Subgroups — C₅×#13180 #11935900

Order 1280.

294 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2159	C₂	1	2	yes	maximal
2	C₂×C₅×#364	C₂	1	1	yes	maximal, characteristic
3	C₅×#1846	C₂	1	1	yes	maximal, characteristic
4	C₅×#1772	C₂	1	1	yes	maximal, characteristic
5	C₅×#1387	C₂	1	2	yes	maximal
6	C₂×C₅×M₃₂	C₂²	1	2	yes	—
7	C₂×C₅×#108	C₂²	1	1	yes	characteristic
8	C₅×#427	C₂²	1	1	yes	characteristic
9	C₂×C₅×#107	C₂²	1	2	yes	—
10	C₂×C₅×#106	—	2	1	no	—
11	C₅×#405	—	2	1	no	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×#382	—	2	1	no	—
14	C₅×#367	—	2	2	no	—
15	C₅×#366	—	2	2	no	—
16	C₅×#364	C₄	1	4	yes	complemented
17	C₅×#339	—	2	1	no	—
18	#13180	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂×C₅×M₁₆	—	2	1	no	—
20	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
22	C₅×#117	—	2	1	no	—
23	C₂×C₅×C₄⋊C₄	—	2	2	no	—
24	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
25	C₅×M₃₂	—	2	2	no	—
26	C₅×M₃₂	C₂×C₄	1	4	yes	—
27	C₂×C₅×C₁₆	—	2	2	no	—
28	C₅×#108	—	2	1	no	—
29	C₅×#108	Q₈	1	2	yes	—
30	C₅×#107	—	2	2	no	—
31	C₅×#107	—	2	2	no	—
32	C₅×#107	D₄	1	2	yes	—
33	C₅×#106	—	2	1	no	—
34	C₅×#106	—	4	1	no	—
35	C₅×#106	—	4	1	no	—
36	C₅×#97	—	4	1	no	—
37	C₄×C₅×C₈	—	4	1	no	—
38	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
39	C₄×C₅×C₈	Q₈	1	2	yes	—
40	C₅×#95	—	4	2	no	—
41	#2159	C₂×C₅	1	2	yes	—
42	C₂×#364	C₂×C₅	1	1	yes	characteristic
43	#1846	C₂×C₅	1	1	yes	characteristic
44	#1772	C₂×C₅	1	1	yes	characteristic
45	#1387	C₂×C₅	1	2	yes	—
46	C₂²×C₄×C₅	—	2	1	no	—
47	C₂²×C₄×C₅	—	4	2	no	—
48	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
49	C₅×M₁₆	—	2	2	no	—
50	C₅×M₁₆	—	4	1	no	—
51	C₂×C₅×C₈	—	4	1	no	—
52	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
53	C₂×C₅×C₈	D₈	1	1	yes	characteristic
54	C₂×C₅×C₈	D₈	1	1	yes	characteristic
55	C₂×C₅×C₈	Q₁₆	1	2	yes	—
56	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
58	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₅×C₈	#41	1	1	yes	characteristic
61	C₂×C₅×C₈	#41	1	1	yes	characteristic
62	C₅×C₄⋊C₄	—	2	2	no	—
63	C₅×C₄⋊C₄	—	4	2	no	—
64	C₅×C₄⋊C₄	—	4	2	no	—
65	C₅×C₂²⋊C₄	—	4	1	no	—
66	C₄²×C₅	—	2	1	no	—
67	C₄²×C₅	—	4	1	no	—
68	C₄²×C₅	C₄⋊C₄	1	2	yes	—
69	C₅×C₁₆	—	2	4	no	—
70	C₂×M₃₂	C₂²×C₅	1	2	yes	—
71	C₂×#108	C₂²×C₅	1	1	yes	characteristic
72	#427	C₂²×C₅	1	1	yes	characteristic
73	C₂×#107	C₂²×C₅	1	2	yes	—
74	C₂×#106	—	2	1	no	—
75	#405	—	2	1	no	—
76	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
77	#382	—	2	1	no	—
78	#367	—	2	2	no	—
79	#366	—	2	2	no	—
80	#364	C₄×C₅	1	4	yes	complemented
81	#339	—	2	1	no	—
82	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₂×C₄×C₅	—	2	2	no	—
84	C₂×C₄×C₅	—	4	2	no	—
85	C₂×C₄×C₅	—	4	2	no	—
86	C₂×C₄×C₅	—	8	4	no	—
87	C₂×C₄×C₅	—	8	2	no	—
88	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅	#121	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
93	C₂×C₄×C₅	#137	1	1	yes	characteristic
94	C₅×C₈	—	4	2	no	—
95	C₅×C₈	#107	1	2	yes	—
96	C₅×C₈	#107	1	2	yes	—
97	C₅×C₈	C₄×D₄	1	1	yes	characteristic
98	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
99	C₅×C₈	#125	1	1	yes	characteristic
100	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×M₁₆	—	2	1	no	—
102	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
103	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
104	#117	—	2	1	no	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
