Subgroups — C₅×#9876 #11932596

Order 1280.

486 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#125	C₂	1	1	yes	maximal, characteristic
2	C₅×#1842	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×#125	C₄	1	1	yes	characteristic
6	C₂×C₅×C₄⋊C₈	—	2	2	no	—
7	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₅×#97	—	2	2	no	—
9	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
10	C₅×#389	C₄	1	2	yes	—
11	C₅×#387	—	2	2	no	—
12	C₅×#382	C₄	1	1	yes	characteristic
13	C₄×C₅×C₄⋊C₄	—	2	2	no	—
14	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
15	C₄³×C₅	C₂²	1	1	yes	characteristic
16	#9876	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂²×C₅×C₈	—	2	2	no	—
18	C₂²×C₅×C₈	—	2	2	no	—
19	C₅×#125	—	2	4	no	—
20	C₂×C₅×C₄⋊C₄	—	2	2	no	—
21	C₂×C₅×C₄⋊C₄	—	2	2	no	—
22	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
23	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₅	—	2	2	no	—
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	D₄	1	2	yes	—
28	C₂×C₄²×C₅	D₄	1	2	yes	—
29	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
30	C₅×C₄⋊C₈	—	2	4	no	—
31	C₅×C₄⋊C₈	—	4	4	no	—
32	C₅×#97	—	2	8	no	—
33	C₄×C₅×C₈	—	2	2	no	—
34	C₄×C₅×C₈	Q₈	1	4	yes	—
35	C₅×#95	—	2	2	no	—
36	C₅×#95	—	2	2	no	—
37	C₅×#95	—	2	2	no	—
38	C₄×#125	C₂×C₅	1	1	yes	characteristic
39	#1842	C₂×C₅	1	1	yes	characteristic
40	C₄×#97	C₂×C₅	1	1	yes	characteristic
41	#1279	C₂×C₅	1	4	yes	—
42	C₂²×C₄×C₅	—	2	2	no	—
43	C₂²×C₄×C₅	—	2	2	no	—
44	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
49	C₂×C₅×C₈	—	2	8	no	—
50	C₂×C₅×C₈	—	2	4	no	—
51	C₂×C₅×C₈	—	4	2	no	—
52	C₂×C₅×C₈	—	4	4	no	—
53	C₅×C₄⋊C₄	—	2	8	no	—
54	C₅×C₄⋊C₄	—	2	2	no	—
55	C₅×C₄⋊C₄	—	2	4	no	—
56	C₅×C₄⋊C₄	—	4	4	no	—
57	C₅×C₄⋊C₄	M₁₆	1	4	yes	—
58	C₄²×C₅	—	2	4	no	—
59	C₄²×C₅	—	2	2	no	—
60	C₄²×C₅	—	2	2	no	—
61	C₄²×C₅	—	2	2	no	—
62	C₄²×C₅	—	2	2	no	—
63	C₄²×C₅	—	4	2	no	—
64	C₄²×C₅	—	4	2	no	—
65	C₄²×C₅	C₂×Q₈	1	2	yes	—
66	C₄²×C₅	#41	1	2	yes	—
67	C₂×#125	C₄×C₅	1	1	yes	characteristic
68	C₂×C₄⋊C₈	—	2	2	no	—
69	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
70	C₂×#97	—	2	2	no	—
71	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
72	#389	C₄×C₅	1	2	yes	—
73	#387	—	2	2	no	—
74	#382	C₄×C₅	1	1	yes	characteristic
75	C₄×C₄⋊C₄	—	2	2	no	—
76	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
77	C₄³	C₂²×C₅	1	1	yes	characteristic
78	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅	—	2	2	no	—
80	C₂×C₄×C₅	—	2	2	no	—
81	C₂×C₄×C₅	—	2	2	no	—
82	C₂×C₄×C₅	—	2	2	no	—
83	C₂×C₄×C₅	—	2	4	no	—
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	—	2	2	no	—
86	C₂×C₄×C₅	—	4	2	no	—
87	C₂×C₄×C₅	—	4	2	no	—
88	C₂×C₄×C₅	—	4	2	no	—
89	C₂×C₄×C₅	—	4	2	no	—
90	C₂×C₄×C₅	—	4	2	no	—
91	C₂×C₄×C₅	#104	1	4	yes	—
92	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
93	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
94	C₂×C₄×C₅	#124	1	1	yes	characteristic
95	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₅	#131	1	2	yes	—
98	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
99	C₂×C₄×C₅	#137	1	2	yes	—
100	C₅×C₈	—	2	8	no	—
101	C₅×C₈	—	8	2	no	—
102	C₂²×C₈	—	2	2	no	—
103	C₂²×C₈	—	2	2	no	—
