Subgroups — C₅×#10845 #11933565

Order 1280.

774 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#423	C₂	1	2	yes	maximal
2	C₂²×C₅×#97	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#341	C₂	1	4	yes	maximal
4	C₂²×C₅×M₁₆	C₂²	1	4	yes	—
5	C₂³×C₅×C₈	C₂²	1	2	yes	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₅×#423	—	2	4	no	—
8	C₅×#423	C₄	1	8	yes	—
9	C₂×C₅×C₄⋊C₈	—	2	4	no	—
10	C₂×C₅×#97	—	2	4	no	—
11	C₂×C₅×#97	C₄	1	4	yes	—
12	C₅×#341	—	2	8	no	—
13	#10845	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂³×C₄×C₅	Q₈	1	2	yes	—
15	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
16	C₂×C₅×M₁₆	—	2	4	no	—
17	C₂×C₅×M₁₆	—	2	8	no	—
18	C₂×C₅×M₁₆	—	2	8	no	—
19	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
20	C₂²×C₅×C₈	—	2	4	no	—
21	C₂²×C₅×C₈	—	2	4	no	—
22	C₂²×C₅×C₈	—	2	4	no	—
23	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₅×C₈	D₄	1	4	yes	—
25	C₂²×C₅×C₈	Q₈	1	4	yes	—
26	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₅	D₄	1	4	yes	—
29	C₂×C₄²×C₅	D₄	1	2	yes	—
30	C₅×C₄⋊C₈	—	2	16	no	—
31	C₅×#97	—	2	8	no	—
32	C₂×#423	C₂×C₅	1	2	yes	—
33	C₂²×#97	C₂×C₅	1	1	yes	characteristic
34	C₂×#341	C₂×C₅	1	4	yes	—
35	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₅	—	4	4	no	—
37	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
38	C₂²×C₄×C₅	C₄⋊C₄	1	8	yes	—
39	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
42	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
45	C₂²×C₄×C₅	#41	1	4	yes	—
46	C₅×M₁₆	—	2	16	no	—
47	C₅×M₁₆	—	4	8	no	—
48	C₂×C₅×C₈	—	2	4	no	—
49	C₂×C₅×C₈	—	2	4	no	—
50	C₂×C₅×C₈	—	2	8	no	—
51	C₂×C₅×C₈	—	2	8	no	—
52	C₂×C₅×C₈	—	2	8	no	—
53	C₂×C₅×C₈	—	4	4	no	—
54	C₂×C₅×C₈	—	4	4	no	—
55	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
56	C₄²×C₅	C₄²	1	4	yes	—
57	C₄²×C₅	C₂²⋊C₄	1	4	yes	—
58	C₄²×C₅	C₂²⋊C₄	1	8	yes	—
59	C₂²×M₁₆	C₂²×C₅	1	4	yes	—
60	C₂³×C₈	C₂²×C₅	1	2	yes	—
61	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
62	#423	—	2	4	no	—
63	#423	C₄×C₅	1	8	yes	—
64	C₂×C₄⋊C₈	—	2	4	no	—
65	C₂×#97	—	2	4	no	—
66	C₂×#97	C₄×C₅	1	4	yes	—
67	#341	—	2	8	no	—
68	C₂³×C₅	—	2	4	no	—
69	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
70	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂³×C₅	#125	1	2	yes	—
72	C₂³×C₅	C₄⋊Q₈	1	2	yes	—
73	C₂×C₄×C₅	—	2	8	no	—
74	C₂×C₄×C₅	—	4	8	no	—
75	C₂×C₄×C₅	#95	1	8	yes	—
76	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
77	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
79	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
80	C₂×C₄×C₅	#117	1	2	yes	—
81	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
82	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
83	C₂×C₄×C₅	#121	1	4	yes	—
84	C₂×C₄×C₅	C₂²⋊Q₈	1	4	yes	—
85	C₂×C₄×C₅	#124	1	2	yes	—
86	C₂×C₄×C₅	#127	1	1	yes	characteristic
87	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
88	C₅×C₈	—	2	8	no	—
89	C₅×C₈	—	4	8	no	—
90	C₂³×C₄	C₅×Q₈	1	2	yes	—
91	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
92	C₂×M₁₆	—	2	4	no	—
93	C₂×M₁₆	—	2	8	no	—
94	C₂×M₁₆	—	2	8	no	—
95	C₂×M₁₆	C₂×C₄×C₅	1	8	yes	—
96	C₂²×C₈	—	2	4	no	—
97	C₂²×C₈	—	2	4	no	—
98	C₂²×C₈	—	2	4	no	—
99	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
100	C₂²×C₈	C₅×D₄	1	4	yes	—
101	C₂²×C₈	C₅×Q₈	1	4	yes	—
