Subgroups — #13105072

Order 1600.

514 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6045	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄²×C₅×D₅	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717036	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#6039	C₂	1	2	yes	maximal, complemented
6	#11716986	C₂	1	2	yes	maximal, complemented
7	#11716983	C₂	1	1	yes	maximal, characteristic, complemented
8	#11716975	C₂	1	1	yes	maximal, characteristic
9	#11716974	C₂	1	1	yes	maximal, characteristic
10	#11716973	C₂	1	2	yes	maximal
11	#11716969	C₂	1	2	yes	maximal
12	C₂×#6045	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
16	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic
17	C₂×#6039	C₂²	1	2	yes	—
18	C₄²×C₅²	C₂²	1	1	yes	characteristic
19	C₄⋊#1039	C₂²	1	1	yes	characteristic
20	#87806	C₂²	1	2	yes	—
21	C₄×#1039	C₂²	1	2	yes	—
22	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
23	C₅×#683	C₂²	1	2	yes	—
24	C₄×C₅×Dic₅	C₂²	1	2	yes	—
25	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
26	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
27	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
28	#87781	C₂²	1	2	yes	—
29	#87780	C₂²	1	2	yes	—
30	#87780	C₂²	1	2	yes	—
31	#87780	C₂²	1	2	yes	—
32	#87780	C₂²	1	2	yes	—
33	#87778	C₂²	1	2	yes	—
34	#87778	C₂²	1	2	yes	—
35	#65912	—	5	1	no	maximal
36	#65804	—	5	1	no	maximal
37	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
41	C₂×#1039	C₂³	1	2	yes	—
42	C₂×#1039	C₂³	1	2	yes	—
43	#6045	—	2	2	no	—
44	C₂×C₅×Dic₅	C₂³	1	2	yes	—
45	C₂×C₅×Dic₅	C₂³	1	2	yes	—
46	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
47	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
48	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
49	C₄×C₅×D₅	—	2	2	no	—
50	C₄×C₅×D₅	—	2	2	no	—
51	C₄×C₅×D₅	Q₈	1	4	yes	complemented
52	C₅×Dic₁₀	—	2	2	no	—
53	#6039	—	2	4	no	—
54	C₄×C₅×Q₈	—	5	1	no	—
55	C₂×C₄²×C₅	—	5	1	no	—
56	#4092	—	5	1	no	—
57	#4090	—	5	1	no	—
58	#4072	—	5	2	no	—
59	#4070	—	5	1	no	—
60	#4070	—	5	1	no	—
61	#4070	—	5	2	no	—
62	#4043	—	5	1	no	—
63	#4042	—	5	2	no	—
64	#4035	—	5	2	no	—
65	#4034	—	5	1	no	—
66	#4018	—	5	2	no	—
67	C₄²×D₅	D₅	1	1	yes	characteristic
68	#4016	—	5	2	no	—
69	#4015	—	5	1	no	—
70	#4015	—	5	1	no	—
71	C₄×Dic₁₀	—	5	1	no	—
72	C₄×Dic₁₀	—	5	1	no	—
73	C₄×Dic₁₀	—	5	2	no	—
74	C₄×Dic₁₀	—	5	2	no	—
75	C₄×Dic₁₀	—	10	2	no	—
76	C₄×Dic₁₀	D₅	1	1	yes	characteristic, complemented
77	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
78	C₂×C₅×D₅	—	2	1	no	—
79	C₂×C₅×D₅	—	2	1	no	—
80	C₂×C₅×D₅	C₂×Q₈	1	2	yes	—
81	C₄×C₅²	—	2	1	no	—
82	C₄×C₅²	C₂×Q₈	1	2	yes	—
83	C₄×C₅²	#41	1	2	yes	—
84	#1039	—	2	2	no	—
85	#1039	—	2	2	no	—
86	C₅×Dic₅	—	2	2	no	—
87	C₅×Dic₅	—	2	2	no	—
88	C₅×Dic₅	—	2	1	no	—
89	C₅×Dic₅	C₂×Q₈	1	2	yes	—
90	C₅×Dic₅	#41	1	2	yes	—
91	C₂×C₅×Q₈	—	5	1	no	—
92	C₂²×C₄×C₅	—	5	1	no	—
93	C₂²×C₄×C₅	—	5	1	no	—
94	C₂²×C₄×C₅	—	5	1	no	—
95	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
96	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
97	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
98	C₂×Dic₁₀	—	5	1	no	—
99	C₂×Dic₁₀	—	5	1	no	—
