Subgroups — C₅×D₅×C₂²⋊C₈ #13101986

Order 1600.

454 conjugacy classes of subgroups.

The subgroup lattice has 247 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₈×D₅	C₂	1	2	yes	maximal, complemented
3	C₅²×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×C₂²⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#3952	C₂	1	2	yes	maximal, complemented
6	C₂³×C₅×D₅	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
8	C₂²×C₅×Dic₅	C₄	1	1	yes	characteristic
9	C₂×C₄×C₅×D₅	—	2	4	no	—
10	C₂×C₄×C₅×D₅	C₄	1	1	yes	characteristic
11	C₂×C₄×C₅×D₅	C₄	1	1	yes	characteristic
12	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
14	C₂×C₅²×C₈	C₂²	1	2	yes	complemented
15	C₂×C₅×#168	C₂²	1	2	yes	complemented
16	C₅×C₈×D₅	—	2	4	no	—
17	C₂×C₅×C₂²⋊C₈	—	5	1	no	maximal
18	D₅×C₂²⋊C₈	C₅	1	1	yes	maximal, characteristic, direct product factor
19	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅×D₅	—	2	2	no	—
21	C₂²×C₅×D₅	—	2	2	no	—
22	C₂²×C₅×D₅	C₈	1	4	yes	complemented
23	C₂²×C₅×D₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×D₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₅²	—	2	2	no	—
26	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
28	C₂×C₅×Dic₅	—	2	2	no	—
29	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
30	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
31	C₄×C₅×D₅	—	2	2	no	—
32	C₄×C₅×D₅	—	2	2	no	—
33	C₄×C₅×D₅	—	2	2	no	—
34	C₄×C₅×D₅	D₄	1	4	yes	—
35	C₅²×C₈	—	2	2	no	—
36	C₅×#168	—	2	2	no	—
37	C₂³×C₄×C₅	—	5	1	no	—
38	C₂²×C₄×D₅	C₂×C₅	1	1	yes	characteristic
39	C₂²×C₅×C₈	—	5	2	no	—
40	C₂×C₈×D₅	C₂×C₅	1	2	yes	complemented
41	C₅×C₂²⋊C₈	—	2	2	no	—
42	C₅×C₂²⋊C₈	—	5	1	no	—
43	C₅×C₂²⋊C₈	—	5	2	no	—
44	C₅×C₂²⋊C₈	D₅	1	1	yes	characteristic, direct product factor
45	C₅×C₂²⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
46	C₂²⋊#168	C₂×C₅	1	1	yes	characteristic, complemented
47	#3952	C₂×C₅	1	2	yes	complemented
48	C₂²×C₅²	—	2	1	no	—
49	C₂²×C₅²	—	2	1	no	—
50	C₂²×C₅²	C₂×C₈	1	2	yes	complemented
51	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₅×D₅	—	2	8	no	—
53	C₂×C₅×D₅	—	2	1	no	—
54	C₂×C₅×D₅	—	2	1	no	—
55	C₂×C₅×D₅	C₂²⋊C₄	1	2	yes	—
56	C₂×C₅×D₅	C₂×C₈	1	2	yes	—
57	C₂×C₅×D₅	C₂×C₈	1	2	yes	complemented
58	C₂×C₅×D₅	M₁₆	1	2	yes	—
59	C₄×C₅²	—	2	1	no	—
