Subgroups — C₅×D₅×C₄⋊C₈ #13102096

Order 1600.

314 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₈×D₅	C₂	1	2	yes	maximal
2	C₄²×C₅×D₅	C₂	1	1	yes	maximal, characteristic
3	C₅²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#3946	C₂	1	2	yes	maximal, complemented
5	C₅×C₄⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₅×D₅	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×D₅	C₄	1	1	yes	characteristic
8	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
9	C₂×C₅²×C₈	C₂²	1	2	yes	—
10	C₄²×C₅²	C₂²	1	1	yes	characteristic
11	C₄×C₅×Dic₅	C₄	1	1	yes	characteristic
12	C₄×C₅×Dic₅	C₄	1	1	yes	characteristic
13	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
14	C₂×C₅×#168	C₂²	1	2	yes	—
15	C₅×C₈×D₅	—	2	4	no	—
16	C₂×C₅×C₄⋊C₈	—	5	1	no	maximal
17	D₅×C₄⋊C₈	C₅	1	1	yes	maximal, characteristic, direct product factor
18	C₂²×C₅×D₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
22	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
25	C₄×C₅×D₅	—	2	2	no	—
26	C₄×C₅×D₅	C₈	1	4	yes	complemented
27	C₄×C₅×D₅	D₄	1	2	yes	—
28	C₄×C₅×D₅	Q₈	1	2	yes	—
29	C₅²×C₈	—	2	2	no	—
30	C₅×#168	—	2	2	no	—
31	C₂²×C₅×C₈	—	5	2	no	—
32	C₂×C₄²×C₅	—	5	1	no	—
33	C₂×C₈×D₅	C₂×C₅	1	2	yes	—
34	C₄²×D₅	C₂×C₅	1	1	yes	characteristic
35	C₅×C₄⋊C₈	—	2	2	no	—
36	C₅×C₄⋊C₈	—	5	2	no	—
37	C₅×C₄⋊C₈	—	5	1	no	—
38	C₅×C₄⋊C₈	D₅	1	1	yes	characteristic, direct product factor
39	C₅×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
40	#3946	C₂×C₅	1	2	yes	complemented
41	C₄⋊#168	C₂×C₅	1	1	yes	characteristic, complemented
42	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₅×D₅	C₄⋊C₄	1	2	yes	—
44	C₂×C₅×D₅	C₂×C₈	1	2	yes	—
45	C₂×C₅×D₅	M₁₆	1	2	yes	—
46	C₄×C₅²	—	2	1	no	—
47	C₄×C₅²	C₂×C₈	1	2	yes	complemented
48	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
49	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
50	C₅×Dic₅	—	2	1	no	—
51	C₅×Dic₅	C₄⋊C₄	1	2	yes	—
52	C₅×Dic₅	C₂×C₈	1	2	yes	complemented
53	C₂²×C₄×C₅	—	5	1	no	—
54	C₂²×C₄×C₅	—	5	1	no	—
55	C₂²×C₄×C₅	—	5	1	no	—
56	C₂×C₄×D₅	C₄×C₅	1	1	yes	characteristic
57	C₂×C₄×D₅	C₄×C₅	1	1	yes	characteristic
58	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
59	C₂×C₅×C₈	—	2	4	no	—
60	C₂×C₅×C₈	—	5	2	no	—
61	C₂×C₅×C₈	—	10	4	no	—
62	C₂×C₅×C₈	C₂×D₅	1	2	yes	—
63	C₂×C₅×C₈	C₂²×C₅	1	2	yes	—
64	C₄²×C₅	—	2	2	no	—
65	C₄²×C₅	—	5	1	no	—
66	C₄²×C₅	—	5	1	no	—
67	C₄²×C₅	—	5	1	no	—
68	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
69	C₄²×C₅	C₂²×C₅	1	1	yes	characteristic
70	C₄×Dic₅	C₄×C₅	1	1	yes	characteristic
