Subgroups — C₅×#8685 #11931405

Order 1280.

362 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	C₅×#2960	C₂	1	1	yes	maximal, characteristic
2	C₅×#1848	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#335	C₂	1	1	yes	maximal, characteristic
4	C₅×#1550	C₂	1	2	yes	maximal
5	C₅×#1292	C₂	1	2	yes	maximal
6	C₅×#533	C₂²	1	1	yes	characteristic
7	C₅×#446	—	2	1	no	—
8	C₅×#446	C₄	1	2	yes	—
9	C₅×C₈×Q₈	—	2	1	no	—
10	C₅×C₈×Q₈	C₄	1	2	yes	—
11	C₂×C₅×#108	—	2	1	no	—
12	C₂×C₅×#108	—	2	1	no	—
13	C₅×#424	—	2	1	no	—
14	C₅×#423	C₂²	1	2	yes	—
15	C₅×#423	C₂²	1	1	yes	characteristic
16	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
17	C₅×#405	—	2	1	no	—
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₅×#335	—	2	1	no	—
20	C₅×#335	—	2	1	no	—
21	C₅×#333	—	2	2	no	—
22	C₅×#332	—	2	2	no	—
23	#8685	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂×C₅×M₁₆	—	2	1	no	—
25	C₂×C₅×M₁₆	—	2	2	no	—
26	C₂²×C₅×C₈	—	2	2	no	—
27	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
29	C₅×C₄⋊Q₈	D₄	1	1	yes	characteristic
30	C₅×C₄⋊Q₈	D₄	1	1	yes	characteristic
31	C₅×#125	—	2	1	no	—
32	C₅×C₂²⋊Q₈	—	2	1	no	—
33	C₅×C₂²⋊Q₈	Q₈	1	2	yes	—
34	C₄×C₅×Q₈	—	2	1	no	—
35	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
36	C₅×#117	—	2	1	no	—
37	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
38	C₅×#108	—	4	2	no	—
39	C₅×#108	—	4	2	no	—
40	C₅×C₄⋊C₈	—	2	2	no	—
41	C₅×C₄⋊C₈	—	2	2	no	—
42	C₅×C₄⋊C₈	—	2	1	no	—
43	C₅×C₄⋊C₈	—	2	2	no	—
44	C₅×C₄⋊C₈	—	2	2	no	—
45	C₅×C₄⋊C₈	—	2	2	no	—
46	C₅×C₄⋊C₈	—	2	2	no	—
47	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
48	C₅×#97	—	2	2	no	—
49	C₄×C₅×C₈	—	2	2	no	—
50	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
51	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
52	C₄×C₅×C₈	D₄	1	1	yes	characteristic
53	C₄×C₅×C₈	D₄	1	1	yes	characteristic
54	#2960	C₂×C₅	1	1	yes	characteristic
55	#1848	C₂×C₅	1	1	yes	characteristic
56	C₂×#335	C₂×C₅	1	1	yes	characteristic
57	#1550	C₂×C₅	1	2	yes	—
58	#1292	C₂×C₅	1	2	yes	—
59	C₂×C₅×Q₈	—	2	1	no	—
60	C₂×C₅×Q₈	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₅	#41	1	1	yes	characteristic
64	C₅×M₁₆	—	4	1	no	—
65	C₅×M₁₆	—	4	1	no	—
66	C₅×M₁₆	—	4	2	no	—
67	C₅×M₁₆	—	4	2	no	—
68	C₂×C₅×C₈	—	2	1	no	—
69	C₂×C₅×C₈	—	2	1	no	—
70	C₂×C₅×C₈	—	2	2	no	—
71	C₂×C₅×C₈	—	4	2	no	—
72	C₂×C₅×C₈	—	4	2	no	—
73	C₂×C₅×C₈	—	8	2	no	—
74	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	SD₁₆	1	2	yes	—
79	C₂×C₅×C₈	Q₁₆	1	2	yes	—
80	C₅×C₄⋊C₄	—	2	2	no	—
81	C₅×C₄⋊C₄	—	2	1	no	—
82	C₅×C₄⋊C₄	—	2	1	no	—
83	C₅×C₄⋊C₄	—	2	1	no	—
84	C₅×C₄⋊C₄	—	2	2	no	—
85	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
86	C₅×C₂²⋊C₄	—	2	2	no	—
87	C₄²×C₅	—	2	2	no	—
88	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅	#41	1	1	yes	characteristic
92	#533	C₂²×C₅	1	1	yes	characteristic
93	#446	—	2	1	no	—
94	#446	C₄×C₅	1	2	yes	—
95	C₈×Q₈	—	2	1	no	—
96	C₈×Q₈	C₄×C₅	1	2	yes	—
97	C₂×#108	—	2	1	no	—
98	C₂×#108	—	2	1	no	—
99	#424	—	2	1	no	—
100	#423	C₂²×C₅	1	2	yes	—
101	#423	C₂²×C₅	1	1	yes	characteristic
102	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
103	#405	—	2	1	no	—
104	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
105	#335	—	2	1	no	—
106	#335	—	2	1	no	—
107	#333	—	2	2	no	—
108	#332	—	2	2	no	—
109	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
110	C₅×Q₈	—	4	2	no	—
111	C₅×Q₈	—	4	1	no	—
112	C₅×Q₈	—	4	1	no	—
113	C₂×C₄×C₅	—	2	1	no	—
114	C₂×C₄×C₅	—	2	1	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	4	1	no	—
118	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×C₄⋊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	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
126	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₅	#137	1	1	yes	characteristic
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	#103	1	2	yes	—
134	C₅×C₈	#103	1	2	yes	—
135	C₂×M₁₆	—	2	1	no	—
136	C₂×M₁₆	—	2	2	no	—
