Subgroups — C₅×#11658 #11934378

Order 1280.

366 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₅×#2162	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	C₂²×C₅×Q₁₆	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×Q₈	—	2	1	no	—
7	C₄×C₅×Q₁₆	—	2	1	no	—
8	C₄×C₅×Q₁₆	—	2	1	no	—
9	C₄×C₅×Q₁₆	C₄	1	2	yes	—
10	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
11	C₂×C₅×#107	C₄	1	1	yes	characteristic
12	C₅×#423	—	2	1	no	—
13	C₂×C₅×#103	—	2	1	no	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×C₄⋊C₁₆	—	2	1	no	—
16	C₅×C₂²⋊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	#11658	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×Q₈	—	2	1	no	—
22	C₂×C₅×Q₁₆	—	2	4	no	—
23	C₂×C₅×Q₁₆	—	2	1	no	—
24	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₅×M₁₆	—	4	1	no	—
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
28	C₄×C₅×Q₈	—	2	2	no	—
29	C₄×C₅×Q₈	—	2	2	no	—
30	C₄×C₅×Q₈	—	4	1	no	—
31	C₄×C₅×Q₈	—	4	1	no	—
32	C₂×C₅×C₄⋊C₄	—	2	1	no	—
33	C₂×C₅×C₄⋊C₄	—	4	1	no	—
34	C₂×C₄²×C₅	—	4	1	no	—
35	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
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	C₂×C₄×Q₁₆	C₂×C₅	1	1	yes	characteristic
48	#2162	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	4	no	—
52	C₂×C₅×Q₈	—	2	2	no	—
53	C₂×C₅×Q₈	—	4	2	no	—
54	C₂×C₅×Q₈	—	4	1	no	—
55	C₂²×C₄×C₅	—	4	1	no	—
56	C₂²×C₄×C₅	—	4	1	no	—
57	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
60	C₅×Q₁₆	—	2	4	no	—
61	C₅×Q₁₆	—	2	2	no	—
62	C₅×Q₁₆	M₁₆	1	4	yes	complemented
63	C₅×M₁₆	—	8	2	no	—
64	C₂×C₅×C₈	—	2	2	no	—
65	C₂×C₅×C₈	—	4	1	no	—
66	C₂×C₅×C₈	—	4	2	no	—
67	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
71	C₅×C₄⋊C₄	—	2	2	no	—
72	C₅×C₄⋊C₄	—	4	1	no	—
73	C₅×C₄⋊C₄	—	4	2	no	—
74	C₅×C₄⋊C₄	—	4	2	no	—
75	C₄²×C₅	—	4	2	no	—
76	C₄²×C₅	—	4	2	no	—
77	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₅	D₈	1	1	yes	characteristic
79	C₄²×C₅	SD₁₆	1	1	yes	characteristic
80	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
81	C₅×C₁₆	—	4	2	no	—
82	C₂²×Q₁₆	C₄×C₅	1	1	yes	characteristic
83	C₂×C₄×Q₈	—	2	1	no	—
84	C₄×Q₁₆	—	2	1	no	—
85	C₄×Q₁₆	—	2	1	no	—
86	C₄×Q₁₆	C₄×C₅	1	2	yes	—
87	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
88	C₂×#107	C₄×C₅	1	1	yes	characteristic
89	#423	—	2	1	no	—
90	C₂×#103	—	2	1	no	—
91	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
92	C₄⋊C₁₆	—	2	1	no	—
93	C₂²⋊C₁₆	—	2	1	no	—
94	C₄×C₁₆	C₂²×C₅	1	2	yes	—
95	#335	—	2	1	no	—
96	#335	C₂²×C₅	1	2	yes	—
97	C₂³×C₅	#102	1	1	yes	characteristic
98	C₅×Q₈	—	2	4	no	—
99	C₅×Q₈	—	4	4	no	—
100	C₅×Q₈	—	4	2	no	—
101	C₂×C₄×C₅	—	2	1	no	—
102	C₂×C₄×C₅	—	2	1	no	—
103	C₂×C₄×C₅	—	4	1	no	—
104	C₂×C₄×C₅	—	4	4	no	—
105	C₂×C₄×C₅	—	4	2	no	—
106	C₂×C₄×C₅	—	4	2	no	—
107	C₂×C₄×C₅	—	8	2	no	—
108	C₂×C₄×C₅	#102	1	1	yes	characteristic
109	C₂×C₄×C₅	#102	1	1	yes	characteristic
110	C₂×C₄×C₅	#102	1	1	yes	characteristic
111	C₂×C₄×C₅	SD₃₂	1	2	yes	—
112	C₂×C₄×C₅	Q₃₂	1	2	yes	—
113	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
116	C₅×C₈	—	2	1	no	—
117	C₅×C₈	—	2	1	no	—
118	C₅×C₈	—	2	1	no	—
119	C₅×C₈	—	8	2	no	—
120	C₅×C₈	C₂×M₁₆	1	2	yes	—
121	C₂²×Q₈	—	2	1	no	—
122	C₂×Q₁₆	—	2	4	no	—
123	C₂×Q₁₆	—	2	1	no	—
124	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
