Subgroups — #13101463

Order 1600.

432 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#1039	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₂×#87780	C₂	1	2	yes	maximal
5	#11716620	C₂	1	2	yes	maximal
6	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
7	C₂²×#1039	C₂²	1	2	yes	—
8	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
9	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
10	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
11	C₄⋊#1039	—	2	2	no	—
12	C₅×C₄⋊Dic₅	C₄	1	4	yes	complemented
13	C₄×C₅×Dic₅	—	2	2	no	—
14	#87780	—	2	4	no	—
15	#65274	—	5	1	no	maximal
16	#65217	—	5	1	no	maximal
17	C₂³×C₅²	C₂³	1	1	yes	characteristic
18	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₅²	D₄	1	2	yes	—
20	C₂×C₄×C₅²	Q₈	1	2	yes	—
21	C₂×#1039	—	2	2	no	—
22	C₂×#1039	—	2	2	no	—
23	C₂×#1039	—	2	2	no	—
24	C₂×C₅×Dic₅	—	2	2	no	—
25	C₂×C₅×Dic₅	—	2	1	no	—
26	C₂×C₅×Dic₅	—	2	1	no	—
27	C₂×C₅×Dic₅	—	2	2	no	—
28	C₂×C₅×Dic₅	—	2	1	no	—
29	C₂×C₅×Dic₅	—	2	1	no	—
30	C₂×C₅×Dic₅	C₂×C₄	1	4	yes	—
31	C₂×C₅×Dic₅	D₄	1	2	yes	—
32	C₂×C₅×Dic₅	Q₈	1	2	yes	—
33	C₂×C₅×C₄⋊C₄	—	5	1	no	—
34	C₂×C₄²×C₅	—	5	1	no	—
35	C₂×C₄⋊Dic₅	—	5	1	no	—
36	C₂×C₄⋊Dic₅	—	5	1	no	—
37	C₂×C₄⋊Dic₅	—	5	2	no	—
38	C₂×C₄⋊Dic₅	—	10	2	no	—
39	C₂×C₄⋊Dic₅	D₅	1	1	yes	characteristic
40	C₂×#683	—	5	2	no	—
41	C₂×C₄×Dic₅	D₅	1	1	yes	characteristic
42	#3963	—	5	2	no	—
43	#3963	—	5	2	no	—
44	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₅²	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₅²	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₅²	#41	1	1	yes	characteristic
50	C₂²×C₅²	#41	1	1	yes	characteristic
51	C₄×C₅²	C₄⋊C₄	1	4	yes	complemented
52	#1039	—	4	2	no	—
53	C₅×Dic₅	—	2	4	no	—
54	C₅×Dic₅	—	2	2	no	—
55	C₅×Dic₅	—	2	2	no	—
56	C₂²×C₄×C₅	—	2	2	no	—
57	C₂²×C₄×C₅	—	5	1	no	—
58	C₂²×C₄×C₅	—	5	1	no	—
59	C₂²×C₄×C₅	—	5	2	no	—
60	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
62	C₂²×Dic₅	—	5	2	no	—
63	C₂²×Dic₅	—	5	2	no	—
64	C₂²×Dic₅	—	10	4	no	—
65	C₂²×Dic₅	C₂×D₅	1	1	yes	characteristic
66	C₂²×Dic₅	C₂×D₅	1	1	yes	characteristic
67	C₂²×Dic₅	C₂×D₅	1	2	yes	—
68	C₅×C₄⋊C₄	—	5	4	no	—
69	C₄²×C₅	—	10	2	no	—
70	C₄⋊Dic₅	—	10	8	no	—
71	C₄⋊Dic₅	—	10	2	no	—
72	C₄⋊Dic₅	—	10	2	no	—
73	C₄⋊Dic₅	—	10	4	no	—
74	C₄⋊Dic₅	Dic₅	1	4	yes	complemented
75	#683	—	10	4	no	—
76	C₄×Dic₅	—	2	2	no	—
77	#389	—	25	1	no	Sylow
78	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₅²	C₄×D₄	1	1	yes	characteristic
80	C₂×C₅²	C₄×Q₈	1	1	yes	characteristic
81	C₂×C₅²	#121	1	1	yes	characteristic
82	C₂×C₅²	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂×C₅²	#125	1	1	yes	characteristic
