Subgroups — #13106429

Order 1600.

518 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717266	C₂	1	1	yes	maximal, characteristic
2	C₅×Q₈×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4090	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717036	C₂	1	1	yes	maximal, characteristic, complemented
5	#11717009	C₂	1	2	yes	maximal
6	C₄×#6036	C₂	1	1	yes	maximal, characteristic
7	#11716990	C₂	1	1	yes	maximal, characteristic
8	#11716964	C₂	1	2	yes	maximal, complemented
9	Dic₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
10	#11716955	C₂	1	2	yes	maximal, complemented
11	#11716954	C₂	1	2	yes	maximal
12	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
14	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic
15	C₂×#6039	C₂²	1	2	yes	—
16	C₂×#6036	C₂²	1	1	yes	characteristic
17	C₂×#6036	C₂²	1	1	yes	characteristic
18	#87808	C₂²	1	2	yes	—
19	C₄⋊#1039	C₂²	1	1	yes	characteristic
20	#87806	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	2	yes	—
24	C₅×#683	C₂²	1	2	yes	—
25	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
26	C₄×C₅×Dic₅	C₂²	1	2	yes	—
27	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
28	#87780	C₂²	1	2	yes	—
29	#87780	C₂²	1	2	yes	—
30	#87780	C₂²	1	2	yes	—
31	#87779	C₂²	1	2	yes	—
32	Dic₅²	C₂²	1	2	yes	—
33	Dic₅²	C₂²	1	1	yes	characteristic
34	Dic₅²	C₂²	1	1	yes	characteristic
35	#66062	—	5	1	no	maximal
36	#65912	—	5	1	no	maximal
37	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
38	C₅²×Q₈	—	2	2	no	—
39	C₂×C₄×C₅²	C₂³	1	2	yes	—
40	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
41	C₂×#1039	C₂³	1	2	yes	—
42	C₂×#1039	C₂³	1	1	yes	characteristic
43	C₄×C₅⋊D₅	—	2	2	no	—
44	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
45	C₂×C₅×Dic₅	C₂³	1	2	yes	—
46	C₂×C₅×Dic₅	C₂³	1	2	yes	—
47	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
48	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
49	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
50	C₅×Dic₁₀	—	2	2	no	—
51	#6039	—	2	4	no	—
52	#6036	—	2	2	no	—
53	#6036	Q₈	1	4	yes	complemented
54	C₅×C₄⋊Q₈	—	5	1	no	—
55	C₄×C₅×Q₈	—	5	1	no	—
56	#4092	—	5	1	no	—
57	#4092	—	5	1	no	—
58	#4092	—	5	1	no	—
59	#4092	—	10	2	no	—
60	Q₈×Dic₅	—	5	1	no	—
61	Q₈×Dic₅	D₅	1	1	yes	characteristic, complemented
62	#4090	D₅	1	1	yes	characteristic, complemented
63	#4043	—	5	1	no	—
64	#4043	—	5	2	no	—
65	#4042	—	5	2	no	—
66	#4038	—	5	2	no	—
67	#4036	—	5	2	no	—
68	#4035	—	5	2	no	—
69	#4034	—	5	1	no	—
70	#4033	—	5	2	no	—
71	#4018	—	5	2	no	—
72	C₄²×D₅	—	5	1	no	—
73	C₄²×D₅	—	5	1	no	—
74	#4015	—	5	1	no	—
75	C₄×Dic₁₀	—	5	1	no	—
76	C₄×Dic₁₀	—	5	2	no	—
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	2	no	—
82	C₄×C₅²	#41	1	2	yes	—
83	#1039	—	2	2	no	—
84	#1039	—	2	1	no	—
85	C₅×Dic₅	—	2	1	no	—
86	C₅×Dic₅	—	2	2	no	—
87	C₅×Dic₅	—	2	2	no	—
88	C₅×Dic₅	C₂×Q₈	1	2	yes	—
89	C₅×Dic₅	C₂×Q₈	1	2	yes	—
90	C₅×Dic₅	#41	1	2	yes	—
91	C₂×C₅×Q₈	—	2	2	no	—
92	C₂×C₅×Q₈	—	5	1	no	—
93	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
