅×C₈	D₄	1	1	yes	characteristic
47	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
48	#3170	C₂×C₅	1	1	yes	characteristic
49	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic
50	#1567	C₂×C₅	1	1	yes	characteristic
51	#1365	C₂×C₅	1	2	yes	complemented
52	#1362	C₂×C₅	1	2	yes	complemented
53	C₂×C₅×Q₈	—	2	2	no	—
54	C₂×C₅×Q₈	—	4	1	no	—
55	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
58	C₅×Q₁₆	—	4	2	no	—
59	C₅×Q₁₆	—	4	1	no	—
60	C₅×Q₁₆	—	4	1	no	—
61	C₅×M₁₆	—	8	2	no	—
62	C₂×C₅×C₈	—	2	1	no	—
63	C₂×C₅×C₈	—	2	1	no	—
64	C₂×C₅×C₈	—	4	1	no	—
65	C₂×C₅×C₈	—	4	2	no	—
66	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
70	C₅×C₄⋊C₄	—	2	2	no	—
71	C₅×C₄⋊C₄	—	4	2	no	—
72	C₅×C₄⋊C₄	—	4	2	no	—
73	C₅×C₄⋊C₄	—	4	1	no	—
74	C₅×C₄⋊C₄	—	4	1	no	—
75	C₅×C₄⋊C₄	—	4	1	no	—
76	C₅×C₂²⋊C₄	—	4	2	no	—
77	C₄²×C₅	—	4	2	no	—
78	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²×C₅	D₈	1	1	yes	characteristic
80	C₄²×C₅	SD₁₆	1	1	yes	characteristic
81	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
82	C₅×C₁₆	—	4	2	no	—
83	#533	—	2	1	no	—
84	C₂²×C₁₆	—	2	1	no	—
85	#500	C₂²×C₅	1	1	yes	characteristic
86	#494	C₂²×C₅	1	1	yes	characteristic
87	#487	—	2	1	no	—
88	#467	—	2	1	no	—
89	#467	C₄×C₅	1	2	yes	—
90	C₄×Q₁₆	—	2	1	no	—
91	C₄×Q₁₆	C₄×C₅	1	2	yes	—
92	#423	—	2	1	no	—
93	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
94	C₄⋊C₁₆	—	2	1	no	—
95	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
96	#335	—	2	1	no	—
97	#335	C₂²×C₅	1	2	yes	—
98	C₂³×C₅	#102	1	1	yes	characteristic
99	C₅×Q₈	—	4	2	no	—
100	C₅×Q₈	—	4	2	no	—
101	C₅×Q₈	—	8	2	no	—
102	C₂×C₄×C₅	—	2	1	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	4	1	no	—
105	C₂×C₄×C₅	—	4	2	no	—
106	C₂×C₄×C₅	—	4	2	no	—
107	C₂×C₄×C₅	#102	1	1	yes	characteristic
108	C₂×C₄×C₅	#102	1	1	yes	characteristic
109	C₂×C₄×C₅	#102	1	1	yes	characteristic
110	C₂×C₄×C₅	SD₃₂	1	2	yes	—
111	C₂×C₄×C₅	Q₃₂	1	2	yes	—
112	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
115	C₅×C₈	—	2	1	no	—
116	C₅×C₈	—	2	1	no	—
117	C₅×C₈	—	2	1	no	—
118	C₅×C₈	—	8	2	no	—
119	C₅×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₅×C₈	#137	1	1	yes	characteristic
121	C₂×Q₁₆	—	2	1	no	—
122	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
123	C₂×M₁₆	—	4	1	no	—
124	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
125	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
126	C₄⋊Q₈	—	2	1	no	—
127	C₄⋊Q₈	—	2	1	no	—
128	#125	—	4	1	no	—
129	C₂²⋊Q₈	—	2	2	no	—
130	C₂²⋊Q₈	—	4	1	no	—
131	C₄×Q₈	—	2	2	no	—
132	C₄×Q₈	—	4	1	no	—
133	#117	—	4	1	no	—
134	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	4	2	no	—
137	#107	—	2	1	no	—
138	#107	C₂×C₄×C₅	1	2	yes	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	4	1	no	—
141	#103	—	2	2	no	—
142	#103	—	2	2	no	—
143	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
144	C₄×C₈	C₅×D₄	1	1	yes	characteristic
145	C₄×C₈	C₅×D₄	1	1	yes	characteristic
146	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
147	C₂²×C₅	—	2	1	no	—
148	C₂²×C₅	—	2	1	no	—
149	C₂²×C₅	#358	1	2	yes	—
150	C₂²×C₅	C₂×#102	1	1	yes	characteristic