84	C₂×C₅²	C₄⋊Q₈	1	1	yes	characteristic
85	C₂³×C₅	—	2	2	no	—
86	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
87	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
88	C₂×C₄×C₅	—	2	4	no	—
89	C₂×C₄×C₅	—	2	4	no	—
90	C₂×C₄×C₅	—	2	4	no	—
91	C₂×C₄×C₅	—	5	4	no	—
92	C₂×C₄×C₅	—	5	2	no	—
93	C₂×C₄×C₅	—	5	2	no	—
94	C₂×C₄×C₅	—	10	2	no	—
95	C₂×C₄×C₅	—	10	1	no	—
96	C₂×C₄×C₅	—	10	1	no	—
97	C₂×C₄×C₅	—	10	2	no	—
98	C₂×C₄×C₅	—	10	1	no	—
99	C₂×C₄×C₅	—	10	1	no	—
100	C₂×C₄×C₅	Dic₁₀	1	2	yes	—
101	C₂×C₄×C₅	Dic₁₀	1	2	yes	—
102	C₂×C₄×C₅	C₄×D₅	1	2	yes	—
103	C₂×C₄×C₅	D₂₀	1	2	yes	—
104	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	—
105	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
106	C₂×Dic₅	—	2	1	no	—
107	C₂×Dic₅	—	2	1	no	—
108	C₂×Dic₅	—	2	2	no	—
109	C₂×Dic₅	—	2	2	no	—
110	C₂×Dic₅	—	2	1	no	—
111	C₂×Dic₅	—	2	1	no	—
112	C₂×Dic₅	—	10	8	no	—
113	C₂×Dic₅	—	10	2	no	—
114	C₂×Dic₅	—	10	2	no	—
115	C₂×Dic₅	—	10	2	no	—
116	C₂×Dic₅	—	10	2	no	—
117	C₂×Dic₅	—	10	2	no	—
118	C₂×Dic₅	—	10	2	no	—
119	C₂×Dic₅	—	20	4	no	—
120	C₂×Dic₅	—	20	4	no	—
121	C₂×Dic₅	Dic₁₀	1	2	yes	—
122	C₂×Dic₅	D₂₀	1	2	yes	—
123	C₂×Dic₅	C₂×Dic₅	1	4	yes	—
124	C₂×C₄⋊C₄	—	5	1	no	—
125	C₂×C₄⋊C₄	—	25	2	no	—
126	C₂×C₄⋊C₄	—	25	1	no	—
127	C₂×C₄²	—	5	1	no	—
128	#95	—	25	2	no	—
129	C₅²	#389	1	1	yes	characteristic, complemented, Sylow
130	C₂²×C₅	—	2	2	no	—
131	C₂²×C₅	—	2	2	no	—
132	C₂²×C₅	—	2	2	no	—
133	C₂²×C₅	—	2	2	no	—
134	C₂²×C₅	—	2	2	no	—
135	C₂²×C₅	—	2	2	no	—
136	C₂²×C₅	—	2	2	no	—
137	C₂²×C₅	C₂×Dic₁₀	1	1	yes	characteristic
138	C₂²×C₅	C₂×Dic₁₀	1	1	yes	characteristic
139	C₂²×C₅	C₂×Dic₁₀	1	1	yes	characteristic
140	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
141	C₂²×C₅	C₂×D₂₀	1	1	yes	characteristic
142	C₂²×C₅	C₂×D₂₀	1	1	yes	characteristic
143	C₂²×C₅	#709	1	1	yes	characteristic
144	C₂²×C₅	#709	1	1	yes	characteristic
145	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
146	C₂²×C₅	#711	1	1	yes	characteristic
147	C₂²×C₅	Q₈×D₅	1	1	yes	characteristic
148	C₂²×C₅	#713	1	1	yes	characteristic
149	C₂²×C₅	C₂²×Dic₅	1	1	yes	characteristic
150	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
151	C₄×C₅	—	2	8	no	—
152	C₄×C₅	—	10	4	no	—
153	C₄×C₅	—	10	2	no	—
154	C₄×C₅	—	10	2	no	—
155	C₄×C₅	#683	1	4	yes	complemented
156	C₄×C₅	C₄⋊Dic₅	1	4	yes	complemented
157	Dic₅	—	2	2	no	—
158	Dic₅	—	2	2	no	—
159	Dic₅	—	2	4	no	—
160	Dic₅	—	20	8	no	—
161	Dic₅	—	20	2	no	—
162	Dic₅	—	20	2	no	—
163	C₂²×C₄	—	5	1	no	—
164	C₂²×C₄	—	5	2	no	—
165	C₂²×C₄	—	5	1	no	—
166	C₂²×C₄	—	25	2	no	—
167	C₂²×C₄	D₅²	1	1	yes	characteristic
168	C₄⋊C₄	—	5	4	no	—
