Subgroups — C₂²×C₅×#499 #11983373

Order 1280.

1966 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	C₂²×C₅×#125	C₂	1	2	yes	maximal
2	C₂×C₅×#499	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₅×#107	C₂	1	2	yes	maximal
4	C₂²×C₅×#106	C₂	1	2	yes	maximal
5	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×#125	—	2	12	no	—
9	C₂×C₅×#125	C₂²	1	24	yes	—
10	C₂²×C₅×C₄⋊C₄	—	2	4	no	—
11	C₂²×C₅×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₅×#499	C₂²	1	64	yes	direct product factor
14	C₂×C₅×#107	C₂²	1	24	yes	—
15	C₂×C₅×#106	C₂²	1	24	yes	—
16	C₂×C₄×C₅×C₈	C₂²	1	12	yes	—
17	C₂²×#499	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₅	—	2	4	no	—
19	C₂³×C₄×C₅	D₄	1	2	yes	—
20	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂²×C₅×C₈	Q₈	1	8	yes	—
22	C₂²×C₅×C₈	Q₈	1	8	yes	—
23	C₂²×C₅×C₈	C₂³	1	6	yes	—
24	C₂²×C₅×C₈	C₂³	1	6	yes	—
25	C₅×#125	—	2	48	no	—
26	C₅×#125	C₂³	1	32	yes	—
27	C₂×C₅×C₄⋊C₄	—	2	48	no	—
28	C₂×C₅×C₄⋊C₄	—	2	12	no	—
29	C₂×C₅×C₄⋊C₄	C₂³	1	24	yes	—
30	C₂×C₄²×C₅	D₄	1	6	yes	—
31	C₂×C₄²×C₅	C₂³	1	6	yes	—
32	C₅×#107	C₂³	1	32	yes	—
33	C₅×#106	C₂³	1	32	yes	—
34	C₄×C₅×C₈	C₂³	1	16	yes	—
35	C₂²×#125	C₂×C₅	1	2	yes	—
36	C₂×#499	C₂×C₅	1	24	yes	direct product factor
37	C₂²×#107	C₂×C₅	1	2	yes	—
38	C₂²×#106	C₂×C₅	1	2	yes	—
39	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
40	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅	—	2	8	no	—
42	C₂²×C₄×C₅	—	2	24	no	—
43	C₂²×C₄×C₅	—	4	16	no	—
44	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
45	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
46	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
47	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
48	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
49	C₂×C₅×C₈	C₂×Q₈	1	24	yes	—
50	C₂×C₅×C₈	C₂×Q₈	1	24	yes	—
51	C₂×C₅×C₈	C₂⁴	1	4	yes	—
52	C₂×C₅×C₈	C₂⁴	1	4	yes	—
53	C₅×C₄⋊C₄	—	2	64	no	—
54	C₅×C₄⋊C₄	—	2	24	no	—
55	C₅×C₄⋊C₄	C₂⁴	1	16	yes	—
56	C₄²×C₅	C₂×D₄	1	12	yes	—
57	C₄²×C₅	C₂⁴	1	4	yes	—
58	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
59	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
60	C₂×#125	—	2	12	no	—
61	C₂×#125	C₂²×C₅	1	24	yes	—
62	C₂²×C₄⋊C₄	—	2	4	no	—
63	C₂²×C₄⋊C₄	C₂²×C₅	1	4	yes	—
64	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
65	#499	C₂²×C₅	1	64	yes	direct product factor
66	C₂×#107	C₂²×C₅	1	24	yes	—
67	C₂×#106	C₂²×C₅	1	24	yes	—
68	C₂×C₄×C₈	C₂²×C₅	1	12	yes	—
69	C₂³×C₅	C₄⋊Q₈	1	4	yes	—
70	C₂³×C₅	#135	1	8	yes	—
71	C₂³×C₅	C₂²×D₄	1	3	yes	—
72	C₂×C₄×C₅	—	2	24	no	—
73	C₂×C₄×C₅	—	2	16	no	—
74	C₂×C₄×C₅	—	4	48	no	—
75	C₂×C₄×C₅	C₄⋊Q₈	1	12	yes	—
76	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
77	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
78	C₂×C₄×C₅	C₂²×Q₈	1	6	yes	—
79	C₂×C₄×C₅	C₂²×Q₈	1	6	yes	—
80	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
81	C₅×C₈	C₂²×Q₈	1	8	yes	—
82	C₅×C₈	C₂²×Q₈	1	8	yes	—
83	C₂³×C₄	—	2	4	no	—
84	C₂³×C₄	C₅×D₄	1	2	yes	—
85	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