71	C₄×Dic₅	C₄×C₅	1	1	yes	characteristic
72	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic
73	C₂×#168	C₂²×C₅	1	2	yes	—
74	C₈×D₅	—	2	4	no	—
75	C₂×C₄⋊C₈	—	5	1	no	Sylow
76	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₅²	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₅²	C₂×M₁₆	1	1	yes	characteristic
79	C₅×D₅	C₄⋊C₈	1	4	yes	direct product factor
80	C₂³×C₅	—	5	1	no	—
81	C₂²×D₅	C₂×C₄×C₅	1	1	yes	characteristic
82	C₂×C₄×C₅	—	2	2	no	—
83	C₂×C₄×C₅	—	2	2	no	—
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	—	5	1	no	—
86	C₂×C₄×C₅	—	5	1	no	—
87	C₂×C₄×C₅	—	5	4	no	—
88	C₂×C₄×C₅	—	5	2	no	—
89	C₂×C₄×C₅	—	5	2	no	—
90	C₂×C₄×C₅	—	5	1	no	—
91	C₂×C₄×C₅	—	10	2	no	—
92	C₂×C₄×C₅	C₄×D₅	1	1	yes	characteristic
93	C₂×C₄×C₅	C₄×D₅	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
98	C₂×Dic₅	C₂×C₄×C₅	1	1	yes	characteristic
99	C₂×Dic₅	C₂×C₄×C₅	1	1	yes	characteristic
100	C₂×Dic₅	C₂×C₄×C₅	1	1	yes	characteristic
101	C₄×D₅	—	2	2	no	—
102	C₄×D₅	C₅×C₈	1	4	yes	complemented
103	C₄×D₅	C₅×D₄	1	2	yes	—
104	C₄×D₅	C₅×Q₈	1	2	yes	—
105	C₅×C₈	—	2	2	no	—
106	C₅×C₈	—	2	2	no	—
107	C₅×C₈	—	4	4	no	—
108	C₅×C₈	—	10	2	no	—
109	#168	—	2	2	no	—
110	C₂²×C₈	—	5	2	no	—
111	C₂×C₄²	—	5	1	no	—
112	C₄⋊C₈	—	5	2	no	—
113	C₄⋊C₈	—	5	1	no	—
114	C₄⋊C₈	C₅×D₅	1	1	yes	characteristic, direct product factor
115	C₅²	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₅	—	2	2	no	—
117	C₂²×C₅	—	5	2	no	—
118	C₂²×C₅	—	5	2	no	—
119	C₂²×C₅	—	5	2	no	—
120	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
121	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
122	C₂×D₅	C₅×C₄⋊C₄	1	2	yes	—
123	C₂×D₅	C₂×C₅×C₈	1	2	yes	—
124	C₂×D₅	C₅×M₁₆	1	2	yes	—
125	C₄×C₅	—	2	1	no	—
126	C₄×C₅	—	2	1	no	—
127	C₄×C₅	—	2	2	no	—
128	C₄×C₅	—	2	2	no	—
129	C₄×C₅	—	2	4	no	—
130	C₄×C₅	—	4	2	no	—
131	C₄×C₅	—	5	2	no	—
132	C₄×C₅	—	5	2	no	—
133	C₄×C₅	—	10	1	no	—
134	C₄×C₅	C₈×D₅	1	2	yes	complemented
135	C₄×C₅	C₂×C₅×C₈	1	2	yes	complemented
136	C₄×C₅	D₄×D₅	1	1	yes	characteristic
137	C₄×C₅	Q₈×D₅	1	1	yes	characteristic
138	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
139	C₄×C₅	C₂×C₅×Q₈	1	1	yes	characteristic
140	Dic₅	—	2	1	no	—
141	Dic₅	C₅×C₄⋊C₄	1	2	yes	—
142	Dic₅	C₂×C₅×C₈	1	2	yes	complemented
143	C₂²×C₄	—	5	1	no	—
144	C₂²×C₄	—	5	1	no	—
145	C₂²×C₄	—	5	1	no	—
146	C₂×C₈	—	5	2	no	—
147	C₂×C₈	—	10	4	no	—
148	C₂×C₈	C₂×C₅×D₅	1	2	yes	—
149	C₄²	—	5	1	no	—
150	C₄²	—	5	1	no	—
151	C₄²	—	5	1	no	—
152	C₄²	C₂×C₅×D₅	1	1	yes	characteristic
153	C₂×C₅	—	2	2	no	—
154	C₂×C₅	—	2	2	no	—
155	C₂×C₅	—	2	2	no	—
156	C₂×C₅	—	5	4	no	—
