Subgroups — D₅×#1376 #12014151

Order 1280.

670 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	C₂×D₅×M₃₂	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	C₅×#1376	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598301	C₂	1	2	yes	maximal, complemented
5	#10598149	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₅	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₅	C₄	1	1	yes	characteristic
9	C₂×C₅×M₃₂	C₂²	1	2	yes	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
13	C₂×#3944	C₂²	1	2	yes	—
14	C₂×C₄×#168	C₄	1	1	yes	characteristic
15	D₅×M₃₂	—	2	4	no	—
16	D₅×M₃₂	C₄	1	8	yes	complemented
17	C₂×#3930	—	2	2	no	—
18	C₂×D₅×C₁₆	—	2	2	no	—
19	C₄×C₈×D₅	—	2	4	no	—
20	C₂×#1376	—	5	1	no	maximal, Sylow
21	C₂²×C₄×D₅	—	2	1	no	—
22	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₅	—	2	1	no	—
27	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×#168	C₂×C₄	1	1	yes	characteristic
29	C₂²×#168	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×D₅	—	2	2	no	—
31	C₂×C₈×D₅	—	2	2	no	—
32	C₂×C₈×D₅	C₈	1	2	yes	—
33	C₂×C₈×D₅	C₈	1	2	yes	—
34	C₂×C₈×D₅	C₂×C₄	1	4	yes	—
35	C₂×C₈×D₅	D₄	1	1	yes	characteristic
36	C₂×C₈×D₅	D₄	1	1	yes	characteristic
37	C₂×C₈×D₅	D₄	1	4	yes	—
38	C₂×C₈×D₅	Q₈	1	1	yes	characteristic
39	C₂×C₈×D₅	Q₈	1	1	yes	characteristic
40	C₄²×D₅	—	2	2	no	—
41	C₄²×D₅	C₈	1	4	yes	—
42	C₅×M₃₂	—	2	2	no	—
43	C₅×M₃₂	C₂×C₄	1	4	yes	complemented
44	C₂×C₅×C₁₆	—	2	2	no	—
45	C₄×C₅×C₈	—	2	2	no	—
46	C₈×Dic₅	—	2	2	no	—
47	C₈×Dic₅	—	2	2	no	—
48	#3944	—	2	2	no	—
49	#3944	C₂×C₄	1	4	yes	complemented
50	C₂×#672	—	2	2	no	—
51	C₄×#168	—	2	2	no	—
52	#3930	—	2	8	no	—
53	D₅×C₁₆	—	2	8	no	—
54	C₂²×M₃₂	—	5	2	no	—
55	C₂²×C₄×C₈	—	5	1	no	—
56	#1376	—	5	2	no	—
57	#1376	—	5	1	no	—
58	#1376	D₅	1	1	yes	characteristic, direct product factor
59	C₂³×D₅	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₅	—	2	1	no	—
61	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂²×Dic₅	—	2	1	no	—
63	C₂²×Dic₅	M₁₆	1	1	yes	characteristic
64	C₂×C₄×D₅	—	2	8	no	—
65	C₂×C₄×D₅	—	2	4	no	—
66	C₂×C₄×D₅	C₄²	1	2	yes	—
67	C₂×C₄×D₅	C₂²⋊C₄	1	2	yes	—
68	C₂×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
71	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
73	C₂×C₄×D₅	M₁₆	1	2	yes	—
74	C₂×C₅×C₈	—	2	1	no	—
75	C₂×C₅×C₈	—	2	1	no	—
76	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
77	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
78	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
79	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
81	C₄²×C₅	—	2	1	no	—
82	C₄²×C₅	C₂×C₈	1	2	yes	—
83	C₄×Dic₅	—	2	1	no	—
84	C₄×Dic₅	—	2	4	no	—
85	C₄×Dic₅	C₂×C₈	1	2	yes	—
86	C₂×#168	—	2	1	no	—
87	C₂×#168	—	2	1	no	—
88	C₂×#168	C₄²	1	2	yes	—
89	C₂×#168	C₂²⋊C₄	1	2	yes	—
90	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
91	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
92	C₂×#168	C₂×C₈	1	2	yes	—
93	C₈×D₅	—	2	4	no	—
94	C₈×D₅	—	2	4	no	—
95	C₈×D₅	C₂²⋊C₄	1	4	yes	—
96	C₈×D₅	C₂²⋊C₄	1	4	yes	—
97	C₈×D₅	C₄⋊C₄	1	4	yes	—
98	C₈×D₅	C₄⋊C₄	1	4	yes	—
99	C₅×C₁₆	—	2	4	no	—
100	#672	—	2	4	no	—
101	C₂³×C₈	—	5	1	no	—
102	C₂³×C₈	—	5	1	no	—
103	C₂²×C₄²	—	5	1	no	—
104	C₂×M₃₂	—	5	2	no	—
105	C₂×M₃₂	—	5	8	no	—
