Subgroups — C₅×#12915 #11935635

Order 1280.

374 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×#3407	C₂	1	1	yes	maximal, characteristic
2	C₅×#2151	C₂	1	2	yes	maximal, complemented
3	C₅×#2113	C₂	1	1	yes	maximal, characteristic
4	C₅×#1757	C₂	1	1	yes	maximal, characteristic
5	C₅×#1378	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#131	C₂²	1	1	yes	characteristic
7	C₅×#504	—	2	4	no	—
8	C₂×C₅×M₃₂	—	2	2	no	—
9	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₅×M₃₂	C₂²	1	2	yes	—
11	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
12	C₅×#413	C₄	1	1	yes	characteristic
13	C₅×#410	C₄	1	1	yes	characteristic
14	C₅×#405	C₂²	1	2	yes	—
15	C₅×C₄⋊C₁₆	—	2	4	no	—
16	C₅×C₂²⋊C₁₆	—	2	2	no	—
17	C₅×#344	C₄	1	2	yes	—
18	#12915	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
20	C₅×#131	—	2	2	no	—
21	C₅×#131	—	2	2	no	—
22	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
24	C₂×C₅×M₁₆	Q₈	1	2	yes	—
25	C₂²×C₅×C₈	D₄	1	2	yes	—
26	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₅×#117	C₂×C₄	1	2	yes	—
28	C₅×M₃₂	—	2	4	no	—
29	C₅×M₃₂	—	2	2	no	—
30	C₅×M₃₂	—	2	2	no	—
31	C₅×M₃₂	—	2	2	no	—
32	C₂×C₅×C₁₆	—	2	4	no	—
33	C₂×C₅×C₁₆	—	2	1	no	—
34	C₂×C₅×C₁₆	—	2	1	no	—
35	C₂×C₅×C₁₆	—	2	2	no	—
36	C₂×C₅×C₁₆	D₄	1	4	yes	—
37	C₅×#101	—	2	1	no	—
38	C₅×#100	—	2	1	no	—
39	C₅×#99	C₈	1	4	yes	—
40	C₅×#97	—	2	2	no	—
41	C₄×C₅×C₈	—	2	2	no	—
42	#3407	C₂×C₅	1	1	yes	characteristic
43	#2151	C₂×C₅	1	2	yes	complemented
44	#2113	C₂×C₅	1	1	yes	characteristic
45	#1757	C₂×C₅	1	1	yes	characteristic
46	#1378	C₂×C₅	1	2	yes	complemented
47	C₅×#41	—	2	2	no	—
48	C₅×#41	—	2	2	no	—
49	C₂×C₅×Q₈	M₁₆	1	1	yes	characteristic
50	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
51	C₂×C₅×D₄	M₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₅×M₁₆	—	2	2	no	—
55	C₅×M₁₆	—	2	4	no	—
56	C₅×M₁₆	—	2	2	no	—
57	C₅×M₁₆	—	2	2	no	—
58	C₂×C₅×C₈	—	2	2	no	—
59	C₂×C₅×C₈	—	2	2	no	—
60	C₂×C₅×C₈	—	2	2	no	—
61	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
62	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
63	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₅×C₈	#41	1	2	yes	—
66	C₅×C₄⋊C₄	—	2	2	no	—
67	C₅×C₂²⋊C₄	C₂×C₈	1	4	yes	—
68	C₄²×C₅	—	2	2	no	—
69	C₅×C₁₆	—	2	4	no	—
70	C₅×C₁₆	—	4	1	no	—
71	C₅×C₁₆	—	4	1	no	—
72	C₂×#131	C₂²×C₅	1	1	yes	characteristic
73	#504	—	2	4	no	—
74	C₂×M₃₂	—	2	2	no	—
75	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
76	C₂×M₃₂	C₂²×C₅	1	2	yes	—
77	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
78	#413	C₄×C₅	1	1	yes	characteristic
79	#410	C₄×C₅	1	1	yes	characteristic
80	#405	C₂²×C₅	1	2	yes	—
81	C₄⋊C₁₆	—	2	4	no	—
82	C₂²⋊C₁₆	—	2	2	no	—
83	#344	C₄×C₅	1	2	yes	—
84	C₂³×C₅	C₄⋊C₈	1	2	yes	—
85	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
86	C₅×Q₈	—	2	2	no	—
87	C₅×D₄	—	2	4	no	—
88	C₅×D₄	—	2	2	no	—
89	C₂×C₄×C₅	—	2	2	no	—
90	C₂×C₄×C₅	—	2	2	no	—
91	C₂×C₄×C₅	—	2	4	no	—
92	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
93	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₅	#131	1	2	yes	—
97	C₅×C₈	—	2	2	no	—
98	C₅×C₈	—	4	2	no	—
99	C₅×C₈	#120	1	1	yes	characteristic
100	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
101	C₅×C₈	#123	1	1	yes	characteristic
102	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
103	#131	—	2	2	no	—
104	#131	—	2	2	no	—
105	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