104	#125	—	2	4	no	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
108	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
109	C₂×C₄²	—	2	2	no	—
110	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
111	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
112	C₂×C₄²	C₅×D₄	1	2	yes	—
113	C₂×C₄²	C₅×D₄	1	2	yes	—
114	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
115	C₄⋊C₈	—	2	4	no	—
116	C₄⋊C₈	—	4	4	no	—
117	#97	—	2	8	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	C₅×Q₈	1	4	yes	—
120	#95	—	2	2	no	—
121	#95	—	2	2	no	—
122	#95	—	2	2	no	—
123	C₂²×C₅	#391	1	1	yes	characteristic
124	C₂²×C₅	#407	1	1	yes	characteristic
125	C₂²×C₅	#408	1	1	yes	characteristic
126	C₂²×C₅	#417	1	2	yes	—
127	C₂²×C₅	C₂×#104	1	2	yes	—
128	C₄×C₅	—	2	2	no	—
129	C₄×C₅	—	2	4	no	—
130	C₄×C₅	—	4	2	no	—
131	C₄×C₅	—	4	4	no	—
132	C₄×C₅	—	8	2	no	—
133	C₄×C₅	#446	1	4	yes	—
134	C₄×C₅	#490	1	1	yes	characteristic
135	C₄×C₅	#491	1	1	yes	characteristic
136	C₄×C₅	#501	1	2	yes	—
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
140	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
141	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
144	C₂×C₈	—	2	8	no	—
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	4	2	no	—
147	C₂×C₈	—	4	4	no	—
148	C₄⋊C₄	—	2	8	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₄⋊C₄	C₅×M₁₆	1	4	yes	—
153	C₄²	—	2	2	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	2	4	no	—
158	C₄²	—	4	2	no	—
159	C₄²	—	4	2	no	—
160	C₄²	C₂×C₅×Q₈	1	2	yes	—
161	C₄²	C₅×#41	1	2	yes	—
162	C₂×C₅	#1483	1	2	yes	—
163	C₂×C₅	#1489	1	2	yes	—
164	C₂×C₅	#1985	1	1	yes	characteristic
165	C₂×C₅	#1992	1	1	yes	characteristic
166	C₂×C₅	#1994	1	1	yes	characteristic
167	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	C₅×#104	1	4	yes	—
181	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
182	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
183	C₂×C₄	C₅×#124	1	1	yes	characteristic
184	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
185	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
186	C₂×C₄	C₅×#131	1	2	yes	—
187	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
188	C₂×C₄	C₅×#137	1	2	yes	—
189	C₈	—	2	8	no	—
190	C₈	—	8	2	no	—
191	C₅	#9876	1	1	yes	characteristic, direct product factor, Sylow
192	C₂²	C₅×#391	1	1	yes	characteristic
193	C₂²	C₅×#407	1	1	yes	characteristic
194	C₂²	C₅×#408	1	1	yes	characteristic
195	C₂²	C₅×#417	1	2	yes	—
196	C₂²	C₂×C₅×#104	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	2	4	no	—
199	C₄	—	4	2	no	—
200	C₄	—	4	4	no	—
201	C₄	—	8	2	no	—
202	C₄	C₅×#446	1	4	yes	—
203	C₄	C₅×#490	1	1	yes	characteristic
204	C₄	C₅×#491	1	1	yes	characteristic
205	C₄	C₅×#501	1	2	yes	—
206	C₂	C₅×#1483	1	2	yes	—
207	C₂	C₅×#1489	1	2	yes	—
208	C₂	C₅×#1985	1	1	yes	characteristic
209	C₂	C₅×#1992	1	1	yes	characteristic
210	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
16. #9876	C₅	direct (N × H)
191. C₅	#9876	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×#125	2
C₅×#1842	2
C₄×C₅×#97	2
C₅×#1279	2
#9876	5

Subgroups — C5×#9876 #11932596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9876 #11932596