Subgroups — C₅×#7217 #11929937

Order 1280.

226 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2162	C₂	1	1	yes	maximal, characteristic
2	C₅×#1834	C₂	1	1	yes	maximal, characteristic
3	C₅×#1567	C₂	1	1	yes	maximal, characteristic
4	C₅×#423	—	2	1	no	—
5	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₅×#384	—	2	1	no	—
7	C₅×C₄⋊C₁₆	—	2	1	no	—
8	C₅×C₂²⋊C₁₆	—	2	1	no	—
9	C₄×C₅×C₁₆	C₄	1	2	yes	complemented
10	C₅×#340	—	2	1	no	—
11	C₅×#340	C₄	1	2	yes	—
12	C₅×#335	—	2	1	no	—
13	C₅×#335	C₄	1	2	yes	complemented
14	#7217	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂×C₅×M₁₆	—	4	1	no	—
16	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
18	C₂×C₅×C₄⋊C₄	—	2	2	no	—
19	C₂×C₅×C₄⋊C₄	—	4	1	no	—
20	C₂×C₅×C₄⋊C₄	—	4	1	no	—
21	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
22	C₂×C₅×C₁₆	—	2	2	no	—
23	C₅×C₄⋊C₈	—	2	2	no	—
24	C₅×C₄⋊C₈	—	4	1	no	—
25	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₅×C₈	D₄	1	1	yes	characteristic
28	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
29	#2162	C₂×C₅	1	1	yes	characteristic
30	#1834	C₂×C₅	1	1	yes	characteristic
31	#1567	C₂×C₅	1	1	yes	characteristic
32	C₂²×C₄×C₅	—	4	2	no	—
33	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
36	C₅×M₁₆	—	8	1	no	—
37	C₅×M₁₆	—	8	1	no	—
38	C₂×C₅×C₈	—	4	1	no	—
39	C₂×C₅×C₈	—	4	1	no	—
40	C₂×C₅×C₈	—	4	2	no	—
41	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
42	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₅×C₄⋊C₄	—	4	2	no	—
46	C₅×C₄⋊C₄	—	4	2	no	—
47	C₅×C₄⋊C₄	—	8	2	no	—
48	C₅×C₄⋊C₄	—	8	1	no	—
49	C₅×C₄⋊C₄	—	8	1	no	—
50	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
51	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₅	SD₁₆	1	1	yes	characteristic
53	C₄²×C₅	Q₁₆	1	1	yes	characteristic
54	C₅×C₁₆	—	4	2	no	—
55	#423	—	2	1	no	—
56	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
57	#384	—	2	1	no	—
58	C₄⋊C₁₆	—	2	1	no	—
59	C₂²⋊C₁₆	—	2	1	no	—
60	C₄×C₁₆	C₄×C₅	1	2	yes	complemented
61	#340	—	2	1	no	—
62	#340	C₄×C₅	1	2	yes	—
63	#335	—	2	1	no	—
64	#335	C₄×C₅	1	2	yes	complemented
65	C₂³×C₅	#102	1	1	yes	characteristic
66	C₂×C₄×C₅	—	2	1	no	—
67	C₂×C₄×C₅	—	2	1	no	—
68	C₂×C₄×C₅	—	2	2	no	—
69	C₂×C₄×C₅	—	8	2	no	—
70	C₂×C₄×C₅	—	8	2	no	—
71	C₂×C₄×C₅	—	8	2	no	—
72	C₂×C₄×C₅	#95	1	1	yes	characteristic
73	C₂×C₄×C₅	#102	1	1	yes	characteristic
74	C₂×C₄×C₅	#103	1	1	yes	characteristic
75	C₂×C₄×C₅	#103	1	1	yes	characteristic
76	C₂×C₄×C₅	#106	1	1	yes	characteristic
77	C₂×C₄×C₅	#107	1	1	yes	characteristic
78	C₂×C₄×C₅	D₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅	SD₃₂	1	1	yes	characteristic
80	C₂×C₄×C₅	SD₃₂	1	1	yes	characteristic
81	C₂×C₄×C₅	Q₃₂	1	1	yes	characteristic
82	C₅×C₈	—	2	1	no	—
83	C₅×C₈	—	2	1	no	—
84	C₅×C₈	—	2	1	no	—
85	C₅×C₈	—	2	1	no	—
86	C₅×C₈	—	8	2	no	—
87	C₂×M₁₆	—	4	1	no	—
88	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
89	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	2	2	no	—
91	C₂×C₄⋊C₄	—	4	1	no	—
92	C₂×C₄⋊C₄	—	4	1	no	—
93	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
94	C₂×C₁₆	—	2	2	no	—
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	4	1	no	—
97	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
98	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
99	C₄×C₈	C₅×D₄	1	1	yes	characteristic