60	C₄×C₅²	C₂×D₄	1	2	yes	—
61	C₅×Dic₅	—	2	1	no	—
62	C₅×Dic₅	C₂²⋊C₄	1	2	yes	—
63	C₂⁴×C₅	—	5	1	no	—
64	C₂³×D₅	C₄×C₅	1	1	yes	characteristic
65	C₂²×C₄×C₅	—	2	2	no	—
66	C₂²×C₄×C₅	—	5	1	no	—
67	C₂²×C₄×C₅	—	5	1	no	—
68	C₂²×C₄×C₅	—	5	1	no	—
69	C₂²×C₄×C₅	—	5	1	no	—
70	C₂²×C₄×C₅	—	5	1	no	—
71	C₂²×C₄×C₅	—	10	4	no	—
72	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
74	C₂²×Dic₅	C₄×C₅	1	1	yes	characteristic
75	C₂×C₄×D₅	—	2	4	no	—
76	C₂×C₄×D₅	C₄×C₅	1	1	yes	characteristic
77	C₂×C₄×D₅	C₄×C₅	1	1	yes	characteristic
78	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
79	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	—	5	2	no	—
82	C₂×C₅×C₈	—	10	4	no	—
83	C₂×C₅×C₈	C₂×D₅	1	2	yes	complemented
84	C₂×C₅×C₈	C₂²×C₅	1	2	yes	complemented
85	C₂×#168	C₂²×C₅	1	2	yes	complemented
86	C₈×D₅	—	2	4	no	—
87	C₂×C₂²⋊C₈	—	5	1	no	Sylow
88	C₂×C₅²	—	2	2	no	—
89	C₂×C₅²	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₅²	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₅²	C₂×M₁₆	1	1	yes	characteristic
92	C₅×D₅	—	2	2	no	—
93	C₅×D₅	C₂²⋊C₈	1	4	yes	direct product factor
94	C₂³×C₅	—	2	2	no	—
95	C₂³×C₅	—	5	1	no	—
96	C₂³×C₅	—	5	4	no	—
97	C₂³×C₅	—	5	1	no	—
98	C₂³×C₅	—	10	2	no	—
99	C₂³×C₅	—	10	2	no	—
100	C₂³×C₅	C₄×D₅	1	1	yes	characteristic
101	C₂³×C₅	C₂×C₄×C₅	1	1	yes	characteristic
102	C₂²×D₅	—	2	2	no	—
103	C₂²×D₅	—	2	2	no	—
104	C₂²×D₅	C₅×C₈	1	4	yes	complemented
105	C₂²×D₅	C₂×C₄×C₅	1	1	yes	characteristic
106	C₂²×D₅	C₂×C₄×C₅	1	1	yes	characteristic
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	2	2	no	—
110	C₂×C₄×C₅	—	2	2	no	—
111	C₂×C₄×C₅	—	4	4	no	—
112	C₂×C₄×C₅	—	5	1	no	—
113	C₂×C₄×C₅	—	5	4	no	—
114	C₂×C₄×C₅	—	5	1	no	—
115	C₂×C₄×C₅	—	10	2	no	—
116	C₂×C₄×C₅	—	10	2	no	—
117	C₂×C₄×C₅	—	10	2	no	—
118	C₂×C₄×C₅	—	10	2	no	—
119	C₂×C₄×C₅	C₄×D₅	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
123	C₂×Dic₅	—	2	2	no	—
124	C₂×Dic₅	C₂×C₄×C₅	1	1	yes	characteristic
125	C₂×Dic₅	C₂×C₄×C₅	1	1	yes	characteristic
126	C₄×D₅	—	2	2	no	—
127	C₄×D₅	—	2	2	no	—
128	C₄×D₅	—	2	2	no	—
129	C₄×D₅	C₅×D₄	1	4	yes	—
130	C₅×C₈	—	2	2	no	—
131	C₅×C₈	—	2	2	no	—
132	C₅×C₈	—	4	4	no	—
133	C₅×C₈	—	10	2	no	—
134	#168	—	2	2	no	—
135	C₂³×C₄	—	5	1	no	—
136	C₂²×C₈	—	5	2	no	—
137	C₂²⋊C₈	—	5	1	no	—
138	C₂²⋊C₈	—	5	2	no	—
