Subgroups — #13105522

Order 1600.

512 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717208	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#4038	C₂	1	1	yes	maximal, characteristic
3	C₅×#4036	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717037	C₂	1	1	yes	maximal, characteristic, complemented
5	#11717015	C₂	1	2	yes	maximal
6	#11717014	C₂	1	2	yes	maximal
7	C₄×D₅×Dic₅	C₂	1	1	yes	maximal, characteristic
8	#11716988	C₂	1	1	yes	maximal, characteristic, complemented
9	#11716962	C₂	1	2	yes	maximal, complemented
10	#11716959	C₂	1	1	yes	maximal, characteristic
11	#11716955	C₂	1	2	yes	maximal, complemented
12	C₂×#6045	C₂²	1	2	yes	—
13	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
14	C₂×#6039	C₂²	1	2	yes	—
15	C₂×D₅×Dic₅	C₂²	1	1	yes	characteristic
16	C₂×D₅×Dic₅	C₂²	1	1	yes	characteristic
17	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₄⋊#1039	C₂²	1	1	yes	characteristic
19	#87806	C₂²	1	2	yes	—
20	C₄×#1039	C₂²	1	1	yes	characteristic
21	C₅×#685	C₂²	1	2	yes	—
22	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
23	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
24	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
25	C₅×#683	C₂²	1	2	yes	—
26	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
27	C₄×C₅×Dic₅	C₂²	1	2	yes	—
28	#87781	C₂²	1	2	yes	—
29	#87781	C₂²	1	2	yes	—
30	#87780	C₂²	1	2	yes	—
31	#87778	C₂²	1	2	yes	—
32	Dic₅²	C₂²	1	1	yes	characteristic
33	Dic₅²	C₂²	1	2	yes	—
34	Dic₅²	C₂²	1	1	yes	characteristic
35	#66022	—	5	1	no	maximal
36	#65851	—	5	1	no	maximal
37	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₅²	C₂³	1	2	yes	—
39	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
40	C₂×#1039	C₂³	1	2	yes	—
41	C₂×#1039	C₂³	1	1	yes	characteristic
42	C₂×#1039	C₂³	1	1	yes	characteristic
43	#6045	—	2	4	no	—
44	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
45	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
46	C₂×C₅×Dic₅	C₂³	1	2	yes	—
47	C₂×C₅×Dic₅	C₂³	1	2	yes	—
48	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
49	C₄×C₅×D₅	Q₈	1	4	yes	complemented
50	#6039	—	2	4	no	—
51	D₅×Dic₅	—	2	2	no	—
52	D₅×Dic₅	—	2	2	no	—
53	C₅×#125	—	5	1	no	—
54	C₅×#117	—	5	1	no	—
55	#4072	—	5	1	no	—
56	C₂×C₄×Dic₅	—	5	1	no	—
57	#4042	—	5	2	no	—
58	#4042	—	5	2	no	—
59	#4038	—	5	1	no	—
60	#4038	D₅	1	1	yes	characteristic