94	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
95	C₂×C₄×D₅	—	5	1	no	—
96	C₂×C₄×D₅	—	5	1	no	—
97	C₂×C₄×D₅	—	5	1	no	—
98	C₂×C₄×D₅	—	5	1	no	—
99	C₂×C₄×D₅	—	5	1	no	—
100	C₂×C₄×D₅	—	5	1	no	—
101	C₂×C₄×D₅	—	10	2	no	—
102	C₂×Dic₁₀	—	5	2	no	—
103	C₂×Dic₁₀	—	5	2	no	—
104	C₂×Dic₁₀	C₂×D₅	1	1	yes	characteristic
105	C₅×C₄⋊C₄	—	5	1	no	—
106	C₅×C₄⋊C₄	—	5	2	no	—
107	C₅×C₄⋊C₄	—	5	2	no	—
108	C₅×C₄⋊C₄	—	5	2	no	—
109	C₄²×C₅	—	5	1	no	—
110	C₄²×C₅	—	5	2	no	—
111	C₄²×C₅	—	5	1	no	—
112	#685	—	5	2	no	—
113	#685	—	5	2	no	—
114	#685	—	5	2	no	—
115	#685	—	5	2	no	—
116	#685	—	10	4	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	—	10	2	no	—
123	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
124	C₄⋊Dic₅	C₂×D₅	1	2	yes	—
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	—	10	4	no	—
131	#683	C₂×D₅	1	2	yes	—
132	#683	C₂×D₅	1	2	yes	—
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	1	no	—
138	C₄×Dic₅	—	5	2	no	—
139	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
140	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
141	C₄×Dic₅	C₂×D₅	1	2	yes	—
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	C₂²×D₅	—	5	1	no	—
148	C₂²×D₅	—	5	1	no	—
149	C₂²×D₅	—	10	2	no	—
150	C₅×Q₈	—	2	2	no	—
151	C₅×Q₈	—	2	2	no	—
152	C₅×Q₈	—	4	4	no	—
153	C₅×Q₈	—	10	2	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	4	no	—
156	C₂×C₄×C₅	—	5	1	no	—
157	C₂×C₄×C₅	—	5	1	no	—
158	C₂×C₄×C₅	—	5	1	no	—
159	C₂×C₄×C₅	—	5	1	no	—
160	C₂×C₄×C₅	—	5	2	no	—
161	C₂×C₄×C₅	—	5	2	no	—
162	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
163	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
164	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
166	C₂×Dic₅	—	5	1	no	—
167	C₂×Dic₅	—	5	1	no	—
168	C₂×Dic₅	—	5	2	no	—
169	C₂×Dic₅	—	5	2	no	—
170	C₂×Dic₅	—	10	2	no	—
171	C₂×Dic₅	—	10	4	no	—
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	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
177	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
178	C₄×D₅	—	5	4	no	—
179	C₄×D₅	—	5	4	no	—
180	C₄×D₅	—	10	2	no	—
181	C₄×D₅	—	10	2	no	—
182	C₄×D₅	—	10	2	no	—
183	C₄×D₅	—	10	2	no	—
184	C₄×D₅	—	20	4	no	—
185	Dic₁₀	—	2	2	no	—
186	Dic₁₀	—	10	4	no	—
187	Dic₁₀	—	10	4	no	—
188	C₄⋊Q₈	—	5	1	no	—
189	C₄⋊Q₈	—	25	1	no	—
190	#125	—	25	2	no	—
191	C₂²⋊Q₈	—	25	2	no	—
192	C₂²⋊Q₈	—	25	1	no	—
193	C₂²⋊Q₈	—	25	1	no	—
194	C₄×Q₈	—	5	1	no	—
195	C₄×Q₈	—	25	2	no	—
196	C₄×Q₈	—	25	1	no	—
197	#117	—	25	2	no	—
198	C₂×C₄²	—	25	1	no	—
199	C₅²	#533	1	1	yes	characteristic, complemented, Sylow
200	C₂²×C₅	—	2	2	no	—
201	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
202	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
203	C₂×D₅	—	5	2	no	—
204	C₂×D₅	—	5	2	no	—
205	C₂×D₅	—	10	4	no	—
206	C₂×D₅	—	10	1	no	—
207	C₂×D₅	—	10	1	no	—
208	C₂×D₅	—	10	1	no	—
209	C₂×D₅	—	10	1	no	—
210	C₂×D₅	—	20	2	no	—
211	C₂×D₅	—	20	2	no	—
212	C₄×C₅	—	2	2	no	—
213	C₄×C₅	—	2	2	no	—
214	C₄×C₅	—	2	4	no	—
215	C₄×C₅	—	4	4	no	—
216	C₄×C₅	—	5	2	no	—
217	C₄×C₅	—	5	2	no	—
218	C₄×C₅	—	5	2	no	—
219	C₄×C₅	—	10	2	no	—