106	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
107	C₂×M₁₆	C₅×Q₈	1	2	yes	—
108	C₂²×C₈	C₅×D₄	1	2	yes	—
109	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
110	#117	C₂×C₄×C₅	1	2	yes	—
111	M₃₂	—	2	4	no	—
112	M₃₂	—	2	2	no	—
113	M₃₂	—	2	2	no	—
114	M₃₂	—	2	2	no	—
115	C₂×C₁₆	—	2	4	no	—
116	C₂×C₁₆	—	2	1	no	—
117	C₂×C₁₆	—	2	1	no	—
118	C₂×C₁₆	—	2	2	no	—
119	C₂×C₁₆	C₅×D₄	1	4	yes	—
120	#101	—	2	1	no	—
121	#100	—	2	1	no	—
122	#99	C₅×C₈	1	4	yes	—
123	#97	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₂²×C₅	—	2	2	no	—
126	C₂²×C₅	—	4	1	no	—
127	C₂²×C₅	—	4	2	no	—
128	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
129	C₂²×C₅	C₈×D₄	1	2	yes	—
130	C₄×C₅	—	2	2	no	—
131	C₄×C₅	—	4	4	no	—
132	C₄×C₅	#385	1	1	yes	characteristic
133	C₄×C₅	#424	1	1	yes	characteristic
134	C₄×C₅	#435	1	2	yes	—
135	#41	—	2	2	no	—
136	#41	—	2	2	no	—
137	C₂×Q₈	C₅×M₁₆	1	1	yes	characteristic
138	C₂×D₄	C₂×C₅×C₈	1	2	yes	—
139	C₂×D₄	C₅×M₁₆	1	1	yes	characteristic
140	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
141	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
142	M₁₆	—	2	2	no	—
143	M₁₆	—	2	4	no	—
144	M₁₆	—	2	2	no	—
145	M₁₆	—	2	2	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
150	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
151	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
152	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
153	C₂×C₈	C₅×#41	1	2	yes	—
154	C₄⋊C₄	—	2	2	no	—
155	C₂²⋊C₄	C₂×C₅×C₈	1	4	yes	—
156	C₄²	—	2	2	no	—
157	C₁₆	—	2	4	no	—
158	C₁₆	—	4	1	no	—
159	C₁₆	—	4	1	no	—
160	C₂×C₅	—	2	2	no	—
161	C₂×C₅	—	2	1	no	—
162	C₂×C₅	—	4	2	no	—
163	C₂×C₅	#1850	1	1	yes	characteristic
164	C₂³	C₅×C₄⋊C₈	1	2	yes	—
165	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
166	Q₈	—	2	2	no	—
167	D₄	—	2	4	no	—
168	D₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	C₅×C₄⋊C₈	1	2	yes	—
173	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
175	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
176	C₂×C₄	C₅×#131	1	2	yes	—
177	C₈	—	2	2	no	—
178	C₈	—	4	2	no	—
179	C₈	C₅×#120	1	1	yes	characteristic
180	C₈	C₅×C₂²⋊Q₈	1	2	yes	—
181	C₈	C₅×#123	1	1	yes	characteristic
182	C₅	#12915	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²	—	2	2	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	4	2	no	—
186	C₂²	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
187	C₂²	C₅×C₈×D₄	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	4	4	no	—
190	C₄	C₅×#385	1	1	yes	characteristic
191	C₄	C₅×#424	1	1	yes	characteristic
192	C₄	C₅×#435	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	—	2	1	no	—
195	C₂	—	4	2	no	—
196	C₂	C₅×#1850	1	1	yes	characteristic

	subgroup	G/H
center	C₅×C₈	#120
commutator	C₂³	C₂²×C₅×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1850
prime radical	C₂×C₅×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₅×C₈ ◁ C₂²×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2151	C₂	semidirect
5. C₅×#1378	C₂	semidirect
18. #12915	C₅	direct (N × H)
43. #2151	C₂×C₅	semidirect
46. #1378	C₂×C₅	semidirect
182. C₅	#12915	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×#3407	2
C₅×#2151	2
C₅×#2113	2
C₅×#1757	2
C₅×#1378	2
#12915	5

Subgroups — C5×#12915 #11935635

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12915 #11935635