Subgroups — C₅×#7601 #11930321

Order 1280.

492 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#124	C₂	1	1	yes	maximal, characteristic
2	C₅×#1831	C₂	1	2	yes	maximal, complemented
3	C₅×#1272	C₂	1	2	yes	maximal, complemented
4	C₅×#1270	C₂	1	2	yes	maximal
5	C₂×C₅×#124	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×Q₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×D₄	C₄	1	1	yes	characteristic
8	C₂×C₅×C₄⋊C₈	—	2	2	no	—
9	C₅×#391	C₂²	1	1	yes	characteristic
10	C₅×#390	C₂²	1	1	yes	characteristic
11	C₅×#388	—	2	2	no	—
12	C₅×#382	C₄	1	1	yes	characteristic
13	C₄×C₅×C₂²⋊C₄	—	2	2	no	—
14	C₄³×C₅	C₂²	1	1	yes	characteristic
15	C₅×#336	—	2	2	no	—
16	C₅×#336	C₂²	1	4	yes	—
17	#7601	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅×D₄	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₅	—	2	1	no	—
21	C₂²×C₅×C₈	—	2	4	no	—
22	C₅×#124	—	2	2	no	—
23	C₅×#124	C₈	1	4	yes	complemented
24	C₄×C₅×Q₈	—	4	2	no	—
25	C₄×C₅×D₄	—	4	2	no	—
26	C₂×C₅×C₄⋊C₄	—	2	1	no	—
27	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
28	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
29	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
30	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
31	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
32	C₂×C₄²×C₅	—	2	1	no	—
33	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	D₄	1	2	yes	—
36	C₂×C₄²×C₅	D₄	1	2	yes	—
37	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
38	C₅×C₄⋊C₈	—	4	4	no	—
39	C₅×#95	—	2	2	no	—
40	C₅×#95	—	2	2	no	—
41	C₄×#124	C₂×C₅	1	1	yes	characteristic
42	#1831	C₂×C₅	1	2	yes	complemented
43	#1272	C₂×C₅	1	2	yes	complemented
44	#1270	C₂×C₅	1	2	yes	—
45	C₂⁴×C₅	—	2	1	no	—
46	C₂×C₅×Q₈	—	2	1	no	—
47	C₂×C₅×Q₈	—	4	1	no	—
48	C₂×C₅×Q₈	—	4	1	no	—
49	C₂×C₅×Q₈	C₂×C₈	1	2	yes	—
50	C₂×C₅×D₄	—	2	1	no	—
51	C₂×C₅×D₄	—	4	1	no	—
52	C₂×C₅×D₄	—	4	1	no	—
53	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₅	—	2	1	no	—
55	C₂²×C₄×C₅	—	2	1	no	—
56	C₂²×C₄×C₅	—	2	1	no	—
57	C₂²×C₄×C₅	—	4	4	no	—
58	C₂²×C₄×C₅	—	4	2	no	—
59	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
64	C₂×C₅×C₈	—	4	4	no	—
65	C₂×C₅×C₈	—	4	4	no	—
66	C₂×C₅×C₈	—	4	4	no	—
67	C₅×C₄⋊C₄	—	4	1	no	—
68	C₅×C₄⋊C₄	—	4	1	no	—
69	C₅×C₄⋊C₄	—	4	2	no	—
70	C₅×C₂²⋊C₄	—	2	8	no	—
71	C₅×C₂²⋊C₄	—	4	2	no	—
72	C₅×C₂²⋊C₄	—	4	2	no	—
73	C₅×C₂²⋊C₄	—	4	2	no	—
74	C₄²×C₅	—	2	2	no	—
75	C₄²×C₅	—	2	2	no	—
76	C₄²×C₅	—	2	2	no	—
77	C₄²×C₅	—	4	2	no	—
78	C₄²×C₅	—	4	1	no	—
79	C₄²×C₅	—	4	2	no	—
80	C₄²×C₅	C₂×C₈	1	2	yes	—
81	C₄²×C₅	M₁₆	1	2	yes	—
82	C₂×#124	C₄×C₅	1	1	yes	characteristic
83	C₂×C₄×Q₈	C₄×C₅	1	1	yes	characteristic
84	C₂×C₄×D₄	C₄×C₅	1	1	yes	characteristic