220	C₄×C₅	—	10	2	no	—
221	C₄×C₅	—	10	1	no	—
222	C₄×C₅	#709	1	2	yes	—
223	C₄×C₅	#711	1	2	yes	—
224	Dic₅	—	2	2	no	—
225	Dic₅	—	2	2	no	—
226	Dic₅	—	2	1	no	—
227	Dic₅	—	10	1	no	—
228	Dic₅	—	10	1	no	—
229	Dic₅	—	10	2	no	—
230	Dic₅	—	10	2	no	—
231	Dic₅	—	20	2	no	—
232	Dic₅	—	20	4	no	—
233	Dic₅	Q₈×D₅	1	2	yes	—
234	Dic₅	Q₈×D₅	1	2	yes	—
235	Dic₅	#713	1	2	yes	—
236	C₂×Q₈	—	5	1	no	—
237	C₂×Q₈	—	25	2	no	—
238	C₂×Q₈	D₅²	1	1	yes	characteristic
239	C₂²×C₄	—	25	1	no	—
240	C₂²×C₄	—	25	1	no	—
241	C₂²×C₄	—	25	1	no	—
242	C₄⋊C₄	—	5	2	no	—
243	C₄⋊C₄	—	5	2	no	—
244	C₄⋊C₄	—	5	1	no	—
245	C₄⋊C₄	—	5	2	no	—
246	C₄⋊C₄	—	25	2	no	—
247	C₄⋊C₄	—	25	2	no	—
248	C₄⋊C₄	—	25	2	no	—
249	C₄⋊C₄	—	25	2	no	—
250	C₄⋊C₄	—	25	1	no	—
251	C₂²⋊C₄	—	25	2	no	—
252	C₂²⋊C₄	—	25	2	no	—
253	C₄²	—	5	1	no	—
254	C₄²	—	5	1	no	—
255	C₄²	—	5	2	no	—
256	C₄²	—	25	1	no	—
257	C₄²	—	25	1	no	—
258	C₄²	—	25	2	no	—
259	C₂×C₅	—	2	2	no	—
260	C₂×C₅	—	2	2	no	—
261	C₂×C₅	—	2	2	no	—
262	C₂×C₅	C₂×#709	1	1	yes	characteristic
263	C₂×C₅	C₂×#711	1	1	yes	characteristic
264	C₂×C₅	C₂×Q₈×D₅	1	1	yes	characteristic
265	C₂×C₅	C₂×Q₈×D₅	1	1	yes	characteristic
266	C₂×C₅	C₂×#713	1	1	yes	characteristic
267	C₂×C₅	D₅×#41	1	1	yes	characteristic
268	D₅	—	10	2	no	—
269	D₅	—	10	2	no	—
270	D₅	—	20	4	no	—
271	C₂³	—	25	1	no	—
272	Q₈	—	2	2	no	—
273	Q₈	—	10	2	no	—
274	Q₈	—	50	4	no	—
275	C₂×C₄	—	5	1	no	—
276	C₂×C₄	—	5	1	no	—
277	C₂×C₄	—	5	1	no	—
278	C₂×C₄	—	5	1	no	—
279	C₂×C₄	—	5	2	no	—
280	C₂×C₄	—	5	2	no	—
281	C₂×C₄	—	25	4	no	—
282	C₂×C₄	—	25	1	no	—
283	C₂×C₄	—	25	2	no	—
284	C₂×C₄	—	50	2	no	—
285	C₂×C₄	—	50	2	no	—
286	C₂×C₄	C₂×D₅²	1	2	yes	—
287	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
288	C₅	—	2	2	no	—
289	C₅	#65912	1	1	yes	characteristic, complemented
290	C₅	#66062	1	1	yes	characteristic, complemented
291	C₂²	—	25	2	no	—
292	C₂²	—	50	1	no	—
293	C₂²	—	50	1	no	—
294	C₂²	C₂²×D₅²	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	5	2	no	—
297	C₄	—	5	2	no	—
298	C₄	—	5	2	no	—
299	C₄	—	10	1	no	—
300	C₄	—	10	2	no	—
301	C₄	—	10	2	no	—
302	C₄	—	50	2	no	—
303	C₄	—	50	1	no	—
304	C₄	#87879	1	2	yes	—
305	C₂	—	50	2	no	—
306	C₂	Q₈×D₅²	1	1	yes	characteristic
307	C₂	D₅×#713	1	1	yes	characteristic
308	C₂	C₂×#87879	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅²
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₂×C₅²×Q₈	C₂²
prime radical	C₂²×C₅⋊D₅	C₂³
hypercenter	C₂×Q₈	D₅²
hypocenter	C₅²	#533
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
2. C₅×Q₈×Dic₅	C₂	semidirect
3. C₅×#4090	C₂	semidirect
4. #11717036	C₂	semidirect
8. #11716964	C₂	semidirect
9. Dic₅×Dic₁₀	C₂	semidirect
10. #11716955	C₂	semidirect
53. #6036	Q₈	semidirect
61. Q₈×Dic₅	D₅	semidirect
62. #4090	D₅	semidirect
199. C₅²	#533	semidirect
289. C₅	#65912	semidirect
290. C₅	#66062	semidirect

Maximal subgroups

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

subgroup	index
#11717266	2
C₅×Q₈×Dic₅	2
C₅×#4090	2
#11717036	2
#11717009	2
C₄×#6036	2
#11716990	2
#11716964	2
Dic₅×Dic₁₀	2
#11716955	2
#11716954	2
#66062	5
#65912	5