Subgroups — C₅×#9877 #11932597

Order 1280.

518 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#1831	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#97	C₂	1	1	yes	maximal, characteristic
4	C₅×#1279	C₂	1	4	yes	maximal
5	C₂×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×Q₈	—	2	1	no	—
7	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₅×#97	—	2	1	no	—
9	C₂×C₅×#97	—	2	1	no	—
10	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
11	C₅×#391	—	2	1	no	—
12	C₅×#389	—	2	1	no	—
13	C₅×#389	C₄	1	2	yes	—
14	C₅×#384	C₄	1	1	yes	characteristic
15	C₄×C₅×C₄⋊C₄	—	2	1	no	—
16	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
17	C₄³×C₅	C₂²	1	1	yes	characteristic
18	C₅×#336	—	2	1	no	—
19	C₅×#336	—	2	1	no	—
20	#9877	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×Q₈	—	2	1	no	—
22	C₂²×C₅×C₈	—	2	2	no	—
23	C₂²×C₅×C₈	—	2	2	no	—
24	C₅×C₄⋊Q₈	—	2	4	no	—
25	C₄×C₅×Q₈	—	4	4	no	—
26	C₂×C₅×C₄⋊C₄	—	2	1	no	—
27	C₂×C₅×C₄⋊C₄	—	2	1	no	—
28	C₂×C₅×C₄⋊C₄	—	2	1	no	—
29	C₂×C₅×C₄⋊C₄	—	2	2	no	—
30	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅	—	2	2	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	1	yes	characteristic
36	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₅×C₄⋊C₈	—	2	4	no	—
41	C₅×#97	—	2	4	no	—
42	C₅×#97	—	2	4	no	—
43	C₄×C₅×C₈	—	2	2	no	—
44	C₄×C₅×C₈	Q₈	1	4	yes	complemented
45	C₅×#95	—	2	2	no	—
46	C₅×#95	—	2	2	no	—
47	C₄×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
48	#1831	C₂×C₅	1	1	yes	characteristic
49	C₄×#97	C₂×C₅	1	1	yes	characteristic
50	#1279	C₂×C₅	1	4	yes	—
51	C₂×C₅×Q₈	—	2	4	no	—
52	C₂×C₅×Q₈	—	4	2	no	—
53	C₂×C₅×Q₈	—	4	2	no	—
54	C₂²×C₄×C₅	—	2	2	no	—
55	C₂²×C₄×C₅	—	2	2	no	—
56	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₅×C₈	—	2	4	no	—
64	C₂×C₅×C₈	—	2	4	no	—
65	C₂×C₅×C₈	—	2	4	no	—
66	C₂×C₅×C₈	—	4	2	no	—
67	C₂×C₅×C₈	—	4	2	no	—
68	C₂×C₅×C₈	—	4	2	no	—
69	C₅×C₄⋊C₄	—	2	4	no	—
70	C₅×C₄⋊C₄	—	2	2	no	—
71	C₅×C₄⋊C₄	—	2	4	no	—
72	C₅×C₄⋊C₄	—	4	2	no	—
73	C₅×C₄⋊C₄	—	4	4	no	—
74	C₅×C₄⋊C₄	—	4	2	no	—
75	C₅×C₄⋊C₄	M₁₆	1	4	yes	—
76	C₄²×C₅	—	2	2	no	—
77	C₄²×C₅	—	2	2	no	—
78	C₄²×C₅	—	2	2	no	—
79	C₄²×C₅	—	2	2	no	—
80	C₄²×C₅	—	2	2	no	—
81	C₄²×C₅	—	2	2	no	—
82	C₄²×C₅	—	4	4	no	—
83	C₄²×C₅	C₂×Q₈	1	2	yes	—
84	C₄²×C₅	#41	1	2	yes	—
85	C₂×C₄⋊Q₈	C₄×C₅	1	1	yes	characteristic
86	C₂×C₄×Q₈	—	2	1	no	—
87	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
88	C₂×#97	—	2	1	no	—
89	C₂×#97	—	2	1	no	—
90	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
91	#391	—	2	1	no	—
92	#389	—	2	1	no	—
93	#389	C₄×C₅	1	2	yes	—
94	#384	C₄×C₅	1	1	yes	characteristic
95	C₄×C₄⋊C₄	—	2	1	no	—
96	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
97	C₄³	C₂²×C₅	1	1	yes	characteristic
98	#336	—	2	1	no	—
99	#336	—	2	1	no	—
100	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₅×Q₈	—	4	8	no	—
102	C₂×C₄×C₅	—	2	1	no	—
103	C₂×C₄×C₅	—	2	1	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	—	2	4	no	—
109	C₂×C₄×C₅	—	2	1	no	—
110	C₂×C₄×C₅	—	2	1	no	—
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	2	1	no	—
114	C₂×C₄×C₅	—	2	1	no	—
115	C₂×C₄×C₅	—	4	2	no	—
116	C₂×C₄×C₅	—	4	2	no	—
117	C₂×C₄×C₅	—	4	2	no	—
118	C₂×C₄×C₅	—	4	2	no	—
119	C₂×C₄×C₅	—	4	2	no	—
120	C₂×C₄×C₅	#104	1	2	yes	—
121	C₂×C₄×C₅	#104	1	2	yes	—
122	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₅	#124	1	1	yes	characteristic
125	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
127	C₂×C₄×C₅	#131	1	2	yes	—
128	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
129	C₂×C₄×C₅	#137	1	2	yes	—
130	C₅×C₈	—	2	8	no	—
131	C₅×C₈	—	8	2	no	—
132	C₂²×Q₈	—	2	1	no	—
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	2	2	no	—
135	C₄⋊Q₈	—	2	4	no	—
136	C₄×Q₈	—	4	4	no	—
137	C₂×C₄⋊C₄	—	2	1	no	—
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	2	1	no	—
141	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
