Subgroups — C₅×#8647 #11931367

Order 1280.

438 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#1834	C₂	1	1	yes	maximal, characteristic
3	C₅×#1803	C₂	1	2	yes	maximal
4	C₂×C₅×#334	C₂	1	1	yes	maximal, characteristic
5	C₅×#1296	C₂	1	2	yes	maximal
6	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
7	C₅×C₈×Q₈	—	2	2	no	—
8	C₅×C₈×Q₈	C₄	1	4	yes	complemented
9	C₂×C₅×C₄⋊C₈	—	2	1	no	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₅×#103	—	2	2	no	—
13	C₂×C₄×C₅×C₈	—	2	1	no	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×#384	C₂²	1	2	yes	—
16	C₅×#340	—	2	1	no	—
17	C₅×#340	—	2	1	no	—
18	C₅×#340	—	2	2	no	—
19	C₅×#334	—	2	2	no	—
20	#8647	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	—	2	1	no	—
23	C₂²×C₅×C₈	—	2	2	no	—
24	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
26	C₄×C₅×Q₈	—	2	1	no	—
27	C₄×C₅×Q₈	—	2	1	no	—
28	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
29	C₄×C₅×Q₈	Q₈	1	2	yes	—
30	C₂×C₅×C₄⋊C₄	—	2	1	no	—
31	C₂×C₅×C₄⋊C₄	—	2	2	no	—
32	C₂×C₅×C₄⋊C₄	—	2	2	no	—
33	C₂×C₅×C₄⋊C₄	—	2	2	no	—
34	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	—	2	1	no	—
36	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
37	C₅×C₄⋊C₈	—	2	4	no	—
38	C₅×C₄⋊C₈	—	2	1	no	—
39	C₅×C₄⋊C₈	—	2	2	no	—
40	C₅×C₄⋊C₈	—	2	2	no	—
41	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
42	C₅×#103	—	4	4	no	—
43	C₄×C₅×C₈	—	2	4	no	—
44	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
45	C₄×C₅×C₈	D₄	1	2	yes	—
46	C₂×C₈×Q₈	C₂×C₅	1	1	yes	characteristic
47	#1834	C₂×C₅	1	1	yes	characteristic
48	#1803	C₂×C₅	1	2	yes	—
49	C₂×#334	C₂×C₅	1	1	yes	characteristic
50	#1296	C₂×C₅	1	2	yes	—
51	C₂×C₅×Q₈	—	2	1	no	—
52	C₂×C₅×Q₈	—	2	2	no	—
53	C₂×C₅×Q₈	C₄⋊C₄	1	2	yes	—
54	C₂×C₅×Q₈	SD₁₆	1	4	yes	—
55	C₂²×C₄×C₅	—	2	1	no	—
56	C₂²×C₄×C₅	—	2	1	no	—
57	C₂²×C₄×C₅	—	4	2	no	—
58	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	#41	1	1	yes	characteristic
61	C₂×C₅×C₈	—	2	2	no	—
62	C₂×C₅×C₈	—	2	2	no	—
63	C₂×C₅×C₈	—	2	2	no	—
64	C₂×C₅×C₈	—	4	2	no	—
65	C₂×C₅×C₈	—	4	2	no	—
66	C₂×C₅×C₈	—	4	2	no	—
67	C₂×C₅×C₈	—	4	2	no	—
68	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
69	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
70	C₅×C₄⋊C₄	—	2	2	no	—
71	C₅×C₄⋊C₄	—	2	1	no	—
72	C₅×C₄⋊C₄	—	2	2	no	—
73	C₅×C₄⋊C₄	—	4	2	no	—
74	C₅×C₄⋊C₄	—	4	2	no	—
75	C₅×C₄⋊C₄	—	4	2	no	—
76	C₅×C₄⋊C₄	—	4	2	no	—
77	C₅×C₄⋊C₄	—	8	2	no	—
78	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
79	C₄²×C₅	—	2	2	no	—
80	C₄²×C₅	—	2	2	no	—
81	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
84	C₄²×C₅	#41	1	1	yes	characteristic
85	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
86	C₈×Q₈	—	2	2	no	—
87	C₈×Q₈	C₄×C₅	1	4	yes	complemented
88	C₂×C₄⋊C₈	—	2	1	no	—
89	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
90	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
91	C₂×#103	—	2	2	no	—
92	C₂×C₄×C₈	—	2	1	no	—
93	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
94	#384	C₂²×C₅	1	2	yes	—
95	#340	—	2	1	no	—
96	#340	—	2	1	no	—
97	#340	—	2	2	no	—
98	#334	—	2	2	no	—
99	C₂³×C₅	#121	1	1	yes	characteristic
100	C₅×Q₈	—	2	2	no	—
101	C₅×Q₈	—	2	4	no	—
102	C₅×Q₈	#106	1	4	yes	—
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	—	2	4	no	—
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	—	2	1	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	4	1	no	—
109	C₂×C₄×C₅	—	4	1	no	—
110	C₂×C₄×C₅	—	8	2	no	—
111	C₂×C₄×C₅	—	8	2	no	—
112	C₂×C₄×C₅	—	8	2	no	—
113	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅	#117	1	1	yes	characteristic
116	C₂×C₄×C₅	#121	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
120	C₂×C₄×C₅	#135	1	2	yes	—
121	C₂×C₄×C₅	#135	1	2	yes	—
122	C₂×C₄×C₅	#137	1	2	yes	—
123	C₅×C₈	—	2	2	no	—
124	C₅×C₈	—	2	2	no	—
125	C₅×C₈	—	4	2	no	—