100	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
101	C₂²×C₅	—	2	1	no	—
102	C₂²×C₅	—	2	1	no	—
103	C₂²×C₅	#340	1	1	yes	characteristic
104	C₂²×C₅	#357	1	1	yes	characteristic
105	C₂²×C₅	#358	1	1	yes	characteristic
106	C₄×C₅	—	2	1	no	—
107	C₄×C₅	—	2	1	no	—
108	C₄×C₅	—	16	2	no	—
109	C₄×C₅	H(3,ℤ/4)	1	1	yes	characteristic
110	C₄×C₅	#344	1	1	yes	characteristic
111	C₄×C₅	#357	1	1	yes	characteristic
112	C₄×C₅	#358	1	1	yes	characteristic
113	C₄×C₅	#365	1	2	yes	—
114	C₄×C₅	#366	1	1	yes	characteristic
115	C₄×C₅	#367	1	1	yes	characteristic
116	C₂²×C₄	—	4	2	no	—
117	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
118	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
119	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
120	M₁₆	—	8	1	no	—
121	M₁₆	—	8	1	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	—	4	1	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	C₄²×C₅	1	1	yes	characteristic
126	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	8	2	no	—
132	C₄⋊C₄	—	8	1	no	—
133	C₄⋊C₄	—	8	1	no	—
134	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
135	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
136	C₄²	C₅×SD₁₆	1	1	yes	characteristic
137	C₄²	C₅×Q₁₆	1	1	yes	characteristic
138	C₁₆	—	4	2	no	—
139	C₂×C₅	—	2	1	no	—
140	C₂×C₅	—	2	1	no	—
141	C₂×C₅	#1286	1	1	yes	characteristic
142	C₂×C₅	#1380	1	1	yes	characteristic
143	C₂×C₅	#1383	1	1	yes	characteristic
144	C₂³	C₅×#102	1	1	yes	characteristic
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	8	2	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	C₅×#95	1	1	yes	characteristic
152	C₂×C₄	C₅×#102	1	1	yes	characteristic
153	C₂×C₄	C₅×#103	1	1	yes	characteristic
154	C₂×C₄	C₅×#103	1	1	yes	characteristic
155	C₂×C₄	C₅×#106	1	1	yes	characteristic
156	C₂×C₄	C₅×#107	1	1	yes	characteristic
157	C₂×C₄	C₅×D₁₆	1	1	yes	characteristic
158	C₂×C₄	C₅×SD₃₂	1	1	yes	characteristic
159	C₂×C₄	C₅×SD₃₂	1	1	yes	characteristic
160	C₂×C₄	C₅×Q₃₂	1	1	yes	characteristic
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	8	2	no	—
166	C₅	#7217	1	1	yes	characteristic, direct product factor, Sylow
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	C₅×#340	1	1	yes	characteristic
170	C₂²	C₅×#357	1	1	yes	characteristic
171	C₂²	C₅×#358	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	16	2	no	—
175	C₄	C₅×H(3,ℤ/4)	1	1	yes	characteristic
176	C₄	C₅×#344	1	1	yes	characteristic
177	C₄	C₅×#357	1	1	yes	characteristic
178	C₄	C₅×#358	1	1	yes	characteristic
179	C₄	C₅×#365	1	2	yes	—
180	C₄	C₅×#366	1	1	yes	characteristic
181	C₄	C₅×#367	1	1	yes	characteristic
182	C₂	—	2	1	no	—
183	C₂	—	2	1	no	—
184	C₂	C₅×#1286	1	1	yes	characteristic
185	C₂	C₅×#1380	1	1	yes	characteristic
186	C₂	C₅×#1383	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#340
commutator	C₂×C₈	C₄²×C₅
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅
prime radical	C₂³×C₅	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
9. C₄×C₅×C₁₆	C₄	semidirect
13. C₅×#335	C₄	semidirect
14. #7217	C₅	direct (N × H)
60. C₄×C₁₆	C₄×C₅	semidirect
64. #335	C₄×C₅	semidirect
166. C₅	#7217	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×#2162	2
C₅×#1834	2
C₅×#1567	2
#7217	5

Subgroups — C5×#7217 #11929937

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7217 #11929937