Subgroups — C₂×C₅×#2844 #11959408

Order 1280.

1870 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2844	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₅×#400	C₂	1	6	yes	maximal
3	C₂×C₅×#395	C₂	1	6	yes	maximal
4	C₂×C₅×#382	C₂	1	3	yes	maximal
5	C₂³×C₅×Q₈	C₂²	1	2	yes	—
6	C₂²×C₅×C₄⋊C₄	C₂²	1	3	yes	—
7	C₂²×C₅×C₄⋊C₄	C₂²	1	3	yes	—
8	C₂²×C₅×C₄⋊C₄	C₂²	1	6	yes	—
9	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₅×#400	C₂²	1	48	yes	—
11	C₅×#395	C₂²	1	48	yes	—
12	C₅×#382	C₂²	1	24	yes	—
13	C₂×C₅×#95	C₂²	1	12	yes	—
14	C₂×C₅×#95	C₂²	1	6	yes	—
15	C₂×C₅×#95	C₂²	1	2	yes	—
16	C₂×#2844	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂²×C₅×Q₈	—	4	12	no	—
18	C₂²×C₅×Q₈	C₂³	1	8	yes	—
19	C₂³×C₄×C₅	C₂³	1	6	yes	—
20	C₂³×C₄×C₅	C₂³	1	6	yes	—
21	C₂³×C₄×C₅	C₂³	1	3	yes	—
22	C₂×C₅×C₄⋊C₄	—	4	6	no	—
23	C₂×C₅×C₄⋊C₄	—	4	6	no	—
24	C₂×C₅×C₄⋊C₄	—	4	12	no	—
25	C₂×C₅×C₄⋊C₄	C₂³	1	12	yes	—
26	C₂×C₅×C₄⋊C₄	C₂³	1	12	yes	—
27	C₂×C₅×C₄⋊C₄	C₂³	1	24	yes	—
28	C₂×C₄²×C₅	Q₈	1	8	yes	complemented
29	C₂×C₄²×C₅	C₂³	1	4	yes	—
30	C₅×#95	C₂³	1	48	yes	—
31	C₅×#95	C₂³	1	24	yes	—
32	C₅×#95	C₂³	1	8	yes	—
33	#2844	C₂×C₅	1	16	yes	direct product factor
34	C₂×#400	C₂×C₅	1	6	yes	—
35	C₂×#395	C₂×C₅	1	6	yes	—
36	C₂×#382	C₂×C₅	1	3	yes	—
37	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
38	C₂×C₅×Q₈	—	4	24	no	—
39	C₂×C₅×Q₈	—	4	32	no	—
40	C₂²×C₄×C₅	—	2	6	no	—
41	C₂²×C₄×C₅	—	2	6	no	—
42	C₂²×C₄×C₅	—	2	12	no	—
43	C₂²×C₄×C₅	—	2	12	no	—
44	C₂²×C₄×C₅	—	2	24	no	—
45	C₂²×C₄×C₅	—	2	24	no	—
46	C₂²×C₄×C₅	C₂×Q₈	1	12	yes	—
47	C₂²×C₄×C₅	C₂⁴	1	12	yes	—
48	C₂²×C₄×C₅	C₂⁴	1	12	yes	—
49	C₂²×C₄×C₅	C₂⁴	1	6	yes	—
50	C₅×C₄⋊C₄	—	4	12	no	—
51	C₅×C₄⋊C₄	—	4	12	no	—
52	C₅×C₄⋊C₄	—	4	24	no	—
53	C₄²×C₅	C₂×Q₈	1	16	yes	complemented
54	C₂³×Q₈	C₂²×C₅	1	2	yes	—
55	C₂²×C₄⋊C₄	C₂²×C₅	1	6	yes	—
56	C₂²×C₄⋊C₄	C₂²×C₅	1	3	yes	—
57	C₂²×C₄⋊C₄	C₂²×C₅	1	3	yes	—
58	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
59	#400	C₂²×C₅	1	48	yes	—
60	#395	C₂²×C₅	1	48	yes	—
61	#382	C₂²×C₅	1	24	yes	—
62	C₂×#95	C₂²×C₅	1	6	yes	—
63	C₂×#95	C₂²×C₅	1	12	yes	—
64	C₂×#95	C₂²×C₅	1	2	yes	—
65	C₂³×C₅	C₂²×Q₈	1	2	yes	—
66	C₂³×C₅	H(5,𝔽₂)	1	6	yes	—
67	C₂³×C₅	2₋¹⁺⁴	1	6	yes	—
68	C₂³×C₅	C₂⁵	1	1	yes	characteristic
69	C₅×Q₈	—	4	32	no	—
70	C₂×C₄×C₅	—	2	24	no	—
71	C₂×C₄×C₅	—	2	6	no	—
72	C₂×C₄×C₅	—	2	6	no	—
73	C₂×C₄×C₅	—	2	12	no	—
74	C₂×C₄×C₅	—	2	12	no	—
75	C₂×C₄×C₅	—	2	24	no	—
76	C₂×C₄×C₅	—	2	24	no	—
77	C₂×C₄×C₅	—	4	24	no	—
78	C₂×C₄×C₅	—	4	24	no	—
79	C₂×C₄×C₅	C₂²×Q₈	1	12	yes	—
80	C₂²×Q₈	—	4	12	no	—
81	C₂²×Q₈	C₂³×C₅	1	8	yes	—
82	C₂³×C₄	C₂³×C₅	1	6	yes	—
83	C₂³×C₄	C₂³×C₅	1	6	yes	—
84	C₂³×C₄	C₂³×C₅	1	3	yes	—
