Subgroups — C₅×#65918 #13106738

Order 1600.

494 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#4092	C₂	1	2	yes	maximal, complemented
3	C₅×#4044	C₂	1	4	yes	maximal, complemented
4	C₅×#4038	C₂	1	2	yes	maximal, complemented
5	C₅×#4021	C₂	1	2	yes	maximal
6	C₄×C₅×D₂₀	C₂	1	1	yes	maximal, characteristic
7	C₅×#4015	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₅×Dic₁₀	C₂	1	2	yes	maximal, complemented
9	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
12	C₂×C₅×Dic₁₀	C₂²	1	2	yes	—
13	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
15	C₄²×C₅²	C₂²	1	2	yes	—
16	C₄²×C₅²	C₂²	1	1	yes	characteristic
17	C₅×#685	C₂²	1	4	yes	—
18	C₅×#685	C₂²	1	2	yes	—
19	C₅×#685	C₂²	1	4	yes	—
20	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
21	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
22	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
23	C₅×#683	C₂²	1	4	yes	—
24	C₄×C₅×Dic₅	C₂²	1	4	yes	—
25	C₅×#556	—	5	1	no	maximal
26	#65918	C₅	1	1	yes	maximal, characteristic, direct product factor
27	C₂²×C₅×D₅	C₂³	1	2	yes	—
28	C₅²×Q₈	—	2	2	no	—
29	C₂×C₄×C₅²	C₂³	1	2	yes	—
30	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₅²	C₂³	1	2	yes	—
32	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
34	C₂×C₅×Dic₅	C₂³	1	4	yes	—
35	C₂×C₅×Dic₅	C₂³	1	2	yes	—
36	C₅×D₂₀	—	2	2	no	—
37	C₄×C₅×D₅	—	2	4	no	—
38	C₅×Dic₁₀	—	2	4	no	—
39	C₅×C₄⋊Q₈	—	5	1	no	—
40	C₅×#126	—	5	4	no	—
41	C₅×#124	—	5	2	no	—
42	C₅×C₂²⋊Q₈	—	5	2	no	—
43	C₄×C₅×Q₈	—	2	2	no	—
44	C₄×C₅×Q₈	—	5	2	no	—
45	C₄×C₅×Q₈	D₅	1	1	yes	characteristic, complemented
46	C₄×C₅×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
47	C₄×C₅×D₄	—	5	1	no	—
48	C₅×#117	—	5	2	no	—
49	#4092	C₂×C₅	1	2	yes	complemented
50	#4044	C₂×C₅	1	4	yes	complemented
51	#4038	C₂×C₅	1	2	yes	complemented
52	#4021	C₂×C₅	1	2	yes	—
53	C₄×D₂₀	C₂×C₅	1	1	yes	characteristic
54	#4015	C₂×C₅	1	1	yes	characteristic, complemented
55	C₄×Dic₁₀	C₂×C₅	1	2	yes	complemented
56	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
57	C₂×C₅×D₅	—	2	2	no	—
58	C₂×C₅×D₅	—	2	2	no	—
59	C₂×C₅×D₅	—	2	2	no	—
60	C₄×C₅²	—	2	2	no	—
61	C₄×C₅²	—	2	2	no	—
62	C₄×C₅²	—	2	1	no	—
63	C₄×C₅²	#41	1	2	yes	—
64	C₄×C₅²	#41	1	2	yes	—
65	C₅×Dic₅	—	2	4	no	—
66	C₅×Dic₅	—	2	2	no	—
67	C₂×C₅×Q₈	—	2	2	no	—
68	C₂×C₅×Q₈	—	5	2	no	—
69	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
70	C₂×C₅×Q₈	C₂²×C₅	1	1	yes	characteristic
71	C₂×C₅×D₄	—	5	1	no	—
72	C₂²×C₄×C₅	—	5	2	no	—
73	C₂×D₂₀	C₂²×C₅	1	1	yes	characteristic
74	C₂×C₄×D₅	C₂²×C₅	1	2	yes	—
75	C₂×Dic₁₀	C₂²×C₅	1	2	yes	—
76	C₅×C₄⋊C₄	—	2	2	no	—
77	C₅×C₄⋊C₄	—	2	4	no	—
78	C₅×C₄⋊C₄	—	5	2	no	—
79	C₅×C₄⋊C₄	—	5	1	no	—
80	C₅×C₄⋊C₄	—	5	4	no	—
81	C₅×C₄⋊C₄	—	5	2	no	—
82	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
83	C₅×C₄⋊C₄	C₂×D₅	1	2	yes	—
84	C₅×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
85	C₅×C₄⋊C₄	C₂²×C₅	1	2	yes	—
86	C₅×C₂²⋊C₄	—	5	4	no	—
87	C₅×C₂²⋊C₄	—	5	2	no	—
88	C₅×C₂²⋊C₄	—	5	4	no	—
89	C₄²×C₅	—	2	2	no	—
90	C₄²×C₅	—	2	4	no	—
91	C₄²×C₅	—	5	4	no	—
92	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
93	C₄²×C₅	C₂×D₅	1	2	yes	—
94	C₄²×C₅	C₂²×C₅	1	2	yes	—
95	C₄²×C₅	C₂²×C₅	1	1	yes	characteristic
96	#685	C₂²×C₅	1	2	yes	—
97	#685	C₂²×C₅	1	4	yes	—
98	#685	C₂²×C₅	1	4	yes	—
99	C₄⋊Dic₅	C₂²×C₅	1	1	yes	characteristic
100	C₄⋊Dic₅	C₂²×C₅	1	2	yes	—
101	C₄⋊Dic₅	C₂²×C₅	1	2	yes	—
102	#683	C₂²×C₅	1	4	yes	—
103	C₄×Dic₅	C₂²×C₅	1	4	yes	—
104	#556	—	5	1	no	Sylow
105	C₂×C₅²	C₂×#41	1	1	yes	characteristic
106	C₂×C₅²	C₂×#41	1	1	yes	characteristic
107	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
108	C₅×D₅	—	4	2	no	—
109	C₂³×C₅	—	5	2	no	—
110	C₂²×D₅	C₂³×C₅	1	2	yes	—
