Subgroups — C₅×#32808 #11955528

Order 1280.

732 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2862	C₂	1	2	yes	maximal
2	C₅×#2056	C₂	1	4	yes	maximal
3	C₅×#2024	C₂	1	4	yes	maximal
4	C₅×#2016	C₂	1	4	yes	maximal
5	C₅×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×Q₁₆	C₂²	1	4	yes	—
7	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₅×#103	C₂²	1	8	yes	—
11	C₅×#391	—	2	2	no	—
12	C₅×#391	—	2	2	no	—
13	C₅×#391	C₂²	1	4	yes	—
14	C₅×#391	C₂²	1	4	yes	—
15	C₅×#382	—	2	2	no	—
16	C₅×#382	—	2	2	no	—
17	C₄³×C₅	C₂²	1	1	yes	characteristic
18	C₅×#340	C₂²	1	4	yes	—
19	C₅×#336	C₂²	1	4	yes	—
20	#32808	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×Q₈	—	2	2	no	—
22	C₂²×C₅×Q₈	C₂³	1	4	yes	—
23	C₂×C₅×Q₁₆	—	4	8	no	—
24	C₂×C₅×Q₁₆	—	4	4	no	—
25	C₂²×C₅×C₈	C₂³	1	4	yes	—
26	C₅×C₄⋊Q₈	—	4	4	no	—
27	C₅×C₄⋊Q₈	—	4	4	no	—
28	C₂×C₅×C₄⋊C₄	—	2	2	no	—
29	C₂×C₅×C₄⋊C₄	—	2	2	no	—
30	C₂×C₅×C₄⋊C₄	—	2	2	no	—
31	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
32	C₂×C₄²×C₅	D₄	1	2	yes	—
33	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂³	1	2	yes	—
37	C₅×C₄⋊C₈	—	4	2	no	—
38	C₅×#103	—	4	8	no	—
39	C₅×#95	—	2	4	no	—
40	C₅×#95	—	2	4	no	—
41	C₅×#95	—	2	4	no	—
42	C₅×#95	—	2	4	no	—
43	#2862	C₂×C₅	1	2	yes	—
44	#2056	C₂×C₅	1	4	yes	—
45	#2024	C₂×C₅	1	4	yes	—
46	#2016	C₂×C₅	1	4	yes	—
47	#1831	C₂×C₅	1	1	yes	characteristic
48	C₂×C₅×Q₈	—	2	8	no	—
49	C₂×C₅×Q₈	—	2	4	no	—
50	C₂×C₅×Q₈	—	2	8	no	—
51	C₂×C₅×Q₈	—	2	4	no	—
52	C₂×C₅×Q₈	—	4	4	no	—
53	C₂×C₅×Q₈	—	4	4	no	—
54	C₂×C₅×Q₈	—	4	4	no	—
55	C₂²×C₄×C₅	—	2	4	no	—
56	C₂²×C₄×C₅	—	2	4	no	—
57	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
62	C₅×Q₁₆	—	4	16	no	—
63	C₂×C₅×C₈	—	2	8	no	—
64	C₂×C₅×C₈	—	2	4	no	—
65	C₅×C₄⋊C₄	—	2	8	no	—
66	C₅×C₄⋊C₄	—	4	4	no	—
67	C₅×C₄⋊C₄	—	4	4	no	—
68	C₅×C₄⋊C₄	—	4	4	no	—
69	C₄²×C₅	—	2	4	no	—
70	C₄²×C₅	—	2	4	no	—
71	C₄²×C₅	—	2	1	no	—
72	C₄²×C₅	—	2	1	no	—
73	C₄²×C₅	—	4	1	no	—
74	C₄²×C₅	Q₁₆	1	4	yes	complemented
75	C₂²×Q₁₆	C₂²×C₅	1	4	yes	—
76	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
77	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
78	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
79	C₂×#103	C₂²×C₅	1	8	yes	—
80	#391	—	2	2	no	—
81	#391	—	2	2	no	—
82	#391	C₂²×C₅	1	4	yes	—
83	#391	C₂²×C₅	1	4	yes	—
84	#382	—	2	2	no	—
85	#382	—	2	2	no	—
86	C₄³	C₂²×C₅	1	1	yes	characteristic
87	#340	C₂²×C₅	1	4	yes	—
88	#336	C₂²×C₅	1	4	yes	—
89	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
90	C₅×Q₈	—	4	8	no	—
91	C₅×Q₈	—	4	8	no	—
92	C₅×Q₈	—	8	8	no	—
93	C₂×C₄×C₅	—	2	2	no	—
94	C₂×C₄×C₅	—	2	2	no	—
95	C₂×C₄×C₅	—	2	1	no	—
96	C₂×C₄×C₅	—	2	2	no	—
97	C₂×C₄×C₅	—	2	1	no	—
98	C₂×C₄×C₅	—	2	2	no	—
99	C₂×C₄×C₅	—	2	1	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	4	4	no	—
102	C₂×C₄×C₅	—	4	4	no	—
103	C₂×C₄×C₅	—	4	4	no	—
104	C₂×C₄×C₅	—	4	4	no	—
105	C₂×C₄×C₅	—	4	4	no	—
106	C₂×C₄×C₅	—	4	4	no	—
107	C₂×C₄×C₅	C₂×Q₁₆	1	4	yes	—
108	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
109	C₂×C₄×C₅	#137	1	4	yes	—
110	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
111	C₂×C₄×C₅	H(5,𝔽₂)	1	4	yes	—
112	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
113	C₅×C₈	—	4	4	no	—