85	C₂×C₄⋊C₈	—	2	2	no	—
86	#391	C₂²×C₅	1	1	yes	characteristic
87	#390	C₂²×C₅	1	1	yes	characteristic
88	#388	—	2	2	no	—
89	#382	C₄×C₅	1	1	yes	characteristic
90	C₄×C₂²⋊C₄	—	2	2	no	—
91	C₄³	C₂²×C₅	1	1	yes	characteristic
92	#336	—	2	2	no	—
93	#336	C₂²×C₅	1	4	yes	—
94	C₂³×C₅	—	2	2	no	—
95	C₂³×C₅	—	4	1	no	—
96	C₂³×C₅	—	4	1	no	—
97	C₂³×C₅	—	4	2	no	—
98	C₂³×C₅	—	4	2	no	—
99	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₅×Q₈	—	4	2	no	—
101	C₅×Q₈	—	4	2	no	—
102	C₅×D₄	—	4	2	no	—
103	C₅×D₄	—	4	2	no	—
104	C₂×C₄×C₅	—	2	2	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	—	2	2	no	—
109	C₂×C₄×C₅	—	2	2	no	—
110	C₂×C₄×C₅	—	2	2	no	—
111	C₂×C₄×C₅	—	2	2	no	—
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	4	1	no	—
115	C₂×C₄×C₅	—	4	1	no	—
116	C₂×C₄×C₅	—	4	1	no	—
117	C₂×C₄×C₅	—	4	1	no	—
118	C₂×C₄×C₅	—	4	2	no	—
119	C₂×C₄×C₅	—	4	2	no	—
120	C₂×C₄×C₅	—	8	4	no	—
121	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
122	C₂×C₄×C₅	#100	1	2	yes	—
123	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×M₁₆	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	—	8	4	no	—
128	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
129	C₂²×D₄	C₂×C₄×C₅	1	1	yes	characteristic
130	C₂³×C₄	—	2	1	no	—
131	C₂²×C₈	—	2	4	no	—
132	#124	—	2	2	no	—
133	#124	C₅×C₈	1	4	yes	complemented
134	C₄×Q₈	—	4	2	no	—
135	C₄×D₄	—	4	2	no	—
136	C₂×C₄⋊C₄	—	2	1	no	—
137	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
138	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
139	C₂×C₂²⋊C₄	—	2	2	no	—
140	C₂×C₂²⋊C₄	—	2	1	no	—
141	C₂×C₂²⋊C₄	—	2	2	no	—
142	C₂×C₄²	—	2	1	no	—
143	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
144	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
145	C₂×C₄²	C₅×D₄	1	2	yes	—
146	C₂×C₄²	C₅×D₄	1	2	yes	—
147	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
148	C₄⋊C₈	—	4	4	no	—
149	#95	—	2	2	no	—
150	#95	—	2	2	no	—
151	C₂²×C₅	—	4	2	no	—
152	C₂²×C₅	—	4	4	no	—
153	C₂²×C₅	—	8	1	no	—
154	C₂²×C₅	—	8	2	no	—
155	C₂²×C₅	—	8	1	no	—
156	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
157	C₂²×C₅	#410	1	1	yes	characteristic
158	C₂²×C₅	C₂×#100	1	1	yes	characteristic
159	C₂²×C₅	#417	1	2	yes	—
160	C₂²×C₅	#421	1	2	yes	—
161	C₄×C₅	—	2	1	no	—
162	C₄×C₅	—	2	1	no	—
163	C₄×C₅	—	2	4	no	—
164	C₄×C₅	—	2	2	no	—
165	C₄×C₅	—	2	2	no	—
166	C₄×C₅	—	2	2	no	—
167	C₄×C₅	—	4	1	no	—
168	C₄×C₅	—	4	2	no	—
169	C₄×C₅	—	8	1	no	—
170	C₄×C₅	—	8	1	no	—
171	C₂⁴	—	2	1	no	—
172	C₂×Q₈	—	2	1	no	—
173	C₂×Q₈	—	4	1	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂×Q₈	C₂×C₅×C₈	1	2	yes	—
176	C₂×D₄	—	2	1	no	—
177	C₂×D₄	—	4	1	no	—
178	C₂×D₄	—	4	1	no	—