111	C₅×Q₈	—	2	8	no	—
112	C₅×Q₈	—	2	2	no	—
113	C₅×Q₈	—	2	2	no	—
114	C₅×Q₈	—	10	4	no	—
115	C₅×D₄	—	10	2	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	2	4	no	—
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	4	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	5	2	no	—
122	C₂×C₄×C₅	—	5	4	no	—
123	C₂×C₄×C₅	—	10	4	no	—
124	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
125	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
127	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
130	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
131	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
134	C₂×Dic₅	C₂³×C₅	1	2	yes	—
135	C₂×Dic₅	C₂³×C₅	1	4	yes	—
136	D₂₀	—	2	2	no	—
137	C₄×D₅	—	2	4	no	—
138	Dic₁₀	—	2	4	no	—
139	C₄⋊Q₈	—	5	1	no	—
140	#126	—	5	4	no	—
141	#124	—	5	2	no	—
142	C₂²⋊Q₈	—	5	2	no	—
143	C₄×Q₈	—	5	2	no	—
144	C₄×Q₈	C₅×D₅	1	1	yes	characteristic, complemented
145	C₄×D₄	—	5	1	no	—
146	#117	—	5	2	no	—
147	C₅²	#556	1	1	yes	characteristic, complemented, Sylow
148	C₂²×C₅	—	2	2	no	—
149	C₂²×C₅	—	10	2	no	—
150	C₂²×C₅	—	10	2	no	—
151	C₂²×C₅	—	10	2	no	—
152	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
153	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
154	C₂×D₅	—	2	2	no	—
155	C₂×D₅	—	2	2	no	—
156	C₂×D₅	—	2	2	no	—
157	C₄×C₅	—	2	8	no	—
158	C₄×C₅	—	2	2	no	—
159	C₄×C₅	—	2	2	no	—
160	C₄×C₅	—	2	1	no	—
161	C₄×C₅	—	2	2	no	—
162	C₄×C₅	—	2	2	no	—
163	C₄×C₅	—	2	1	no	—
164	C₄×C₅	—	2	4	no	—
165	C₄×C₅	—	2	4	no	—
166	C₄×C₅	—	4	4	no	—
167	C₄×C₅	—	4	2	no	—
168	C₄×C₅	—	10	2	no	—
169	C₄×C₅	—	10	4	no	—
170	C₄×C₅	#709	1	2	yes	—
171	C₄×C₅	#713	1	2	yes	—
172	C₄×C₅	C₅×#41	1	2	yes	—
173	C₄×C₅	C₅×#41	1	2	yes	—
174	Dic₅	—	2	4	no	—
175	Dic₅	—	2	2	no	—
176	C₂×Q₈	—	5	2	no	—
177	C₂×Q₈	C₂×C₅×D₅	1	1	yes	characteristic
178	C₂×D₄	—	5	1	no	—
179	C₂²×C₄	—	5	2	no	—
180	C₄⋊C₄	—	5	2	no	—
181	C₄⋊C₄	—	5	1	no	—
182	C₄⋊C₄	—	5	4	no	—
183	C₄⋊C₄	—	5	2	no	—
184	C₄⋊C₄	C₂×C₅×D₅	1	1	yes	characteristic
185	C₄⋊C₄	C₂×C₅×D₅	1	2	yes	—
186	C₂²⋊C₄	—	5	4	no	—
187	C₂²⋊C₄	—	5	2	no	—
188	C₂²⋊C₄	—	5	4	no	—
189	C₄²	—	5	4	no	—
190	C₄²	C₂×C₅×D₅	1	1	yes	characteristic
191	C₄²	C₂×C₅×D₅	1	2	yes	—
192	C₂×C₅	—	2	2	no	—
193	C₂×C₅	—	2	2	no	—
194	C₂×C₅	—	2	2	no	—
195	C₂×C₅	—	20	2	no	—
196	C₂×C₅	C₂×#709	1	1	yes	characteristic
197	C₂×C₅	C₂×#713	1	1	yes	characteristic
198	C₂×C₅	#4150	1	1	yes	characteristic
199	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
200	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
201	C₂×C₅	C₅×2₋¹⁺⁴	1	1	yes	characteristic
202	D₅	—	4	2	no	—
203	C₂³	—	5	2	no	—
204	Q₈	—	2	2	no	—
205	Q₈	—	10	4	no	—
206	D₄	—	10	2	no	—
207	C₂×C₄	—	5	2	no	—
208	C₂×C₄	—	5	4	no	—
209	C₂×C₄	—	10	4	no	—
210	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
211	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
213	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
214	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
215	C₅	—	2	2	no	—
216	C₅	#65918	1	1	yes	characteristic, direct product factor
217	C₅	C₅×#556	1	1	yes	characteristic, complemented
218	C₂²	—	10	2	no	—
219	C₂²	—	10	2	no	—
220	C₂²	—	10	2	no	—
221	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	2	2	no	—
224	C₄	—	2	1	no	—
225	C₄	—	10	2	no	—
226	C₄	—	10	4	no	—
227	C₄	C₅×#709	1	2	yes	—
228	C₄	C₅×#713	1	2	yes	—
229	C₂	—	20	2	no	—
230	C₂	C₂×C₅×#709	1	1	yes	characteristic
231	C₂	C₂×C₅×#713	1	1	yes	characteristic
232	C₂	C₅×#4150	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₄×C₅²×Q₈	C₂
socle	C₂²×C₅²	C₂⁴
prime radical	C₂×C₅×D₂₀	C₂²
hypercenter	C₄×C₅×Q₈	D₅
hypocenter	C₅	C₅×#556
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×Q₈