61	#4036	—	5	1	no	—
62	#4036	D₅	1	1	yes	characteristic, complemented
63	#4034	—	5	1	no	—
64	#4034	—	5	1	no	—
65	#4034	—	5	1	no	—
66	#4034	—	5	1	no	—
67	#4034	—	10	2	no	—
68	#4033	—	5	2	no	—
69	#4033	—	5	2	no	—
70	#4024	—	5	2	no	—
71	#4024	—	5	2	no	—
72	C₄²×D₅	—	5	1	no	—
73	#4016	—	5	1	no	—
74	C₄×Dic₁₀	—	5	2	no	—
75	C₄×Dic₁₀	—	5	2	no	—
76	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
77	C₂×C₅×D₅	—	2	1	no	—
78	C₂×C₅×D₅	—	2	1	no	—
79	C₂×C₅×D₅	C₂×Q₈	1	2	yes	—
80	C₄×C₅²	—	2	2	no	—
81	C₄×C₅²	C₂×Q₈	1	2	yes	—
82	#1039	—	2	2	no	—
83	#1039	#41	1	2	yes	—
84	#1039	#41	1	2	yes	—
85	C₅×Dic₅	—	2	1	no	—
86	C₅×Dic₅	—	2	1	no	—
87	C₅×Dic₅	—	2	2	no	—
88	C₅×Dic₅	—	2	2	no	—
89	C₅×Dic₅	C₂×Q₈	1	2	yes	—
90	C₂²×C₄×C₅	—	5	1	no	—
91	C₂²×Dic₅	—	5	1	no	—
92	C₂²×Dic₅	—	5	1	no	—
93	C₂×C₄×D₅	—	5	1	no	—
94	C₂×C₄×D₅	—	5	1	no	—
95	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
96	C₂×Dic₁₀	—	5	2	no	—
97	C₂×Dic₁₀	—	5	2	no	—
98	C₂×Dic₁₀	—	5	2	no	—
99	C₂×Dic₁₀	—	5	2	no	—
100	C₂×Dic₁₀	—	10	4	no	—
101	C₅×C₄⋊C₄	—	2	2	no	—
102	C₅×C₄⋊C₄	—	5	1	no	—
103	C₅×C₄⋊C₄	—	5	2	no	—
104	C₅×C₄⋊C₄	—	5	2	no	—
105	C₅×C₄⋊C₄	—	5	1	no	—
106	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
107	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
108	C₅×C₂²⋊C₄	—	5	2	no	—
109	C₄²×C₅	—	5	2	no	—
110	C₄²×C₅	—	5	1	no	—
111	C₂²⋊Dic₅	—	5	2	no	—
112	#685	—	5	2	no	—
113	#685	C₂×D₅	1	2	yes	—
114	C₄⋊Dic₅	—	5	1	no	—
115	C₄⋊Dic₅	—	5	1	no	—
116	C₄⋊Dic₅	—	5	2	no	—
117	C₄⋊Dic₅	—	10	2	no	—
118	C₄⋊Dic₅	C₂×D₅	1	2	yes	—
119	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
120	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
121	#683	—	5	2	no	—
122	#683	—	5	2	no	—
123	#683	—	5	2	no	—
124	#683	—	5	2	no	—
125	#683	—	5	2	no	—
126	#683	—	5	2	no	—
127	#683	—	5	2	no	—
128	#683	—	10	4	no	—
129	#683	C₂×D₅	1	2	yes	—
130	C₄×Dic₅	—	5	1	no	—
131	C₄×Dic₅	—	5	1	no	—
132	C₄×Dic₅	—	5	1	no	—
133	C₄×Dic₅	—	5	1	no	—
134	C₄×Dic₅	—	5	2	no	—
135	C₄×Dic₅	—	5	1	no	—
136	C₄×Dic₅	—	5	1	no	—
137	C₄×Dic₅	—	5	2	no	—
138	C₄×Dic₅	—	10	2	no	—
139	C₄×Dic₅	C₂×D₅	1	2	yes	—
140	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
141	#533	—	25	1	no	Sylow
142	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₅²	C₂×#41	1	1	yes	characteristic
144	C₂×C₅²	C₂×#41	1	1	yes	characteristic
145	C₅×D₅	—	2	2	no	—
146	C₂³×C₅	—	5	1	no	—
