Subgroups — C₂×C₅×#1847 #11943518

Order 1280.

1230 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#107	C₂	1	2	yes	maximal
2	C₂²×C₅×#106	C₂	1	2	yes	maximal
3	C₂²×C₅×#97	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#384	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×#382	C₂	1	1	yes	maximal, characteristic
6	C₅×#1847	C₂	1	8	yes	maximal, direct product factor
7	C₂³×C₅×C₈	C₂²	1	2	yes	—
8	C₂²×C₅×C₄⋊C₄	—	2	2	no	—
9	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
10	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₅×#107	—	2	8	no	—
14	C₂×C₅×#107	C₂²	1	8	yes	—
15	C₂×C₅×#106	—	2	8	no	—
16	C₂×C₅×#106	C₂²	1	8	yes	—
17	C₂×C₅×#97	C₄	1	8	yes	complemented
18	C₂×C₅×#97	C₂²	1	4	yes	—
19	C₅×#384	—	2	2	no	—
20	C₅×#384	C₂²	1	4	yes	—
21	C₅×#382	—	2	2	no	—
22	C₅×#382	C₂²	1	4	yes	—
23	C₂×C₅×#95	—	2	2	no	—
24	C₂×#1847	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂³×C₄×C₅	—	2	1	no	—
26	C₂³×C₄×C₅	—	2	1	no	—
27	C₂³×C₄×C₅	—	2	2	no	—
28	C₂³×C₄×C₅	D₄	1	2	yes	—
29	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂×C₄	1	8	yes	—
31	C₂²×C₅×C₈	D₄	1	8	yes	—
32	C₂²×C₅×C₈	Q₈	1	8	yes	—
33	C₂²×C₅×C₈	C₂³	1	4	yes	—
34	C₂×C₅×C₄⋊C₄	—	2	4	no	—
35	C₂×C₅×C₄⋊C₄	—	2	4	no	—
36	C₂×C₅×C₄⋊C₄	—	2	4	no	—
37	C₂×C₅×C₄⋊C₄	—	2	4	no	—
38	C₂×C₅×C₄⋊C₄	—	2	1	no	—
39	C₂×C₅×C₄⋊C₄	—	2	1	no	—
40	C₂×C₅×C₄⋊C₄	—	2	2	no	—
41	C₂×C₅×C₄⋊C₄	—	2	8	no	—
42	C₂×C₅×C₄⋊C₄	—	4	8	no	—
43	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
44	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
45	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
46	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₅	D₄	1	2	yes	—
48	C₂×C₄²×C₅	Q₈	1	4	yes	—
49	C₂×C₄²×C₅	C₂³	1	2	yes	—
50	C₅×#107	—	2	16	no	—
51	C₅×#106	—	2	16	no	—
52	C₅×#97	C₂×C₄	1	16	yes	complemented
53	C₅×#95	—	2	8	no	—
54	C₂²×#107	C₂×C₅	1	2	yes	—
55	C₂²×#106	C₂×C₅	1	2	yes	—
56	C₂²×#97	C₂×C₅	1	1	yes	characteristic
57	C₂×#384	C₂×C₅	1	1	yes	characteristic
58	C₂×#382	C₂×C₅	1	1	yes	characteristic
59	#1847	C₂×C₅	1	8	yes	direct product factor
60	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	—	2	4	no	—
62	C₂²×C₄×C₅	—	2	4	no	—
63	C₂²×C₄×C₅	—	2	4	no	—
64	C₂²×C₄×C₅	—	2	2	no	—
65	C₂²×C₄×C₅	—	2	2	no	—
66	C₂²×C₄×C₅	—	4	4	no	—
67	C₂²×C₄×C₅	—	4	2	no	—
68	C₂²×C₄×C₅	—	4	2	no	—
69	C₂²×C₄×C₅	—	4	8	no	—
70	C₂²×C₄×C₅	—	4	4	no	—
71	C₂²×C₄×C₅	—	4	4	no	—
72	C₂²×C₄×C₅	C₄⋊C₄	1	8	yes	—
73	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
78	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
80	C₂×C₅×C₈	C₄⋊C₄	1	32	yes	complemented
81	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
82	C₂×C₅×C₈	C₂×D₄	1	8	yes	—
83	C₂×C₅×C₈	C₂×Q₈	1	8	yes	—
84	C₅×C₄⋊C₄	—	2	4	no	—
85	C₅×C₄⋊C₄	—	2	4	no	—
86	C₅×C₄⋊C₄	—	2	4	no	—
87	C₅×C₄⋊C₄	—	2	4	no	—
88	C₅×C₄⋊C₄	—	4	16	no	—
89	C₅×C₄⋊C₄	—	4	2	no	—
90	C₅×C₄⋊C₄	—	4	4	no	—
91	C₅×C₄⋊C₄	—	4	2	no	—
92	C₄²×C₅	C₄⋊C₄	1	8	yes	—
93	C₄²×C₅	C₂²×C₄	1	4	yes	—
94	C₄²×C₅	C₂×Q₈	1	4	yes	—
95	C₂³×C₈	C₂²×C₅	1	2	yes	—
96	C₂²×C₄⋊C₄	—	2	2	no	—
97	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
98	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
99	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
