Subgroups — C₅×#32856 #11955576

Order 1280.

510 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2647	C₂	1	2	yes	maximal
2	C₅×#2101	C₂	1	4	yes	maximal
3	C₅×#2087	C₂	1	2	yes	maximal
4	C₅×#2086	C₂	1	2	yes	maximal
5	C₅×#2083	C₂	1	4	yes	maximal
6	C₅×#1832	C₂	1	1	yes	maximal, characteristic
7	C₂×C₅×#107	C₂²	1	2	yes	—
8	C₂×C₅×#107	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₅×#400	—	2	4	no	—
12	C₅×#398	—	2	4	no	—
13	C₅×#389	C₂²	1	4	yes	—
14	C₅×#389	C₂²	1	4	yes	—
15	C₅×#384	C₂²	1	2	yes	—
16	C₅×#382	C₂²	1	2	yes	—
17	C₅×#376	C₂²	1	1	yes	characteristic
18	C₅×#340	C₂²	1	4	yes	—
19	C₅×#340	C₂²	1	4	yes	—
20	C₅×#340	C₂²	1	2	yes	—
21	C₅×#340	C₂²	1	2	yes	—
22	C₅×#336	C₂²	1	4	yes	—
23	#32856	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅×C₈	C₂³	1	2	yes	—
25	C₂²×C₅×C₈	C₂³	1	2	yes	—
26	C₂×C₅×C₄⋊C₄	—	2	4	no	—
27	C₂×C₅×C₄⋊C₄	—	2	4	no	—
28	C₂×C₅×C₄⋊C₄	—	2	4	no	—
29	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
30	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
31	C₂×C₄²×C₅	C₂³	1	2	yes	—
32	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
33	C₅×#107	—	4	2	no	—
34	C₅×#107	—	4	2	no	—
35	C₅×C₄⋊C₈	—	4	1	no	—
36	C₅×C₄⋊C₈	—	4	1	no	—
37	C₅×#95	—	2	4	no	—
38	C₅×#95	—	2	4	no	—
39	C₅×#95	—	2	4	no	—
40	C₅×#95	D₄	1	2	yes	—
41	C₅×#95	D₄	1	2	yes	—
42	#2647	C₂×C₅	1	2	yes	—
43	#2101	C₂×C₅	1	4	yes	—
44	#2087	C₂×C₅	1	2	yes	—
45	#2086	C₂×C₅	1	2	yes	—
46	#2083	C₂×C₅	1	4	yes	—
47	#1832	C₂×C₅	1	1	yes	characteristic
48	C₂²×C₄×C₅	—	2	4	no	—
49	C₂²×C₄×C₅	—	2	4	no	—
50	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
52	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
54	C₂×C₅×C₈	—	2	2	no	—
55	C₂×C₅×C₈	—	2	2	no	—
56	C₂×C₅×C₈	—	2	4	no	—
57	C₂×C₅×C₈	—	2	4	no	—
58	C₅×C₄⋊C₄	—	2	4	no	—
59	C₅×C₄⋊C₄	—	2	4	no	—
60	C₅×C₄⋊C₄	—	2	4	no	—
61	C₅×C₄⋊C₄	—	2	4	no	—
62	C₅×C₄⋊C₄	—	8	4	no	—
63	C₅×C₄⋊C₄	—	8	4	no	—
64	C₅×C₄⋊C₄	—	8	4	no	—
65	C₄²×C₅	—	2	1	no	—
66	C₄²×C₅	—	2	2	no	—
67	C₄²×C₅	—	2	2	no	—
68	C₄²×C₅	—	2	1	no	—
69	C₂×#107	C₂²×C₅	1	2	yes	—
70	C₂×#107	C₂²×C₅	1	2	yes	—
71	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
72	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
73	#400	—	2	4	no	—
74	#398	—	2	4	no	—
75	#389	C₂²×C₅	1	4	yes	—
76	#389	C₂²×C₅	1	4	yes	—
77	#384	C₂²×C₅	1	2	yes	—
78	#382	C₂²×C₅	1	2	yes	—
79	#376	C₂²×C₅	1	1	yes	characteristic
80	#340	C₂²×C₅	1	4	yes	—
81	#340	C₂²×C₅	1	4	yes	—
82	#340	C₂²×C₅	1	2	yes	—
83	#340	C₂²×C₅	1	2	yes	—
84	#336	C₂²×C₅	1	4	yes	—
85	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₄×C₅	—	2	4	no	—
87	C₂×C₄×C₅	—	2	1	no	—
88	C₂×C₄×C₅	—	2	1	no	—
89	C₂×C₄×C₅	—	2	4	no	—
90	C₂×C₄×C₅	—	2	2	no	—
91	C₂×C₄×C₅	—	2	4	no	—
92	C₂×C₄×C₅	—	2	2	no	—
93	C₂×C₄×C₅	—	4	4	no	—
94	C₂×C₄×C₅	—	4	4	no	—
95	C₂×C₄×C₅	—	4	4	no	—
96	C₂×C₄×C₅	—	4	4	no	—
97	C₂×C₄×C₅	—	4	4	no	—
98	C₂×C₄×C₅	—	4	4	no	—
99	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
