Subgroups — C₅×#7209 #11929929

Order 1280.

222 conjugacy classes of subgroups.

The subgroup lattice has 176 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	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₅×C₁₆	C₄	1	2	yes	—
5	C₅×#431	C₄	1	1	yes	characteristic
6	C₄×C₅×M₁₆	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₅×C₄⋊C₁₆	—	2	1	no	—
9	C₅×C₄⋊C₁₆	—	2	1	no	—
10	C₅×C₄⋊C₁₆	—	2	1	no	—
11	C₅×C₂²⋊C₁₆	C₄	1	2	yes	—
12	C₄×C₅×C₁₆	—	2	1	no	—
13	C₅×#322	—	2	1	no	—
14	C₅×C₈²	—	2	1	no	—
15	#7209	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂×C₅×M₁₆	C₈	1	2	yes	—
17	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂×C₅×C₁₆	—	2	2	no	—
21	C₂×C₅×C₁₆	—	2	2	no	—
22	C₂×C₅×C₁₆	—	4	2	no	—
23	C₅×C₄⋊C₈	—	2	1	no	—
24	C₅×C₂²⋊C₈	C₈	1	2	yes	—
25	C₅×#97	—	2	1	no	—
26	C₄×C₅×C₈	—	2	1	no	—
27	C₄×C₅×C₈	—	2	1	no	—
28	C₄×C₅×C₈	D₄	1	1	yes	characteristic
29	C₄×C₅×C₈	D₄	1	1	yes	characteristic
30	C₄×C₅×C₈	D₄	1	1	yes	characteristic
31	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
32	#2162	C₂×C₅	1	1	yes	characteristic
33	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic
34	C₈×M₁₆	C₂×C₅	1	1	yes	characteristic
35	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
38	C₅×M₁₆	—	2	2	no	—
39	C₅×M₁₆	C₁₆	1	4	yes	complemented
40	C₂×C₅×C₈	—	2	2	no	—
41	C₂×C₅×C₈	—	2	1	no	—
42	C₂×C₅×C₈	—	2	1	no	—
43	C₂×C₅×C₈	—	2	2	no	—
44	C₂×C₅×C₈	—	4	1	no	—
45	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
50	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₅	M₁₆	1	1	yes	characteristic
52	C₄²×C₅	M₁₆	1	1	yes	characteristic
53	C₅×C₁₆	—	4	2	no	—
54	C₅×C₁₆	—	4	2	no	—
55	C₂²×C₁₆	C₄×C₅	1	2	yes	—
56	#431	C₄×C₅	1	1	yes	characteristic
57	C₄×M₁₆	C₄×C₅	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
59	C₄⋊C₁₆	—	2	1	no	—
60	C₄⋊C₁₆	—	2	1	no	—
61	C₄⋊C₁₆	—	2	1	no	—
62	C₂²⋊C₁₆	C₄×C₅	1	2	yes	—
63	C₄×C₁₆	—	2	1	no	—
64	#322	—	2	1	no	—
65	C₈²	—	2	1	no	—
66	C₂³×C₅	C₄×C₈	1	1	yes	characteristic
67	C₂×C₄×C₅	—	2	1	no	—
68	C₂×C₄×C₅	—	2	1	no	—
69	C₂×C₄×C₅	—	4	1	no	—
70	C₂×C₄×C₅	#95	1	1	yes	characteristic
71	C₂×C₄×C₅	#97	1	1	yes	characteristic
72	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
73	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
74	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
75	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
76	C₂×C₄×C₅	C₂×C₁₆	1	2	yes	—
77	C₂×C₄×C₅	M₃₂	1	2	yes	—
78	C₅×C₈	—	2	1	no	—
79	C₅×C₈	—	2	1	no	—
80	C₅×C₈	—	2	1	no	—
81	C₅×C₈	—	2	4	no	—
82	C₅×C₈	—	4	2	no	—
83	C₅×C₈	#100	1	1	yes	characteristic
84	C₅×C₈	#101	1	1	yes	characteristic
85	C₂×M₁₆	C₅×C₈	1	2	yes	—
86	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
87	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
88	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
89	C₂×C₁₆	—	2	2	no	—
90	C₂×C₁₆	—	2	2	no	—
91	C₂×C₁₆	—	4	2	no	—
92	C₄⋊C₈	—	2	1	no	—
