Subgroups — C₅×#8649 #11931369

Order 1280.

438 conjugacy classes of subgroups.

The subgroup lattice has 268 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×#335	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×#335	—	2	2	no	—
20	#8649	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×#335	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	1	no	—
53	C₂×C₅×Q₈	—	2	1	no	—
54	C₂×C₅×Q₈	C₄⋊C₄	1	2	yes	—
55	C₂×C₅×Q₈	D₈	1	2	yes	—
56	C₂×C₅×Q₈	Q₁₆	1	2	yes	—
57	C₂²×C₄×C₅	—	2	1	no	—
58	C₂²×C₄×C₅	—	2	1	no	—
59	C₂²×C₄×C₅	—	4	2	no	—
60	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	#41	1	1	yes	characteristic
63	C₂×C₅×C₈	—	2	2	no	—
64	C₂×C₅×C₈	—	2	2	no	—
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	—	4	2	no	—
67	C₂×C₅×C₈	—	4	2	no	—
68	C₂×C₅×C₈	—	4	2	no	—
69	C₂×C₅×C₈	—	4	2	no	—
70	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
71	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
72	C₅×C₄⋊C₄	—	2	2	no	—
73	C₅×C₄⋊C₄	—	2	1	no	—
74	C₅×C₄⋊C₄	—	2	2	no	—
75	C₅×C₄⋊C₄	—	4	2	no	—
76	C₅×C₄⋊C₄	—	4	2	no	—
77	C₅×C₄⋊C₄	—	4	2	no	—
78	C₅×C₄⋊C₄	—	4	2	no	—
79	C₅×C₄⋊C₄	—	8	2	no	—
80	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
81	C₄²×C₅	—	2	2	no	—
82	C₄²×C₅	—	2	2	no	—
83	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
86	C₄²×C₅	#41	1	1	yes	characteristic
87	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
88	C₈×Q₈	—	2	2	no	—
89	C₈×Q₈	C₄×C₅	1	4	yes	complemented
90	C₂×C₄⋊C₈	—	2	1	no	—
91	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
92	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
93	C₂×#103	—	2	2	no	—
94	C₂×C₄×C₈	—	2	1	no	—
95	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
96	#384	C₂²×C₅	1	2	yes	—
97	#340	—	2	1	no	—
98	#340	—	2	1	no	—
99	#340	—	2	2	no	—
100	#335	—	2	2	no	—
101	C₂³×C₅	#121	1	1	yes	characteristic
102	C₅×Q₈	—	2	2	no	—
103	C₅×Q₈	—	2	2	no	—
104	C₅×Q₈	—	2	2	no	—
105	C₅×Q₈	#107	1	4	yes	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	2	2	no	—
110	C₂×C₄×C₅	—	2	2	no	—
111	C₂×C₄×C₅	—	4	2	no	—
112	C₂×C₄×C₅	—	8	2	no	—
113	C₂×C₄×C₅	—	8	2	no	—
114	C₂×C₄×C₅	—	8	2	no	—
115	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅	#117	1	1	yes	characteristic
118	C₂×C₄×C₅	#121	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
123	C₂×C₄×C₅	#135	1	2	yes	—
124	C₂×C₄×C₅	#135	1	1	yes	characteristic
125	C₂×C₄×C₅	#135	1	1	yes	characteristic
126	C₂×C₄×C₅	#137	1	2	yes	—
127	C₅×C₈	—	2	2	no	—
128	C₅×C₈	—	2	2	no	—
129	C₅×C₈	—	4	2	no	—
130	C₅×C₈	—	8	2	no	—
131	C₂²×Q₈	C₅×D₄	1	1	yes	characteristic
132	C₂²×C₈	—	2	1	no	—
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
135	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
136	C₄×Q₈	—	2	1	no	—
137	C₄×Q₈	—	2	1	no	—
138	C₄×Q₈	C₂×C₄×C₅	1	2	yes	—
139	C₄×Q₈	C₅×Q₈	1	2	yes	—
140	C₂×C₄⋊C₄	—	2	1	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	—	2	2	no	—
144	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
145	C₂×C₄²	—	2	1	no	—
146	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
147	C₄⋊C₈	—	2	4	no	—
148	C₄⋊C₈	—	2	1	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
152	#103	—	4	4	no	—
153	C₄×C₈	—	2	4	no	—
154	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
155	C₄×C₈	C₅×D₄	1	2	yes	—