102	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
103	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
104	C₂×C₄²	C₅×D₄	1	4	yes	—
105	C₂×C₄²	C₅×D₄	1	2	yes	—
106	C₄⋊C₈	—	2	16	no	—
107	#97	—	2	8	no	—
108	C₂²×C₅	—	2	4	no	—
109	C₂²×C₅	—	2	4	no	—
110	C₂²×C₅	—	2	4	no	—
111	C₂²×C₅	C₂×#95	1	1	yes	characteristic
112	C₂²×C₅	#382	1	1	yes	characteristic
113	C₂²×C₅	#384	1	1	yes	characteristic
114	C₂²×C₅	#389	1	4	yes	—
115	C₂²×C₅	#430	1	4	yes	—
116	C₄×C₅	—	2	8	no	—
117	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
118	C₄×C₅	#380	1	2	yes	—
119	C₄×C₅	#390	1	2	yes	—
120	C₄×C₅	#391	1	2	yes	—
121	C₂⁴	C₂×C₅×Q₈	1	1	yes	characteristic
122	C₂²×C₄	—	4	4	no	—
123	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
124	C₂²×C₄	C₅×C₄⋊C₄	1	8	yes	—
125	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
126	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
127	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
128	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
129	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
130	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
131	C₂²×C₄	C₅×#41	1	4	yes	—
132	M₁₆	—	2	16	no	—
133	M₁₆	—	4	8	no	—
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	2	8	no	—
137	C₂×C₈	—	2	8	no	—
138	C₂×C₈	—	2	8	no	—
139	C₂×C₈	—	4	4	no	—
140	C₂×C₈	—	4	4	no	—
141	C₂×C₈	C₅×C₄⋊C₄	1	8	yes	—
142	C₄²	C₄²×C₅	1	4	yes	—
143	C₄²	C₅×C₂²⋊C₄	1	4	yes	—
144	C₄²	C₅×C₂²⋊C₄	1	8	yes	—
145	C₂×C₅	—	2	4	no	—
146	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
147	C₂×C₅	#1738	1	2	yes	—
148	C₂×C₅	#1838	1	2	yes	—
149	C₂×C₅	#1849	1	2	yes	—
150	C₂³	—	2	4	no	—
151	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
152	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
153	C₂³	C₅×#125	1	2	yes	—
154	C₂³	C₅×C₄⋊Q₈	1	2	yes	—
155	C₂×C₄	—	2	8	no	—
156	C₂×C₄	—	4	8	no	—
157	C₂×C₄	C₅×#95	1	8	yes	—
158	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
159	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
161	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
162	C₂×C₄	C₅×#117	1	2	yes	—
163	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
164	C₂×C₄	C₄×C₅×Q₈	1	4	yes	—
165	C₂×C₄	C₅×#121	1	4	yes	—
166	C₂×C₄	C₅×C₂²⋊Q₈	1	4	yes	—
167	C₂×C₄	C₅×#124	1	2	yes	—
168	C₂×C₄	C₅×#127	1	1	yes	characteristic
169	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
170	C₈	—	2	8	no	—
171	C₈	—	4	8	no	—
172	C₅	#10845	1	1	yes	characteristic, direct product factor, Sylow
173	C₂²	—	2	4	no	—
174	C₂²	—	2	4	no	—
175	C₂²	—	2	4	no	—
176	C₂²	C₂×C₅×#95	1	1	yes	characteristic
177	C₂²	C₅×#382	1	1	yes	characteristic
178	C₂²	C₅×#384	1	1	yes	characteristic
179	C₂²	C₅×#389	1	4	yes	—
180	C₂²	C₅×#430	1	4	yes	—
181	C₄	—	2	8	no	—
182	C₄	C₄×C₅×C₄⋊C₄	1	2	yes	—
183	C₄	C₅×#380	1	2	yes	—
184	C₄	C₅×#390	1	2	yes	—
185	C₄	C₅×#391	1	2	yes	—
186	C₂	—	2	4	no	—
187	C₂	C₅×C₄⋊#95	1	1	yes	characteristic
188	C₂	C₅×#1738	1	2	yes	—
189	C₂	C₅×#1838	1	2	yes	—
190	C₂	C₅×#1849	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
13. #10845	C₅	direct (N × H)
172. C₅	#10845	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅×#423	2
C₂²×C₅×#97	2
C₂×C₅×#341	2
#10845	5

Subgroups — C5×#10845 #11933565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10845 #11933565