Subgroups — C₅×#8619 #11931339

Order 1280.

378 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#446	C₂	1	1	yes	maximal, characteristic
2	C₅×#1801	C₂	1	2	yes	maximal
3	C₅×#1567	C₂	1	1	yes	maximal, characteristic
4	C₅×#1566	C₂	1	1	yes	maximal, characteristic
5	C₅×#1292	C₂	1	2	yes	maximal
6	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
7	C₅×#446	—	2	1	no	—
8	C₅×#446	—	2	1	no	—
9	C₅×#446	—	2	1	no	—
10	C₅×#446	—	2	1	no	—
11	C₅×#423	C₂²	1	2	yes	—
12	C₅×#423	C₂²	1	2	yes	—
13	C₂×C₅×C₄⋊C₈	C₄	1	2	yes	—
14	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₅×#101	—	2	2	no	—
16	C₂×C₅×#97	C₄	1	2	yes	—
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₅×#343	—	2	2	no	—
19	C₅×#335	—	2	1	no	—
20	C₅×#335	—	2	1	no	—
21	C₅×#334	—	2	1	no	—
22	C₅×#334	—	2	1	no	—
23	#8619	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
25	C₂×C₅×M₁₆	—	2	2	no	—
26	C₂×C₅×M₁₆	—	2	2	no	—
27	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₄×C₅×Q₈	—	2	1	no	—
31	C₄×C₅×Q₈	—	2	1	no	—
32	C₄×C₅×Q₈	—	2	1	no	—
33	C₄×C₅×Q₈	—	2	1	no	—
34	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
35	C₂×C₅×C₄⋊C₄	Q₈	1	2	yes	—
36	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
38	C₅×C₄⋊C₈	—	2	2	no	—
39	C₅×C₄⋊C₈	—	2	2	no	—
40	C₅×C₄⋊C₈	—	2	1	no	—
41	C₅×C₄⋊C₈	—	2	1	no	—
42	C₅×C₄⋊C₈	—	2	2	no	—
43	C₅×C₄⋊C₈	—	2	2	no	—
44	C₅×C₄⋊C₈	—	2	2	no	—
45	C₅×C₄⋊C₈	—	2	2	no	—
46	C₅×#101	—	4	4	no	—
47	C₅×#97	—	2	2	no	—
48	C₅×#97	—	2	2	no	—
49	C₄×C₅×C₈	D₄	1	1	yes	characteristic
50	C₄×C₅×C₈	D₄	1	1	yes	characteristic
51	C₄×C₅×C₈	D₄	1	1	yes	characteristic
52	C₄×C₅×C₈	D₄	1	1	yes	characteristic
53	C₂×#446	C₂×C₅	1	1	yes	characteristic
54	#1801	C₂×C₅	1	2	yes	—
55	#1567	C₂×C₅	1	1	yes	characteristic
56	#1566	C₂×C₅	1	1	yes	characteristic
57	#1292	C₂×C₅	1	2	yes	—
58	C₂×C₅×Q₈	—	2	1	no	—
59	C₂×C₅×Q₈	—	2	1	no	—
60	C₂×C₅×Q₈	—	2	2	no	—
61	C₂×C₅×Q₈	—	2	2	no	—
62	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
67	C₅×M₁₆	—	4	2	no	—
68	C₅×M₁₆	—	4	2	no	—
69	C₅×M₁₆	—	4	2	no	—
70	C₅×M₁₆	—	4	2	no	—
71	C₂×C₅×C₈	—	2	2	no	—
72	C₂×C₅×C₈	—	2	2	no	—
73	C₂×C₅×C₈	—	4	1	no	—
74	C₂×C₅×C₈	—	4	2	no	—
75	C₂×C₅×C₈	—	4	2	no	—
76	C₂×C₅×C₈	—	4	2	no	—
77	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₅×C₄⋊C₄	—	2	1	no	—
82	C₅×C₄⋊C₄	—	2	1	no	—
83	C₅×C₄⋊C₄	—	2	2	no	—
84	C₅×C₄⋊C₄	—	2	2	no	—
85	C₄²×C₅	—	2	2	no	—
86	C₄²×C₅	—	2	2	no	—
87	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₅	#41	1	1	yes	characteristic
90	C₄²×C₅	#41	1	1	yes	characteristic
91	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
92	#446	—	2	1	no	—
93	#446	—	2	1	no	—
94	#446	—	2	1	no	—
95	#446	—	2	1	no	—
96	#423	C₂²×C₅	1	2	yes	—
97	#423	C₂²×C₅	1	2	yes	—
98	C₂×C₄⋊C₈	C₄×C₅	1	2	yes	—
99	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
100	C₂×#101	—	2	2	no	—
101	C₂×#97	C₄×C₅	1	2	yes	—
102	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
103	#343	—	2	2	no	—
104	#335	—	2	1	no	—
105	#335	—	2	1	no	—
106	#334	—	2	1	no	—
107	#334	—	2	1	no	—
108	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₅×Q₈	—	4	2	no	—
110	C₅×Q₈	—	4	1	no	—
111	C₅×Q₈	—	4	1	no	—
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	2	1	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	4	no	—
117	C₂×C₄×C₅	—	4	1	no	—
118	C₂×C₄×C₅	—	4	2	no	—
119	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅	#117	1	1	yes	characteristic
121	C₂×C₄×C₅	#121	1	1	yes	characteristic
122	C₂×C₄×C₅	#121	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
126	C₂×C₄×C₅	#137	1	2	yes	—
127	C₅×C₈	—	2	1	no	—
128	C₅×C₈	—	2	1	no	—
129	C₅×C₈	—	2	1	no	—
130	C₅×C₈	—	4	2	no	—
131	C₅×C₈	—	8	2	no	—
132	C₅×C₈	—	8	2	no	—
133	C₅×C₈	#101	1	2	yes	complemented
134	C₂²×Q₈	C₅×D₄	1	1	yes	characteristic