157	C₂×C₅	D₅×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₅	C₂×C₈×D₅	1	1	yes	characteristic
159	C₂×C₅	D₅×M₁₆	1	1	yes	characteristic
160	C₂×C₅	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₅	C₂²×C₅×C₈	1	1	yes	characteristic
162	C₂×C₅	C₂×C₅×M₁₆	1	1	yes	characteristic
163	D₅	C₅×C₄⋊C₈	1	4	yes	direct product factor
164	C₂³	—	5	1	no	—
165	C₂×C₄	—	5	1	no	—
166	C₂×C₄	—	5	1	no	—
167	C₂×C₄	—	5	2	no	—
168	C₂×C₄	—	5	2	no	—
169	C₂×C₄	—	5	4	no	—
170	C₂×C₄	—	5	1	no	—
171	C₂×C₄	—	10	2	no	—
172	C₂×C₄	C₄×C₅×D₅	1	1	yes	characteristic
173	C₂×C₄	C₄×C₅×D₅	1	1	yes	characteristic
174	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	10	2	no	—
177	C₅	—	2	2	no	—
178	C₅	D₅×C₄⋊C₈	1	1	yes	characteristic, direct product factor
179	C₅	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic, complemented
180	C₂²	—	5	2	no	—
181	C₂²	—	5	2	no	—
182	C₂²	—	5	2	no	—
183	C₂²	C₂×C₄×C₅×D₅	1	1	yes	characteristic
184	C₄	—	2	1	no	—
185	C₄	—	5	2	no	—
186	C₄	—	5	2	no	—
187	C₄	—	10	1	no	—
188	C₄	C₅×C₈×D₅	1	2	yes	complemented
189	C₄	C₅×D₄×D₅	1	1	yes	characteristic
190	C₄	C₅×Q₈×D₅	1	1	yes	characteristic
191	C₂	—	5	4	no	—
192	C₂	C₅×D₅×C₄⋊C₄	1	1	yes	characteristic
193	C₂	C₂×C₅×C₈×D₅	1	1	yes	characteristic
194	C₂	C₅×D₅×M₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₅
commutator	C₂×C₅	C₂²×C₅×C₈
Frattini	C₂×C₄	C₂²×C₅×D₅
Fitting	C₅²×C₄⋊C₈	C₂
socle	C₂²×C₅²	C₂²×C₄
prime radical	C₂²×C₅×D₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂³×C₅
hypercenter	C₅×C₄⋊C₈	D₅
hypocenter	C₅	C₂×C₅×C₄⋊C₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₅×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₅²×C₄⋊C₈	C₂	semidirect
4. C₅×#3946	C₂	semidirect
5. C₅×C₄⋊#168	C₂	semidirect
17. D₅×C₄⋊C₈	C₅	direct (N × H)
26. C₄×C₅×D₅	C₈	semidirect
38. C₅×C₄⋊C₈	D₅	direct (N × H)
39. C₅×C₄⋊C₈	C₂×C₅	semidirect
40. #3946	C₂×C₅	semidirect
41. C₄⋊#168	C₂×C₅	semidirect
47. C₄×C₅²	C₂×C₈	semidirect
52. C₅×Dic₅	C₂×C₈	semidirect
79. C₅×D₅	C₄⋊C₈	direct (N × H)
102. C₄×D₅	C₅×C₈	semidirect
114. C₄⋊C₈	C₅×D₅	direct (N × H)
115. C₅²	C₂×C₄⋊C₈	semidirect
134. C₄×C₅	C₈×D₅	semidirect
135. C₄×C₅	C₂×C₅×C₈	semidirect
142. Dic₅	C₂×C₅×C₈	semidirect
163. D₅	C₅×C₄⋊C₈	direct (N × H)
178. C₅	D₅×C₄⋊C₈	direct (N × H)
179. C₅	C₂×C₅×C₄⋊C₈	semidirect
188. C₄	C₅×C₈×D₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×C₈×D₅	2
C₄²×C₅×D₅	2
C₅²×C₄⋊C₈	2
C₅×#3946	2
C₅×C₄⋊#168	2
C₂×C₅×C₄⋊C₈	5
D₅×C₄⋊C₈	5

Subgroups — C5×D5×C4⋊C8 #13102096

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₅×C₄⋊C₈ #13102096