Subgroups — C₅×#12904 #11935624

Order 1280.

318 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#365	C₂	1	4	yes	maximal
2	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₅×#1848	C₂	1	1	yes	maximal, characteristic
4	C₅×#1846	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×M₃₂	C₂²	1	2	yes	—
6	C₂×C₅×#108	—	2	1	no	—
7	C₂×C₅×#108	C₂²	1	2	yes	—
8	C₂×C₅×#107	—	2	1	no	—
9	C₂×C₅×#106	C₂²	1	2	yes	—
10	C₅×#423	—	2	1	no	—
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₅×#382	—	2	1	no	—
13	C₅×#365	—	2	8	no	—
14	C₅×#346	C₄	1	2	yes	complemented
15	C₄×C₅×C₁₆	C₄	1	2	yes	complemented
16	#12904	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂×C₅×M₁₆	—	2	2	no	—
18	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
20	C₂×C₅×C₄⋊C₄	—	2	2	no	—
21	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
22	C₅×M₃₂	D₄	1	4	yes	—
23	C₅×M₃₂	Q₈	1	4	yes	—
24	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
25	C₅×#108	—	2	4	no	—
26	C₅×#108	—	4	2	no	—
27	C₅×#107	—	4	2	no	—
28	C₅×#106	—	2	4	no	—
29	C₅×C₄⋊C₈	—	4	2	no	—
30	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
33	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
34	C₅×#95	—	4	2	no	—
35	C₂×#365	C₂×C₅	1	4	yes	—
36	C₄×M₃₂	C₂×C₅	1	1	yes	characteristic
37	#1848	C₂×C₅	1	1	yes	characteristic
38	#1846	C₂×C₅	1	1	yes	characteristic
39	C₂²×C₄×C₅	—	4	2	no	—
40	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
41	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
42	C₅×M₁₆	—	4	2	no	—
43	C₅×M₁₆	—	4	2	no	—
44	C₂×C₅×C₈	—	4	2	no	—
45	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₅×C₈	SD₁₆	1	2	yes	—
48	C₂×C₅×C₈	SD₁₆	1	2	yes	—
49	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
51	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
52	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
53	C₅×C₄⋊C₄	—	4	2	no	—
54	C₅×C₄⋊C₄	—	4	2	no	—
55	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
56	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
57	C₄²×C₅	SD₁₆	1	2	yes	—
58	C₅×C₁₆	C₄⋊C₄	1	8	yes	—
59	C₂×M₃₂	C₂²×C₅	1	2	yes	—
60	C₂×#108	—	2	1	no	—
61	C₂×#108	C₂²×C₅	1	2	yes	—
62	C₂×#107	—	2	1	no	—
63	C₂×#106	C₂²×C₅	1	2	yes	—
64	#423	—	2	1	no	—
65	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
66	#382	—	2	1	no	—
67	#365	—	2	8	no	—
68	#346	C₄×C₅	1	2	yes	complemented
69	C₄×C₁₆	C₄×C₅	1	2	yes	complemented
70	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅	—	4	2	no	—
72	C₂×C₄×C₅	—	8	4	no	—
73	C₂×C₄×C₅	—	8	2	no	—
74	C₂×C₄×C₅	#106	1	2	yes	—
75	C₂×C₄×C₅	#106	1	2	yes	—
76	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅	#127	1	1	yes	characteristic
78	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
82	C₅×C₈	—	8	2	no	—
83	C₅×C₈	#106	1	2	yes	—
84	C₅×C₈	#106	1	2	yes	—
85	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
86	C₅×C₈	C₄⋊Q₈	1	2	yes	—
87	C₂×M₁₆	—	2	2	no	—
88	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
89	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	2	2	no	—
91	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
92	M₃₂	C₅×D₄	1	4	yes	—
93	M₃₂	C₅×Q₈	1	4	yes	—
94	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	—
95	#108	—	2	4	no	—
96	#108	—	4	2	no	—
97	#107	—	4	2	no	—
98	#106	—	2	4	no	—
99	C₄⋊C₈	—	4	2	no	—
100	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
101	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
102	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
103	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
104	#95	—	4	2	no	—
105	C₂²×C₅	—	2	2	no	—
106	C₂²×C₅	C₂×#106	1	1	yes	characteristic
107	C₂²×C₅	#492	1	1	yes	characteristic
108	C₂²×C₅	#498	1	1	yes	characteristic
109	C₄×C₅	—	2	2	no	—
110	C₄×C₅	—	2	2	no	—
111	C₄×C₅	—	16	2	no	—
112	C₄×C₅	#384	1	1	yes	characteristic
113	C₄×C₅	C₂×#106	1	1	yes	characteristic
114	C₄×C₅	#492	1	1	yes	characteristic
115	C₄×C₅	#498	1	1	yes	characteristic
116	C₂²×C₄	—	4	2	no	—
117	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
118	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
119	M₁₆	—	4	2	no	—
120	M₁₆	—	4	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₈	C₅×SD₁₆	1	2	yes	—
125	C₂×C₈	C₅×SD₁₆	1	2	yes	—
126	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
127	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
128	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
129	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
133	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
134	C₄²	C₅×SD₁₆	1	2	yes	—
135	C₁₆	C₅×C₄⋊C₄	1	8	yes	—
136	C₂×C₅	—	2	1	no	—
137	C₂×C₅	—	2	1	no	—
138	C₂×C₅	#1844	1	1	yes	characteristic
139	C₂×C₅	#2155	1	2	yes	—
140	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	8	4	no	—
143	C₂×C₄	—	8	2	no	—
144	C₂×C₄	C₅×#106	1	2	yes	—
145	C₂×C₄	C₅×#106	1	2	yes	—
146	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄	C₅×#127	1	1	yes	characteristic
148	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
149	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
150	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
151	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
152	C₈	—	8	2	no	—
153	C₈	C₅×#106	1	2	yes	—
154	C₈	C₅×#106	1	2	yes	—
155	C₈	C₂×C₅×C₄⋊C₄	1	2	yes	—
156	C₈	C₅×C₄⋊Q₈	1	2	yes	—
157	C₅	#12904	1	1	yes	characteristic, direct product factor, Sylow
158	C₂²	—	2	2	no	—
159	C₂²	C₂×C₅×#106	1	1	yes	characteristic
160	C₂²	C₅×#492	1	1	yes	characteristic
161	C₂²	C₅×#498	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	16	2	no	—
165	C₄	C₅×#384	1	1	yes	characteristic
166	C₄	C₂×C₅×#106	1	1	yes	characteristic
167	C₄	C₅×#492	1	1	yes	characteristic
168	C₄	C₅×#498	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	C₅×#1844	1	1	yes	characteristic
172	C₂	C₅×#2155	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂³×C₅	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₅×#346	C₄	semidirect
15. C₄×C₅×C₁₆	C₄	semidirect
16. #12904	C₅	direct (N × H)
68. #346	C₄×C₅	semidirect
69. C₄×C₁₆	C₄×C₅	semidirect
157. C₅	#12904	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×#365	2
C₄×C₅×M₃₂	2
C₅×#1848	2
C₅×#1846	2
#12904	5

Subgroups — C5×#12904 #11935624

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12904 #11935624