135	C₂×M₁₆	—	2	2	no	—
136	C₂×M₁₆	—	2	2	no	—
137	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
138	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
139	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
140	C₄×Q₈	—	2	1	no	—
141	C₄×Q₈	—	2	1	no	—
142	C₄×Q₈	—	2	1	no	—
143	C₄×Q₈	—	2	1	no	—
144	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
145	C₂×C₄⋊C₄	C₅×Q₈	1	2	yes	—
146	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
147	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	1	no	—
151	C₄⋊C₈	—	2	1	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	2	no	—
156	#101	—	4	4	no	—
157	#97	—	2	2	no	—
158	#97	—	2	2	no	—
159	C₄×C₈	C₅×D₄	1	1	yes	characteristic
160	C₄×C₈	C₅×D₄	1	1	yes	characteristic
161	C₄×C₈	C₅×D₄	1	1	yes	characteristic
162	C₄×C₈	C₅×D₄	1	1	yes	characteristic
163	C₂²×C₅	—	2	1	no	—
164	C₂²×C₅	—	2	1	no	—
165	C₂²×C₅	#380	1	1	yes	characteristic
166	C₂²×C₅	#428	1	2	yes	—
167	C₄×C₅	—	2	1	no	—
168	C₄×C₅	—	2	1	no	—
169	C₄×C₅	—	2	1	no	—
170	C₄×C₅	—	4	2	no	—
171	C₄×C₅	—	4	2	no	—
172	C₄×C₅	C₂×#101	1	1	yes	characteristic
173	C₄×C₅	#413	1	1	yes	characteristic
174	C₄×C₅	#468	1	2	yes	—
175	C₄×C₅	#469	1	1	yes	characteristic
176	C₄×C₅	#470	1	1	yes	characteristic
177	C₂×Q₈	—	2	1	no	—
178	C₂×Q₈	—	2	1	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
182	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
183	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
186	M₁₆	—	4	2	no	—
187	M₁₆	—	4	2	no	—
188	M₁₆	—	4	2	no	—
189	M₁₆	—	4	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	4	1	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
197	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
198	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
199	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
207	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
208	C₄²	C₅×#41	1	1	yes	characteristic
209	C₄²	C₅×#41	1	1	yes	characteristic
210	C₂×C₅	—	2	2	no	—
211	C₂×C₅	#1801	1	1	yes	characteristic
212	C₂×C₅	#1809	1	1	yes	characteristic
213	C₂×C₅	#1813	1	1	yes	characteristic
214	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
215	Q₈	—	4	2	no	—
216	Q₈	—	4	1	no	—
217	Q₈	—	4	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
226	C₂×C₄	C₅×#117	1	1	yes	characteristic
227	C₂×C₄	C₅×#121	1	1	yes	characteristic
228	C₂×C₄	C₅×#121	1	1	yes	characteristic
229	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
230	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
231	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
232	C₂×C₄	C₅×#137	1	2	yes	—
233	C₈	—	2	1	no	—
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	4	2	no	—
237	C₈	—	8	2	no	—
238	C₈	—	8	2	no	—
239	C₈	C₅×#101	1	2	yes	complemented
240	C₅	#8619	1	1	yes	characteristic, direct product factor, Sylow
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	C₅×#380	1	1	yes	characteristic
244	C₂²	C₅×#428	1	2	yes	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	2	no	—
250	C₄	C₂×C₅×#101	1	1	yes	characteristic
251	C₄	C₅×#413	1	1	yes	characteristic
252	C₄	C₅×#468	1	2	yes	—
253	C₄	C₅×#469	1	1	yes	characteristic
254	C₄	C₅×#470	1	1	yes	characteristic
255	C₂	—	2	2	no	—
256	C₂	C₅×#1801	1	1	yes	characteristic
257	C₂	C₅×#1809	1	1	yes	characteristic
258	C₂	C₅×#1813	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#380
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
23. #8619	C₅	direct (N × H)
133. C₅×C₈	#101	semidirect
239. C₈	C₅×#101	semidirect
240. C₅	#8619	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#446	2
C₅×#1801	2
C₅×#1567	2
C₅×#1566	2
C₅×#1292	2
#8619	5

Subgroups — C5×#8619 #11931339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8619 #11931339