Subgroups — C₅×#7988 #11930708

Order 1280.

598 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×#2920	C₂	1	2	yes	maximal
2	C₂²×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#1290	C₂	1	4	yes	maximal, complemented
4	C₂³×C₅×C₈	C₄	1	2	yes	—
5	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
6	C₅×#432	C₄	1	4	yes	—
7	C₅×#432	C₂²	1	4	yes	—
8	C₅×#431	—	2	2	no	—
9	C₅×#431	—	2	2	no	—
10	C₅×#423	—	2	2	no	—
11	C₂×C₅×C₄⋊C₈	—	2	2	no	—
12	C₂×C₅×C₄⋊C₈	—	2	2	no	—
13	C₂×C₅×C₄⋊C₈	C₄	1	2	yes	—
14	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
15	C₅×#407	—	2	2	no	—
16	C₄×C₅×M₁₆	C₄	1	4	yes	—
17	#7988	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
21	C₂×C₅×M₁₆	—	2	4	no	—
22	C₂²×C₅×C₈	—	2	2	no	—
23	C₂²×C₅×C₈	—	2	2	no	—
24	C₂²×C₅×C₈	—	4	4	no	—
25	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅	D₄	1	2	yes	—
31	C₂×C₄²×C₅	D₄	1	2	yes	—
32	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
34	C₅×C₄⋊C₈	—	2	4	no	—
35	C₅×C₄⋊C₈	—	2	4	no	—
36	C₅×C₄⋊C₈	—	2	4	no	—
37	C₅×C₄⋊C₈	—	2	4	no	—
38	C₅×C₂²⋊C₈	—	2	4	no	—
39	C₅×C₂²⋊C₈	—	2	4	no	—
40	C₅×#97	C₂×C₄	1	8	yes	—
41	C₄×C₅×C₈	—	2	4	no	—
42	#2920	C₂×C₅	1	2	yes	—
43	C₂²×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
44	#1290	C₂×C₅	1	4	yes	complemented
45	C₂⁴×C₅	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅	—	2	2	no	—
47	C₂²×C₄×C₅	—	2	2	no	—
48	C₂²×C₄×C₅	—	2	4	no	—
49	C₂²×C₄×C₅	—	4	2	no	—
50	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
64	C₅×M₁₆	—	4	8	no	—
65	C₂×C₅×C₈	—	2	4	no	—
66	C₂×C₅×C₈	—	2	8	no	—
67	C₂×C₅×C₈	—	4	4	no	—
68	C₂×C₅×C₈	—	4	4	no	—
69	C₂×C₅×C₈	—	4	4	no	—
70	C₄²×C₅	C₄²	1	2	yes	—
71	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₅	C₄⋊C₄	1	2	yes	—
75	C₄²×C₅	C₄⋊C₄	1	2	yes	—
76	C₄²×C₅	C₂²×C₄	1	2	yes	—
77	C₄²×C₅	C₂×D₄	1	2	yes	—
78	C₂³×C₈	C₄×C₅	1	2	yes	—
79	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
80	#432	C₄×C₅	1	4	yes	—
81	#432	C₂²×C₅	1	4	yes	—
82	#431	—	2	2	no	—
83	#431	—	2	2	no	—
84	#423	—	2	2	no	—
85	C₂×C₄⋊C₈	—	2	2	no	—
86	C₂×C₄⋊C₈	—	2	2	no	—
87	C₂×C₄⋊C₈	C₄×C₅	1	2	yes	—
88	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
89	#407	—	2	2	no	—
90	C₄×M₁₆	C₄×C₅	1	4	yes	—
91	C₂³×C₅	—	2	2	no	—
92	C₂³×C₅	—	2	2	no	—
93	C₂³×C₅	—	2	2	no	—
94	C₂³×C₅	#95	1	1	yes	characteristic
95	C₂³×C₅	#95	1	1	yes	characteristic
96	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅	—	2	4	no	—
98	C₂×C₄×C₅	—	2	2	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	2	4	no	—
101	C₂×C₄×C₅	—	2	4	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	—	2	1	no	—
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	—	4	2	no	—
109	C₂×C₄×C₅	—	4	1	no	—
110	C₂×C₄×C₅	—	4	1	no	—
111	C₂×C₄×C₅	#95	1	2	yes	—
112	C₂×C₄×C₅	#95	1	1	yes	characteristic
113	C₂×C₄×C₅	#95	1	2	yes	—
114	C₂×C₄×C₅	#95	1	1	yes	characteristic
115	C₂×C₄×C₅	#100	1	2	yes	—
116	C₂×C₄×C₅	#101	1	2	yes	—
117	C₂×C₄×C₅	#108	1	4	yes	—
118	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
120	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
122	C₅×C₈	—	4	8	no	—
123	C₅×C₈	—	4	4	no	—
124	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
125	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
126	C₂³×C₄	C₅×Q₈	1	1	yes	characteristic
127	C₂×M₁₆	—	2	4	no	—
128	C₂²×C₈	—	2	2	no	—
129	C₂²×C₈	—	2	2	no	—
130	C₂²×C₈	—	4	4	no	—
131	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
132	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
133	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
134	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
135	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
136	C₂×C₄²	C₅×D₄	1	2	yes	—
137	C₂×C₄²	C₅×D₄	1	2	yes	—
138	C₂×C₄²	C₅×Q₈	1	1	yes	characteristic
139	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
140	C₄⋊C₈	—	2	4	no	—
141	C₄⋊C₈	—	2	4	no	—