100	C₂×Dic₁₀	—	5	2	no	—
101	C₂×Dic₁₀	—	5	2	no	—
102	C₂×Dic₁₀	—	10	2	no	—
103	C₂×Dic₁₀	C₂×D₅	1	1	yes	characteristic
104	C₅×C₄⋊C₄	—	5	1	no	—
105	C₅×C₄⋊C₄	—	5	2	no	—
106	C₄²×C₅	—	2	2	no	—
107	C₄²×C₅	—	5	1	no	—
108	C₄²×C₅	—	5	1	no	—
109	C₄²×C₅	—	5	1	no	—
110	C₄²×C₅	—	5	2	no	—
111	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
112	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
113	C₂²⋊Dic₅	—	5	2	no	—
114	C₂²⋊Dic₅	—	5	2	no	—
115	#685	—	5	2	no	—
116	#685	—	5	2	no	—
117	C₄⋊Dic₅	—	5	1	no	—
118	C₄⋊Dic₅	—	5	1	no	—
119	C₄⋊Dic₅	—	5	2	no	—
120	C₄⋊Dic₅	—	5	2	no	—
121	C₄⋊Dic₅	—	5	2	no	—
122	C₄⋊Dic₅	—	5	2	no	—
123	C₄⋊Dic₅	—	10	2	no	—
124	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
125	#683	—	5	2	no	—
126	#683	—	5	2	no	—
127	#683	—	5	2	no	—
128	#683	—	5	2	no	—
129	#683	—	5	2	no	—
130	#683	—	5	2	no	—
131	#683	—	5	2	no	—
132	#683	—	5	2	no	—
133	#683	—	10	4	no	—
134	#683	C₂×D₅	1	2	yes	—
135	C₄×Dic₅	—	5	2	no	—
136	C₄×Dic₅	—	5	2	no	—
137	C₄×Dic₅	—	10	4	no	—
138	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
139	C₄×Dic₅	C₂×D₅	1	2	yes	—
140	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
141	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
142	#533	—	25	1	no	Sylow
143	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
144	C₂×C₅²	C₂×#41	1	1	yes	characteristic
145	C₂×C₅²	C₂×#41	1	1	yes	characteristic
146	C₅×D₅	—	2	2	no	—
147	C₂³×C₅	—	5	1	no	—
148	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
149	C₅×Q₈	—	10	2	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	5	1	no	—
154	C₂×C₄×C₅	—	5	1	no	—
155	C₂×C₄×C₅	—	5	4	no	—
156	C₂×C₄×C₅	—	5	1	no	—
157	C₂×C₄×C₅	—	5	2	no	—
158	C₂×C₄×C₅	—	5	2	no	—
159	C₂×C₄×C₅	—	10	2	no	—
160	C₂×C₄×C₅	—	10	2	no	—
161	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
167	C₂×Dic₅	—	5	2	no	—
168	C₂×Dic₅	—	5	2	no	—
169	C₂×Dic₅	—	5	2	no	—
170	C₂×Dic₅	—	5	2	no	—
171	C₂×Dic₅	—	10	4	no	—
172	C₂×Dic₅	—	10	4	no	—
173	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
174	C₂×Dic₅	C₂²×D₅	1	2	yes	—
175	C₂×Dic₅	C₂²×D₅	1	2	yes	—
176	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
177	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
178	C₄×D₅	—	2	2	no	—
179	C₄×D₅	—	2	2	no	—
180	C₄×D₅	Dic₁₀	1	4	yes	complemented
181	Dic₁₀	—	2	2	no	—
182	Dic₁₀	—	10	4	no	—
183	Dic₁₀	—	10	8	no	—
184	Dic₁₀	—	10	2	no	—
185	Dic₁₀	—	10	2	no	—
186	Dic₁₀	—	10	4	no	—
187	C₄⋊Q₈	—	25	1	no	—
188	C₄⋊Q₈	—	25	1	no	—
189	#125	—	25	2	no	—
190	C₂²⋊Q₈	—	25	2	no	—
191	C₂²⋊Q₈	—	25	1	no	—
192	C₂²⋊Q₈	—	25	1	no	—
193	C₄×Q₈	—	5	1	no	—
194	C₄×Q₈	—	25	2	no	—
195	C₄×Q₈	—	25	1	no	—
196	#117	—	25	2	no	—
197	C₂×C₄²	—	5	1	no	—
198	C₅²	#533	1	1	yes	characteristic, complemented, Sylow
199	C₂²×C₅	—	2	2	no	—
200	C₂²×C₅	—	5	2	no	—
201	C₂²×C₅	—	10	1	no	—
202	C₂²×C₅	—	10	1	no	—
203	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
204	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
205	C₂×D₅	—	2	1	no	—
206	C₂×D₅	—	2	1	no	—
207	C₂×D₅	C₂×Dic₁₀	1	2	yes	—
208	C₄×C₅	—	2	1	no	—
209	C₄×C₅	—	2	1	no	—
210	C₄×C₅	—	2	4	no	—
211	C₄×C₅	—	2	4	no	—
212	C₄×C₅	—	4	2	no	—
213	C₄×C₅	—	5	2	no	—
214	C₄×C₅	—	5	2	no	—