100	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
101	C₂×#107	—	2	8	no	—
102	C₂×#107	C₂²×C₅	1	8	yes	—
103	C₂×#106	—	2	8	no	—
104	C₂×#106	C₂²×C₅	1	8	yes	—
105	C₂×#97	C₄×C₅	1	8	yes	complemented
106	C₂×#97	C₂²×C₅	1	4	yes	—
107	#384	—	2	2	no	—
108	#384	C₂²×C₅	1	4	yes	—
109	#382	—	2	2	no	—
110	#382	C₂²×C₅	1	4	yes	—
111	C₂×#95	—	2	2	no	—
112	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
113	C₂³×C₅	#127	1	2	yes	—
114	C₂³×C₅	C₄⋊Q₈	1	2	yes	—
115	C₂³×C₅	AGL(1,ℤ/8)	1	4	yes	—
116	C₂³×C₅	#137	1	4	yes	—
117	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅	—	2	4	no	—
119	C₂×C₄×C₅	—	2	4	no	—
120	C₂×C₄×C₅	—	2	16	no	—
121	C₂×C₄×C₅	—	4	4	no	—
122	C₂×C₄×C₅	—	4	2	no	—
123	C₂×C₄×C₅	—	4	2	no	—
124	C₂×C₄×C₅	—	4	8	no	—
125	C₂×C₄×C₅	—	4	4	no	—
126	C₂×C₄×C₅	—	4	4	no	—
127	C₂×C₄×C₅	—	8	8	no	—
128	C₂×C₄×C₅	—	8	4	no	—
129	C₂×C₄×C₅	—	8	4	no	—
130	C₂×C₄×C₅	C₂×C₄⋊C₄	1	8	yes	—
131	C₂×C₄×C₅	C₂×C₄⋊C₄	1	8	yes	—
132	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×C₄×C₅	#127	1	2	yes	—
134	C₂×C₄×C₅	C₄⋊Q₈	1	8	yes	—
135	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
136	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
138	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
139	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
140	C₅×C₈	C₂×C₄⋊C₄	1	16	yes	complemented
141	C₂³×C₄	—	2	2	no	—
142	C₂³×C₄	—	2	1	no	—
143	C₂³×C₄	—	2	1	no	—
144	C₂³×C₄	C₅×D₄	1	2	yes	—
145	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
146	C₂²×C₈	C₂×C₄×C₅	1	8	yes	—
147	C₂²×C₈	C₅×D₄	1	8	yes	—
148	C₂²×C₈	C₅×Q₈	1	8	yes	—
149	C₂²×C₈	C₂³×C₅	1	4	yes	—
150	C₂×C₄⋊C₄	—	2	4	no	—
151	C₂×C₄⋊C₄	—	2	4	no	—
152	C₂×C₄⋊C₄	—	2	4	no	—
153	C₂×C₄⋊C₄	—	2	4	no	—
154	C₂×C₄⋊C₄	—	2	8	no	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	4	8	no	—
159	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
160	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
161	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
162	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
163	C₂×C₄²	C₅×D₄	1	2	yes	—
164	C₂×C₄²	C₅×Q₈	1	4	yes	—
165	C₂×C₄²	C₂³×C₅	1	2	yes	—
166	#107	—	2	16	no	—
167	#106	—	2	16	no	—
168	#97	C₂×C₄×C₅	1	16	yes	complemented
169	#95	—	2	8	no	—
170	C₂²×C₅	#384	1	4	yes	—
171	C₂²×C₅	#428	1	8	yes	—
172	C₂²×C₅	#496	1	4	yes	—
173	C₂²×C₅	#497	1	4	yes	—
174	C₂²×C₅	#501	1	8	yes	—
175	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
176	C₂²×C₅	C₂×#127	1	1	yes	characteristic
177	C₂²×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
178	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
179	C₂²×C₅	C₂×#137	1	2	yes	—
180	C₄×C₅	—	2	8	no	—
181	C₄×C₅	—	8	4	no	—
182	C₄×C₅	—	8	2	no	—
183	C₄×C₅	—	8	2	no	—
184	C₄×C₅	#384	1	4	yes	—
185	C₄×C₅	C₂²×C₄⋊C₄	1	2	yes	—
186	C₄×C₅	C₂×C₄⋊Q₈	1	2	yes	—
187	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	4	4	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	8	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	—	4	4	no	—
199	C₂²×C₄	C₅×C₄⋊C₄	1	8	yes	—
200	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
201	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
202	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
204	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