86	C₂²×C₈	C₅×Q₈	1	8	yes	—
87	C₂²×C₈	C₅×Q₈	1	8	yes	—
88	C₂²×C₈	C₂³×C₅	1	6	yes	—
89	C₂²×C₈	C₂³×C₅	1	6	yes	—
90	#125	—	2	48	no	—
91	#125	C₂³×C₅	1	32	yes	—
92	C₂×C₄⋊C₄	—	2	48	no	—
93	C₂×C₄⋊C₄	—	2	12	no	—
94	C₂×C₄⋊C₄	C₂³×C₅	1	24	yes	—
95	C₂×C₄²	C₅×D₄	1	6	yes	—
96	C₂×C₄²	C₂³×C₅	1	6	yes	—
97	#107	C₂³×C₅	1	32	yes	—
98	#106	C₂³×C₅	1	32	yes	—
99	C₄×C₈	C₂³×C₅	1	16	yes	—
100	C₂²×C₅	#499	1	16	yes	direct product factor
101	C₂²×C₅	C₂×C₄⋊Q₈	1	6	yes	—
102	C₂²×C₅	C₂×#135	1	12	yes	—
103	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
104	C₄×C₅	—	2	8	no	—
105	C₄×C₅	—	4	16	no	—
106	C₄×C₅	C₂×C₄⋊Q₈	1	6	yes	—
107	C₄×C₅	C₂³×Q₈	1	1	yes	characteristic
108	C₄×C₅	C₂³×Q₈	1	1	yes	characteristic
109	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
110	C₂²×C₄	—	2	8	no	—
111	C₂²×C₄	—	2	24	no	—
112	C₂²×C₄	—	4	16	no	—
113	C₂²×C₄	C₂×C₅×D₄	1	12	yes	—
114	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
115	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
116	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
117	C₂²×C₄	C₂⁴×C₅	1	3	yes	—
118	C₂×C₈	C₂×C₅×Q₈	1	24	yes	—
119	C₂×C₈	C₂×C₅×Q₈	1	24	yes	—
120	C₂×C₈	C₂⁴×C₅	1	4	yes	—
121	C₂×C₈	C₂⁴×C₅	1	4	yes	—
122	C₄⋊C₄	—	2	64	no	—
123	C₄⋊C₄	—	2	24	no	—
124	C₄⋊C₄	C₂⁴×C₅	1	16	yes	—
125	C₄²	C₂×C₅×D₄	1	12	yes	—
126	C₄²	C₂⁴×C₅	1	4	yes	—
127	C₂×C₅	C₂×#499	1	12	yes	direct product factor
128	C₂×C₅	C₂²×C₄⋊Q₈	1	1	yes	characteristic
129	C₂×C₅	C₂²×#135	1	2	yes	—
130	C₂³	C₅×C₄⋊Q₈	1	4	yes	—
131	C₂³	C₅×#135	1	8	yes	—
132	C₂³	C₂²×C₅×D₄	1	3	yes	—
133	C₂×C₄	—	2	24	no	—
134	C₂×C₄	—	2	16	no	—
135	C₂×C₄	—	4	48	no	—
136	C₂×C₄	C₅×C₄⋊Q₈	1	12	yes	—
137	C₂×C₄	C₂²×C₅×D₄	1	8	yes	—
138	C₂×C₄	C₂²×C₅×D₄	1	3	yes	—
139	C₂×C₄	C₂²×C₅×Q₈	1	6	yes	—
140	C₂×C₄	C₂²×C₅×Q₈	1	6	yes	—
141	C₂×C₄	C₂⁵×C₅	1	1	yes	characteristic
142	C₈	C₂²×C₅×Q₈	1	8	yes	—
143	C₈	C₂²×C₅×Q₈	1	8	yes	—
144	C₅	C₂²×#499	1	1	yes	characteristic, direct product factor, Sylow
145	C₂²	C₅×#499	1	16	yes	direct product factor
146	C₂²	C₂×C₅×C₄⋊Q₈	1	6	yes	—
147	C₂²	C₂×C₅×#135	1	12	yes	—
148	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
149	C₄	—	2	8	no	—
150	C₄	—	4	16	no	—
151	C₄	C₂×C₅×C₄⋊Q₈	1	6	yes	—
152	C₄	C₂³×C₅×Q₈	1	1	yes	characteristic
153	C₄	C₂³×C₅×Q₈	1	1	yes	characteristic
154	C₂	C₂×C₅×#499	1	12	yes	direct product factor
155	C₂	C₂²×C₅×C₄⋊Q₈	1	1	yes	characteristic
156	C₂	C₂²×C₅×#135	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#499	C₂	direct (N × H)
13. C₅×#499	C₂²	direct (N × H)
17. C₂²×#499	C₅	direct (N × H)
36. C₂×#499	C₂×C₅	direct (N × H)
65. #499	C₂²×C₅	direct (N × H)
100. C₂²×C₅	#499	direct (N × H)
127. C₂×C₅	C₂×#499	direct (N × H)
144. C₅	C₂²×#499	direct (N × H)
145. C₂²	C₅×#499	direct (N × H)
154. C₂	C₂×C₅×#499	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×#125	2
C₂×C₅×#499	2
C₂²×C₅×#107	2
C₂²×C₅×#106	2
C₂²×C₄×C₅×C₈	2
C₂²×#499	5

Subgroups — C22×C5×#499 #11983373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#499 #11983373