156	C₂²×C₅	#380	1	1	yes	characteristic
157	C₂²×C₅	#464	1	2	yes	—
158	C₂²×C₅	#466	1	1	yes	characteristic
159	C₂²×C₅	#467	1	1	yes	characteristic
160	C₂²×C₅	#471	1	2	yes	—
161	C₄×C₅	—	2	4	no	—
162	C₄×C₅	—	2	1	no	—
163	C₄×C₅	—	2	1	no	—
164	C₄×C₅	—	4	2	no	—
165	C₄×C₅	—	16	2	no	—
166	C₄×C₅	#416	1	1	yes	characteristic
167	C₄×C₅	#419	1	1	yes	characteristic
168	C₄×C₅	C₂×#107	1	1	yes	characteristic
169	C₄×C₅	#427	1	1	yes	characteristic
170	C₄×C₅	#466	1	1	yes	characteristic
171	C₄×C₅	#467	1	1	yes	characteristic
172	C₄×C₅	#479	1	2	yes	—
173	C₂×Q₈	—	2	1	no	—
174	C₂×Q₈	—	2	1	no	—
175	C₂×Q₈	—	2	1	no	—
176	C₂×Q₈	C₅×C₄⋊C₄	1	2	yes	—
177	C₂×Q₈	C₅×D₈	1	2	yes	—
178	C₂×Q₈	C₅×Q₁₆	1	2	yes	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
184	C₂²×C₄	C₅×#41	1	1	yes	characteristic
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
193	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	8	2	no	—
202	C₄⋊C₄	C₅×C₄⋊C₄	1	2	yes	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
206	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
207	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
208	C₄²	C₅×#41	1	1	yes	characteristic
209	C₂×C₅	#1682	1	2	yes	—
210	C₂×C₅	#1684	1	2	yes	—
211	C₂×C₅	#1805	1	1	yes	characteristic
212	C₂×C₅	#1808	1	1	yes	characteristic
213	C₂×C₅	#1815	1	1	yes	characteristic
214	C₂³	C₅×#121	1	1	yes	characteristic
215	Q₈	—	2	2	no	—
216	Q₈	—	2	2	no	—
217	Q₈	—	2	2	no	—
218	Q₈	C₅×#107	1	4	yes	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	—	8	2	no	—
227	C₂×C₄	—	8	2	no	—
228	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
229	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
230	C₂×C₄	C₅×#117	1	1	yes	characteristic
231	C₂×C₄	C₅×#121	1	1	yes	characteristic
232	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
233	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
234	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
235	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
236	C₂×C₄	C₅×#135	1	2	yes	—
237	C₂×C₄	C₅×#135	1	1	yes	characteristic
238	C₂×C₄	C₅×#135	1	1	yes	characteristic
239	C₂×C₄	C₅×#137	1	2	yes	—
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	—	4	2	no	—
243	C₈	—	8	2	no	—
244	C₅	#8649	1	1	yes	characteristic, direct product factor, Sylow
245	C₂²	C₅×#380	1	1	yes	characteristic
246	C₂²	C₅×#464	1	2	yes	—
247	C₂²	C₅×#466	1	1	yes	characteristic
248	C₂²	C₅×#467	1	1	yes	characteristic
249	C₂²	C₅×#471	1	2	yes	—
250	C₄	—	2	4	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	2	no	—
254	C₄	—	16	2	no	—
255	C₄	C₅×#416	1	1	yes	characteristic
256	C₄	C₅×#419	1	1	yes	characteristic
257	C₄	C₂×C₅×#107	1	1	yes	characteristic
258	C₄	C₅×#427	1	1	yes	characteristic
259	C₄	C₅×#466	1	1	yes	characteristic
260	C₄	C₅×#467	1	1	yes	characteristic
261	C₄	C₅×#479	1	2	yes	—
262	C₂	C₅×#1682	1	2	yes	—
263	C₂	C₅×#1684	1	2	yes	—
264	C₂	C₅×#1805	1	1	yes	characteristic
265	C₂	C₅×#1808	1	1	yes	characteristic
266	C₂	C₅×#1815	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. #8649	C₅	direct (N × H)
89. C₈×Q₈	C₄×C₅	semidirect
244. C₅	#8649	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×#335	2
C₅×#1296	2
#8649	5

Subgroups — C5×#8649 #11931369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8649 #11931369