Subgroups — C₅×#12907 #11935627

Order 1280.

342 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#367	C₂	1	2	yes	maximal
2	C₂×C₅×#366	C₂	1	2	yes	maximal
3	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₅×#1846	C₂	1	2	yes	maximal
5	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
6	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₅×#107	C₂²	1	4	yes	—
8	C₂×C₅×#106	—	2	2	no	—
9	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₅×#382	—	2	2	no	—
11	C₅×#367	—	2	4	no	—
12	C₅×#366	—	2	4	no	—
13	C₄×C₅×C₁₆	C₄	1	4	yes	complemented
14	#12907	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
16	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
17	C₂×C₅×C₄⋊C₄	—	2	4	no	—
18	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
19	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₅×C₁₆	D₄	1	2	yes	—
22	C₂×C₅×C₁₆	D₄	1	2	yes	—
23	C₂×C₅×C₁₆	Q₈	1	2	yes	—
24	C₂×C₅×C₁₆	Q₈	1	2	yes	—
25	C₅×#107	—	2	4	no	—
26	C₅×#107	—	2	4	no	—
27	C₅×#106	—	4	4	no	—
28	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₄×C₅×C₈	Q₈	1	2	yes	—
30	C₅×#95	—	4	4	no	—
31	C₂×#367	C₂×C₅	1	2	yes	—
32	C₂×#366	C₂×C₅	1	2	yes	—
33	C₂×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic
34	#1846	C₂×C₅	1	2	yes	—
35	C₂²×C₄×C₅	—	4	4	no	—
36	C₂²×C₄×C₅	D₈	1	2	yes	—
37	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
38	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
39	C₂×C₅×C₈	D₈	1	2	yes	—
40	C₂×C₅×C₈	Q₁₆	1	2	yes	—
41	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
43	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
44	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
47	C₅×C₄⋊C₄	—	4	4	no	—
48	C₅×C₄⋊C₄	—	4	4	no	—
49	C₄²×C₅	C₄⋊C₄	1	2	yes	—
50	C₄²×C₅	Q₁₆	1	2	yes	—
51	C₅×C₁₆	C₄⋊C₄	1	4	yes	—
52	C₅×C₁₆	C₄⋊C₄	1	4	yes	—
53	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
54	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
55	C₂×#107	C₂²×C₅	1	4	yes	—
56	C₂×#106	—	2	2	no	—
57	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
58	#382	—	2	2	no	—
59	#367	—	2	4	no	—
60	#366	—	2	4	no	—
61	C₄×C₁₆	C₄×C₅	1	4	yes	complemented
62	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₅	—	2	4	no	—
64	C₂×C₄×C₅	—	8	4	no	—
65	C₂×C₄×C₅	—	8	8	no	—
66	C₂×C₄×C₅	#107	1	4	yes	—
67	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅	#127	1	1	yes	characteristic
69	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
73	C₅×C₈	#107	1	4	yes	—
74	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
77	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
78	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
79	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
80	C₂×C₄⋊C₄	—	2	4	no	—
81	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
82	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
83	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
84	C₂×C₁₆	C₅×D₄	1	2	yes	—
85	C₂×C₁₆	C₅×D₄	1	2	yes	—
86	C₂×C₁₆	C₅×Q₈	1	2	yes	—
87	C₂×C₁₆	C₅×Q₈	1	2	yes	—
88	#107	—	2	4	no	—
89	#107	—	2	4	no	—
90	#106	—	4	4	no	—
91	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
92	C₄×C₈	C₅×Q₈	1	2	yes	—
93	#95	—	4	4	no	—
94	C₂²×C₅	C₂×#107	1	1	yes	characteristic
95	C₂²×C₅	#493	1	1	yes	characteristic
96	C₂²×C₅	#500	1	1	yes	characteristic
97	C₂²×C₅	#508	1	4	yes	—
98	C₄×C₅	—	2	4	no	—
99	C₄×C₅	—	16	4	no	—
100	C₄×C₅	#384	1	1	yes	characteristic
101	C₄×C₅	C₂×#107	1	1	yes	characteristic
102	C₄×C₅	#494	1	1	yes	characteristic
103	C₄×C₅	#500	1	1	yes	characteristic
104	C₂²×C₄	—	4	4	no	—
105	C₂²×C₄	C₅×D₈	1	2	yes	—
106	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
107	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
108	C₂×C₈	C₅×D₈	1	2	yes	—
109	C₂×C₈	C₅×Q₁₆	1	2	yes	—
110	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
111	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
112	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
113	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
114	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
115	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
116	C₄⋊C₄	—	4	4	no	—
117	C₄⋊C₄	—	4	4	no	—
118	C₄²	C₅×C₄⋊C₄	1	2	yes	—
119	C₄²	C₅×Q₁₆	1	2	yes	—
120	C₁₆	C₅×C₄⋊C₄	1	4	yes	—
121	C₁₆	C₅×C₄⋊C₄	1	4	yes	—
122	C₂×C₅	#1845	1	1	yes	characteristic
123	C₂×C₅	#2158	1	2	yes	—
124	C₂×C₅	#2249	1	2	yes	—
125	C₂×C₅	#2253	1	2	yes	—
126	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	8	4	no	—
129	C₂×C₄	—	8	8	no	—
130	C₂×C₄	C₅×#107	1	4	yes	—
131	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄	C₅×#127	1	1	yes	characteristic
133	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
134	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
135	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
136	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
137	C₈	C₅×#107	1	4	yes	—
138	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
139	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
140	C₈	C₅×C₄⋊Q₈	1	1	yes	characteristic
141	C₈	C₅×C₄⋊Q₈	1	1	yes	characteristic
142	C₅	#12907	1	1	yes	characteristic, direct product factor, Sylow
143	C₂²	C₂×C₅×#107	1	1	yes	characteristic
144	C₂²	C₅×#493	1	1	yes	characteristic
145	C₂²	C₅×#500	1	1	yes	characteristic
146	C₂²	C₅×#508	1	4	yes	—
147	C₄	—	2	4	no	—
148	C₄	—	16	4	no	—
149	C₄	C₅×#384	1	1	yes	characteristic
150	C₄	C₂×C₅×#107	1	1	yes	characteristic
151	C₄	C₅×#494	1	1	yes	characteristic
152	C₄	C₅×#500	1	1	yes	characteristic
153	C₂	C₅×#1845	1	1	yes	characteristic
154	C₂	C₅×#2158	1	2	yes	—
155	C₂	C₅×#2249	1	2	yes	—
156	C₂	C₅×#2253	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
13. C₄×C₅×C₁₆	C₄	semidirect
14. #12907	C₅	direct (N × H)
61. C₄×C₁₆	C₄×C₅	semidirect
142. C₅	#12907	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×#367	2
C₂×C₅×#366	2
C₂×C₄×C₅×C₁₆	2
C₅×#1846	2
#12907	5

Subgroups — C5×#12907 #11935627

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12907 #11935627