85	C₂×C₄⋊C₄	—	4	6	no	—
86	C₂×C₄⋊C₄	—	4	6	no	—
87	C₂×C₄⋊C₄	—	4	12	no	—
88	C₂×C₄⋊C₄	C₂³×C₅	1	12	yes	—
89	C₂×C₄⋊C₄	C₂³×C₅	1	12	yes	—
90	C₂×C₄⋊C₄	C₂³×C₅	1	24	yes	—
91	C₂×C₄²	C₅×Q₈	1	8	yes	complemented
92	C₂×C₄²	C₂³×C₅	1	4	yes	—
93	#95	C₂³×C₅	1	48	yes	—
94	#95	C₂³×C₅	1	24	yes	—
95	#95	C₂³×C₅	1	8	yes	—
96	C₂²×C₅	#544	1	12	yes	—
97	C₂²×C₅	H(3,𝔽₄)	1	4	yes	—
98	C₂²×C₅	#563	1	12	yes	—
99	C₂²×C₅	C₂³×Q₈	1	1	yes	characteristic
100	C₂²×C₅	C₂×H(5,𝔽₂)	1	3	yes	—
101	C₂²×C₅	C₂×2₋¹⁺⁴	1	3	yes	—
102	C₄×C₅	—	2	12	no	—
103	C₄×C₅	—	4	12	no	—
104	C₄×C₅	—	4	12	no	—
105	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
106	C₂×Q₈	—	4	24	no	—
107	C₂×Q₈	—	4	32	no	—
108	C₂²×C₄	—	2	6	no	—
109	C₂²×C₄	—	2	6	no	—
110	C₂²×C₄	—	2	12	no	—
111	C₂²×C₄	—	2	12	no	—
112	C₂²×C₄	—	2	24	no	—
113	C₂²×C₄	—	2	24	no	—
114	C₂²×C₄	C₂×C₅×Q₈	1	12	yes	—
115	C₂²×C₄	C₂⁴×C₅	1	12	yes	—
116	C₂²×C₄	C₂⁴×C₅	1	12	yes	—
117	C₂²×C₄	C₂⁴×C₅	1	6	yes	—
118	C₄⋊C₄	—	4	12	no	—
119	C₄⋊C₄	—	4	12	no	—
120	C₄⋊C₄	—	4	24	no	—
121	C₄²	C₂×C₅×Q₈	1	16	yes	complemented
122	C₂×C₅	#2844	1	8	yes	direct product factor
123	C₂×C₅	C₂×#544	1	3	yes	—
124	C₂×C₅	C₂×H(3,𝔽₄)	1	1	yes	characteristic
125	C₂×C₅	C₂×#563	1	3	yes	—
126	C₂³	C₂²×C₅×Q₈	1	2	yes	—
127	C₂³	C₅×H(5,𝔽₂)	1	6	yes	—
128	C₂³	C₅×2₋¹⁺⁴	1	6	yes	—
129	C₂³	C₂⁵×C₅	1	1	yes	characteristic
130	Q₈	—	4	32	no	—
131	C₂×C₄	—	2	24	no	—
132	C₂×C₄	—	2	6	no	—
133	C₂×C₄	—	2	6	no	—
134	C₂×C₄	—	2	12	no	—
135	C₂×C₄	—	2	12	no	—
136	C₂×C₄	—	2	24	no	—
137	C₂×C₄	—	2	24	no	—
138	C₂×C₄	—	4	24	no	—
139	C₂×C₄	—	4	24	no	—
140	C₂×C₄	C₂²×C₅×Q₈	1	12	yes	—
141	C₅	C₂×#2844	1	1	yes	characteristic, direct product factor, Sylow
142	C₂²	C₅×#544	1	12	yes	—
143	C₂²	C₅×H(3,𝔽₄)	1	4	yes	—
144	C₂²	C₅×#563	1	12	yes	—
145	C₂²	C₂³×C₅×Q₈	1	1	yes	characteristic
146	C₂²	C₂×C₅×H(5,𝔽₂)	1	3	yes	—
147	C₂²	C₂×C₅×2₋¹⁺⁴	1	3	yes	—
148	C₄	—	2	12	no	—
149	C₄	—	4	12	no	—
150	C₄	—	4	12	no	—
151	C₂	C₅×#2844	1	8	yes	direct product factor
152	C₂	C₂×C₅×#544	1	3	yes	—
153	C₂	C₂×C₅×H(3,𝔽₄)	1	1	yes	characteristic
154	C₂	C₂×C₅×#563	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#2844	C₂	direct (N × H)
16. C₂×#2844	C₅	direct (N × H)
28. C₂×C₄²×C₅	Q₈	semidirect
33. #2844	C₂×C₅	direct (N × H)
53. C₄²×C₅	C₂×Q₈	semidirect
91. C₂×C₄²	C₅×Q₈	semidirect
121. C₄²	C₂×C₅×Q₈	semidirect
122. C₂×C₅	#2844	direct (N × H)
141. C₅	C₂×#2844	direct (N × H)
151. C₂	C₅×#2844	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×#2844	2
C₂×C₅×#400	2
C₂×C₅×#395	2
C₂×C₅×#382	2
C₂×#2844	5

Subgroups — C2×C5×#2844 #11959408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2844 #11959408