179	C₂×D₄	C₂×C₅×C₈	1	2	yes	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	4	4	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
186	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
187	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
188	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
190	C₂×C₈	—	4	4	no	—
191	C₂×C₈	—	4	4	no	—
192	C₂×C₈	—	4	4	no	—
193	C₄⋊C₄	—	4	1	no	—
194	C₄⋊C₄	—	4	1	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₂²⋊C₄	—	2	8	no	—
197	C₂²⋊C₄	—	4	2	no	—
198	C₂²⋊C₄	—	4	2	no	—
199	C₂²⋊C₄	—	4	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	4	2	no	—
204	C₄²	—	4	1	no	—
205	C₄²	—	4	2	no	—
206	C₄²	C₂×C₅×C₈	1	2	yes	—
207	C₄²	C₅×M₁₆	1	2	yes	—
208	C₂×C₅	—	8	2	no	—
209	C₂×C₅	#1460	1	1	yes	characteristic
210	C₂×C₅	#1477	1	2	yes	—
211	C₂×C₅	#1512	1	2	yes	—
212	C₂×C₅	#1533	1	1	yes	characteristic
213	C₂×C₅	#1560	1	1	yes	characteristic
214	C₂³	—	2	2	no	—
215	C₂³	—	4	1	no	—
216	C₂³	—	4	1	no	—
217	C₂³	—	4	2	no	—
218	C₂³	—	4	2	no	—
219	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
220	Q₈	—	4	2	no	—
221	Q₈	—	4	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	8	4	no	—
241	C₂×C₄	C₅×C₂²⋊C₈	1	4	yes	—
242	C₂×C₄	C₅×#100	1	2	yes	—
243	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
244	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
245	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
246	C₂×C₄	C₅×#137	1	2	yes	—
247	C₈	—	8	4	no	—
248	C₅	#7601	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²	—	4	2	no	—
250	C₂²	—	4	4	no	—
251	C₂²	—	8	2	no	—
252	C₂²	—	8	1	no	—
253	C₂²	—	8	1	no	—
254	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
255	C₂²	C₅×#410	1	1	yes	characteristic
256	C₂²	C₂×C₅×#100	1	1	yes	characteristic
257	C₂²	C₅×#417	1	2	yes	—
258	C₂²	C₅×#421	1	2	yes	—
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	4	no	—
265	C₄	—	4	1	no	—
266	C₄	—	4	2	no	—
267	C₄	—	8	1	no	—
268	C₄	—	8	1	no	—
269	C₂	—	8	2	no	—
270	C₂	C₅×#1460	1	1	yes	characteristic
271	C₂	C₅×#1477	1	2	yes	—
272	C₂	C₅×#1512	1	2	yes	—
273	C₂	C₅×#1533	1	1	yes	characteristic
274	C₂	C₅×#1560	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂²×C₅×D₄	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1831	C₂	semidirect
3. C₅×#1272	C₂	semidirect
17. #7601	C₅	direct (N × H)
23. C₅×#124	C₈	semidirect
42. #1831	C₂×C₅	semidirect
43. #1272	C₂×C₅	semidirect
133. #124	C₅×C₈	semidirect
248. C₅	#7601	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅×#124	2
C₅×#1831	2
C₅×#1272	2
C₅×#1270	2
#7601	5

Subgroups — C5×#7601 #11930321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7601 #11930321