169	C₄⋊C₄	—	50	4	no	—
170	C₄⋊C₄	—	50	2	no	—
171	C₄²	—	10	2	no	—
172	C₂×C₅	—	2	2	no	—
173	C₂×C₅	—	2	2	no	—
174	C₂×C₅	—	2	2	no	—
175	C₂×C₅	—	2	2	no	—
176	C₂×C₅	—	2	2	no	—
177	C₂×C₅	—	2	2	no	—
178	C₂×C₅	—	2	2	no	—
179	C₂×C₅	C₄×Dic₁₀	1	1	yes	characteristic
180	C₂×C₅	#4015	1	1	yes	characteristic
181	C₂×C₅	#4016	1	1	yes	characteristic
182	C₂×C₅	C₄×D₂₀	1	1	yes	characteristic
183	C₂×C₅	#4034	1	1	yes	characteristic
184	C₂×C₅	#4036	1	1	yes	characteristic
185	C₂×C₅	#4041	1	1	yes	characteristic
186	C₂×C₅	#4043	1	1	yes	characteristic
187	C₂×C₅	C₂×#683	1	1	yes	characteristic
188	C₂×C₅	#4070	1	1	yes	characteristic
189	C₂×C₅	C₂×C₄⋊Dic₅	1	1	yes	characteristic
190	C₂×C₅	#4076	1	1	yes	characteristic
191	C₂×C₅	D₄×Dic₅	1	1	yes	characteristic
192	C₂×C₅	Q₈×Dic₅	1	1	yes	characteristic
193	C₂³	C₂×D₅²	1	1	yes	characteristic
194	C₂×C₄	—	5	2	no	—
195	C₂×C₄	—	5	2	no	—
196	C₂×C₄	—	5	4	no	—
197	C₂×C₄	—	10	1	no	—
198	C₂×C₄	—	10	2	no	—
199	C₂×C₄	—	10	1	no	—
200	C₂×C₄	—	10	2	no	—
201	C₂×C₄	—	10	1	no	—
202	C₂×C₄	—	10	1	no	—
203	C₂×C₄	—	50	2	no	—
204	C₂×C₄	—	50	2	no	—
205	C₂×C₄	—	50	2	no	—
206	C₂×C₄	D₅×Dic₅	1	2	yes	—
207	C₂×C₄	#6038	1	2	yes	—
208	C₂×C₄	#6039	1	2	yes	—
209	C₅	—	2	2	no	—
210	C₅	#65217	1	1	yes	characteristic, complemented
211	C₅	#65274	1	1	yes	characteristic, complemented
212	C₂²	D₅×Dic₁₀	1	1	yes	characteristic
213	C₂²	#87870	1	1	yes	characteristic
214	C₂²	#87874	1	1	yes	characteristic
215	C₂²	D₅×D₂₀	1	1	yes	characteristic
216	C₂²	C₂×D₅×Dic₅	1	1	yes	characteristic
217	C₂²	C₂×#6038	1	1	yes	characteristic
218	C₂²	C₂×#6039	1	1	yes	characteristic
219	C₄	—	10	2	no	—
220	C₄	—	10	4	no	—
221	C₄	—	10	2	no	—
222	C₄	—	100	2	no	—
223	C₄	#87780	1	4	yes	complemented
224	C₂	Dic₅×Dic₁₀	1	1	yes	characteristic
225	C₂	#11716975	1	1	yes	characteristic
226	C₂	#11716988	1	1	yes	characteristic
227	C₂	Dic₅×D₂₀	1	1	yes	characteristic
228	C₂	#11717027	1	1	yes	characteristic
229	C₂	#11717036	1	1	yes	characteristic
230	C₂	C₂×#87780	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂×D₅²
commutator	C₂²×C₅²	C₂²×C₄
Fitting	C₂²×C₄×C₅²	C₂²
socle, prime radical, cosocle	C₂³×C₅²	C₂³
hypercenter	C₂²×C₄	D₅²
hypocenter	C₅²	#389
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
12. C₅×C₄⋊Dic₅	C₄	semidirect
51. C₄×C₅²	C₄⋊C₄	semidirect
74. C₄⋊Dic₅	Dic₅	semidirect
129. C₅²	#389	semidirect
155. C₄×C₅	#683	semidirect
156. C₄×C₅	C₄⋊Dic₅	semidirect
210. C₅	#65217	semidirect
211. C₅	#65274	semidirect
223. C₄	#87780	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#1039	2
C₂×C₅×C₄⋊Dic₅	2
C₂×C₄×C₅×Dic₅	2
C₂×#87780	2
#11716620	2
#65274	5
#65217	5