205	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
206	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
207	C₂×C₈	C₅×C₄⋊C₄	1	32	yes	complemented
208	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
209	C₂×C₈	C₂×C₅×D₄	1	8	yes	—
210	C₂×C₈	C₂×C₅×Q₈	1	8	yes	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	4	16	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄²	C₅×C₄⋊C₄	1	8	yes	—
220	C₄²	C₂²×C₄×C₅	1	4	yes	—
221	C₄²	C₂×C₅×Q₈	1	4	yes	—
222	C₂×C₅	#1847	1	8	yes	direct product factor
223	C₂×C₅	C₂×#384	1	1	yes	characteristic
224	C₂×C₅	C₂×#428	1	2	yes	—
225	C₂×C₅	C₂×#496	1	1	yes	characteristic
226	C₂×C₅	C₂×#497	1	1	yes	characteristic
227	C₂×C₅	C₂×#501	1	2	yes	—
228	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
229	C₂³	C₅×#127	1	2	yes	—
230	C₂³	C₅×C₄⋊Q₈	1	2	yes	—
231	C₂³	C₅×AGL(1,ℤ/8)	1	4	yes	—
232	C₂³	C₅×#137	1	4	yes	—
233	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
234	C₂×C₄	—	2	16	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	8	8	no	—
244	C₂×C₄	—	8	4	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	C₂×C₅×C₄⋊C₄	1	8	yes	—
247	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
248	C₂×C₄	C₂×C₅×C₄⋊C₄	1	8	yes	—
249	C₂×C₄	C₅×#127	1	2	yes	—
250	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
251	C₂×C₄	C₅×C₄⋊Q₈	1	8	yes	—
252	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
253	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
254	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
255	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
256	C₈	C₂×C₅×C₄⋊C₄	1	16	yes	complemented
257	C₅	C₂×#1847	1	1	yes	characteristic, direct product factor, Sylow
258	C₂²	C₅×#384	1	4	yes	—
259	C₂²	C₅×#428	1	8	yes	—
260	C₂²	C₅×#496	1	4	yes	—
261	C₂²	C₅×#497	1	4	yes	—
262	C₂²	C₅×#501	1	8	yes	—
263	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
264	C₂²	C₂×C₅×#127	1	1	yes	characteristic
265	C₂²	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
266	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	2	yes	—
267	C₂²	C₂×C₅×#137	1	2	yes	—
268	C₄	—	2	8	no	—
269	C₄	—	8	4	no	—
270	C₄	—	8	2	no	—
271	C₄	—	8	2	no	—
272	C₄	C₅×#384	1	4	yes	—
273	C₄	C₂²×C₅×C₄⋊C₄	1	2	yes	—
274	C₄	C₂×C₅×C₄⋊Q₈	1	2	yes	—
275	C₂	C₅×#1847	1	8	yes	direct product factor
276	C₂	C₂×C₅×#384	1	1	yes	characteristic
277	C₂	C₂×C₅×#428	1	2	yes	—
278	C₂	C₂×C₅×#496	1	1	yes	characteristic
279	C₂	C₂×C₅×#497	1	1	yes	characteristic
280	C₂	C₂×C₅×#501	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×#1847	C₂	direct (N × H)
17. C₂×C₅×#97	C₄	semidirect
24. C₂×#1847	C₅	direct (N × H)
52. C₅×#97	C₂×C₄	semidirect
59. #1847	C₂×C₅	direct (N × H)
80. C₂×C₅×C₈	C₄⋊C₄	semidirect
105. C₂×#97	C₄×C₅	semidirect
140. C₅×C₈	C₂×C₄⋊C₄	semidirect
168. #97	C₂×C₄×C₅	semidirect
207. C₂×C₈	C₅×C₄⋊C₄	semidirect
222. C₂×C₅	#1847	direct (N × H)
256. C₈	C₂×C₅×C₄⋊C₄	semidirect
257. C₅	C₂×#1847	direct (N × H)
275. C₂	C₅×#1847	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×#107	2
C₂²×C₅×#106	2
C₂²×C₅×#97	2
C₂×C₅×#384	2
C₂×C₅×#382	2
C₅×#1847	2
C₂×#1847	5

Subgroups — C2×C5×#1847 #11943518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1847 #11943518