147	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	4	no	—
150	C₂×C₄×C₅	—	5	1	no	—
151	C₂×C₄×C₅	—	5	4	no	—
152	C₂×C₄×C₅	—	5	1	no	—
153	C₂×C₄×C₅	—	5	1	no	—
154	C₂×C₄×C₅	—	5	2	no	—
155	C₂×C₄×C₅	—	5	2	no	—
156	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
157	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
158	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
160	C₂×Dic₅	—	5	1	no	—
161	C₂×Dic₅	—	5	1	no	—
162	C₂×Dic₅	—	5	1	no	—
163	C₂×Dic₅	—	5	1	no	—
164	C₂×Dic₅	—	5	2	no	—
165	C₂×Dic₅	—	5	2	no	—
166	C₂×Dic₅	—	10	2	no	—
167	C₂×Dic₅	—	10	2	no	—
168	C₂×Dic₅	—	10	4	no	—
169	C₂×Dic₅	—	10	2	no	—
170	C₂×Dic₅	—	10	2	no	—
171	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
172	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
173	C₂×Dic₅	C₂²×D₅	1	2	yes	—
174	C₂×Dic₅	C₂²×D₅	1	2	yes	—
175	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
176	C₄×D₅	—	10	2	no	—
177	C₄×D₅	—	10	2	no	—
178	C₄×D₅	Dic₁₀	1	4	yes	complemented
179	Dic₁₀	—	10	4	no	—
180	Dic₁₀	—	10	4	no	—
181	Dic₁₀	—	10	4	no	—
182	Dic₁₀	—	10	4	no	—
183	Dic₁₀	—	20	8	no	—
184	C₄⋊Q₈	—	25	1	no	—
185	C₄⋊Q₈	—	25	1	no	—
186	#125	—	5	1	no	—
187	#125	—	25	1	no	—
188	C₂²⋊Q₈	—	25	2	no	—
189	C₂²⋊Q₈	—	25	2	no	—
190	C₄×Q₈	—	25	2	no	—
191	C₄×Q₈	—	25	2	no	—
192	#117	—	5	1	no	—
193	#117	—	25	1	no	—
194	C₂×C₄²	—	25	1	no	—
195	C₅²	#533	1	1	yes	characteristic, complemented, Sylow
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	—	5	2	no	—
198	C₂²×C₅	—	10	1	no	—
199	C₂²×C₅	—	10	1	no	—
200	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
201	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
202	C₂×D₅	—	2	1	no	—
203	C₂×D₅	—	2	1	no	—
204	C₂×D₅	C₂×Dic₁₀	1	2	yes	—
205	C₄×C₅	—	2	2	no	—
206	C₄×C₅	—	2	2	no	—
207	C₄×C₅	—	2	4	no	—
208	C₄×C₅	—	4	4	no	—
209	C₄×C₅	—	5	2	no	—
210	C₄×C₅	—	10	2	no	—
211	C₄×C₅	—	10	2	no	—
212	C₄×C₅	—	10	1	no	—
213	C₄×C₅	—	10	1	no	—
214	C₄×C₅	C₂×Dic₁₀	1	2	yes	—
215	C₄×C₅	Q₈×D₅	1	2	yes	—
216	Dic₅	—	2	2	no	—
217	Dic₅	—	2	2	no	—
218	Dic₅	—	2	1	no	—
219	Dic₅	—	2	1	no	—
220	Dic₅	—	5	2	no	—
221	Dic₅	—	5	2	no	—
222	Dic₅	—	5	2	no	—
223	Dic₅	—	5	2	no	—
224	Dic₅	—	10	4	no	—
225	Dic₅	—	10	4	no	—
226	Dic₅	—	10	2	no	—
227	Dic₅	—	10	2	no	—
228	Dic₅	—	20	4	no	—
229	Dic₅	C₂×Dic₁₀	1	2	yes	—
230	C₂×Q₈	—	25	2	no	—
231	C₂×Q₈	—	25	2	no	—
232	C₂²×C₄	—	5	1	no	—
233	C₂²×C₄	—	25	1	no	—