114	C₂²×Q₈	—	2	2	no	—
115	C₂²×Q₈	C₂³×C₅	1	4	yes	—
116	C₂×Q₁₆	—	4	8	no	—
117	C₂×Q₁₆	—	4	4	no	—
118	C₂²×C₈	C₂³×C₅	1	4	yes	—
119	C₄⋊Q₈	—	4	4	no	—
120	C₄⋊Q₈	—	4	4	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
125	C₂×C₄²	C₅×D₄	1	2	yes	—
126	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
127	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
128	C₂×C₄²	C₂³×C₅	1	2	yes	—
129	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
130	C₄⋊C₈	—	4	2	no	—
131	#103	—	4	8	no	—
132	#95	—	2	4	no	—
133	#95	—	2	4	no	—
134	#95	—	2	4	no	—
135	#95	—	2	4	no	—
136	C₂²×C₅	#535	1	2	yes	—
137	C₂²×C₅	#537	1	1	yes	characteristic
138	C₂²×C₅	C₂²×Q₁₆	1	1	yes	characteristic
139	C₂²×C₅	C₂×#137	1	2	yes	—
140	C₂²×C₅	#575	1	1	yes	characteristic
141	C₄×C₅	—	2	4	no	—
142	C₄×C₅	—	2	4	no	—
143	C₄×C₅	—	2	2	no	—
144	C₄×C₅	—	4	1	no	—
145	C₄×C₅	—	8	4	no	—
146	C₄×C₅	—	8	4	no	—
147	C₄×C₅	H(3,𝔽₄)	1	2	yes	—
148	C₄×C₅	#562	1	2	yes	—
149	C₂×Q₈	—	2	8	no	—
150	C₂×Q₈	—	2	8	no	—
151	C₂×Q₈	—	2	4	no	—
152	C₂×Q₈	—	2	4	no	—
153	C₂×Q₈	—	4	4	no	—
154	C₂×Q₈	—	4	4	no	—
155	C₂×Q₈	—	4	4	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
160	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
161	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
162	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
163	Q₁₆	—	4	16	no	—
164	C₂×C₈	—	2	8	no	—
165	C₂×C₈	—	2	4	no	—
166	C₄⋊C₄	—	2	8	no	—
167	C₄⋊C₄	—	4	4	no	—
168	C₄⋊C₄	—	4	4	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₄²	—	2	4	no	—
171	C₄²	—	2	1	no	—
172	C₄²	—	2	4	no	—
173	C₄²	—	2	1	no	—
174	C₄²	—	4	1	no	—
175	C₄²	C₅×Q₁₆	1	4	yes	complemented
176	C₂×C₅	#2819	1	1	yes	characteristic
177	C₂×C₅	#3209	1	2	yes	—
178	C₂×C₅	#3215	1	2	yes	—
179	C₂×C₅	#3249	1	1	yes	characteristic
180	C₂×C₅	#3255	1	1	yes	characteristic
181	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
182	Q₈	—	4	8	no	—
183	Q₈	—	4	8	no	—
184	Q₈	—	8	8	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	C₂×C₅×Q₁₆	1	2	yes	—
200	C₂×C₄	C₂×C₅×Q₁₆	1	4	yes	—
201	C₂×C₄	C₅×#137	1	4	yes	—
202	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
203	C₂×C₄	C₅×H(5,𝔽₂)	1	4	yes	—
204	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
205	C₈	—	4	4	no	—
206	C₅	#32808	1	1	yes	characteristic, direct product factor, Sylow
207	C₂²	C₅×#535	1	2	yes	—
208	C₂²	C₅×#537	1	1	yes	characteristic
209	C₂²	C₂²×C₅×Q₁₆	1	1	yes	characteristic
210	C₂²	C₂×C₅×#137	1	2	yes	—
211	C₂²	C₅×#575	1	1	yes	characteristic
212	C₄	—	2	2	no	—
213	C₄	—	2	4	no	—
214	C₄	—	2	4	no	—
215	C₄	—	4	1	no	—
216	C₄	—	8	4	no	—
217	C₄	—	8	4	no	—
218	C₄	C₅×H(3,𝔽₄)	1	2	yes	—
219	C₄	C₅×#562	1	2	yes	—
220	C₂	C₅×#2819	1	1	yes	characteristic
221	C₂	C₅×#3209	1	2	yes	—
222	C₂	C₅×#3215	1	2	yes	—
223	C₂	C₅×#3249	1	1	yes	characteristic
224	C₂	C₅×#3255	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
20. #32808	C₅	direct (N × H)
74. C₄²×C₅	Q₁₆	semidirect
175. C₄²	C₅×Q₁₆	semidirect
206. C₅	#32808	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×#2862	2
C₅×#2056	2
C₅×#2024	2
C₅×#2016	2
C₅×#1831	2
#32808	5

Subgroups — C5×#32808 #11955528

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#32808 #11955528