125	C₂×M₁₆	—	4	1	no	—
126	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
127	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
128	C₄×Q₈	—	2	2	no	—
129	C₄×Q₈	—	2	2	no	—
130	C₄×Q₈	—	4	1	no	—
131	C₄×Q₈	—	4	1	no	—
132	C₂×C₄⋊C₄	—	2	1	no	—
133	C₂×C₄⋊C₄	—	4	1	no	—
134	C₂×C₄²	—	4	1	no	—
135	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
136	C₂×C₁₆	—	2	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	—	4	4	no	—
155	C₄×C₅	—	8	2	no	—
156	C₄×C₅	#358	1	2	yes	—
157	C₄×C₅	#407	1	1	yes	characteristic
158	C₄×C₅	#421	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	4	no	—
162	C₂×Q₈	—	2	2	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂²×C₄	—	4	1	no	—
166	C₂²×C₄	—	4	1	no	—
167	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
168	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
169	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
170	Q₁₆	—	2	4	no	—
171	Q₁₆	—	2	2	no	—
172	Q₁₆	C₅×M₁₆	1	4	yes	complemented
173	M₁₆	—	8	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
178	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
179	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
180	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	4	1	no	—
184	C₄⋊C₄	—	4	2	no	—
185	C₄²	—	4	2	no	—
186	C₄²	—	4	2	no	—
187	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
188	C₄²	C₅×D₈	1	1	yes	characteristic
189	C₄²	C₅×SD₁₆	1	1	yes	characteristic
190	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
191	C₁₆	—	4	2	no	—
192	C₂×C₅	—	2	2	no	—
193	C₂×C₅	#1486	1	1	yes	characteristic
194	C₂×C₅	C₂×#358	1	1	yes	characteristic
195	C₂×C₅	#2143	1	1	yes	characteristic
196	C₂³	C₅×#102	1	1	yes	characteristic
197	Q₈	—	2	4	no	—
198	Q₈	—	4	4	no	—
199	Q₈	—	4	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	8	2	no	—
207	C₂×C₄	C₅×#102	1	1	yes	characteristic
208	C₂×C₄	C₅×#102	1	1	yes	characteristic
209	C₂×C₄	C₅×#102	1	1	yes	characteristic
210	C₂×C₄	C₅×SD₃₂	1	2	yes	—
211	C₂×C₄	C₅×Q₃₂	1	2	yes	—
212	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
213	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
214	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	8	2	no	—
219	C₈	C₂×C₅×M₁₆	1	2	yes	—
220	C₅	#11658	1	1	yes	characteristic, direct product factor, Sylow
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	C₅×#358	1	2	yes	—
224	C₂²	C₂×C₅×#102	1	1	yes	characteristic
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	4	4	no	—
229	C₄	—	8	2	no	—
230	C₄	C₅×#358	1	2	yes	—
231	C₄	C₅×#407	1	1	yes	characteristic
232	C₄	C₅×#421	1	1	yes	characteristic
233	C₄	C₂×C₅×SD₃₂	1	1	yes	characteristic
234	C₄	C₂×C₅×Q₃₂	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	C₅×#1486	1	1	yes	characteristic
237	C₂	C₂×C₅×#358	1	1	yes	characteristic
238	C₂	C₅×#2143	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. #11658	C₅	direct (N × H)
50. #1332	C₂×C₅	semidirect
62. C₅×Q₁₆	M₁₆	semidirect
172. Q₁₆	C₅×M₁₆	semidirect
220. C₅	#11658	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×C₄×C₅×Q₁₆	2
C₅×#2162	2
C₅×#1567	2
C₅×#1332	2
#11658	5

Subgroups — C5×#11658 #11934378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11658 #11934378