100	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
101	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
102	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
103	C₅×C₈	—	4	2	no	—
104	C₅×C₈	—	4	2	no	—
105	C₂²×C₈	C₂³×C₅	1	2	yes	—
106	C₂²×C₈	C₂³×C₅	1	2	yes	—
107	C₂×C₄⋊C₄	—	2	4	no	—
108	C₂×C₄⋊C₄	—	2	4	no	—
109	C₂×C₄⋊C₄	—	2	4	no	—
110	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
111	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
112	C₂×C₄²	C₂³×C₅	1	2	yes	—
113	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
114	#107	—	4	2	no	—
115	#107	—	4	2	no	—
116	C₄⋊C₈	—	4	1	no	—
117	C₄⋊C₈	—	4	1	no	—
118	#95	—	2	4	no	—
119	#95	—	2	4	no	—
120	#95	—	2	4	no	—
121	#95	C₅×D₄	1	2	yes	—
122	#95	C₅×D₄	1	2	yes	—
123	C₂²×C₅	#536	1	2	yes	—
124	C₂²×C₅	#537	1	1	yes	characteristic
125	C₂²×C₅	#576	1	2	yes	—
126	C₂²×C₅	#578	1	2	yes	—
127	C₄×C₅	—	4	4	no	—
128	C₄×C₅	—	4	1	no	—
129	C₄×C₅	—	4	1	no	—
130	C₄×C₅	—	8	4	no	—
131	C₄×C₅	—	8	4	no	—
132	C₄×C₅	#563	1	1	yes	characteristic
133	C₄×C₅	#563	1	1	yes	characteristic
134	C₄×C₅	#564	1	2	yes	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
139	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
140	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	2	4	no	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	8	4	no	—
150	C₄⋊C₄	—	8	4	no	—
151	C₄⋊C₄	—	8	4	no	—
152	C₄²	—	2	1	no	—
153	C₄²	—	2	2	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	1	no	—
156	C₂×C₅	#2822	1	1	yes	characteristic
157	C₂×C₅	#3243	1	4	yes	—
158	C₂×C₅	#3259	1	1	yes	characteristic
159	C₂×C₅	#3261	1	1	yes	characteristic
160	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
175	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
176	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
177	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
178	C₈	—	4	2	no	—
179	C₈	—	4	2	no	—
180	C₅	#32856	1	1	yes	characteristic, direct product factor, Sylow
181	C₂²	C₅×#536	1	2	yes	—
182	C₂²	C₅×#537	1	1	yes	characteristic
183	C₂²	C₅×#576	1	2	yes	—
184	C₂²	C₅×#578	1	2	yes	—
185	C₄	—	4	4	no	—
186	C₄	—	4	1	no	—
187	C₄	—	4	1	no	—
188	C₄	—	8	4	no	—
189	C₄	—	8	4	no	—
190	C₄	C₅×#563	1	1	yes	characteristic
191	C₄	C₅×#563	1	1	yes	characteristic
192	C₄	C₅×#564	1	2	yes	—
193	C₂	C₅×#2822	1	1	yes	characteristic
194	C₂	C₅×#3243	1	4	yes	—
195	C₂	C₅×#3259	1	1	yes	characteristic
196	C₂	C₅×#3261	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 ◁ C₅×#376 ◁ G

Complements

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

N	H ≅ G/N	type
23. #32856	C₅	direct (N × H)
180. C₅	#32856	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×#2647	2
C₅×#2101	2
C₅×#2087	2
C₅×#2086	2
C₅×#2083	2
C₅×#1832	2
#32856	5

Subgroups — C5×#32856 #11955576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#32856 #11955576