93	C₂²⋊C₈	C₅×C₈	1	2	yes	—
94	#97	—	2	1	no	—
95	C₄×C₈	—	2	1	no	—
96	C₄×C₈	—	2	1	no	—
97	C₄×C₈	C₅×D₄	1	1	yes	characteristic
98	C₄×C₈	C₅×D₄	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	#336	1	1	yes	characteristic
104	C₂²×C₅	C₄×C₁₆	1	1	yes	characteristic
105	C₂²×C₅	#346	1	1	yes	characteristic
106	C₄×C₅	—	2	1	no	—
107	C₄×C₅	—	2	1	no	—
108	C₄×C₅	—	2	1	no	—
109	C₄×C₅	#343	1	1	yes	characteristic
110	C₄×C₅	#344	1	1	yes	characteristic
111	C₄×C₅	C₂²⋊C₁₆	1	2	yes	—
112	C₄×C₅	C₄⋊C₁₆	1	2	yes	—
113	C₂²×C₄	C₄²×C₅	1	1	yes	characteristic
114	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
115	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
116	M₁₆	—	2	2	no	—
117	M₁₆	C₅×C₁₆	1	4	yes	complemented
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
127	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
128	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
129	C₄²	C₅×M₁₆	1	1	yes	characteristic
130	C₄²	C₅×M₁₆	1	1	yes	characteristic
131	C₁₆	—	4	2	no	—
132	C₁₆	—	4	2	no	—
133	C₂×C₅	—	2	1	no	—
134	C₂×C₅	—	2	1	no	—
135	C₂×C₅	#1283	1	1	yes	characteristic
136	C₂×C₅	#1374	1	1	yes	characteristic
137	C₂×C₅	#1378	1	1	yes	characteristic
138	C₂³	C₄×C₅×C₈	1	1	yes	characteristic
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	C₅×#95	1	1	yes	characteristic
143	C₂×C₄	C₅×#97	1	1	yes	characteristic
144	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
145	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
146	C₂×C₄	C₅×C₄⋊C₈	1	1	yes	characteristic
147	C₂×C₄	C₅×C₄⋊C₈	1	1	yes	characteristic
148	C₂×C₄	C₂×C₅×C₁₆	1	2	yes	—
149	C₂×C₄	C₅×M₃₂	1	2	yes	—
150	C₈	—	2	1	no	—
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	2	4	no	—
154	C₈	—	4	2	no	—
155	C₈	C₅×#100	1	1	yes	characteristic
156	C₈	C₅×#101	1	1	yes	characteristic
157	C₅	#7209	1	1	yes	characteristic, direct product factor, Sylow
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	C₅×#336	1	1	yes	characteristic
161	C₂²	C₄×C₅×C₁₆	1	1	yes	characteristic
162	C₂²	C₅×#346	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	C₅×#343	1	1	yes	characteristic
167	C₄	C₅×#344	1	1	yes	characteristic
168	C₄	C₅×C₂²⋊C₁₆	1	2	yes	—
169	C₄	C₅×C₄⋊C₁₆	1	2	yes	—
170	C₂	—	2	1	no	—
171	C₂	—	2	1	no	—
172	C₂	C₅×#1283	1	1	yes	characteristic
173	C₂	C₅×#1374	1	1	yes	characteristic
174	C₂	C₅×#1378	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅×C₈	C₂²⋊C₄
commutator	C₂²	C₄×C₅×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅
socle	C₂²×C₅	#336
prime radical	C₂³×C₅	C₄×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
15. #7209	C₅	direct (N × H)
39. C₅×M₁₆	C₁₆	semidirect
117. M₁₆	C₅×C₁₆	semidirect
157. C₅	#7209	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
C₂×C₅×C₄⋊C₁₆	2
C₅×C₈×M₁₆	2
#7209	5

Subgroups — C5×#7209 #11929929

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7209 #11929929