Complements

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

N	H ≅ G/N	type
1. C₄×C₅²×Q₈	C₂	semidirect
2. C₅×#4092	C₂	semidirect
3. C₅×#4044	C₂	semidirect
4. C₅×#4038	C₂	semidirect
7. C₅×#4015	C₂	semidirect
8. C₄×C₅×Dic₁₀	C₂	semidirect
26. #65918	C₅	direct (N × H)
45. C₄×C₅×Q₈	D₅	semidirect
46. C₄×C₅×Q₈	C₂×C₅	semidirect
49. #4092	C₂×C₅	semidirect
50. #4044	C₂×C₅	semidirect
51. #4038	C₂×C₅	semidirect
54. #4015	C₂×C₅	semidirect
55. C₄×Dic₁₀	C₂×C₅	semidirect
144. C₄×Q₈	C₅×D₅	semidirect
147. C₅²	#556	semidirect
216. C₅	#65918	direct (N × H)
217. C₅	C₅×#556	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅²×Q₈	2
C₅×#4092	2
C₅×#4044	2
C₅×#4038	2
C₅×#4021	2
C₄×C₅×D₂₀	2
C₅×#4015	2
C₄×C₅×Dic₁₀	2
C₅×#556	5
#65918	5

Subgroups — C5×#65918 #13106738

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65918 #13106738