151	C₄×C₅	—	2	1	no	—
152	C₄×C₅	—	2	1	no	—
153	C₄×C₅	—	2	1	no	—
154	C₄×C₅	—	8	2	no	—
155	C₄×C₅	—	8	2	no	—
156	C₄×C₅	#358	1	2	yes	—
157	C₄×C₅	#413	1	1	yes	characteristic
158	C₄×C₅	#417	1	1	yes	characteristic
159	C₄×C₅	C₂×SD₃₂	1	1	yes	characteristic
160	C₄×C₅	C₂×Q₃₂	1	1	yes	characteristic
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	4	1	no	—
163	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
164	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
165	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
166	Q₁₆	—	4	2	no	—
167	Q₁₆	—	4	1	no	—
168	Q₁₆	—	4	1	no	—
169	M₁₆	—	8	2	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	4	1	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
176	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
177	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	4	1	no	—
180	C₄⋊C₄	—	4	1	no	—
181	C₄⋊C₄	—	4	1	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₂²⋊C₄	—	4	2	no	—
185	C₄²	—	4	2	no	—
186	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
187	C₄²	C₅×D₈	1	1	yes	characteristic
188	C₄²	C₅×SD₁₆	1	1	yes	characteristic
189	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
190	C₁₆	—	4	2	no	—
191	C₂×C₅	—	2	2	no	—
192	C₂×C₅	#1547	1	1	yes	characteristic
193	C₂×C₅	C₂×#358	1	1	yes	characteristic
194	C₂×C₅	#2148	1	1	yes	characteristic
195	C₂³	C₅×#102	1	1	yes	characteristic
196	Q₈	—	4	2	no	—
197	Q₈	—	4	2	no	—
198	Q₈	—	8	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	C₅×#102	1	1	yes	characteristic
205	C₂×C₄	C₅×#102	1	1	yes	characteristic
206	C₂×C₄	C₅×#102	1	1	yes	characteristic
207	C₂×C₄	C₅×SD₃₂	1	2	yes	—
208	C₂×C₄	C₅×Q₃₂	1	2	yes	—
209	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
210	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
211	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
212	C₈	—	2	1	no	—
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	—	8	2	no	—
216	C₈	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
217	C₈	C₅×#137	1	1	yes	characteristic
218	C₅	#11983	1	1	yes	characteristic, direct product factor, Sylow
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	C₅×#358	1	2	yes	—
222	C₂²	C₂×C₅×#102	1	1	yes	characteristic
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	8	2	no	—
227	C₄	—	8	2	no	—
228	C₄	C₅×#358	1	2	yes	—
229	C₄	C₅×#413	1	1	yes	characteristic
230	C₄	C₅×#417	1	1	yes	characteristic
231	C₄	C₂×C₅×SD₃₂	1	1	yes	characteristic
232	C₄	C₂×C₅×Q₃₂	1	1	yes	characteristic
233	C₂	—	2	2	no	—
234	C₂	C₅×#1547	1	1	yes	characteristic
235	C₂	C₂×C₅×#358	1	1	yes	characteristic
236	C₂	C₅×#2148	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₂³×C₅	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1365	C₂	semidirect
5. C₅×#1362	C₂	semidirect
21. #11983	C₅	direct (N × H)
51. #1365	C₂×C₅	semidirect
52. #1362	C₂×C₅	semidirect
218. C₅	#11983	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×#3170	2
C₂×C₅×C₄⋊C₁₆	2
C₅×#1567	2
C₅×#1365	2
C₅×#1362	2
#11983	5

Subgroups — C5×#11983 #11934703

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11983 #11934703