142	C₄⋊C₈	—	2	4	no	—
143	C₄⋊C₈	—	2	4	no	—
144	C₂²⋊C₈	—	2	4	no	—
145	C₂²⋊C₈	—	2	4	no	—
146	#97	C₂×C₄×C₅	1	8	yes	—
147	C₄×C₈	—	2	4	no	—
148	C₂²×C₅	—	2	8	no	—
149	C₂²×C₅	—	2	1	no	—
150	C₂²×C₅	—	2	1	no	—
151	C₂²×C₅	—	2	1	no	—
152	C₂²×C₅	—	2	1	no	—
153	C₂²×C₅	—	2	1	no	—
154	C₂²×C₅	—	2	1	no	—
155	C₂²×C₅	H(3,ℤ/4)	1	2	yes	—
156	C₂²×C₅	#343	1	2	yes	—
157	C₂²×C₅	C₂×#95	1	1	yes	characteristic
158	C₂²×C₅	C₂×#108	1	2	yes	—
159	C₄×C₅	—	2	2	no	—
160	C₄×C₅	—	2	2	no	—
161	C₄×C₅	—	2	2	no	—
162	C₄×C₅	—	2	1	no	—
163	C₄×C₅	—	2	1	no	—
164	C₄×C₅	#341	1	4	yes	—
165	C₄×C₅	#343	1	2	yes	—
166	C₄×C₅	C₂×#100	1	1	yes	characteristic
167	C₄×C₅	C₂×#101	1	1	yes	characteristic
168	C₂⁴	C₅×C₄⋊C₄	1	1	yes	characteristic
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	C₄²×C₅	1	1	yes	characteristic
174	C₂²×C₄	C₄²×C₅	1	1	yes	characteristic
175	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
176	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
177	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
178	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
179	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
180	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
181	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
182	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
183	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
184	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
187	M₁₆	—	4	8	no	—
188	C₂×C₈	—	2	8	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	4	4	no	—
191	C₂×C₈	—	4	4	no	—
192	C₂×C₈	—	4	4	no	—
193	C₄²	C₄²×C₅	1	2	yes	—
194	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
195	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
196	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
197	C₄²	C₅×C₄⋊C₄	1	2	yes	—
198	C₄²	C₅×C₄⋊C₄	1	2	yes	—
199	C₄²	C₂²×C₄×C₅	1	2	yes	—
200	C₄²	C₂×C₅×D₄	1	2	yes	—
201	C₂×C₅	—	2	2	no	—
202	C₂×C₅	—	2	2	no	—
203	C₂×C₅	—	2	2	no	—
204	C₂×C₅	C₂×H(3,ℤ/4)	1	1	yes	characteristic
205	C₂×C₅	C₂×#341	1	1	yes	characteristic
206	C₂×C₅	C₂×#343	1	1	yes	characteristic
207	C₂³	—	2	2	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	C₅×#95	1	1	yes	characteristic
211	C₂³	C₅×#95	1	1	yes	characteristic
212	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	C₅×#95	1	2	yes	—
228	C₂×C₄	C₅×#95	1	1	yes	characteristic
229	C₂×C₄	C₅×#95	1	2	yes	—
230	C₂×C₄	C₅×#95	1	1	yes	characteristic
231	C₂×C₄	C₅×#100	1	2	yes	—
232	C₂×C₄	C₅×#101	1	2	yes	—
233	C₂×C₄	C₅×#108	1	4	yes	—
234	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
235	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
236	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
238	C₈	—	4	8	no	—
239	C₈	—	4	4	no	—
240	C₅	#7988	1	1	yes	characteristic, direct product factor, Sylow
241	C₂²	—	2	8	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	C₅×H(3,ℤ/4)	1	2	yes	—
249	C₂²	C₅×#343	1	2	yes	—
250	C₂²	C₂×C₅×#95	1	1	yes	characteristic
251	C₂²	C₂×C₅×#108	1	2	yes	—
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	C₅×#341	1	4	yes	—
258	C₄	C₅×#343	1	2	yes	—
259	C₄	C₂×C₅×#100	1	1	yes	characteristic
260	C₄	C₂×C₅×#101	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	2	2	no	—
263	C₂	—	2	2	no	—
264	C₂	C₂×C₅×H(3,ℤ/4)	1	1	yes	characteristic
265	C₂	C₂×C₅×#341	1	1	yes	characteristic
266	C₂	C₂×C₅×#343	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#95
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	C₂×#95
prime radical	C₂⁴×C₅	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1290	C₂	semidirect
17. #7988	C₅	direct (N × H)
44. #1290	C₂×C₅	semidirect
240. C₅	#7988	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×#2920	2
C₂²×C₅×C₄⋊C₈	2
C₅×#1290	2
#7988	5

Subgroups — C5×#7988 #11930708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7988 #11930708