Subgroups — C₅×#12670 #11935390

Order 1280.

318 conjugacy classes of subgroups.

The subgroup lattice has 184 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×C₄×C₅×C₁₆	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	C₄×C₅×C₁₆	C₄	1	4	yes	complemented
16	C₅×#339	—	2	2	no	—
17	#12670	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×C₄×C₁₆	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	1	yes	characteristic
50	C₂×C₅×C₈	D₈	1	1	yes	characteristic
51	C₂×C₅×C₈	Q₁₆	1	2	yes	—
52	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₅×C₈	#41	1	1	yes	characteristic
57	C₂×C₅×C₈	#41	1	1	yes	characteristic
58	C₅×C₄⋊C₄	—	2	4	no	—
59	C₅×C₂²⋊C₄	—	4	2	no	—
60	C₄²×C₅	—	2	1	no	—
61	C₄²×C₅	—	4	2	no	—
62	C₄²×C₅	C₄⋊C₄	1	2	yes	—
63	C₅×C₁₆	—	2	2	no	—
64	C₅×C₁₆	C₄²	1	2	yes	—
65	C₅×C₁₆	C₄²	1	2	yes	—
66	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
67	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
68	C₂×#108	C₂²×C₅	1	2	yes	—
69	#427	C₂²×C₅	1	2	yes	—
70	#405	—	2	2	no	—
71	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
72	#368	C₄×C₅	1	4	yes	complemented
73	#367	C₄×C₅	1	2	yes	complemented
74	#366	C₄×C₅	1	2	yes	complemented
75	C₄×C₁₆	C₄×C₅	1	4	yes	complemented
76	#339	—	2	2	no	—
77	C₂³×C₅	#135	1	1	yes	characteristic
78	C₂×C₄×C₅	—	2	4	no	—
79	C₂×C₄×C₅	—	2	2	no	—
80	C₂×C₄×C₅	—	4	4	no	—
81	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
86	C₂×C₄×C₅	#135	1	1	yes	characteristic
87	C₅×C₈	—	4	4	no	—
88	C₅×C₈	#107	1	2	yes	—
89	C₅×C₈	#107	1	2	yes	—
90	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
91	C₅×C₈	#117	1	1	yes	characteristic
92	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
93	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
94	C₂×M₁₆	—	2	2	no	—
95	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
96	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
97	#117	—	2	2	no	—
98	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
102	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
103	C₂×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
104	C₂×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
105	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
106	#108	—	2	2	no	—
107	#108	C₂×C₄×C₅	1	4	yes	—
108	#107	C₂×C₄×C₅	1	4	yes	—
109	#106	—	2	2	no	—
110	#97	—	4	2	no	—
111	C₄×C₈	—	4	2	no	—
112	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
113	C₄×C₈	C₅×Q₈	1	2	yes	—
114	C₂²×C₅	—	2	2	no	—
115	C₂²×C₅	#427	1	1	yes	characteristic
116	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
117	C₂²×C₅	C₄×Q₁₆	1	1	yes	characteristic
118	C₄×C₅	—	2	4	no	—
119	C₄×C₅	—	8	4	no	—
120	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
121	C₄×C₅	C₂×#107	1	1	yes	characteristic
122	C₄×C₅	C₄×D₈	1	1	yes	characteristic
123	C₄×C₅	C₄×Q₁₆	1	1	yes	characteristic
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
126	M₁₆	—	2	4	no	—
127	M₁₆	—	4	2	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
130	C₂×C₈	C₅×D₈	1	1	yes	characteristic
131	C₂×C₈	C₅×D₈	1	1	yes	characteristic
132	C₂×C₈	C₅×Q₁₆	1	2	yes	—
133	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
134	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
135	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
136	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
137	C₂×C₈	C₅×#41	1	1	yes	characteristic
138	C₂×C₈	C₅×#41	1	1	yes	characteristic
139	C₄⋊C₄	—	2	4	no	—
140	C₂²⋊C₄	—	4	2	no	—
141	C₄²	—	2	1	no	—
142	C₄²	—	4	2	no	—
143	C₄²	C₅×C₄⋊C₄	1	2	yes	—
144	C₁₆	—	2	2	no	—
145	C₁₆	C₄²×C₅	1	2	yes	—
146	C₁₆	C₄²×C₅	1	2	yes	—
147	C₂×C₅	—	2	1	no	—
148	C₂×C₅	—	2	1	no	—
149	C₂×C₅	C₄×#107	1	1	yes	characteristic
150	C₂×C₅	#2178	1	2	yes	—
151	C₂³	C₅×#135	1	1	yes	characteristic
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
157	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
158	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
159	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
160	C₂×C₄	C₅×#135	1	1	yes	characteristic
161	C₈	—	4	4	no	—
162	C₈	C₅×#107	1	2	yes	—
163	C₈	C₅×#107	1	2	yes	—
164	C₈	C₂×C₄²×C₅	1	1	yes	characteristic
165	C₈	C₅×#117	1	1	yes	characteristic
166	C₈	C₄×C₅×Q₈	1	1	yes	characteristic
167	C₈	C₄×C₅×Q₈	1	1	yes	characteristic
168	C₅	#12670	1	1	yes	characteristic, direct product factor, Sylow
169	C₂²	—	2	2	no	—
170	C₂²	C₅×#427	1	1	yes	characteristic
171	C₂²	C₄×C₅×D₈	1	1	yes	characteristic
172	C₂²	C₄×C₅×Q₁₆	1	1	yes	characteristic
173	C₄	—	2	4	no	—
174	C₄	—	8	4	no	—
175	C₄	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
176	C₄	C₂×C₅×#107	1	1	yes	characteristic
177	C₄	C₄×C₅×D₈	1	1	yes	characteristic
178	C₄	C₄×C₅×Q₁₆	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	C₄×C₅×#107	1	1	yes	characteristic
182	C₂	C₅×#2178	1	2	yes	—

	subgroup	G/H
center	C₂×C₅×C₈	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×C₅×C₈ ◁ 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. C₄×C₅×C₁₆	C₄	semidirect
17. #12670	C₅	direct (N × H)
72. #368	C₄×C₅	semidirect
73. #367	C₄×C₅	semidirect
74. #366	C₄×C₅	semidirect
75. C₄×C₁₆	C₄×C₅	semidirect
168. C₅	#12670	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×C₄×C₅×C₁₆	2
C₅×#1772	2
C₅×#1382	2
#12670	5

Subgroups — C5×#12670 #11935390

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12670 #11935390