234	C₂²×C₄	—	25	1	no	—
235	C₄⋊C₄	—	5	1	no	—
236	C₄⋊C₄	—	5	1	no	—
237	C₄⋊C₄	—	5	2	no	—
238	C₄⋊C₄	—	5	2	no	—
239	C₄⋊C₄	—	25	2	no	—
240	C₄⋊C₄	—	25	2	no	—
241	C₄⋊C₄	—	25	2	no	—
242	C₄⋊C₄	—	25	1	no	—
243	C₄⋊C₄	—	25	2	no	—
244	C₄⋊C₄	D₅²	1	1	yes	characteristic
245	C₂²⋊C₄	—	5	2	no	—
246	C₂²⋊C₄	—	25	2	no	—
247	C₄²	—	5	1	no	—
248	C₄²	—	5	2	no	—
249	C₄²	—	25	1	no	—
250	C₄²	—	25	1	no	—
251	C₄²	—	25	2	no	—
252	C₄²	—	25	1	no	—
253	C₂×C₅	—	2	2	no	—
254	C₂×C₅	—	2	2	no	—
255	C₂×C₅	—	2	2	no	—
256	C₂×C₅	—	10	2	no	—
257	C₂×C₅	C₂²×Dic₁₀	1	1	yes	characteristic
258	C₂×C₅	C₂×Q₈×D₅	1	1	yes	characteristic
259	C₂×C₅	D₅×#41	1	1	yes	characteristic
260	C₂×C₅	D₅×#41	1	1	yes	characteristic
261	C₂×C₅	D₅×#41	1	1	yes	characteristic
262	C₂×C₅	D₅×#41	1	1	yes	characteristic
263	D₅	—	2	2	no	—
264	C₂³	—	5	1	no	—
265	Q₈	—	50	4	no	—
266	Q₈	—	50	4	no	—
267	C₂×C₄	—	5	1	no	—
268	C₂×C₄	—	5	4	no	—
269	C₂×C₄	—	5	1	no	—
270	C₂×C₄	—	5	1	no	—
271	C₂×C₄	—	5	2	no	—
272	C₂×C₄	—	5	2	no	—
273	C₂×C₄	—	25	1	no	—
274	C₂×C₄	—	25	1	no	—
275	C₂×C₄	—	25	2	no	—
276	C₂×C₄	—	50	2	no	—
277	C₂×C₄	—	50	2	no	—
278	C₂×C₄	C₂×D₅²	1	2	yes	—
279	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
280	C₅	—	2	2	no	—
281	C₅	#65851	1	1	yes	characteristic, complemented
282	C₅	#66022	1	1	yes	characteristic, complemented
283	C₂²	—	5	2	no	—
284	C₂²	—	10	1	no	—
285	C₂²	—	10	1	no	—
286	C₂²	C₂²×D₅²	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	5	2	no	—
289	C₄	—	10	1	no	—
290	C₄	—	10	1	no	—
291	C₄	—	10	2	no	—
292	C₄	—	10	2	no	—
293	C₄	—	25	2	no	—
294	C₄	—	25	2	no	—
295	C₄	—	50	2	no	—
296	C₄	D₅×Dic₁₀	1	2	yes	—
297	C₂	—	10	2	no	—
298	C₂	C₂×D₅×Dic₁₀	1	1	yes	characteristic
299	C₂	#11717477	1	1	yes	characteristic
300	C₂	#11717484	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. #11717208	C₂	semidirect
3. C₅×#4036	C₂	semidirect
4. #11717037	C₂	semidirect
8. #11716988	C₂	semidirect
9. #11716962	C₂	semidirect
11. #11716955	C₂	semidirect
49. C₄×C₅×D₅	Q₈	semidirect
62. #4036	D₅	semidirect
178. C₄×D₅	Dic₁₀	semidirect
195. C₅²	#533	semidirect
281. C₅	#65851	semidirect
282. C₅	#66022	semidirect

Maximal subgroups

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

subgroup	index
#11717208	2
C₅×#4038	2
C₅×#4036	2
#11717037	2
#11717015	2
#11717014	2
C₄×D₅×Dic₅	2
#11716988	2
#11716962	2
#11716959	2
#11716955	2
#66022	5
#65851	5