106	C₂×M₃₂	—	10	2	no	—
107	C₂×M₃₂	—	10	4	no	—
108	C₂×M₃₂	C₂×D₅	1	2	yes	—
109	C₂²×C₁₆	—	10	2	no	—
110	C₂×C₄×C₈	—	5	1	no	—
111	C₂×C₄×C₈	—	5	1	no	—
112	C₂×C₄×C₈	—	5	1	no	—
113	C₂×C₄×C₈	—	10	4	no	—
114	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
115	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
116	C₂²×D₅	—	2	4	no	—
117	C₂²×D₅	#97	1	2	yes	—
118	C₂²×D₅	C₂²⋊C₈	1	2	yes	—
119	C₂²×D₅	C₄⋊C₈	1	1	yes	characteristic
120	C₂²×D₅	C₄⋊C₈	1	1	yes	characteristic
121	C₂×C₄×C₅	—	2	4	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
129	C₂×Dic₅	—	2	4	no	—
130	C₂×Dic₅	—	2	2	no	—
131	C₂×Dic₅	#97	1	2	yes	—
132	C₂×Dic₅	C₂²⋊C₈	1	2	yes	—
133	C₂×Dic₅	C₄⋊C₈	1	1	yes	characteristic
134	C₂×Dic₅	C₄⋊C₈	1	1	yes	characteristic
135	C₄×D₅	—	2	16	no	—
136	C₄×D₅	#95	1	2	yes	—
137	C₄×D₅	#95	1	2	yes	—
138	C₄×D₅	C₄×C₈	1	2	yes	—
139	C₄×D₅	C₄×C₈	1	2	yes	—
140	C₄×D₅	C₂²⋊C₈	1	2	yes	—
141	C₄×D₅	C₂²⋊C₈	1	2	yes	—
142	C₄×D₅	C₄⋊C₈	1	2	yes	—
143	C₄×D₅	C₄⋊C₈	1	2	yes	—
144	C₅×C₈	—	2	2	no	—
145	C₅×C₈	C₂×C₂²⋊C₄	1	2	yes	—
146	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
147	#168	—	2	2	no	—
148	#168	#95	1	2	yes	—
149	#168	#95	1	2	yes	—
150	C₂³×C₄	—	5	1	no	—
151	C₂³×C₄	—	10	1	no	—
152	C₂²×C₈	—	5	1	no	—
153	C₂²×C₈	—	5	4	no	—
154	C₂²×C₈	—	5	4	no	—
155	C₂²×C₈	—	5	1	no	—
156	C₂²×C₈	—	5	1	no	—
157	C₂²×C₈	—	5	1	no	—
158	C₂²×C₈	—	5	1	no	—
159	C₂²×C₈	—	5	1	no	—
160	C₂²×C₈	—	5	2	no	—
161	C₂²×C₈	—	5	2	no	—
162	C₂²×C₈	—	10	2	no	—
163	C₂²×C₈	—	10	2	no	—
164	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
165	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
166	C₂×C₄²	—	5	1	no	—
167	C₂×C₄²	—	5	4	no	—
168	C₂×C₄²	—	10	2	no	—
169	C₂×C₄²	—	10	1	no	—
170	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
171	M₃₂	—	2	2	no	—
172	M₃₂	—	5	4	no	—
173	M₃₂	—	10	8	no	—
174	M₃₂	—	10	2	no	—
175	M₃₂	C₄×D₅	1	4	yes	complemented
176	C₂×C₁₆	—	2	2	no	—
177	C₂×C₁₆	—	10	8	no	—
178	C₂×C₁₆	—	10	2	no	—
179	C₄×C₈	—	2	2	no	—
180	C₄×C₈	—	10	2	no	—
181	C₄×C₈	—	10	2	no	—
182	C₄×C₈	—	10	2	no	—
183	C₂²×C₅	—	2	2	no	—
184	C₂²×C₅	C₂×#97	1	1	yes	characteristic
185	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
186	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
187	C₂×D₅	—	2	2	no	—
188	C₂×D₅	—	2	4	no	—
189	C₂×D₅	—	2	4	no	—
190	C₂×D₅	#336	1	2	yes	—
191	C₂×D₅	#336	1	2	yes	—
192	C₂×D₅	#364	1	4	yes	—
193	C₄×C₅	—	2	4	no	—
194	C₄×C₅	C₂×#95	1	1	yes	characteristic
195	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
196	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
197	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
198	Dic₅	—	2	4	no	—
199	Dic₅	#336	1	2	yes	—
200	Dic₅	#336	1	2	yes	—
201	C₂⁴	—	5	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	5	1	no	—
204	C₂²×C₄	—	5	2	no	—
205	C₂²×C₄	—	5	2	no	—
206	C₂²×C₄	—	5	1	no	—
207	C₂²×C₄	—	5	1	no	—
208	C₂²×C₄	—	5	2	no	—
209	C₂²×C₄	—	5	1	no	—
210	C₂²×C₄	—	5	1	no	—
211	C₂²×C₄	—	5	2	no	—
212	C₂²×C₄	—	10	4	no	—
213	C₂²×C₄	—	10	8	no	—
214	C₂²×C₄	—	10	1	no	—
215	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	5	2	no	—
219	C₂×C₈	—	5	2	no	—
220	C₂×C₈	—	5	1	no	—
221	C₂×C₈	—	5	1	no	—
222	C₂×C₈	—	5	4	no	—
223	C₂×C₈	—	5	4	no	—
224	C₂×C₈	—	5	4	no	—
225	C₂×C₈	—	5	4	no	—