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
139	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
140	C₄⋊Q₈	C₅×D₄	1	1	yes	characteristic
141	C₄⋊Q₈	C₅×D₄	1	1	yes	characteristic
142	#125	—	2	1	no	—
143	C₂²⋊Q₈	—	2	1	no	—
144	C₂²⋊Q₈	C₅×Q₈	1	2	yes	—
145	C₄×Q₈	—	2	1	no	—
146	C₄×Q₈	C₂×C₄×C₅	1	2	yes	—
147	#117	—	2	1	no	—
148	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
149	#108	—	4	2	no	—
150	#108	—	4	2	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	1	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
159	#97	—	2	2	no	—
160	C₄×C₈	—	2	2	no	—
161	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
162	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
163	C₄×C₈	C₅×D₄	1	1	yes	characteristic
164	C₄×C₈	C₅×D₄	1	1	yes	characteristic
165	C₂²×C₅	—	2	1	no	—
166	C₂²×C₅	—	2	1	no	—
167	C₂²×C₅	#380	1	1	yes	characteristic
168	C₂²×C₅	#475	1	1	yes	characteristic
169	C₂²×C₅	#477	1	1	yes	characteristic
170	C₄×C₅	—	2	1	no	—
171	C₄×C₅	—	2	1	no	—
172	C₄×C₅	—	2	1	no	—
173	C₄×C₅	—	4	2	no	—
174	C₄×C₅	—	4	2	no	—
175	C₄×C₅	C₂×#103	1	1	yes	characteristic
176	C₄×C₅	#417	1	1	yes	characteristic
177	C₄×C₅	#460	1	1	yes	characteristic
178	C₄×C₅	#462	1	1	yes	characteristic
179	C₄×C₅	#472	1	1	yes	characteristic
180	C₄×C₅	#473	1	1	yes	characteristic
181	C₂×Q₈	—	2	1	no	—
182	C₂×Q₈	C₅×C₄⋊C₄	1	2	yes	—
183	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
185	C₂²×C₄	C₅×#41	1	1	yes	characteristic
186	M₁₆	—	4	1	no	—
187	M₁₆	—	4	1	no	—
188	M₁₆	—	4	2	no	—
189	M₁₆	—	4	2	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	8	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	C₂×C₈	C₅×SD₁₆	1	2	yes	—
201	C₂×C₈	C₅×Q₁₆	1	2	yes	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	C₅×C₄⋊C₄	1	2	yes	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
211	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
212	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
213	C₄²	C₅×#41	1	1	yes	characteristic
214	C₂×C₅	—	2	1	no	—
215	C₂×C₅	—	2	1	no	—
216	C₂×C₅	#1803	1	1	yes	characteristic
217	C₂×C₅	#1813	1	1	yes	characteristic
218	C₂×C₅	#1816	1	1	yes	characteristic
219	C₂³	C₅×C₂²⋊Q₈	1	1	yes	characteristic
220	Q₈	—	4	2	no	—
221	Q₈	—	4	1	no	—
222	Q₈	—	4	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	4	1	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×#121	1	1	yes	characteristic
233	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
234	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
235	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
236	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
237	C₂×C₄	C₅×#137	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	4	2	no	—
241	C₈	—	8	2	no	—
242	C₈	—	8	2	no	—
243	C₈	C₅×#103	1	2	yes	—
244	C₈	C₅×#103	1	2	yes	—
245	C₅	#8685	1	1	yes	characteristic, direct product factor, Sylow
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	C₅×#380	1	1	yes	characteristic
249	C₂²	C₅×#475	1	1	yes	characteristic
250	C₂²	C₅×#477	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	2	no	—
255	C₄	—	4	2	no	—
256	C₄	C₂×C₅×#103	1	1	yes	characteristic
257	C₄	C₅×#417	1	1	yes	characteristic
258	C₄	C₅×#460	1	1	yes	characteristic
259	C₄	C₅×#462	1	1	yes	characteristic
260	C₄	C₅×#472	1	1	yes	characteristic
261	C₄	C₅×#473	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	C₅×#1803	1	1	yes	characteristic
265	C₂	C₅×#1813	1	1	yes	characteristic
266	C₂	C₅×#1816	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#380
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ 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. #8685	C₅	direct (N × H)
245. C₅	#8685	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#2960	2
C₅×#1848	2
C₂×C₅×#335	2
C₅×#1550	2
C₅×#1292	2
#8685	5

Subgroups — C5×#8685 #11931405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8685 #11931405