139	C₂²⋊C₈	C₅×D₅	1	1	yes	characteristic, direct product factor
140	C₅²	C₂×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
141	C₂²×C₅	—	2	1	no	—
142	C₂²×C₅	—	2	1	no	—
143	C₂²×C₅	—	2	1	no	—
144	C₂²×C₅	—	2	1	no	—
145	C₂²×C₅	—	2	2	no	—
146	C₂²×C₅	—	2	4	no	—
147	C₂²×C₅	—	4	2	no	—
148	C₂²×C₅	—	4	2	no	—
149	C₂²×C₅	—	5	2	no	—
150	C₂²×C₅	—	5	2	no	—
151	C₂²×C₅	—	5	2	no	—
152	C₂²×C₅	—	5	2	no	—
153	C₂²×C₅	—	10	8	no	—
154	C₂²×C₅	—	10	1	no	—
155	C₂²×C₅	—	10	1	no	—
156	C₂²×C₅	C₈×D₅	1	2	yes	complemented
157	C₂²×C₅	C₂×C₅×C₈	1	2	yes	complemented
158	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
159	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
160	C₂×D₅	—	2	1	no	—
161	C₂×D₅	—	2	1	no	—
162	C₂×D₅	—	2	8	no	—
163	C₂×D₅	C₅×C₂²⋊C₄	1	2	yes	—
164	C₂×D₅	C₂×C₅×C₈	1	2	yes	—
165	C₂×D₅	C₂×C₅×C₈	1	2	yes	complemented
166	C₂×D₅	C₅×M₁₆	1	2	yes	—
167	C₄×C₅	—	2	1	no	—
168	C₄×C₅	—	2	1	no	—
169	C₄×C₅	—	2	4	no	—
170	C₄×C₅	—	4	2	no	—
171	C₄×C₅	—	5	2	no	—
172	C₄×C₅	—	10	1	no	—
173	C₄×C₅	D₄×D₅	1	2	yes	—
174	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
175	Dic₅	—	2	1	no	—
176	Dic₅	C₅×C₂²⋊C₄	1	2	yes	—
177	C₂⁴	—	5	1	no	—
178	C₂²×C₄	—	5	1	no	—
179	C₂²×C₄	—	5	1	no	—
180	C₂²×C₄	—	5	1	no	—
181	C₂²×C₄	—	5	1	no	—
182	C₂²×C₄	—	5	1	no	—
183	C₂²×C₄	—	10	4	no	—
184	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
185	C₂×C₈	—	5	2	no	—
186	C₂×C₈	—	10	4	no	—
187	C₂×C₈	C₂×C₅×D₅	1	2	yes	complemented
188	C₂×C₅	—	2	2	no	—
189	C₂×C₅	—	2	2	no	—
190	C₂×C₅	—	2	2	no	—
191	C₂×C₅	—	2	2	no	—
192	C₂×C₅	—	2	2	no	—
193	C₂×C₅	—	4	4	no	—
194	C₂×C₅	—	5	4	no	—
195	C₂×C₅	—	10	2	no	—
196	C₂×C₅	D₅×C₂²⋊C₄	1	1	yes	characteristic
197	C₂×C₅	C₂×C₈×D₅	1	1	yes	characteristic
198	C₂×C₅	D₅×M₁₆	1	1	yes	characteristic
199	C₂×C₅	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
200	C₂×C₅	C₂²×C₅×C₈	1	1	yes	characteristic
201	C₂×C₅	C₂×C₅×M₁₆	1	1	yes	characteristic
202	D₅	—	2	2	no	—
203	D₅	C₅×C₂²⋊C₈	1	4	yes	direct product factor
204	C₂³	—	5	1	no	—
205	C₂³	—	5	1	no	—
206	C₂³	—	5	4	no	—
207	C₂³	—	10	2	no	—
208	C₂³	—	10	2	no	—
209	C₂³	C₄×C₅×D₅	1	1	yes	characteristic
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	5	1	no	—
212	C₂×C₄	—	5	4	no	—
213	C₂×C₄	—	5	1	no	—
214	C₂×C₄	—	10	2	no	—
215	C₂×C₄	—	10	2	no	—
216	C₂×C₄	—	10	2	no	—
