Subgroups — C₅×#11672 #11934392

Order 1280.

334 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	C₅×#3170	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅×#1567	C₂	1	1	yes	maximal, characteristic
4	C₅×#1332	C₂	1	4	yes	maximal, complemented
5	C₅×#533	—	2	1	no	—
6	C₂²×C₅×C₁₆	—	2	1	no	—
7	C₅×#500	C₄	1	1	yes	characteristic
8	C₅×#494	C₄	1	1	yes	characteristic
9	C₅×#487	—	2	1	no	—
10	C₅×#467	—	2	1	no	—
11	C₅×#467	C₄	1	2	yes	—
12	C₄×C₅×Q₁₆	—	2	1	no	—
13	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
14	C₅×#423	—	2	1	no	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₄×C₅×C₁₆	—	2	1	no	—
17	C₄×C₅×C₁₆	C₂²	1	2	yes	—
18	C₅×#335	—	2	1	no	—
19	C₅×#335	C₂²	1	2	yes	—
20	#11672	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂×C₅×Q₁₆	—	2	1	no	—
22	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₅×M₁₆	—	4	1	no	—
24	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
26	C₅×C₄⋊Q₈	—	2	1	no	—
27	C₅×C₄⋊Q₈	—	2	1	no	—
28	C₅×#125	—	4	1	no	—
29	C₅×C₂²⋊Q₈	—	2	2	no	—
30	C₅×C₂²⋊Q₈	—	4	1	no	—
31	C₄×C₅×Q₈	—	2	2	no	—
32	C₄×C₅×Q₈	—	4	1	no	—
33	C₅×#117	—	4	1	no	—
34	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
35	C₂×C₅×C₁₆	—	2	4	no	—
36	C₂×C₅×C₁₆	—	2	2	no	—
37	C₅×#107	—	2	1	no	—
38	C₅×#107	C₂×C₄	1	2	yes	—
39	C₅×C₄⋊C₈	—	2	2	no	—
40	C₅×C₄⋊C₈	—	4	1	no	—
41	C₅×#103	—	2	2	no	—
42	C₅×#103	—	2	2	no	—
43	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₅×C₈	D₄	1	1	yes	characteristic
45	C₄×C₅×C₈	D₄	1	1	yes	characteristic
46	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
47	#3170	C₂×C₅	1	1	yes	characteristic
48	C₂×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic
49	#1567	C₂×C₅	1	1	yes	characteristic
50	#1332	C₂×C₅	1	4	yes	complemented
51	C₂×C₅×Q₈	—	2	2	no	—
52	C₂×C₅×Q₈	—	4	1	no	—
53	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
56	C₅×Q₁₆	—	2	4	no	—
57	C₅×Q₁₆	—	4	2	no	—
58	C₅×M₁₆	—	8	2	no	—
59	C₂×C₅×C₈	—	2	2	no	—
60	C₂×C₅×C₈	—	2	2	no	—
61	C₂×C₅×C₈	—	4	2	no	—
62	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
66	C₅×C₄⋊C₄	—	2	2	no	—
67	C₅×C₄⋊C₄	—	4	1	no	—
68	C₅×C₄⋊C₄	—	4	1	no	—
69	C₅×C₄⋊C₄	—	4	2	no	—
70	C₅×C₄⋊C₄	—	4	1	no	—
71	C₅×C₄⋊C₄	—	4	2	no	—
72	C₅×C₂²⋊C₄	—	4	2	no	—
73	C₄²×C₅	—	4	2	no	—
74	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₅	D₈	1	1	yes	characteristic
76	C₄²×C₅	SD₁₆	1	1	yes	characteristic
77	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
78	C₅×C₁₆	—	2	4	no	—
79	#533	—	2	1	no	—
80	C₂²×C₁₆	—	2	1	no	—
81	#500	C₄×C₅	1	1	yes	characteristic
82	#494	C₄×C₅	1	1	yes	characteristic
83	#487	—	2	1	no	—
84	#467	—	2	1	no	—
85	#467	C₄×C₅	1	2	yes	—
86	C₄×Q₁₆	—	2	1	no	—
87	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
88	#423	—	2	1	no	—
89	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
90	C₄×C₁₆	—	2	1	no	—
91	C₄×C₁₆	C₂²×C₅	1	2	yes	—
92	#335	—	2	1	no	—
93	#335	C₂²×C₅	1	2	yes	—
94	C₂³×C₅	#102	1	1	yes	characteristic
95	C₅×Q₈	—	4	2	no	—
96	C₅×Q₈	—	4	2	no	—
97	C₅×Q₈	—	8	2	no	—
98	C₂×C₄×C₅	—	2	2	no	—
99	C₂×C₄×C₅	—	2	1	no	—
100	C₂×C₄×C₅	—	2	1	no	—
101	C₂×C₄×C₅	—	4	2	no	—
102	C₂×C₄×C₅	—	4	2	no	—
103	C₂×C₄×C₅	#102	1	1	yes	characteristic
104	C₂×C₄×C₅	#102	1	1	yes	characteristic
105	C₂×C₄×C₅	#102	1	1	yes	characteristic
106	C₂×C₄×C₅	SD₃₂	1	2	yes	—
107	C₂×C₄×C₅	Q₃₂	1	2	yes	—
108	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
111	C₅×C₈	—	2	1	no	—
112	C₅×C₈	—	2	1	no	—
113	C₅×C₈	—	8	2	no	—
114	C₅×C₈	#104	1	2	yes	—
115	C₅×C₈	#131	1	2	yes	—
116	C₂×Q₁₆	—	2	1	no	—
117	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
118	C₂×M₁₆	—	4	1	no	—
119	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