107	M₃₂	—	2	2	no	—
108	M₃₂	C₂×C₄×C₅	1	4	yes	—
109	C₂×C₁₆	—	2	2	no	—
110	#108	—	2	1	no	—
111	#108	C₅×Q₈	1	2	yes	—
112	#107	—	2	2	no	—
113	#107	—	2	2	no	—
114	#107	C₅×D₄	1	2	yes	—
115	#106	—	2	1	no	—
116	#106	—	4	1	no	—
117	#106	—	4	1	no	—
118	#97	—	4	1	no	—
119	C₄×C₈	—	4	1	no	—
120	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
121	C₄×C₈	C₅×Q₈	1	2	yes	—
122	#95	—	4	2	no	—
123	C₂²×C₅	—	2	2	no	—
124	C₂²×C₅	#428	1	1	yes	characteristic
125	C₂²×C₅	#459	1	1	yes	characteristic
126	C₂²×C₅	#477	1	1	yes	characteristic
127	C₄×C₅	—	4	2	no	—
128	C₄×C₅	—	8	2	no	—
129	C₄×C₅	—	16	2	no	—
130	C₄×C₅	#389	1	1	yes	characteristic
131	C₄×C₅	C₂×#107	1	1	yes	characteristic
132	C₄×C₅	#462	1	1	yes	characteristic
133	C₄×C₅	#474	1	1	yes	characteristic
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	4	2	no	—
136	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
137	M₁₆	—	2	2	no	—
138	M₁₆	—	4	1	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
141	C₂×C₈	C₅×D₈	1	1	yes	characteristic
142	C₂×C₈	C₅×D₈	1	1	yes	characteristic
143	C₂×C₈	C₅×Q₁₆	1	2	yes	—
144	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
145	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
146	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
147	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
148	C₂×C₈	C₅×#41	1	1	yes	characteristic
149	C₂×C₈	C₅×#41	1	1	yes	characteristic
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₂²⋊C₄	—	4	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	4	1	no	—
156	C₄²	C₅×C₄⋊C₄	1	2	yes	—
157	C₁₆	—	2	4	no	—
158	C₂×C₅	—	2	1	no	—
159	C₂×C₅	—	2	1	no	—
160	C₂×C₅	#1947	1	1	yes	characteristic
161	C₂×C₅	#2212	1	2	yes	—
162	C₂³	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	8	4	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₄	C₅×#121	1	1	yes	characteristic
170	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
171	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
172	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
173	C₂×C₄	C₅×#137	1	1	yes	characteristic
174	C₈	—	4	2	no	—
175	C₈	C₅×#107	1	2	yes	—
176	C₈	C₅×#107	1	2	yes	—
177	C₈	C₄×C₅×D₄	1	1	yes	characteristic
178	C₈	C₄×C₅×Q₈	1	1	yes	characteristic
179	C₈	C₅×#125	1	1	yes	characteristic
180	C₈	C₅×C₄⋊Q₈	1	1	yes	characteristic
181	C₅	#13180	1	1	yes	characteristic, direct product factor, Sylow
182	C₂²	—	2	2	no	—
183	C₂²	C₅×#428	1	1	yes	characteristic
184	C₂²	C₅×#459	1	1	yes	characteristic
185	C₂²	C₅×#477	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	8	2	no	—
188	C₄	—	16	2	no	—
189	C₄	C₅×#389	1	1	yes	characteristic
190	C₄	C₂×C₅×#107	1	1	yes	characteristic
191	C₄	C₅×#462	1	1	yes	characteristic
192	C₄	C₅×#474	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	C₅×#1947	1	1	yes	characteristic
196	C₂	C₅×#2212	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	#459
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂³×C₅	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
16. C₅×#364	C₄	semidirect
18. #13180	C₅	direct (N × H)
80. #364	C₄×C₅	semidirect
181. C₅	#13180	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×#2159	2
C₂×C₅×#364	2
C₅×#1846	2
C₅×#1772	2
C₅×#1387	2
#13180	5

Subgroups — C5×#13180 #11935900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13180 #11935900