226	C₂×C₈	—	5	2	no	—
227	C₂×C₈	—	10	4	no	—
228	C₂×C₈	—	10	4	no	—
229	C₂×C₈	—	10	1	no	—
230	C₂×C₈	—	10	1	no	—
231	C₂×C₈	C₈×D₅	1	2	yes	—
232	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
233	C₂×C₈	D₄×D₅	1	2	yes	—
234	C₂×C₈	D₄×D₅	1	1	yes	characteristic
235	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
236	C₄²	—	2	1	no	—
237	C₄²	—	5	2	no	—
238	C₄²	—	10	4	no	—
239	C₄²	—	10	1	no	—
240	C₄²	C₈×D₅	1	2	yes	—
241	C₁₆	—	2	4	no	—
242	C₁₆	—	10	4	no	—
243	C₂×C₅	—	2	1	no	—
244	C₂×C₅	—	2	1	no	—
245	C₂×C₅	C₂×#336	1	1	yes	characteristic
246	C₂×C₅	C₂×#364	1	2	yes	—
247	D₅	—	2	2	no	—
248	D₅	#1376	1	4	yes	direct product factor
249	C₂³	—	5	1	no	—
250	C₂³	—	5	1	no	—
251	C₂³	—	5	2	no	—
252	C₂³	—	5	2	no	—
253	C₂³	—	10	4	no	—
254	C₂³	D₅×M₁₆	1	1	yes	characteristic
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	5	1	no	—
258	C₂×C₄	—	5	1	no	—
259	C₂×C₄	—	5	2	no	—
260	C₂×C₄	—	5	2	no	—
261	C₂×C₄	—	5	2	no	—
262	C₂×C₄	—	5	2	no	—
263	C₂×C₄	—	5	2	no	—
264	C₂×C₄	—	5	2	no	—
265	C₂×C₄	—	5	2	no	—
266	C₂×C₄	—	5	2	no	—
267	C₂×C₄	—	5	2	no	—
268	C₂×C₄	—	5	2	no	—
269	C₂×C₄	—	10	16	no	—
270	C₂×C₄	—	10	4	no	—
271	C₂×C₄	—	10	2	no	—
272	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
273	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
274	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
275	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
276	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
277	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	5	2	no	—
280	C₈	—	5	2	no	—
281	C₈	—	10	2	no	—
282	C₈	D₅×C₂²⋊C₄	1	2	yes	—
283	C₈	D₅×C₄⋊C₄	1	2	yes	—
284	C₅	C₂×#1376	1	1	yes	characteristic, complemented, Sylow
285	C₂²	—	2	2	no	—
286	C₂²	—	5	2	no	—
287	C₂²	—	5	4	no	—
288	C₂²	—	5	2	no	—
289	C₂²	—	10	4	no	—
290	C₂²	—	10	4	no	—
291	C₂²	—	10	2	no	—
292	C₂²	D₅×#97	1	1	yes	characteristic
293	C₂²	D₅×C₂²⋊C₈	1	1	yes	characteristic
294	C₂²	D₅×C₄⋊C₈	1	1	yes	characteristic
295	C₄	—	2	4	no	—
296	C₄	—	5	2	no	—
297	C₄	—	5	2	no	—
298	C₄	—	10	4	no	—
299	C₄	D₅×#95	1	1	yes	characteristic
300	C₄	C₄×C₈×D₅	1	1	yes	characteristic
301	C₄	D₅×C₂²⋊C₈	1	1	yes	characteristic
302	C₄	D₅×C₄⋊C₈	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	5	4	no	—
306	C₂	—	10	2	no	—
307	C₂	D₅×#336	1	1	yes	characteristic
308	C₂	D₅×#364	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₅
commutator	C₄×C₅	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#1376	C₂
socle	C₂²×C₅	C₂×C₄⋊C₈
prime radical	C₂³×D₅	M₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1376	D₅
hypocenter	C₅	C₂×#1376
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

Complements

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

N	H ≅ G/N	type
3. C₅×#1376	C₂	semidirect
4. #10598301	C₂	semidirect
5. #10598149	C₂	semidirect
16. D₅×M₃₂	C₄	semidirect
43. C₅×M₃₂	C₂×C₄	semidirect
49. #3944	C₂×C₄	semidirect
58. #1376	D₅	direct (N × H)
175. M₃₂	C₄×D₅	semidirect
248. D₅	#1376	direct (N × H)
284. C₅	C₂×#1376	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅×M₃₂	2
C₂×C₄×C₈×D₅	2
C₅×#1376	2
#10598301	2
#10598149	2
C₂×#1376	5

Subgroups — D5×#1376 #12014151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×#1376 #12014151