120	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
121	C₄⋊Q₈	—	2	1	no	—
122	C₄⋊Q₈	—	2	1	no	—
123	#125	—	4	1	no	—
124	C₂²⋊Q₈	—	2	2	no	—
125	C₂²⋊Q₈	—	4	1	no	—
126	C₄×Q₈	—	2	2	no	—
127	C₄×Q₈	—	4	1	no	—
128	#117	—	4	1	no	—
129	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
130	C₂×C₁₆	—	2	4	no	—
131	C₂×C₁₆	—	2	2	no	—
132	#107	—	2	1	no	—
133	#107	C₂×C₄×C₅	1	2	yes	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	4	1	no	—
136	#103	—	2	2	no	—
137	#103	—	2	2	no	—
138	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
139	C₄×C₈	C₅×D₄	1	1	yes	characteristic
140	C₄×C₈	C₅×D₄	1	1	yes	characteristic
141	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
142	C₂²×C₅	—	2	1	no	—
143	C₂²×C₅	—	2	1	no	—
144	C₂²×C₅	#358	1	2	yes	—
145	C₂²×C₅	C₂×#102	1	1	yes	characteristic
146	C₄×C₅	—	2	1	no	—
147	C₄×C₅	—	2	1	no	—
148	C₄×C₅	—	8	2	no	—
149	C₄×C₅	—	8	2	no	—
150	C₄×C₅	#358	1	2	yes	—
151	C₄×C₅	#359	1	2	yes	—
152	C₄×C₅	#408	1	1	yes	characteristic
153	C₄×C₅	C₂×#104	1	1	yes	characteristic
154	C₄×C₅	C₂×SD₃₂	1	1	yes	characteristic
155	C₄×C₅	C₂×Q₃₂	1	1	yes	characteristic
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	—	4	1	no	—
158	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
160	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
161	Q₁₆	—	2	4	no	—
162	Q₁₆	—	4	2	no	—
163	M₁₆	—	8	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
170	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	1	no	—
174	C₄⋊C₄	—	4	1	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	1	no	—
177	C₂²⋊C₄	—	4	2	no	—
178	C₄²	—	4	2	no	—
179	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
180	C₄²	C₅×D₈	1	1	yes	characteristic
181	C₄²	C₅×SD₁₆	1	1	yes	characteristic
182	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
183	C₁₆	—	2	4	no	—
184	C₂×C₅	—	2	2	no	—
185	C₂×C₅	#1492	1	1	yes	characteristic
186	C₂×C₅	C₂×#358	1	1	yes	characteristic
187	C₂×C₅	C₂×#359	1	1	yes	characteristic
188	C₂³	C₅×#102	1	1	yes	characteristic
189	Q₈	—	4	2	no	—
190	Q₈	—	4	2	no	—
191	Q₈	—	8	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	C₅×#102	1	1	yes	characteristic
198	C₂×C₄	C₅×#102	1	1	yes	characteristic
199	C₂×C₄	C₅×#102	1	1	yes	characteristic
200	C₂×C₄	C₅×SD₃₂	1	2	yes	—
201	C₂×C₄	C₅×Q₃₂	1	2	yes	—
202	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
203	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
204	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	8	2	no	—
208	C₈	C₅×#104	1	2	yes	—
209	C₈	C₅×#131	1	2	yes	—
210	C₅	#11672	1	1	yes	characteristic, direct product factor, Sylow
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	C₅×#358	1	2	yes	—
214	C₂²	C₂×C₅×#102	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	8	2	no	—
218	C₄	—	8	2	no	—
219	C₄	C₅×#358	1	2	yes	—
220	C₄	C₅×#359	1	2	yes	—
221	C₄	C₅×#408	1	1	yes	characteristic
222	C₄	C₂×C₅×#104	1	1	yes	characteristic
223	C₄	C₂×C₅×SD₃₂	1	1	yes	characteristic
224	C₄	C₂×C₅×Q₃₂	1	1	yes	characteristic
225	C₂	—	2	2	no	—
226	C₂	C₅×#1492	1	1	yes	characteristic
227	C₂	C₂×C₅×#358	1	1	yes	characteristic
228	C₂	C₂×C₅×#359	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#102
prime radical	C₂³×C₅	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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×#1332	C₂	semidirect
20. #11672	C₅	direct (N × H)
50. #1332	C₂×C₅	semidirect
210. C₅	#11672	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×C₄×C₅×C₁₆	2
C₅×#1567	2
C₅×#1332	2
#11672	5

Subgroups — C5×#11672 #11934392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11672 #11934392