Subgroups — C₅×#12671 #11935391

Order 1280.

310 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#368	C₂	1	1	yes	maximal, characteristic
2	C₅×#2158	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic
4	C₅×#1772	C₂	1	2	yes	maximal
5	C₅×#1382	C₂	1	2	yes	maximal
6	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₅×#108	C₂²	1	2	yes	—
9	C₅×#427	C₂²	1	2	yes	—
10	C₅×#405	—	2	2	no	—
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₅×#368	C₄	1	4	yes	complemented
13	C₅×#367	C₄	1	2	yes	complemented
14	C₅×#366	C₄	1	2	yes	complemented
15	C₅×#346	C₄	1	4	yes	complemented
16	C₅×#339	—	2	2	no	—
17	#12671	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂×C₅×M₁₆	—	2	2	no	—
19	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
21	C₅×#117	—	2	2	no	—
22	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
23	C₂×C₅×C₁₆	—	2	1	no	—
24	C₂×C₅×C₁₆	—	2	1	no	—
25	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
30	C₅×#108	—	2	2	no	—
31	C₅×#108	C₂×C₄	1	4	yes	—
32	C₅×#107	C₂×C₄	1	4	yes	—
33	C₅×#106	—	2	2	no	—
34	C₅×#97	—	4	2	no	—
35	C₄×C₅×C₈	—	4	2	no	—
36	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
37	C₄×C₅×C₈	Q₈	1	2	yes	—
38	C₂×#368	C₂×C₅	1	1	yes	characteristic
39	#2158	C₂×C₅	1	1	yes	characteristic
40	C₂×#346	C₂×C₅	1	1	yes	characteristic
41	#1772	C₂×C₅	1	2	yes	—
42	#1382	C₂×C₅	1	2	yes	—
43	C₂²×C₄×C₅	—	2	1	no	—
44	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
45	C₅×M₁₆	—	2	4	no	—
46	C₅×M₁₆	—	4	2	no	—
47	C₂×C₅×C₈	—	4	2	no	—
48	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
49	C₂×C₅×C₈	D₈	1	2	yes	—
50	C₂×C₅×C₈	Q₁₆	1	2	yes	—
51	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₅×C₈	#41	1	1	yes	characteristic
56	C₂×C₅×C₈	#41	1	1	yes	characteristic
57	C₅×C₄⋊C₄	—	2	4	no	—
58	C₅×C₂²⋊C₄	—	4	2	no	—
59	C₄²×C₅	—	2	1	no	—
60	C₄²×C₅	—	4	2	no	—
61	C₄²×C₅	C₄⋊C₄	1	2	yes	—
62	C₅×C₁₆	—	2	2	no	—
63	C₅×C₁₆	C₄²	1	2	yes	—
64	C₅×C₁₆	C₄²	1	2	yes	—
65	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
66	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
67	C₂×#108	C₂²×C₅	1	2	yes	—
68	#427	C₂²×C₅	1	2	yes	—
69	#405	—	2	2	no	—
70	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
71	#368	C₄×C₅	1	4	yes	complemented
72	#367	C₄×C₅	1	2	yes	complemented
73	#366	C₄×C₅	1	2	yes	complemented
74	#346	C₄×C₅	1	4	yes	complemented
75	#339	—	2	2	no	—
76	C₂³×C₅	#135	1	1	yes	characteristic
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	—	4	2	no	—
79	C₂×C₄×C₅	—	4	4	no	—
80	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
85	C₂×C₄×C₅	#135	1	1	yes	characteristic
86	C₅×C₈	—	4	4	no	—
87	C₅×C₈	#107	1	4	yes	—
88	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
89	C₅×C₈	#117	1	1	yes	characteristic
90	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
91	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
92	C₂×M₁₆	—	2	2	no	—
93	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
94	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
95	#117	—	2	2	no	—
96	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
97	C₂×C₁₆	—	2	1	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
100	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
101	C₂×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
102	C₂×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