126	C₅×C₈	—	8	2	no	—
127	C₂²×Q₈	C₅×D₄	1	1	yes	characteristic
128	C₂²×C₈	—	2	1	no	—
129	C₂²×C₈	—	2	2	no	—
130	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
131	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
132	C₄×Q₈	—	2	1	no	—
133	C₄×Q₈	—	2	1	no	—
134	C₄×Q₈	C₂×C₄×C₅	1	2	yes	—
135	C₄×Q₈	C₅×Q₈	1	2	yes	—
136	C₂×C₄⋊C₄	—	2	1	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
141	C₂×C₄²	—	2	1	no	—
142	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
143	C₄⋊C₈	—	2	4	no	—
144	C₄⋊C₈	—	2	1	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
148	#103	—	4	4	no	—
149	C₄×C₈	—	2	4	no	—
150	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
151	C₄×C₈	C₅×D₄	1	2	yes	—
152	C₂²×C₅	#380	1	1	yes	characteristic
153	C₂²×C₅	#464	1	2	yes	—
154	C₂²×C₅	#465	1	2	yes	—
155	C₂²×C₅	#472	1	1	yes	characteristic
156	C₂²×C₅	#473	1	1	yes	characteristic
157	C₄×C₅	—	2	4	no	—
158	C₄×C₅	—	2	1	no	—
159	C₄×C₅	—	2	1	no	—
160	C₄×C₅	—	4	2	no	—
161	C₄×C₅	—	16	2	no	—
162	C₄×C₅	#416	1	1	yes	characteristic
163	C₄×C₅	#419	1	1	yes	characteristic
164	C₄×C₅	C₂×#106	1	1	yes	characteristic
165	C₄×C₅	#427	1	1	yes	characteristic
166	C₄×C₅	#465	1	2	yes	—
167	C₄×C₅	#479	1	2	yes	—
168	C₂×Q₈	—	2	1	no	—
169	C₂×Q₈	—	2	2	no	—
170	C₂×Q₈	C₅×C₄⋊C₄	1	2	yes	—
171	C₂×Q₈	C₅×SD₁₆	1	4	yes	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
177	C₂²×C₄	C₅×#41	1	1	yes	characteristic
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
186	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	8	2	no	—
195	C₄⋊C₄	C₅×C₄⋊C₄	1	2	yes	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
199	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
200	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
201	C₄²	C₅×#41	1	1	yes	characteristic
202	C₂×C₅	#1678	1	2	yes	—
203	C₂×C₅	#1680	1	2	yes	—
204	C₂×C₅	#1805	1	1	yes	characteristic
205	C₂×C₅	#1807	1	1	yes	characteristic
206	C₂×C₅	#1816	1	1	yes	characteristic
207	C₂³	C₅×#121	1	1	yes	characteristic
208	Q₈	—	2	2	no	—
209	Q₈	—	2	4	no	—
210	Q₈	C₅×#106	1	4	yes	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	—	8	2	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
222	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
223	C₂×C₄	C₅×#117	1	1	yes	characteristic
224	C₂×C₄	C₅×#121	1	1	yes	characteristic
225	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
226	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
227	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
228	C₂×C₄	C₅×#135	1	2	yes	—
229	C₂×C₄	C₅×#135	1	2	yes	—
230	C₂×C₄	C₅×#137	1	2	yes	—
231	C₈	—	2	2	no	—
232	C₈	—	2	2	no	—
233	C₈	—	4	2	no	—
234	C₈	—	8	2	no	—
235	C₅	#8647	1	1	yes	characteristic, direct product factor, Sylow
236	C₂²	C₅×#380	1	1	yes	characteristic
237	C₂²	C₅×#464	1	2	yes	—
238	C₂²	C₅×#465	1	2	yes	—
239	C₂²	C₅×#472	1	1	yes	characteristic
240	C₂²	C₅×#473	1	1	yes	characteristic
241	C₄	—	2	4	no	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	4	2	no	—
245	C₄	—	16	2	no	—
246	C₄	C₅×#416	1	1	yes	characteristic
247	C₄	C₅×#419	1	1	yes	characteristic
248	C₄	C₂×C₅×#106	1	1	yes	characteristic
249	C₄	C₅×#427	1	1	yes	characteristic
250	C₄	C₅×#465	1	2	yes	—
251	C₄	C₅×#479	1	2	yes	—
252	C₂	C₅×#1678	1	2	yes	—
253	C₂	C₅×#1680	1	2	yes	—
254	C₂	C₅×#1805	1	1	yes	characteristic
255	C₂	C₅×#1807	1	1	yes	characteristic
256	C₂	C₅×#1816	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	#121
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₅×C₈×Q₈	C₄	semidirect
20. #8647	C₅	direct (N × H)
87. C₈×Q₈	C₄×C₅	semidirect
235. C₅	#8647	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×C₈×Q₈	2
C₅×#1834	2
C₅×#1803	2
C₂×C₅×#334	2
C₅×#1296	2
#8647	5

Subgroups — C5×#8647 #11931367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8647 #11931367