Subgroups — C₅×#8689 #11931409

Order 1280.

346 conjugacy classes of subgroups.

The subgroup lattice has 248 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×#1837	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	C₅×#424	—	2	1	no	—
12	C₅×#423	C₂²	1	2	yes	—
13	C₅×#423	C₂²	1	1	yes	characteristic
14	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
15	C₅×#405	—	2	1	no	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#341	—	2	1	no	—
18	C₅×#341	—	2	1	no	—
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	#8689	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×C₄⋊C₈	—	2	2	no	—
39	C₅×C₄⋊C₈	—	2	2	no	—
40	C₅×C₄⋊C₈	—	2	1	no	—
41	C₅×C₄⋊C₈	—	2	2	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	C₂×C₄	1	2	yes	—
46	C₅×#97	—	2	2	no	—
47	C₄×C₅×C₈	—	2	2	no	—
48	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
49	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₅×C₈	D₄	1	1	yes	characteristic
51	C₄×C₅×C₈	D₄	1	1	yes	characteristic
52	#2960	C₂×C₅	1	1	yes	characteristic
53	#1837	C₂×C₅	1	1	yes	characteristic
54	C₂×#335	C₂×C₅	1	1	yes	characteristic
55	#1550	C₂×C₅	1	2	yes	—
56	#1292	C₂×C₅	1	2	yes	—
57	C₂×C₅×Q₈	—	2	1	no	—
58	C₂×C₅×Q₈	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₅	#41	1	1	yes	characteristic
62	C₅×M₁₆	—	4	2	no	—
63	C₅×M₁₆	—	4	2	no	—
64	C₅×M₁₆	—	4	2	no	—
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	—	2	2	no	—
67	C₂×C₅×C₈	—	2	2	no	—
68	C₂×C₅×C₈	—	4	2	no	—
69	C₂×C₅×C₈	—	4	2	no	—
70	C₂×C₅×C₈	—	8	2	no	—
71	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₅×C₄⋊C₄	—	2	2	no	—
76	C₅×C₄⋊C₄	—	2	1	no	—
77	C₅×C₄⋊C₄	—	2	1	no	—
78	C₅×C₄⋊C₄	—	2	1	no	—
79	C₅×C₄⋊C₄	—	2	2	no	—
80	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
81	C₅×C₂²⋊C₄	—	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×D₄	1	1	yes	characteristic
86	C₄²×C₅	#41	1	1	yes	characteristic
87	#533	C₂²×C₅	1	1	yes	characteristic
88	#446	—	2	1	no	—
89	#446	C₄×C₅	1	2	yes	—
90	C₈×Q₈	—	2	1	no	—
91	C₈×Q₈	C₄×C₅	1	2	yes	—
92	#424	—	2	1	no	—
93	#423	C₂²×C₅	1	2	yes	—
94	#423	C₂²×C₅	1	1	yes	characteristic
95	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
96	#405	—	2	1	no	—
97	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
98	#341	—	2	1	no	—
99	#341	—	2	1	no	—
100	#335	—	2	1	no	—
101	#335	—	2	1	no	—
102	#333	—	2	2	no	—
103	#332	—	2	2	no	—
104	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
105	C₅×Q₈	—	4	2	no	—
106	C₅×Q₈	—	4	1	no	—
107	C₅×Q₈	—	4	1	no	—
108	C₂×C₄×C₅	—	2	1	no	—
109	C₂×C₄×C₅	—	2	1	no	—
110	C₂×C₄×C₅	—	2	2	no	—
111	C₂×C₄×C₅	—	2	2	no	—
112	C₂×C₄×C₅	—	4	1	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	#121	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₅	#135	1	2	yes	—
120	C₂×C₄×C₅	#137	1	2	yes	—
121	C₅×C₈	—	2	1	no	—
122	C₅×C₈	—	2	1	no	—
123	C₅×C₈	—	2	1	no	—
124	C₅×C₈	—	2	1	no	—
125	C₅×C₈	—	4	2	no	—
126	C₅×C₈	—	8	2	no	—
127	C₅×C₈	—	8	2	no	—
128	C₂×M₁₆	—	2	1	no	—
129	C₂×M₁₆	—	2	2	no	—
130	C₂²×C₈	—	2	2	no	—
131	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
132	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
133	C₄⋊Q₈	C₅×D₄	1	1	yes	characteristic
134	C₄⋊Q₈	C₅×D₄	1	1	yes	characteristic
135	#125	—	2	1	no	—
136	C₂²⋊Q₈	—	2	1	no	—