103	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
104	#108	—	2	2	no	—
105	#108	C₂×C₄×C₅	1	4	yes	—
106	#107	C₂×C₄×C₅	1	4	yes	—
107	#106	—	2	2	no	—
108	#97	—	4	2	no	—
109	C₄×C₈	—	4	2	no	—
110	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
111	C₄×C₈	C₅×Q₈	1	2	yes	—
112	C₂²×C₅	—	2	2	no	—
113	C₂²×C₅	#427	1	1	yes	characteristic
114	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
115	C₂²×C₅	C₄×Q₁₆	1	1	yes	characteristic
116	C₄×C₅	—	4	2	no	—
117	C₄×C₅	—	8	4	no	—
118	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
119	C₄×C₅	C₂×#107	1	1	yes	characteristic
120	C₄×C₅	C₄×D₈	1	1	yes	characteristic
121	C₄×C₅	C₄×Q₁₆	1	1	yes	characteristic
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
124	M₁₆	—	2	4	no	—
125	M₁₆	—	4	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
128	C₂×C₈	C₅×D₈	1	2	yes	—
129	C₂×C₈	C₅×Q₁₆	1	2	yes	—
130	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
131	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
132	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
133	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
134	C₂×C₈	C₅×#41	1	1	yes	characteristic
135	C₂×C₈	C₅×#41	1	1	yes	characteristic
136	C₄⋊C₄	—	2	4	no	—
137	C₂²⋊C₄	—	4	2	no	—
138	C₄²	—	2	1	no	—
139	C₄²	—	4	2	no	—
140	C₄²	C₅×C₄⋊C₄	1	2	yes	—
141	C₁₆	—	2	2	no	—
142	C₁₆	C₄²×C₅	1	2	yes	—
143	C₁₆	C₄²×C₅	1	2	yes	—
144	C₂×C₅	—	2	1	no	—
145	C₂×C₅	—	2	1	no	—
146	C₂×C₅	C₄×#107	1	1	yes	characteristic
147	C₂×C₅	#2179	1	2	yes	—
148	C₂³	C₅×#135	1	1	yes	characteristic
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	4	no	—
152	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
154	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
155	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
156	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
157	C₂×C₄	C₅×#135	1	1	yes	characteristic
158	C₈	—	4	4	no	—
159	C₈	C₅×#107	1	4	yes	—
160	C₈	C₂×C₄²×C₅	1	1	yes	characteristic
161	C₈	C₅×#117	1	1	yes	characteristic
162	C₈	C₄×C₅×Q₈	1	1	yes	characteristic
163	C₈	C₄×C₅×Q₈	1	1	yes	characteristic
164	C₅	#12671	1	1	yes	characteristic, direct product factor, Sylow
165	C₂²	—	2	2	no	—
166	C₂²	C₅×#427	1	1	yes	characteristic
167	C₂²	C₄×C₅×D₈	1	1	yes	characteristic
168	C₂²	C₄×C₅×Q₁₆	1	1	yes	characteristic
169	C₄	—	4	2	no	—
170	C₄	—	8	4	no	—
171	C₄	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
172	C₄	C₂×C₅×#107	1	1	yes	characteristic
173	C₄	C₄×C₅×D₈	1	1	yes	characteristic
174	C₄	C₄×C₅×Q₁₆	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	C₄×C₅×#107	1	1	yes	characteristic
178	C₂	C₅×#2179	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₈
commutator	C₈	C₂×C₄²×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₄×D₈
prime radical	C₂³×C₅	#135
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₅×#368	C₄	semidirect
13. C₅×#367	C₄	semidirect
14. C₅×#366	C₄	semidirect
15. C₅×#346	C₄	semidirect
17. #12671	C₅	direct (N × H)
71. #368	C₄×C₅	semidirect
72. #367	C₄×C₅	semidirect
73. #366	C₄×C₅	semidirect
74. #346	C₄×C₅	semidirect
164. C₅	#12671	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#368	2
C₅×#2158	2
C₂×C₅×#346	2
C₅×#1772	2
C₅×#1382	2
#12671	5

Subgroups — C5×#12671 #11935391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12671 #11935391