142	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
143	C₂×C₄²	—	2	2	no	—
144	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
145	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
146	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
147	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
148	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
149	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
150	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
151	C₄⋊C₈	—	2	4	no	—
152	#97	—	2	4	no	—
153	#97	—	2	4	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	C₅×Q₈	1	4	yes	complemented
156	#95	—	2	2	no	—
157	#95	—	2	2	no	—
158	C₂²×C₅	#391	1	1	yes	characteristic
159	C₂²×C₅	#407	1	1	yes	characteristic
160	C₂²×C₅	#408	1	1	yes	characteristic
161	C₂²×C₅	#418	1	1	yes	characteristic
162	C₂²×C₅	#419	1	1	yes	characteristic
163	C₂²×C₅	C₂×#104	1	1	yes	characteristic
164	C₂²×C₅	C₂×#104	1	1	yes	characteristic
165	C₄×C₅	—	2	2	no	—
166	C₄×C₅	—	2	2	no	—
167	C₄×C₅	—	2	2	no	—
168	C₄×C₅	—	4	1	no	—
169	C₄×C₅	—	4	1	no	—
170	C₄×C₅	—	4	4	no	—
171	C₄×C₅	—	8	2	no	—
172	C₄×C₅	#446	1	4	yes	—
173	C₄×C₅	#489	1	2	yes	—
174	C₄×C₅	#501	1	2	yes	—
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
181	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
182	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
183	C₂²×C₄	C₅×C₂²⋊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×D₄	1	1	yes	characteristic
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	4	2	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	4	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	C₅×M₁₆	1	4	yes	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	4	4	no	—
207	C₄²	C₂×C₅×Q₈	1	2	yes	—
208	C₄²	C₅×#41	1	2	yes	—
209	C₂×C₅	#1484	1	1	yes	characteristic
210	C₂×C₅	#1485	1	1	yes	characteristic
211	C₂×C₅	#1490	1	1	yes	characteristic
212	C₂×C₅	#1491	1	1	yes	characteristic
213	C₂×C₅	#1985	1	1	yes	characteristic
214	C₂×C₅	#1993	1	1	yes	characteristic
215	C₂×C₅	#1994	1	1	yes	characteristic
216	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
217	Q₈	—	4	8	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	C₅×#104	1	2	yes	—
237	C₂×C₄	C₅×#104	1	2	yes	—
238	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
239	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
240	C₂×C₄	C₅×#124	1	1	yes	characteristic
241	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
242	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
243	C₂×C₄	C₅×#131	1	2	yes	—
244	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
245	C₂×C₄	C₅×#137	1	2	yes	—
246	C₈	—	2	8	no	—
247	C₈	—	8	2	no	—
248	C₅	#9877	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²	C₅×#391	1	1	yes	characteristic
250	C₂²	C₅×#407	1	1	yes	characteristic
251	C₂²	C₅×#408	1	1	yes	characteristic
252	C₂²	C₅×#418	1	1	yes	characteristic
253	C₂²	C₅×#419	1	1	yes	characteristic
254	C₂²	C₂×C₅×#104	1	1	yes	characteristic
255	C₂²	C₂×C₅×#104	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	4	1	no	—
260	C₄	—	4	1	no	—
261	C₄	—	4	4	no	—
262	C₄	—	8	2	no	—
263	C₄	C₅×#446	1	4	yes	—
264	C₄	C₅×#489	1	2	yes	—
265	C₄	C₅×#501	1	2	yes	—
266	C₂	C₅×#1484	1	1	yes	characteristic
267	C₂	C₅×#1485	1	1	yes	characteristic
268	C₂	C₅×#1490	1	1	yes	characteristic
269	C₂	C₅×#1491	1	1	yes	characteristic
270	C₂	C₅×#1985	1	1	yes	characteristic
271	C₂	C₅×#1993	1	1	yes	characteristic
272	C₂	C₅×#1994	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
20. #9877	C₅	direct (N × H)
44. C₄×C₅×C₈	Q₈	semidirect
155. C₄×C₈	C₅×Q₈	semidirect
248. C₅	#9877	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×C₄⋊Q₈	2
C₅×#1831	2
C₄×C₅×#97	2
C₅×#1279	2
#9877	5

Subgroups — C5×#9877 #11932597

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9877 #11932597