215	C₄×C₅	—	10	2	no	—
216	C₄×C₅	—	10	1	no	—
217	C₄×C₅	—	10	2	no	—
218	C₄×C₅	C₂×Dic₁₀	1	2	yes	—
219	C₄×C₅	#709	1	2	yes	—
220	C₄×C₅	#709	1	2	yes	—
221	C₄×C₅	Q₈×D₅	1	2	yes	—
222	Dic₅	—	2	2	no	—
223	Dic₅	—	2	1	no	—
224	Dic₅	—	2	2	no	—
225	Dic₅	—	10	8	no	—
226	Dic₅	—	10	2	no	—
227	Dic₅	—	10	2	no	—
228	Dic₅	—	10	2	no	—
229	Dic₅	—	10	2	no	—
230	Dic₅	—	20	4	no	—
231	Dic₅	C₂×Dic₁₀	1	2	yes	—
232	Dic₅	#709	1	2	yes	—
233	C₂×Q₈	—	5	1	no	—
234	C₂×Q₈	—	25	2	no	—
235	C₂×Q₈	—	25	1	no	—
236	C₂²×C₄	—	5	1	no	—
237	C₂²×C₄	—	5	1	no	—
238	C₂²×C₄	—	5	1	no	—
239	C₄⋊C₄	—	5	2	no	—
240	C₄⋊C₄	—	5	1	no	—
241	C₄⋊C₄	—	25	2	no	—
242	C₄⋊C₄	—	25	2	no	—
243	C₄⋊C₄	—	25	2	no	—
244	C₄⋊C₄	—	25	2	no	—
245	C₄⋊C₄	—	25	2	no	—
246	C₄⋊C₄	—	25	2	no	—
247	C₄⋊C₄	—	25	1	no	—
248	C₂²⋊C₄	—	25	2	no	—
249	C₂²⋊C₄	—	25	2	no	—
250	C₄²	—	5	1	no	—
251	C₄²	—	5	1	no	—
252	C₄²	—	5	1	no	—
253	C₄²	—	5	2	no	—
254	C₄²	—	25	2	no	—
255	C₄²	D₅²	1	1	yes	characteristic
256	C₂×C₅	—	2	2	no	—
257	C₂×C₅	—	2	2	no	—
258	C₂×C₅	—	2	2	no	—
259	C₂×C₅	—	10	2	no	—
260	C₂×C₅	C₂²×Dic₁₀	1	1	yes	characteristic
261	C₂×C₅	C₂×#709	1	1	yes	characteristic
262	C₂×C₅	C₂×#709	1	1	yes	characteristic
263	C₂×C₅	C₂×#709	1	1	yes	characteristic
264	C₂×C₅	C₂×Q₈×D₅	1	1	yes	characteristic
265	C₂×C₅	D₅×#41	1	1	yes	characteristic
266	D₅	—	2	2	no	—
267	C₂³	—	5	1	no	—
268	Q₈	—	10	2	no	—
269	Q₈	—	50	4	no	—
270	Q₈	—	50	2	no	—
271	C₂×C₄	—	5	1	no	—
272	C₂×C₄	—	5	1	no	—
273	C₂×C₄	—	5	4	no	—
274	C₂×C₄	—	5	1	no	—
275	C₂×C₄	—	5	2	no	—
276	C₂×C₄	—	5	2	no	—
277	C₂×C₄	—	10	2	no	—
278	C₂×C₄	—	10	2	no	—
279	C₂×C₄	—	25	2	no	—
280	C₂×C₄	—	25	2	no	—
281	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
282	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
283	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
284	C₅	—	2	2	no	—
285	C₅	#65804	1	1	yes	characteristic, complemented
286	C₅	#65912	1	1	yes	characteristic, complemented
287	C₂²	—	5	2	no	—
288	C₂²	—	10	1	no	—
289	C₂²	—	10	1	no	—
290	C₂²	C₂²×D₅²	1	1	yes	characteristic
291	C₄	—	2	1	no	—
292	C₄	—	5	2	no	—
293	C₄	—	5	2	no	—
294	C₄	—	10	1	no	—
295	C₄	—	10	2	no	—
296	C₄	—	10	2	no	—
297	C₄	—	50	2	no	—
298	C₄	—	50	2	no	—
299	C₄	D₅×Dic₁₀	1	2	yes	—
300	C₄	#87873	1	2	yes	—
301	C₂	—	10	2	no	—
302	C₂	C₂×D₅×Dic₁₀	1	1	yes	characteristic
303	C₂	C₂×#87873	1	1	yes	characteristic
304	C₂	D₅×#709	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅²
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Frattini	C₂²	C₂²×D₅²
Fitting	C₄²×C₅²	C₂²
prime radical	C₂²×C₅×D₅	C₂³
hypercenter	C₄²	D₅²
hypocenter	C₅²	#533
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
1. C₄×#6045	C₂	semidirect
3. C₄×C₅×Dic₁₀	C₂	semidirect
4. #11717036	C₂	semidirect
5. C₄×#6039	C₂	semidirect
6. #11716986	C₂	semidirect
7. #11716983	C₂	semidirect
51. C₄×C₅×D₅	Q₈	semidirect
76. C₄×Dic₁₀	D₅	semidirect
180. C₄×D₅	Dic₁₀	semidirect
198. C₅²	#533	semidirect
285. C₅	#65804	semidirect
286. C₅	#65912	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6045	2
C₄²×C₅×D₅	2
C₄×C₅×Dic₁₀	2
#11717036	2
C₄×#6039	2
#11716986	2
#11716983	2
#11716975	2
#11716974	2
#11716973	2
#11716969	2
#65912	5
#65804	5