217	C₂×C₄	—	10	2	no	—
218	C₂×C₄	C₄×C₅×D₅	1	1	yes	characteristic
219	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	10	2	no	—
222	C₅	—	2	2	no	—
223	C₅	D₅×C₂²⋊C₈	1	1	yes	characteristic, direct product factor
224	C₅	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic, complemented
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	5	2	no	—
228	C₂²	—	5	2	no	—
229	C₂²	—	5	2	no	—
230	C₂²	—	5	2	no	—
231	C₂²	—	10	8	no	—
232	C₂²	—	10	1	no	—
233	C₂²	—	10	1	no	—
234	C₂²	C₅×C₈×D₅	1	2	yes	complemented
235	C₂²	C₂×C₄×C₅×D₅	1	1	yes	characteristic
236	C₄	—	2	1	no	—
237	C₄	—	5	2	no	—
238	C₄	—	10	1	no	—
239	C₄	C₅×D₄×D₅	1	2	yes	—
240	C₂	—	2	2	no	—
241	C₂	—	5	4	no	—
242	C₂	—	10	2	no	—
243	C₂	C₅×D₅×C₂²⋊C₄	1	1	yes	characteristic
244	C₂	C₂×C₅×C₈×D₅	1	1	yes	characteristic
245	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×C₂²⋊C₈	C₂
socle	C₂²×C₅²	C₂²×C₄
prime radical	C₂³×C₅×D₅	C₄
cosocle	C₂×C₄×C₅	C₂³×C₅
hypercenter	C₅×C₂²⋊C₈	D₅
hypocenter	C₅	C₂×C₅×C₂²⋊C₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₅×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×C₈×D₅	C₂	semidirect
3. C₅²×C₂²⋊C₈	C₂	semidirect
4. C₅×C₂²⋊#168	C₂	semidirect
5. C₅×#3952	C₂	semidirect
14. C₂×C₅²×C₈	C₂²	semidirect
15. C₂×C₅×#168	C₂²	semidirect
18. D₅×C₂²⋊C₈	C₅	direct (N × H)
22. C₂²×C₅×D₅	C₈	semidirect
40. C₂×C₈×D₅	C₂×C₅	semidirect
44. C₅×C₂²⋊C₈	D₅	direct (N × H)
45. C₅×C₂²⋊C₈	C₂×C₅	semidirect
46. C₂²⋊#168	C₂×C₅	semidirect
47. #3952	C₂×C₅	semidirect
50. C₂²×C₅²	C₂×C₈	semidirect
57. C₂×C₅×D₅	C₂×C₈	semidirect
83. C₂×C₅×C₈	C₂×D₅	semidirect
84. C₂×C₅×C₈	C₂²×C₅	semidirect
85. C₂×#168	C₂²×C₅	semidirect
93. C₅×D₅	C₂²⋊C₈	direct (N × H)
104. C₂²×D₅	C₅×C₈	semidirect
139. C₂²⋊C₈	C₅×D₅	direct (N × H)
140. C₅²	C₂×C₂²⋊C₈	semidirect
156. C₂²×C₅	C₈×D₅	semidirect
157. C₂²×C₅	C₂×C₅×C₈	semidirect
165. C₂×D₅	C₂×C₅×C₈	semidirect
187. C₂×C₈	C₂×C₅×D₅	semidirect
203. D₅	C₅×C₂²⋊C₈	direct (N × H)
223. C₅	D₅×C₂²⋊C₈	direct (N × H)
224. C₅	C₂×C₅×C₂²⋊C₈	semidirect
234. C₂²	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
C₂²×C₄×C₅×D₅	2
C₂×C₅×C₈×D₅	2
C₅²×C₂²⋊C₈	2
C₅×C₂²⋊#168	2
C₅×#3952	2
C₂×C₅×C₂²⋊C₈	5
D₅×C₂²⋊C₈	5

Subgroups — C5×D5×C22⋊C8 #13101986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₅×C₂²⋊C₈ #13101986