137	C₂²⋊Q₈	C₅×Q₈	1	2	yes	—
138	C₄×Q₈	—	2	1	no	—
139	C₄×Q₈	C₂×C₄×C₅	1	2	yes	—
140	#117	—	2	1	no	—
141	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	1	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
150	#97	—	2	2	no	—
151	C₄×C₈	—	2	2	no	—
152	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
153	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
154	C₄×C₈	C₅×D₄	1	1	yes	characteristic
155	C₄×C₈	C₅×D₄	1	1	yes	characteristic
156	C₂²×C₅	—	2	1	no	—
157	C₂²×C₅	—	2	1	no	—
158	C₂²×C₅	#380	1	1	yes	characteristic
159	C₂²×C₅	#479	1	2	yes	—
160	C₄×C₅	—	2	1	no	—
161	C₄×C₅	—	2	1	no	—
162	C₄×C₅	—	2	1	no	—
163	C₄×C₅	—	4	2	no	—
164	C₄×C₅	—	4	2	no	—
165	C₄×C₅	#416	1	1	yes	characteristic
166	C₄×C₅	#419	1	1	yes	characteristic
167	C₄×C₅	#464	1	2	yes	—
168	C₄×C₅	#472	1	1	yes	characteristic
169	C₄×C₅	#473	1	1	yes	characteristic
170	C₂×Q₈	—	2	1	no	—
171	C₂×Q₈	C₅×C₄⋊C₄	1	2	yes	—
172	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
174	C₂²×C₄	C₅×#41	1	1	yes	characteristic
175	M₁₆	—	4	2	no	—
176	M₁₆	—	4	2	no	—
177	M₁₆	—	4	2	no	—
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	—	8	2	no	—
184	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
185	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
186	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
187	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	C₅×C₄⋊C₄	1	2	yes	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₄²	—	2	2	no	—
196	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
197	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
198	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
199	C₄²	C₅×#41	1	1	yes	characteristic
200	C₂×C₅	—	2	2	no	—
201	C₂×C₅	#1805	1	1	yes	characteristic
202	C₂×C₅	#1813	1	1	yes	characteristic
203	C₂×C₅	#1816	1	1	yes	characteristic
204	C₂³	C₅×C₂²⋊Q₈	1	1	yes	characteristic
205	Q₈	—	4	2	no	—
206	Q₈	—	4	1	no	—
207	Q₈	—	4	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
214	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
215	C₂×C₄	C₅×#117	1	1	yes	characteristic
216	C₂×C₄	C₅×#121	1	1	yes	characteristic
217	C₂×C₄	C₅×#121	1	1	yes	characteristic
218	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
219	C₂×C₄	C₅×#135	1	2	yes	—
220	C₂×C₄	C₅×#137	1	2	yes	—
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	4	2	no	—
226	C₈	—	8	2	no	—
227	C₈	—	8	2	no	—
228	C₅	#8689	1	1	yes	characteristic, direct product factor, Sylow
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	C₅×#380	1	1	yes	characteristic
232	C₂²	C₅×#479	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	2	no	—
238	C₄	C₅×#416	1	1	yes	characteristic
239	C₄	C₅×#419	1	1	yes	characteristic
240	C₄	C₅×#464	1	2	yes	—
241	C₄	C₅×#472	1	1	yes	characteristic
242	C₄	C₅×#473	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	C₅×#1805	1	1	yes	characteristic
245	C₂	C₅×#1813	1	1	yes	characteristic
246	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. #8689	C₅	direct (N × H)
228. C₅	#8689	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×#1837	2
C₂×C₅×#335	2
C₅×#1550	2
C₅×#1292	2
#8689	5

Subgroups — C5×#8689 #11931409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8689 #11931409